Γ-Limsup estimate for a nonlocal approximation of the Willmore functional

TL;DR

The paper introduces a nonlocal approximation of the Willmore functional using Gamma-convergence and proves a Γ-limsup estimate.

Contribution

The novelty is proposing a nonlocal counterpart to a known local phase-field approximation of the Willmore functional.

Findings

A Γ-limsup estimate is proven for the nonlocal approximation of the Willmore functional.

The analysis uses Fermi coordinates and decay estimates of higher-order derivatives of a nonlocal optimal profile.

Abstract

We propose a possible nonlocal approximation of the Willmore functional, in the sense of Gamma-convergence, based on the first variation of the fractional Allen–Cahn energies, and we prove the corresponding \documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document}Γ-limsup estimate. Our analysis is based on the expansion of the fractional Laplacian in Fermi coordinates and fine estimates on the decay of higher order derivatives of the one-dimensional nonlocal optimal profile. This result is the nonlocal counterpart of that obtained by Bellettini and Paolini, where they proposed a phase-field approximation of the Willmore functional based on the first variation of the (local) Allen–Cahn energies.

Full text

Introduction

In recent years, there has been a growing interest in geometric energies, such as the Area functional or the Willmore functional, the latter being defined by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {W}(\Sigma , \Omega ) = \int _{\Sigma \cap \Omega } H_{\Sigma }^2(y) \, d \mathcal {H}^{d-1}(y), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d \ge 2$$\end{document} is an integer number, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}^d$$\end{document} is an open set, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma \subset \mathbb {R}^d$$\end{document} is a smooth hypersurface, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\Sigma }(y)$$\end{document} is the mean curvature of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} at the point y and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}^{d-1}$$\end{document} stands for the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(d-1)$$\end{document} -dimensional Hausdorff measure in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^d$$\end{document} . Among several questions related to this functional, a relevant problem consists in minimizing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {W}$$\end{document} within all the surfaces of a certain type, for instance with fixed genus [7, 34, 54]; or connected, closed, confined to a prescribed region and with fixed area [23, 24]; or with prescribed isoperimetric ratio [51]. Another problem that generates a lot of interest is the study of the geometric flow associated to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {W}$$\end{document} , see among others [35–37, 55].

While both of these problems can be attacked directly, it could be useful to approximate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {W}$$\end{document} with simpler functionals, solve similar problems for the approximating functionals and then try to pass to the limit to obtain a solution to the original problems. Here, the notion of approximation considered is that of Gamma-convergence, see [21] for the definition and the fundamental properties, which is well suited for the convergence of the minima and the minimizers, and, under additional assumptions, also for the convergence of the flows, see [48, 52].

For the Area functional, this approach has been highly successful when considering the phase-transition regularization given by the Allen–Cahn energies, which are defined for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon > 0$$\end{document} by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {E}_{\varepsilon }(u, \Omega ) = \int _{\Omega } \bigg ( \frac{\varepsilon }{2} \left| \nabla u\right| ^2 + \frac{W(u)}{\varepsilon } \bigg ) \, dx, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W :\mathbb {R}\rightarrow [0,+\infty )$$\end{document} is a double-well potential, e.g. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W(s)=(1-s^2)^2$$\end{document} , which has wells at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=\pm 1$$\end{document} as we assume for convenience. From now on, in any statement concerning W, we will implicitly assume that it satisfies few structural assumptions, see Sect. 2.1 for the details.

In the foundational work [44] (see also [43]) Modica and Mortola proved that for any set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E \subset \mathbb {R}^d$$\end{document} of finite perimeter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Per}(E,\Omega )$$\end{document} inside a Lipschitz domain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} it holds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Gamma (L^1(\Omega ))-\lim _{\varepsilon \rightarrow 0^+} \mathcal {E}_{\varepsilon }(\chi _E, \Omega ) = \sigma \,\textrm{Per}(E,\Omega ), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _E:= \mathbbm {1}_E - \mathbbm {1}_{E^c}$$\end{document} denotes the difference between the characteristic functions of E and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E^c$$\end{document} , while \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document} is a positive constant depending on the potential W. For further details about the notation, see Sect. 2.1.

The convergence of (constrained) critical points of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {E}_\varepsilon $$\end{document} to “generalized” critical points of the Area functional (also when the ambient space is a closed manifold) has been studied by many authors, see for instance [31, 38, 43, 57, 59, 60] and the more recent results in [8, 9, 16, 17, 27, 29, 40, 41]. Moreover, the convergence of the rescaled \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} -gradient flows of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {E}_\varepsilon $$\end{document} to Brakke’s motion by mean curvature has been established in [32].

Now, it is well-known that the mean curvature represents the first variation of the Perimeter. Therefore, it is reasonable to consider a suitable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} -norm of the first variation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {E}_{\varepsilon }$$\end{document} to build an approximation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {W}$$\end{document} . Indeed, starting from a conjecture of De Giorgi stated in [20] (see also [10] for more about this conjecture), Bellettini and Paolini proposed to consider in [12] the following approximation of the Willmore functional:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {W}_{\varepsilon }(u,\Omega ) = \frac{1}{\varepsilon } \int _{\Omega } \bigg ( \varepsilon \Delta u - \frac{W'(u)}{\varepsilon } \bigg )^2 \, dx. \end{aligned}$$\end{document}

In addition, for any open set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E \subset \mathbb {R}^d$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial E \in C^2$$\end{document} , they proved the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} - \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\limsup $$\end{document} estimate:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Gamma (L^1(\mathbb {R}^d))-\limsup _{\varepsilon \rightarrow 0^+} \big ( \mathcal {E}_{\varepsilon } + \mathcal {W}_{\varepsilon } \big )(\chi _E, \mathbb {R}^d) \le \sigma \textrm{Per}(E,\mathbb {R}^d) + \sigma \mathcal {W}(\partial E, \mathbb {R}^d). \end{aligned}$$\end{document}

To this end, they exhibited a family of functions converging to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _E$$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document} for which the limit of the values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {E}_{\varepsilon } + \mathcal {W}_{\varepsilon }$$\end{document} evaluated on this family equals the right-hand side of (1.4). They considered a slight modification of the rescaled transition profiles

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u_\varepsilon (x):=q_0\bigg (\frac{\textrm{dist}_{\partial E}(x)}{\varepsilon }\bigg ), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_0$$\end{document} is the one-dimensional optimal profile (i.e. the unique, up to translations, monotone increasing solution of the Euler–Lagrange equation of (1.1) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=1, \Omega =\mathbb {R}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon =1$$\end{document} ) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{dist}_{\partial E}$$\end{document} is the signed distance function from the boundary of E. The \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} - \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\liminf $$\end{document} estimate completing (1.4) turns out to be more delicate, and it was proved by Röger and Schätzle in [47] when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d \in \{ 2, 3\}$$\end{document} and by Nagase and Tonegawa in [45] for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=2$$\end{document} with different techniques, while being still open in higher dimension.

The aim of the present paper is to prove a nonlocal counterpart of the estimate (1.4) exploiting the approximation of the (local) Perimeter given by the fractional Allen–Cahn energies. Given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in \left( {1}/{2},1\right) $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon > 0$$\end{document} , the s-fractional Allen–Cahn energy is defined by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & \mathcal {F}_{s,\varepsilon }(u, \Omega ) = \varepsilon ^{2s-1} \frac{\gamma _{d,s}}{4} \iint _{\mathbb {R}^d \times \mathbb {R}^d \setminus (\Omega ^c \times \Omega ^c)} \frac{\left| u(x)-u(y)\right| ^2}{\left| x-y\right| ^{d+2s}} \, dx \, dy + \int _{\Omega } \frac{W(u(x))}{\varepsilon } \, dx, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{d,s}$$\end{document} is given by (2.2). The starting point of our analysis is the work by Savin and Valdinoci [49] where, among other results, they established a nonlocal version of (1.2). Specifically, they proved that for any set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E \subset \mathbb {R}^d$$\end{document} of finite perimeter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Per}(E,\Omega )$$\end{document} inside a Lipschitz domain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} it holds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Gamma (L^1_{\textrm{loc}}(\mathbb {R}^d))-\lim _{\varepsilon \rightarrow 0^+} \mathcal {F}_{s,\varepsilon }(\chi _E, \Omega ) = c_{\star } \textrm{Per}(E,\Omega ), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{\star }$$\end{document} is a positive constant depending on s, d, W and the functionals \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_{s,\varepsilon }$$\end{document} are thought to be defined on those functions in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\infty }(\mathbb {R}^d)$$\end{document} that take values between the zeros of W (see [18] for similar results on more general functionals of nonlocal type). This result holds also for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s={1}/{2}$$\end{document} , but with a different scaling in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} , that is with an extra \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| \log (\varepsilon )\right| ^{-1}$$\end{document} factor in front of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_{s,\varepsilon }$$\end{document} . We mention also [2] for related results in the case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s= {1}/{2}$$\end{document} . Similarly to the local case, the estimate from above in the previous theorem is achieved by considering functions of the form of (1.5), at least when the set E meets the boundary of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} transversely, and where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_0$$\end{document} is replaced by the nonlocal one-dimensional optimal profile w. In particular, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w :\mathbb {R}\rightarrow (-1,1)$$\end{document} is defined as the unique, up to translations, monotone increasing solution of the fractional Allen–Cahn equation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (-\partial _{zz})^s w + W'(w) = 0, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\partial _{zz})^s$$\end{document} is the fractional Laplacian of order 2s in dimension one. We refer to Sect. 2.4 for a collection of some well-known properties of w. More in general, the first variation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_{s,\varepsilon }$$\end{document} is represented by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon ^{2s-1}(-\Delta )^s u + \varepsilon ^{-1} W'(u)$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta )^s$$\end{document} denotes the s-fractional Laplacian (see Sect. 2.3 for precise definitions). At this point, it is natural to consider the following nonlocal version of the functional defined by (1.3), that is

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {G}_{s,\varepsilon }(u, \Omega ) = \frac{1}{\varepsilon } \int _{\Omega } \bigg ( \varepsilon ^{2s-1} (-\Delta )^s u + \frac{W'(u)}{\varepsilon } \bigg )^2 \, dx, \end{aligned}$$\end{document}

if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u \in L^{\infty }(\mathbb {R}^d) \cap C^2(\Omega )$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}_{s,\varepsilon }(u, \Omega ):= +\infty $$\end{document} otherwise in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1_{\textrm{loc}}(\mathbb {R}^d)$$\end{document} . Here, the nonlocal behaviour of the functional is encoded in the fractional Laplacian. We are ready to state our main result.

Theorem 1.1

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in \left( {3}/{4},1\right) $$\end{document} . Then, for any bounded open set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E \subset \mathbb {R}^d$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial E \in C^2$$\end{document} it holds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Gamma (L^1_{\textrm{loc}}(\mathbb {R}^d))-\limsup _{\varepsilon \rightarrow 0^+} \big (\mathcal {F}_{s,\varepsilon } + \mathcal {G}_{s,\varepsilon }\big )(\chi _E,\Omega ) \le c_{\star } \textrm{Per}(E, \Omega ) + \kappa _{\star } \mathcal {W}(\partial E, \Omega ), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}^d$$\end{document} is any bounded open set with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega \in C^1$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{\star }$$\end{document} is the constant in (1.6) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa _{\star }$$\end{document} is a positive constant depending only on s and W. In the case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s={3}/{4}$$\end{document} the same conclusion holds if in the definition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}_{s,\varepsilon }$$\end{document} we add an extra \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| \log (\varepsilon )\right| ^{-1}$$\end{document} factor in front of the integral.

We recall the definition of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} - \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\limsup $$\end{document} , more precisely

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & \Gamma (L^1_{\textrm{loc}}(\mathbb {R}^d))-\limsup _{\varepsilon \rightarrow 0^+} \big (\mathcal {F}_{s,\varepsilon } + \mathcal {G}_{s,\varepsilon }\big )(\chi _E,\Omega ) \\ & = \inf \left\{ \limsup _{\varepsilon \rightarrow 0^+} (\mathcal {F}_{s,\varepsilon }+\mathcal {G}_{s,\varepsilon })(u_\varepsilon ,\Omega ) :u_\varepsilon \rightarrow \chi _E \text { in }L^1_{\text {loc}}(\mathbb {R}^d) \right\} . \end{aligned}$$\end{document}

The constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa _\star $$\end{document} is defined by (6.1). We mention that without the regularity assumption on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} an analogous of (1.9) still holds, that is with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{\Omega }}$$\end{document} on the right-hand side (see Remark 6.2). Before discussing the strategy of the proof, some comments are in order.

• We point out that we consider the sum of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_{s,\varepsilon }$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}_{s,\varepsilon }$$\end{document} in order to rule out the behaviour of certain trivial sequences that are uniformly bounded with respect to the functionals \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ \mathcal {G}_{s,\varepsilon } \}$$\end{document} but unbounded for the functionals \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ \mathcal {F}_{s,\varepsilon } \}$$\end{document} . In particular, the ultimate goal is to show that there is a relationship between the limit of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}_{s,\varepsilon }(u_\varepsilon )$$\end{document} and the limit of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_{s,\varepsilon }(u_\varepsilon )$$\end{document} when both are uniformly bounded in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon > 0$$\end{document} . Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_{s,\varepsilon }$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}_{s,\varepsilon }$$\end{document} are non-negative functionals, it is reasonable to consider their sum. An example of a trivial sequence that we want to exclude from our analysis is the following. We know from Rolle’s Theorem that there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_0 \in (-1,1)$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W'(c_0)=0$$\end{document} . For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon > 0$$\end{document} , we consider \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_\varepsilon \equiv c_0$$\end{document} , then it holds that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}_{s,\varepsilon }(u_\varepsilon ) = 0$$\end{document} . On the other hand, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W(c_0) > 0$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{\varepsilon \rightarrow 0^+} \mathcal {F}_{s,\varepsilon }(u_\varepsilon ) = \lim _{\varepsilon \rightarrow 0^+} \frac{W(c_0)}{\varepsilon } \mathcal {L}^d(\Omega ) = +\infty . \end{aligned}$$\end{document}

• The need of a different scaling for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s={3}/{4}$$\end{document} mildly suggests that the previous result might no longer be true when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in \left[ {1}/{2},{3}/{4}\right) $$\end{document} . Maybe, for these values of the parameter s the limit of the functionals \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_{s,\varepsilon }+\mathcal {G}_{s,\varepsilon }$$\end{document} (with a different scaling in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}_{s,\varepsilon }$$\end{document} ) could be equal to the sum of the (local) Perimeter and a combination of the Willmore functional and a nonlocal quantity possibly depending on the nonlocal mean curvature (for the definition see, for example, [1] or [25, Section 6]). We refer to Sect. 6 for further comments.
• On the other hand, when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in \left( 0,{1}/{2}\right) $$\end{document} the situation is fairly different. Already for the functionals \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_{s,\varepsilon }$$\end{document} it is not longer true that they approximate the classical (local) Perimeter. In [49], Savin and Valdinoci showed that the rescaled functionals \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon ^{1-2s} \mathcal {F}_{s,\varepsilon }$$\end{document} approximate the 2s-fractional Perimeter and that any family of minimizers with equibounded energy converges to a minimizer of the 2s-fractional Perimeter in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} . In [42], the previous convergence was extended to more general families \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ u_{\varepsilon } \}$$\end{document} of critical points of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon ^{1-2s} \mathcal {F}_{s,\varepsilon }$$\end{document} having equibounded energy. In particular, the authors proved that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{\varepsilon }$$\end{document} converges in a quite strong sense to the characteristic function of a set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{*}$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial E_{*} \cap \Omega $$\end{document} is a stationary 2s-fractional minimal surface in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} . Roughly speaking, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial E_{*}$$\end{document} has vanishing 2s-fractional mean curvature on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} . Moreover, they obtained compactness results also for family of functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ u_\varepsilon \}$$\end{document} with equibounded energy and a uniform Sobolev bound on their first variations. These results are the nonlocal analogies, in the range \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in \left( 0,{1}/{2}\right) $$\end{document} , of those in [31, 58] in the local case.
• In the case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in \left[ {3}/{4},1\right) $$\end{document} , it would be interesting to know if a full Gamma-convergence result holds, namely, if the functionals \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_{s,\varepsilon } + \mathcal {G}_{s,\varepsilon }$$\end{document} Gamma-converge to the right-hand side of (1.9), at least in small dimensions and within the class of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2$$\end{document} sets, where the corresponding result in the local case is known to be true [45, 47]. More in general, understanding the compactness properties of families of functions having \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_{s,\varepsilon } + \mathcal {G}_{s,\varepsilon }$$\end{document} equibounded would be of interest. However, up to our knowledge the situation is not clear even for families \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ u_\varepsilon \}$$\end{document} of critical points of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_{s,\varepsilon }$$\end{document} having equibounded energy. It is reasonable to expect a similar behaviour to the local case, where the energies of the critical points concentrate towards a “generalized” critical point of the Perimeter, we refer again to [31]. In order to prove Theorem 1.1 we need to exhibit, for any set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E \subset \mathbb {R}^d$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial E$$\end{document} of class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2$$\end{document} , a family of functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ u_{\varepsilon } \} \subset L^{\infty }(\mathbb {R}^d) \cap C^2(\Omega )$$\end{document} converging to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _E$$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1_{\textrm{loc}}(\mathbb {R}^d)$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{\varepsilon \rightarrow 0^+} \mathcal {F}_{s,\varepsilon }(u_{\varepsilon },\Omega ) = c_{\star } \textrm{Per}(E,\Omega ) \quad \text {and} \quad \lim _{\varepsilon \rightarrow 0^+} \mathcal {G}_{s,\varepsilon }(u_{\varepsilon },\Omega ) = \kappa _{\star } \mathcal {W}(\partial E, \Omega ). \end{aligned}$$\end{document}

If E is a smooth set with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{E}} \subset \Omega $$\end{document} , then a natural candidate for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{\varepsilon }(x)$$\end{document} would be \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon (\textrm{dist}_{\partial E}(x))$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon (z) = w\left( {z}/{\varepsilon }\right) $$\end{document} and w is the nonlocal one-dimensional optimal profile (see Sect. 2.4). Indeed, as mentioned earlier, this family of functions is a recovery sequence for the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} -limit of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_{s,\varepsilon }$$\end{document} , meaning that the first identity in (1.10) holds true when considering this family of functions. On the other hand, while the signed distance function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{dist}_{\partial E}$$\end{document} is globally 1-Lipschitz, it is smooth only near the boundary of E, and therefore \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}_{s,\varepsilon }$$\end{document} has infinite value at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon ( \textrm{dist}_{\partial E}(x))$$\end{document} . To overcome this issue, we introduce a suitable extension \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{\partial E} \in C^2(\mathbb {R}^d)$$\end{document} of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{dist}_{\partial E}$$\end{document} outside a fixed small tubular neighbourhood of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial E$$\end{document} (see Sect. 2.6). Accordingly, we consider the family of functions defined by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{\varepsilon }(x):=w_\varepsilon (\beta _{\partial E}(x))$$\end{document} . We point out here that the first identity in (1.10) remains true also when considering this family of functions, this follows by the same argument used in [49]. After doing so, our strategy to prove the second identity in (1.10) is similar to the one adopted in [12]. We split \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}_{s,\varepsilon }(u_\varepsilon , \Omega )$$\end{document} into the sum of two contributions, the first one being the integral in a small tubular neighbourhood of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial E$$\end{document} and the second one being the contribution coming from the complement of this tubular neighbourhood. However, the analysis of these two quantities is technically more involved than in the local case. We also remark that it is also quite different from the approach used in [49] to obtain the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} -limsup estimate for the functionals \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ F_{s,\varepsilon } \}$$\end{document} alone. Indeed, in our case we need a precise description of the fractional Laplacian of the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_\varepsilon $$\end{document} near the interface \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial E$$\end{document} , as well as an accurate asymptotic of the higher-order derivatives of the nonlocal one-dimensional optimal profile w. Both of these aspects were not necessary for the corresponding estimate in [49]. As mentioned shortly before, in order to estimate the contribution of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}_{s,\varepsilon }(u_\varepsilon , \Omega )$$\end{document} away from the interface, we require fine estimates on the decay of the higher order derivatives of w. While such estimates could be known by the experts, we were not able to find them explicitly stated in the literature. For the local one-dimensional optimal profile \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_0$$\end{document} they are well-known (see for instance [11]). In particular, borrowing ideas from [5, 33] we deduce the following result.

Theorem 1.2

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in (0,1)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \ge 2$$\end{document} . In addition to our structural assumptions, suppose that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W \in C^{k+1}(\mathbb {R})$$\end{document} . Letting w be the nonlocal one-dimensional optimal profile, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w\in C^{k} (\mathbb {R})$$\end{document} and there exists a positive constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C = C(s,W,k)$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \partial _x^k w(x)\right| \le \frac{C}{1+ \left| x\right| ^{k+2s}}, \qquad \forall x \in \mathbb {R}. \end{aligned}$$\end{document}

The estimate in the theorem above is coherent with the fact that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w'$$\end{document} is asymptotic to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| x\right| ^{-1-2s}$$\end{document} , as proved in [13, Theorem 2.7]. See also Remark 3.2 for further discussions.

To deal with the contribution coming from a small tubular neighbourhood \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {U}_{\delta }$$\end{document} of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial E$$\end{document} of size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} , it is convenient to use Fermi coordinates (see Sect. 2.7 for the precise definition). More precisely, we identify any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in \mathcal {U}_{\delta }$$\end{document} with a couple (y, z), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y \in \partial E$$\end{document} is the point of minimum distance between x and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial E$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z=\textrm{dist}_{\partial E}(x)$$\end{document} . Using these coordinates the (local) Laplacian can be written as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta = \partial _{zz} - H_z \partial _z + \Delta _{\Sigma _z}, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_z$$\end{document} denotes the mean curvature (with respect to its outer normal) of the hypersurface \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma _z:= \partial \{ x \in \mathcal {U}_{\delta } :\textrm{dist}_{\partial E}(x) > z \}$$\end{document} , which is diffeomorphic and parallel to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial E$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{\Sigma _z}$$\end{document} is the Laplace–Beltrami operator on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma _z$$\end{document} . One advantage of these coordinates is that whenever u is a function that depends only on the distance from the boundary of E, then the computation of its Laplacian considerably simplifies. In contrast, a similar expression to (1.12) for the fractional Laplacian has been derived only recently. Indeed, the first-named author studied Fermi coordinates in the context of the fractional Laplacian in his PhD Thesis and he introduced suitable expansions for the fractional Laplacian in [14, 15], in collaboration with Liu and Wei. Roughly speaking, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in \left( {1}/{2},1\right) $$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_\varepsilon $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon $$\end{document} are defined as above, then we have the following expansion

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (-\Delta )^s u_{\varepsilon }(x) = (-\partial _{zz})^s w_{\varepsilon }(z) + H_{\partial E}(y) L_s[w_{\varepsilon }'](z) + \mathcal {R}_{\varepsilon }(y,z), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{s}$$\end{document} is a suitable operator evaluated at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w'_{\varepsilon }$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_{\varepsilon }(y,z)$$\end{document} is an error term. In addition, the first term on the right-hand side in the previous expansion scales as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon ^{-2s}$$\end{document} , the second one scales as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon ^{1-2s}$$\end{document} , while the remainder term is uniformly bounded in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} . We refer to Sect. 4 for the precise statement and a detailed proof.

In conclusion, we briefly comment about the assumption \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in \left[ {3}/{4},1\right) $$\end{document} . If we neglect the error term in (1.13), then from (1.7) and (1.8) we obtain the following asymptotic

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {G}_{s, \varepsilon }(u_\varepsilon , \mathcal {U}_{\delta }) & \simeq \varepsilon ^{4s-3} \Vert {L_{s}[w_{\varepsilon }']} \Vert ^2_{L^2((-\delta , \delta ))} \mathcal {W}(\partial E, \mathcal {U}_{\delta })\nonumber \\ & =\Vert {L_{s}[w']} \Vert ^2_{L^2\left( \left( -{\delta }/{\varepsilon }, {\delta }/{\varepsilon }\right) \right) } \mathcal {W}(\partial E, \mathcal {U}_{\delta }). \end{aligned}$$\end{document}

Moreover, we will see in Sect. 5 that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{s}[w'] \in L^2(\mathbb {R})$$\end{document} when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in ({3}/{4},1)$$\end{document} , while for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s={3}/{4}$$\end{document} the norm in the right-hand side of (1.14) diverges logarithmically in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} . We refer to Sect. 6 for a more detailed description of the heuristic argument as well as the proof of Theorem 1.1.

The paper is organised as follows. In Sect. 2 we discuss some geometric lemmas and we deduce some useful decay properties of the nonlocal one-dimensional optimal profile w. In Sect. 3 we discuss the proof of Theorem 1.2. Section 4 is entirely devoted to the expansion of the fractional Laplacian in Fermi coordinates. In this section, we follow the presentation of [14], adapting some of their results to our framework. In Sect. 5 we show that the constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa _{\star }$$\end{document} appearing in (1.9) is finite. In Sect. 6 we combine our results to conclude the proof of Theorem 1.1.

Tools

In this section we collect some tools which will be needed. We start by introducing some notation that we keep throughout the manuscript.

Notation

• We denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma ((\mathbb {X},d)) - \lim _{\varepsilon } F_\varepsilon $$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma ((\mathbb {X},d)) - \limsup _{\varepsilon } F_\varepsilon $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Gamma ((\mathbb {X},d)) - \liminf _{\varepsilon } F_\varepsilon $$\end{document} the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} -limit, the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} -limsup, and the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} -liminf, as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0$$\end{document} , of a family of functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_\varepsilon : \mathbb {X} \rightarrow \mathbb {R}\cup \{ +\infty \}$$\end{document} with respect to the metric d on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {X}$$\end{document} . For the definitions see for instance [19].
• We denote by W a double-well potential satisfying the following structural assumptions:

(W1) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W: \mathbb {R}\rightarrow [0, +\infty )$$\end{document} , W is even and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \{W(x) = 0 \} = \{\pm 1\}$$\end{document} ;(W2) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W \in C^{3}(\mathbb {R})$$\end{document} ;(W3) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W''(\pm 1) = \lambda > 0$$\end{document} ;

• we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {W}$$\end{document} the Willmore functional, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {Per}$$\end{document} the Perimeter functional, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_{s,\varepsilon }$$\end{document} the scaled nonlocal Allen–Cahn energy and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}_{s,\varepsilon }$$\end{document} the squared \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} -norm of its first variation (Sect. 1);
• unless otherwise specified, we denote by w the one-dimensional optimal profile (Sect. 2.4) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon (z) = w({z}/{\varepsilon })$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_\varepsilon (x) = w_\varepsilon (\beta _{\Sigma }(x))$$\end{document} (Definition 2.10);
• we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[\cdot ]_{C^{k,\theta }}, \Vert {\cdot } \Vert _{C^{k,\theta }}$$\end{document} the Hölder seminorms and norms (Sect. 2.2);
• we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {F}}_d$$\end{document} the Fourier transform in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^d$$\end{document} (Sect. 2.3);
• we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta )^s$$\end{document} the fractional Laplacian operator of power \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in (0,1)$$\end{document} (see (2.1));
• \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{d,s}$$\end{document} is the constant given by (2.2);
• we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P^{(s)}_d$$\end{document} the fractional heat kernel of power s in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^d$$\end{document} (Sect. 2.5);
• we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{s,\lambda }$$\end{document} the fundamental solution of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta )^s + \lambda \textrm{Id}$$\end{document} (Proposition 3.1);
• we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_\delta ^m = \{ x \in \mathbb {R}^m :\left| x\right| < \delta \}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}^m$$\end{document} the m-dimensional Hausdorff measure;
• we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}^d$$\end{document} a bounded open set;
• we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E \subset \mathbb {R}^d$$\end{document} an open set of class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2$$\end{document} as in Definition 2.11 and we set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma = \partial E$$\end{document} ;
• we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _E = \mathbbm {1}_E - \mathbbm {1}_{E^c}$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbbm {1}_E$$\end{document} stands for the indicator function of the set E;
• we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{dist}_{\Sigma }$$\end{document} the signed distance from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{\Sigma }$$\end{document} the smoothed signed distance (Definition 2.10);
• for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell >0$$\end{document} we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma _\ell = \textrm{dist}_{\Sigma }^{-1}((-\ell ,\ell ))$$\end{document} ;
• we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{\Sigma }: \Sigma _\ell \rightarrow \Sigma $$\end{document} the projection onto \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} (Lemma 2.9);
• given an open set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E \subset \mathbb {R}^d$$\end{document} , for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in \partial E = \Sigma $$\end{document} we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_x \Sigma , N_\Sigma (x), H_\Sigma (x)$$\end{document} the tangent space to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} at x, the inner unit normal to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} at x and the scalar mean curvature of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} at x computed with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-N_\Sigma (x)$$\end{document} , respectively;
• given an open set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E \subset \mathbb {R}^d$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ x_0 \in \partial E = \Sigma $$\end{document} , we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y = \textrm{Id}\times g$$\end{document} a principal parameterization around \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0$$\end{document} , N the inner unit normal vector given by the parameterization g, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_1, \dots , k_{d-1}$$\end{document} the principal coordinates at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi (y,z) = Y(y) + z N(y)$$\end{document} the Fermi coordinated around \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0$$\end{document} (Definition 2.11);
• we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| (Y(y) - z_0 e_d)_\tau \right| $$\end{document} the tangential distance (Definition 2.14);
• we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta _{\varepsilon , \ell }$$\end{document} the function in (5.1);
• we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\star ,\mu _w, \kappa _\star $$\end{document} the constants in (1.6), (5.3) and (6.1).

Function spaces

Given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}^d$$\end{document} open, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \in (0,1]$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \in \mathbb {N}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u: \Omega \rightarrow \mathbb {R}$$\end{document} , we set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & [u]_{C^\theta (\Omega )} = \sup _{x,y \in \Omega , x \ne y} \frac{\left| u(x)-u(y)\right| }{\left| x-y\right| ^\theta }, \quad \Vert {u} \Vert _{C^\theta (\Omega )} = \Vert {u} \Vert _{L^\infty (\Omega )} + [u]_{C^\theta (\Omega )},\\ & [u]_{C^k(\Omega )} = \sum _{\left| \alpha \right| = k}\Vert {\partial ^\alpha u} \Vert _{L^\infty (\Omega )}, \quad \Vert {u} \Vert _{C^k(\Omega )} = \Vert {u} \Vert _{L^\infty (\Omega )} + \sum _{j=1}^k [u]_{C^j(\Omega )},\\ & [u]_{C^{k,\theta }(\Omega )} = \sum _{\left| \alpha \right| = k} [\partial ^\alpha _x u]_{C^\theta (\Omega )}, \quad \Vert {u} \Vert _{C^{k,\theta }(\Omega )} = \Vert {u} \Vert _{C^k(\Omega )} + [u]_{C^{k,\theta }(\Omega )}. \end{aligned}$$\end{document}

The fractional Laplacian

Here we recall the definition and some basic properties of the fractional Laplacian. We refer to [22, 26] for an extensive discussion on the topic. Given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in (0,1)$$\end{document} and a function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u :\mathbb {R}^d \rightarrow \mathbb {R}$$\end{document} globally bounded and of class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2$$\end{document} , the fractional Laplacian and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta )^s u$$\end{document} is defined by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (-\Delta )^s u(x) = \gamma _{d,s} P.V. \int _{\mathbb {R}^d} \frac{u(x) -u(y)}{\left| x-y\right| ^{d+2s}}\, dy = \gamma _{d,s} \lim _{\nu \rightarrow 0} \int _{B_\nu (x)^c} \frac{u(x) -u(y)}{\left| x-y\right| ^{d+2s}}\, dy , \end{aligned}$$\end{document}

where the constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{d,s}$$\end{document} is defined for convenience by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \gamma _{d,s}:= s 2^{2s} \pi ^{-\frac{d}{2}} \frac{\Gamma \left( \frac{d+2s}{2} \right) }{\Gamma (1-s)}. \end{aligned}$$\end{document}

It is easy to check that for globally bounded functions of class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2$$\end{document} it holds that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (-\Delta )^s u(x) = \frac{\gamma _{d,s}}{2} \int _{\mathbb {R}^d} \frac{2 u(x) - u(x+y) - u(x-y)}{\left| y\right| ^{d+2s}} \, dy, \end{aligned}$$\end{document}

We define the Fourier transform by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathscr {F}}_d u(\xi ) = \int _{\mathbb {R}^d} u(x) e^{2 \pi i x \cdot \xi } \, dx, \end{aligned}$$\end{document}

so that the inversion formula reads as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\mathscr {F}}_d^{-1} u ) (x) = ({\mathscr {F}}_d u) (-x)$$\end{document} . With this convention, it can be checked that (see e.g. [26])

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathscr {F}}_d [(-\Delta )^s u] (\xi ) = \left| 2\pi \xi \right| ^{2s} {\mathscr {F}}_d u(\xi ), \end{aligned}$$\end{document}

The definition of the fractional Laplacian extends to tempered distributions by duality. We state and prove the following elementary estimate.

Lemma 2.1

Given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in (0,1)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega ^{\prime } \subset \! \subset \Omega ^{\prime \prime } \subset \! \subset \Omega \subset \mathbb {R}^d$$\end{document} open sets, then for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u \in L^\infty (\mathbb {R}^d) \cap C^{2}(\Omega )$$\end{document} it holds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert {(-\Delta )^s u} \Vert _{L^\infty (\Omega ^{\prime })} \le C(d,s,\Omega ^{\prime }, \Omega ^{\prime \prime }) \left( [u]_{C^{2}(\Omega ^{\prime \prime })} + \Vert {u} \Vert _{L^\infty (\mathbb {R}^d)} \right) . \end{aligned}$$\end{document}

Proof

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta >0$$\end{document} be such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{\delta }(x) \subset \Omega ^{\prime \prime }$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in \Omega ^{\prime }$$\end{document} . Then, up to a multiplicative constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(d,s,\Omega ^{\prime }, \Omega ^{\prime \prime })>0$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| (-\Delta )^s u(x)\right|&\lesssim \int _{B_{\delta }^c(x)} \frac{\left| u(x) -u(y)\right| }{\left| x-y\right| ^{d+2s}} \, dy +\lim _{\nu \rightarrow 0} \left| \int _{B_{\delta }(x) \setminus B_\nu (x)} \frac{u(x) - u(y) - \nabla u(x) \cdot (y-x) }{\left| x-y\right| ^{d+2s}}\, dy\right| \\ &\lesssim \Vert {u} \Vert _{L^\infty (\mathbb {R}^d) } \int _{B_\delta ^c(x)} \frac{1}{\left| x-y\right| ^{d+2s}} \, dy + [u]_{C^2(\Omega ^{\prime \prime })} \int _{B_{\delta }(x)} \frac{1}{\left| x-y\right| ^{d+2s-2}} \, dy \\ &\lesssim \Vert {u} \Vert _{L^\infty (\mathbb {R}^d)} + [u]_{C^2(\Omega ^{\prime \prime })}. \end{aligned}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

The one-dimensional optimal profile

Throughout the whole manuscript, we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w: \mathbb {R}\rightarrow (-1,1)$$\end{document} the unique increasing solution to the fractional Allen–Cahn equation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} (-\partial _{zz} )^s w + W'(w) = 0, \\ w(0)=0, \\ \lim _{z \rightarrow \pm \infty } w(z) = \pm 1. \end{array}\right. } \end{aligned}$$\end{document}

For the sake of clarity, we collect some well-known results about the optimal profile that will be needed in our analysis.

Proposition 2.2

Fix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in (0,1)$$\end{document} and let W be a double-well potential satisfying (W1), (W2), (W3). There exists a unique strictly increasing solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w: \mathbb {R}\rightarrow (-1,1)$$\end{document} to the problem (2.3) Moreover, w is odd, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w \in C^1(\mathbb {R})$$\end{document} and there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(s,W)>0$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| w'(z) \right| \le \frac{C}{1+ \left| z\right| ^{1+2s}}, \qquad \forall z \in \mathbb {R}. \end{aligned}$$\end{document}

We recall that w is built by minimization of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_{s,1}$$\end{document} when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega = \mathbb {R}$$\end{document} . More precisely, w is the unique (up to translation) minimizer of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_{s,1}(\cdot , \mathbb {R})$$\end{document} with respect to perturbation with compact support in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}$$\end{document} , see [46, Theorem 2] for instance. Since W is even, it is readily checked that w is odd. The regularity of w is established by iterating the apriori estimates in [53, Proposition 2.8 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-$$\end{document} 2.9]. As we already mentioned, the decay estimate (2.4) is optimal, see [13, Theorem 2.7]. Integrating (2.4), we find that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| w(z) - \mathrm{sgn\,}(z)\right| \le \frac{C}{1+ \left| z\right| ^{2s}} \qquad \forall z \in \mathbb {R}. \end{aligned}$$\end{document}

Lastly, we recall that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w'$$\end{document} solves the fractional Allen–Cahn equation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (-\partial _{zz})^s w' + W''(w) w' =0. \end{aligned}$$\end{document}

The fractional heat kernel

We consider the solution to the fractional heat equation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t P^{(s)}_d + (-\Delta )^s P^{(s)}_d = 0 & (t, x) \in (0, +\infty ) \times \mathbb {R}^d, \\ P^{(s)}_d(0) = \delta _0 & x \in \mathbb {R}^d, \end{array}\right. } \end{aligned}$$\end{document}

where the initial value is taken in the sense of distributions. More precisely, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P^{(s)}_d$$\end{document} is defined via the Fourier transform.

Definition 2.3

Fix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in (0,1)$$\end{document} . For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(t, x) \in (0, +\infty ) \times \mathbb {R}^d$$\end{document} , we define the fractional heat kernel

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P^{(s)}_d(t,x):= \int _{\mathbb {R}^d} \exp \left( - t \left| 2\pi \xi \right| ^{2s} \right) e^{2 \pi i x \cdot \xi } \, d \xi = {\mathscr {F}}_d^{-1} \left( \exp { \left( - t \left| 2\pi (\cdot )\right| ^{2s}\right) }\right) (x), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {F}}_d$$\end{document} is the Fourier transform.

We mention [26, Chapter 16] and the references therein for a presentation of the topic. By a change of variables, it is easy to check that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P^{(s)}_d (t,x) = t^{-\frac{d}{2s}} P^{(s)}_d \left( 1, x t^{-\frac{1}{2s}}\right) . \end{aligned}$$\end{document}

In analogy with the classical heat kernel, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P^{(s)}_d(t,x) >0$$\end{document} for any (t, x) (see [26, Proposition 16.3] ). Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exp (- t \left| \xi \right| ^{2\,s})$$\end{document} is a rapidly decaying function, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P^{(s)}_d(t,\cdot )$$\end{document} is smooth. However, due to lack of differentiability of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exp (- t \left| \xi \right| ^{2s})$$\end{document} at 0 for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in (0,1)$$\end{document} , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P^{(s)}_d$$\end{document} is not a Schwarz function. In particular, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P^{(s)}_d(1, \cdot )$$\end{document} enjoys a polynomial decay, whereas the classical heat kernel decays exponentially.

Proposition 2.4

([26, Proposition 16.5]) Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in (0,1)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P^{(s)}_d(1, \cdot )$$\end{document} be as in Definition 2.3. For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in \mathbb {R}^d$$\end{document} it holds that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{C_1(d,s)}{1 + \left| x\right| ^{d+2s}} \le P^{(s)}_d(1, x) \le \frac{C_2(d,s)}{1 +\left| x\right| ^{d+2s}}. \end{aligned}$$\end{document}

By the properties of the Fourier transform, for any index \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j = 1, \dots , d$$\end{document} we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _j P^{(s)}_{d}(1,x) = 2 \pi i {\mathscr {F}}_d^{-1}\left( \xi _j \exp \left( - \left| 2 \pi \xi \right| ^{2s} \right) \right) (x). \end{aligned}$$\end{document}

Using the Bochner relation (see e.g. [56, Section 3.2]), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathscr {F}}_d (\xi _j \exp (-\left| \cdot \right| ^{2s}))(x) = i x_j {\mathscr {F}}_{d+2} (\exp (-\left| \cdot \right| ^{2s})) ({\tilde{x}}), \end{aligned}$$\end{document}

where we set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{x}} = (x, 0,0) \in \mathbb {R}^d \times \mathbb {R}\times \mathbb {R}$$\end{document} . Hence, combining (2.7) and Proposition 2.4 we obtain the following result.

Proposition 2.5

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in (0,1)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P^{(s)}_1(1, \cdot )$$\end{document} be as in Definition 2.3. For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \in \mathbb {N}$$\end{document} it holds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \partial _x^k P^{(s)}_1(1,x)\right| \le \frac{C(k,s)}{1+ \left| x\right| ^{k+1+2s}} \qquad \forall x \in \mathbb {R}. \end{aligned}$$\end{document}

Proof

We prove by induction that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial ^k_x P^{(s)}_1 (1, x) = \sum _{i= \lceil \frac{k}{2} \rceil }^k c_{i,k} x^{2i-k} P^{(s)}_{1+2i} (1, {\tilde{x}}), \qquad \forall x \in \mathbb {R} \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{i,k} $$\end{document} are real-valued coefficients, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lceil \cdot \rceil $$\end{document} is the upper integer part and we denote by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P^{(s)}_{d}(1, {\tilde{x}}) = P^{(s)}_{d}(1, (x, 0, \cdots , 0)) \qquad \forall x \in \mathbb {R}. \end{aligned}$$\end{document}

Then, (2.8) follows immediately by (2.9) and Proposition 2.4. To check the validity of (2.9), for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=1$$\end{document} by (2.7) we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _x P^{(s)}_1(1, x) = {\tilde{c}}_{1,1} x P^{(s)}_3(1, {\tilde{x}}). \end{aligned}$$\end{document}

Assume that k is even, being the case of k odd similar. Then, deriving (2.9) we find explicit constants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{c}}_{i,k}$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial ^{k+1}_x P^{(s)}_1(1, x)&= \sum _{i= \lceil \frac{k}{2}\rceil +1 }^k c_{i,k} (2i-k) x^{2i-k-1} P^{(s)}_{1+2i}(1, {\tilde{x}}) + \sum _{i=\lceil \frac{k}{2}\rceil }^k {\tilde{c}}_{i,k} c_{i,k} x^{2(i+1)-k-1} P^{(s)}_{1+2(i+1)} (1, {\tilde{x}}) \\ &= \sum _{i= \lceil \frac{k+1}{2}\rceil }^k c^1_{i,k+1} x^{2i-(k+1)} P^{(s)}_{1+2i}(1, {\tilde{x}}) + \sum _{i=\lceil \frac{k}{2}\rceil +1 }^{k+1} c_{i,k+1}^2 x^{2i-(k+1)} P^{(s)}_{1+2i} (1, {\tilde{x}}). \end{aligned}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Distance function

Throughout this note, we adopt the following notation. Given an open set E, we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma = \partial E$$\end{document} and we let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{dist}_{\Sigma }$$\end{document} be the signed distance from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} , i.e.

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textrm{dist}_{\Sigma } (x) = {\left\{ \begin{array}{ll} \inf \{ \left| x-y\right| :y \in \Sigma \} & \text { if } x \in E, \\ - \inf \{ \left| x-y\right| :y \in \Sigma \} & \text { if } x \in E^c. \end{array}\right. } \end{aligned}$$\end{document}

It is well known that the distance function is 1-Lipschitz continuous and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| \nabla \textrm{dist}_{\Sigma }(y)\right| =1$$\end{document} at any point of differentiability. We refer to [3] and [28, Section 14.6] for an overview of the properties of the distance function.

Definition 2.6

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \ge 2$$\end{document} . We say that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E \subset \mathbb {R}^d$$\end{document} is an open set of class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^k$$\end{document} ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial E \in C^k$$\end{document} in short) if for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0 \in \Sigma = \partial E$$\end{document} there exist \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta > 0$$\end{document} and a map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g :B_{\delta }^{d-1}\rightarrow \mathbb {R}$$\end{document} of class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^k$$\end{document} such that, up to an affine isometry of the ambient space, it holds \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0 =0$$\end{document} and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&T_{x_0} \Sigma = \textrm{Span}(e_1,\dots ,e_{d-1}), \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\Sigma \cap \big (B_\delta ^{d-1} \times (-\delta , \delta ) \big ) = \big \{(y, g(y)) :y \in B_\delta ^{d-1} \big \}, \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&E \cap \big (B_\delta ^{d-1} \times (-\delta , \delta ) \big ) = \big \{(y, t) :y \in B_\delta ^{d-1}, t > g(y) \big \}. \end{aligned}$$\end{document}

Moreover, we say that the map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y:= \textrm{Id} \times g :B^{d-1}_\delta \rightarrow \mathbb {R}^d$$\end{document} is a principal parameterization of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} around \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0$$\end{document} if g satisfies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \nabla ^2 g(0) = \textrm{diag}(k_1, \dots , k_{d-1}), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_1, \dots , k_{d-1}$$\end{document} denote the principal curvatures of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0$$\end{document} computed with respect to the outer unit normal.

Remark 2.7

Given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E \subset \mathbb {R}^d$$\end{document} an open set of class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^k$$\end{document} , then for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0 \in \Sigma $$\end{document} there exists a principal parameterization around \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0$$\end{document} according to Definition 2.6. Indeed, if g satisfies (2.10)–(2.12), then we find a linear isometry of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O: \mathbb {R}^{d-1} \rightarrow \mathbb {R}^{d-1}$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g \circ O$$\end{document} satisfies also (2.13).

Remark 2.8

In the framework of Definition 2.6, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} is locally the graph of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2$$\end{document} function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g :B^{d-1}_{\delta } \rightarrow \mathbb {R}$$\end{document} and we have the following well-known formulas:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & T_{(y,g(y))} \Sigma = \textrm{Span}\big ( (e_1, \partial _1 g(y)),\dots ,(e_{d-1}, \partial _{d-1} g(y)) \big ), \nonumber \\ & N(y) = \frac{(-\nabla g(y), 1)}{\sqrt{1+\left| \nabla g(y)\right| ^2}},\nonumber \\ & H_{\Sigma }(y,g(y)) = \textrm{div} \left( \frac{\nabla g}{\sqrt{ 1 + \left| \nabla g\right| ^2}} \right) = \frac{\Delta g(y)}{\sqrt{ 1 + \left| \nabla g(y)\right| ^2}} - \frac{\nabla ^2 g(y)[\nabla g(y), \nabla g(y)]}{(1+\left| \nabla g(y)\right| ^2)^{3/2}},\nonumber \\ \end{aligned}$$\end{document}

where N(y) denotes the inner unit normal to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} at the point (y, g(y)) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\Sigma }(y,g(y))$$\end{document} is the scalar mean curvature of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} at the point (y, g(y)) computed with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-N(y)$$\end{document} . In addition, if the function g satisfies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} g(0)=0, \qquad \nabla g(0) = 0, \qquad \nabla ^2 g(0) = \textrm{diag}(k_1, \dots , k_{d-1}), \end{aligned}$$\end{document}

then the previous formulas at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 \in \Sigma $$\end{document} simplify to

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} T_{0} \Sigma = \mathbb {R}^{d-1} \times \{ 0 \}, \qquad N(0) = (0,1), \qquad H_{\Sigma }(0) = \Delta g(0) = \sum _{i=1}^{d-1} k_i. \end{aligned}$$\end{document}

We recall some well-known properties of the signed distance function from an hypersurface (see for instance [3, 28, Lemma 14.16], [43, Lemma 3]).

Lemma 2.9

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E \subset \mathbb {R}^d$$\end{document} be a bounded open set of class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^k$$\end{document} for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \ge 2$$\end{document} . Then, there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta >0$$\end{document} depending only on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} with the following properties.

• (i)For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in \Sigma _{\delta }$$\end{document} there exists a unique point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{\Sigma }(x) \in \Sigma $$\end{document} of minimal distance between x and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} . Moreover, the map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _{\Sigma } :\Sigma _\delta \rightarrow \Sigma $$\end{document} is of class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{k-1}$$\end{document} .
• (ii)For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in \Sigma _\delta $$\end{document} it holds that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \nabla \textrm{dist}_{\Sigma }(x) = N(\pi _{\Sigma }(x)) , \end{aligned}$$\end{document}

where N is the inner unit normal to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} . In particular, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{dist}_{\Sigma } \in C^{k}(\Sigma _\delta )$$\end{document} .

We consider a smooth version of the signed distance, which will play a key role in our analysis.

Definition 2.10

Let E be a bounded open set of class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2$$\end{document} and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ 0< \delta < {1}/{5}$$\end{document} be such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$5\delta $$\end{document} satisfies the properties of Lemma 2.9. We denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{\Sigma }: \mathbb {R}^d \rightarrow [-1,1]$$\end{document} any function of class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2(\mathbb {R}^d)$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| \beta _{\Sigma } (x)\right| \in [4\delta , 1]$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in \Sigma _{5\delta } {\setminus } \Sigma _{4\delta }$$\end{document} and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \beta _{\Sigma }(x) = {\left\{ \begin{array}{ll} \textrm{dist}_{\Sigma }(x) & x \in \Sigma _{4\delta }, \\ \mathrm{sgn\,}(\textrm{dist}_{\Sigma }(x)) & x \in \mathbb {R}^d \setminus \Sigma _{5\delta }. \end{array}\right. } \end{aligned}$$\end{document}

Uniform Fermi coordinates

We are interested in sets possessing parameterization with uniform bounds. Following [28, Section 14.6], we introduce a useful notation.

Definition 2.11

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E \subset \mathbb {R}^d$$\end{document} be an open set. We say that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma = \partial E$$\end{document} is of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\delta ,C,k)$$\end{document} -type if E is of class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^k$$\end{document} and there exist positive constants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} , C such that for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0 \in \Sigma $$\end{document} there exists a principal parameterization \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y=\textrm{Id} \times g :B_{\delta }^{d-1} \rightarrow \mathbb {R}^d$$\end{document} around \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0$$\end{document} as in Definition 2.6 such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert {g} \Vert _{C^k\big (B_{\delta }^{d-1}\big )} \le C. \end{aligned}$$\end{document}

Moreover, letting N be the inner unit normal defined by (2.14), we define the Fermi coordinates \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi : B_{\delta }^{d-1} \times (-\delta , \delta ) \rightarrow \mathbb {R}^{d} $$\end{document} around \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0$$\end{document} by setting

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Phi (y,z):= Y(y) + z N(y). \end{aligned}$$\end{document}

Remark 2.12

It is clear that if E is an open set of class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^k$$\end{document} and with compact boundary, then there exist positive constants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta _0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_0$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} is of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\delta , C, k)$$\end{document} -type according to Definition 2.11 for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta \le \delta _0$$\end{document} and for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C \ge C_0$$\end{document} . Since the proof follows by a straightforward compactness argument, we leave the details to the reader.

Remark 2.13

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E \subset \mathbb {R}^d$$\end{document} be an open set of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\delta , C, k)$$\end{document} -type for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C, \delta >0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \ge 3$$\end{document} . With the notation of Definition 2.11, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0 \in \Sigma $$\end{document} the map g can be expanded near \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y=0$$\end{document} as follows

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} g(y)&= \frac{1}{2} \sum _{i=1}^{d-1} k_i y_i^2 + O(\left| y\right| ^3), \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _i g(y)&= k_i y_i +O(\left| y\right| ^2), \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _{ij} g(y)&= \delta _{ij} k_i + O(\left| y\right| ). \end{aligned}$$\end{document}

The reminders satisfy uniform estimates with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0 \in \Sigma $$\end{document} thanks to (2.15).

In view of proving Theorem 4.1, we need to compute integrals involving suitable powers of the Euclidean distance in small domains contained in the tubular neighbourhood of the boundary of a smooth set. Since it is natural to use Fermi coordinates as local charts of the tubular neighbourhood, it is convenient to write the Euclidean distance in terms of the Fermi coordinates.

Definition 2.14

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E \subset \mathbb {R}^d$$\end{document} be an open set of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\delta , C, k)$$\end{document} -type for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C, \delta >0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \ge 3$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0 \in \Sigma _\delta $$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0' = \pi _\Sigma (x_0)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_0 = \textrm{dist}_\Sigma (x_0)$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi $$\end{document} be Fermi coordinates around \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0'$$\end{document} . For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y \in B_\delta ^{d-1}$$\end{document} , we define the (squared) tangential distance from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0$$\end{document} by setting

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| (Y(y) - z_0 e_d)_\tau \right| ^2:= \left| Y(y) - z_0 e_d\right| ^2 - \left| (Y(y) - z_0 e_d)\cdot N(y)\right| ^2. \end{aligned}$$\end{document}

We refer to Remark 2.16 for a geometric interpretation of the tangential distance. We recall some Taylor expansions that will be useful later on, when we integrate on domains contained in the tubular neighbourhood using Fermi coordinates.

Lemma 2.15

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E \subset \mathbb {R}^d$$\end{document} be an open set such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma = \partial E$$\end{document} is of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\delta , C, 3)$$\end{document} -type according to Definition 2.11, for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta , C >0$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0 \in \Sigma _\delta $$\end{document} , let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0' = \pi _\Sigma (x_0)$$\end{document} and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_0 = \textrm{dist}_\Sigma (x_0)$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi $$\end{document} be Fermi coordinates around \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0'$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| (Y(y)-z_0 e_d)_\tau \right| $$\end{document} be the tangential distance given by Definition 2.14. Then, the following estimates hold true:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & Y(y) \cdot N(y) = - \frac{1}{2} \sum _{i=1}^{d-1} k_i y_i^2 + O(\left| y\right| ^3), \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & 1 - N_d(y) = O(\left| y\right| ^2), \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & \nabla \Phi (0,z) = \textrm{diag}(1- z k_1, \dots , 1- z k_{d-1}, 1), \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & \det ( \nabla \Phi (y,z)) = \prod _{i=1}^{d-1}(1- z k_i) + O(\left| y\right| \left| z\right| ) + O(\left| y\right| ^2), \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & \left| (Y(y) - z_0 e_d)_\tau \right| ^2 = \sum _{1=1}^{d-1} y_i^2(1-k_i z_0)^2 + O(\left| z_0\right| \left| y\right| ^3) + O(\left| y\right| ^4). \end{aligned}$$\end{document}

The error terms in (2.20)–(2.24) satisfy uniform bounds with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0 \in \Sigma _\delta \cap E$$\end{document} .

Remark 2.16

With the notation of Definition 2.14, we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0 = \Phi (0, z_0)$$\end{document} . By completing the squares with respect to z, it is easy to check that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \Phi (y,z) -\Phi (0,z_0)\right|&= \left| z-z_0 + Y(y)\cdot N(y) + z_0(1-N_d(y))\right| ^2 + \left| (Y(y) - z_0 e_d)_{\tau }\right| ^2. \end{aligned}$$\end{document}

Moreover, by Lemma 2.15, we infer that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| z- z_0 + Y(y)\cdot N(y) + z_0(1- N_d(y))\right| = \left| z- z_0\right| + O(\left| y\right| ^2), \end{aligned}$$\end{document}

which is the size of the “normal” component of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi (y, z)- \Phi (0, z_0)$$\end{document} . By (2.24), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| (Y(y) - z_0 e_d)_{\tau }\right| $$\end{document} is proportional to the size of the “tangential” variable stretched by the curvature of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _\Sigma (x_0)$$\end{document} .

Proof of Lemma 2.15

The proof is a direct computation and we include it for the reader’s convenience. By (2.16), (2.17), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} Y(y) \cdot N(y)&= \frac{g(y) - \nabla g(y) \cdot y}{\sqrt{1 + \left| \nabla g(y)\right| ^2}} = \left( - \frac{1}{2} \sum _{i=1}^{d-1} y_i^2 k_i + O(\left| y\right| ^3) \right) \left( 1 + O(\left| y\right| ^2) \right) , \end{aligned}$$\end{document}

thus proving (2.20). Similarly, we check (2.21), that is

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 1- N_d(y) = 1- \frac{1}{\sqrt{1 + \left| \nabla g(y)\right| ^2 }} = O(\left| \nabla g(y)\right| ^2) = O(\left| y\right| ^2 ). \end{aligned}$$\end{document}

In order to check (2.22) and (2.23), computing explicitly \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla \Phi (y,z)$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \nabla \Phi (y,z) = \begin{pmatrix} 1+ z \partial _1 N_1 & z \partial _2 N_1 & \cdots & z \partial _{d-1} N_1 & N_1 \\ z \partial _1 N_2 & 1+ z \partial _2 N_2 & \cdots & z \partial _{d-1} N_2 & N_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ z \partial _1 N_{d-1} & z \partial _2 N_{d-1} & \cdots & 1+ z \partial _{d-1} N_{d-1} & N_{d-1} \\ \partial _1 g + z \partial _1 N_d & \partial _2 g + z \partial _{2} N_{d} & \cdots & \partial _{d-1}g + z \partial _{d-1} N_d & N_d \end{pmatrix}, \end{aligned}$$\end{document}

Then, using (2.17), (2.18), for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i, j = 1, \dots , d-1$$\end{document} , we compute

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _i N_j = \frac{\partial _{ik} g \partial _i g \partial _k g - \partial _{ij} g (1 + \left| \nabla g\right| ^2)}{(1+ \left| \nabla g\right| ^2)^{3/2}} = -\delta _{ij} k_i + O(\left| y\right| ),\\ \partial _i g + z \partial _i N_d = \left( 1- \frac{z}{(1+ \left| \nabla g\right| ^2)^{3/2}} \right) \partial _i g = O(\left| y\right| ).\\ N_i = \frac{-\partial _i g}{\sqrt{1 + \left| \nabla g\right| ^2}} = O(\left| y\right| ), \ \ \ N_d = 1+ O(\left| y\right| ^2) \end{aligned}$$\end{document}

where g, N are always computed at y. Hence, we obtain that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \nabla \Phi (y,z) = \begin{pmatrix} 1 - z k_1 + O(\left| y\right| \left| z\right| ) & \cdots & O(\left| y\right| \left| z\right| ) & O(\left| y\right| ) \\ O(\left| y\right| \left| z\right| ) & \cdots & O(\left| y\right| \left| z\right| ) & O(\left| y\right| ) \\ \vdots & \ddots & \vdots & \vdots \\ O(\left| y\right| \left| z\right| ) & \cdots & 1 - z k_{d-1} + O(\left| y\right| \left| z\right| ) & O(\left| y\right| ) \\ O(\left| y\right| ) & \cdots & O(\left| y\right| ) & 1+ O(\left| y\right| ^2) \end{pmatrix}. \end{aligned}$$\end{document}

Evaluating at (0, z), then (2.22) is proved. To compute the determinant, expanding with respect to the last row, it results that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \det (\nabla \Phi (y,z)) = \prod _{i=1}^{d-1} (1- z k_i + O(\left| y\right| \left| z\right| )) + O(\left| y\right| ^2) + O(\left| y\right| \left| z\right| ), \end{aligned}$$\end{document}

thus proving (2.23). To conclude, we check the validity of (2.24). By (2.19), (2.16), (2.17) it holds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\left| (Y(y) - z_0 e_d )_\tau \right| ^2 = \sum _{i=1}^{d-1} y_i^2 + (g(y) - z_0)^2 - \frac{1}{1 + \left| \nabla g(y)\right| ^2} \left( - \sum _{i=1}^{d-1} y_i \partial _i g(y) + (g(y) - z_0) \right) ^2\\&\quad = \left| y\right| ^2 + (g(y)-z_0)^2 - (1- \left| \nabla g(y)\right| ^2 + O(\left| y\right| ^4)) \left( - 2g(y) + O(\left| y\right| ^3) + g(y)-z_0 \right) ^2\\&\quad = \left| y\right| ^2 + (g(y)-z_0)^2 - (g(y)+z_0 + O(\left| y\right| ^3))^2 + \left| \nabla g(y)\right| ^2 (g(y)+z_0 + O(\left| y\right| ^3))^2 + O(\left| y\right| ^4)\\&\quad = \left| y\right| ^2 + (g(y)-z_0)^2 - (g(y)+z_0)^2 + \sum _{i=1}^{d-1} y_i^2 k_i ^2 (g(y)+z_0 + O(\left| y\right| ^3))^2 + O(\left| z_0\right| \left| y\right| ^3)\\&\quad = \left| y\right| ^2 - 4g(y) z_0 + \sum _{i=1}^{d-1}y_i^2 k_i ^2 z_0^2 + O(\left| z_0\right| \left| y\right| ^3) + O(\left| y\right| ^4)\\&\quad = \left| y\right| ^2 - 2 \sum _{i=1}^{d-1} y_i^2 k_i z_0 + \sum _{i=1}^{d-1}y_i^2 k_i ^2 z_0^2 + O(\left| z_0\right| \left| y\right| ^3) + O(\left| y\right| ^4), \end{aligned}$$\end{document}

thus proving (2.24).

The following lemma is needed to justify the change of variables in the proof of Theorem 4.1.

Lemma 2.17

Under the assumptions of Lemma 2.15, there exists a constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _0\ge 1$$\end{document} depending on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} such that for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \ge \Lambda _0$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0 \in \Sigma _{{\delta }/{10\Lambda }}$$\end{document} it holds that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathscr {B}}_{{\delta }/{\Lambda }}^{d-1}(z_0):= \left\{ y \in \mathbb {R}^{d-1} :\left| (\textrm{Id}- z_0 \nabla ^2 g(0)) y\right| < {\delta }/{\Lambda } \right\} \subset B_{{\delta }/{2} }^{d-1}, \end{aligned}$$\end{document}

where as before \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_0 = \textrm{dist}_{\Sigma }(x_0)$$\end{document} . In addition, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y \in {\mathscr {B}}_{{\delta }/{\Lambda } }^{d-1}(z_0)$$\end{document} it holds that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathscr {I}}_{{\delta }/{\Lambda } }(y,z_0):= \left\{ z \in \mathbb {R}:\left| z-z_0 + Y(y)\cdot N(y) + z_0(1-N_d(y))\right| < {\delta }/{\Lambda } \right\} \subset \left( -{\delta }/{2}, {\delta }/{2} \right) . \end{aligned}$$\end{document}

Then, denoting by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {B}_{{\delta }/{\Lambda }} (z_0):= \left\{ (y,z) \in {\mathscr {B}}_{{\delta }/{\Lambda }}^{d-1}(z_0) \times \mathbb {R}:z \in {\mathscr {I}}_{{\delta }/{\Lambda }}(y, z_0) \right\} , \end{aligned}$$\end{document}

it holds that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}_{{\delta }/{\Lambda }}(z_0) \subset B_{{\delta }/{2}}^{d-1}\times \left( -{\delta }/{2}, {\delta }/{2} \right) $$\end{document} . Finally, we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{{\delta }/{10 \Lambda }} (x_0) \subset \Phi (\mathcal {B}_{{\delta }/{\Lambda }}(z_0))$$\end{document} .

Proof

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \ge 1$$\end{document} be a constant, which will be fixed shortly. We remark that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| k_i \right| \le C $$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1, \dots , d-1$$\end{document} (see (2.15)). We know that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| z_0\right| \le {\delta }/{10 \Lambda }$$\end{document} , therefore for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y \in {\mathscr {B}}_{{\delta }/{\Lambda }}^{d-1}(z_0)$$\end{document} , by the triangular inequality, it is clear that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| y\right| \le \frac{\delta }{\Lambda } \left( 1- \frac{C \delta }{10 \Lambda } \right) ^{-1}. \end{aligned}$$\end{document}

Hence, (2.25) is satisfied provided that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\delta }{\Lambda } \left( 1- \frac{C \delta }{10 \Lambda } \right) ^{-1} \le \frac{\delta }{2}. \end{aligned}$$\end{document}

From Lemma 2.15, there exists a constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{C}} > 0$$\end{document} depending on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} such that for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \ge 1$$\end{document} we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| Y(y) \cdot N(y) + z_0( 1- N_d(y) )\right| \le {\bar{C}} \left| y\right| ^2 \qquad \forall y \in {\mathscr {B}}^{d-1}_{{\delta }/{\Lambda }}(z_0). \end{aligned}$$\end{document}

Thus, given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y \in {\mathscr {B}}^{d-1}_{{\delta }/{\Lambda }}(z_0)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ z \in {\mathscr {I}}_{{\delta }/{\Lambda }}(y, z_0)$$\end{document} , by the triangular inequality, (2.28) and (2.30) it holds that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| z\right| \le \frac{\delta }{\Lambda } + \frac{\delta }{10 \Lambda } + {\bar{C}} \frac{\delta ^2}{\Lambda ^2} \left( 1- \frac{C \delta }{10 \Lambda } \right) ^{-2}. \end{aligned}$$\end{document}

Hence, we infer that (2.26) is satisfied provided that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\delta }{\Lambda } + \frac{\delta }{10 \Lambda } + {\bar{C}} \frac{\delta ^2}{\Lambda ^2} \left( 1- \frac{C \delta }{10 \Lambda } \right) ^{-2} \le \frac{\delta }{2}. \end{aligned}$$\end{document}

To summarize, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} satisfies (2.29) and (2.31), then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}_{{\delta }/{\Lambda }}(z_0) \subset B_{{\delta }/{2}}^{d-1}\times \left( -{\delta }/{2}, {\delta }/{2}\right) $$\end{document} . More precisely, using again the triangular inequality, (2.28) and (2.30), it can be checked in the same way that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ B^{d-1}_{{\delta }/{2\Lambda }} \times \left( z_0 - {\delta }/{2\Lambda }, z_0 + {\delta }/{2\Lambda } \right) \subset \mathcal {B}_{{\delta }/{\Lambda }}(z_0) $$\end{document} provided that the following condition is satisfied:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \max \left\{ \frac{\delta }{2\Lambda }\left( 1+ \frac{C \delta }{10 \Lambda } \right) , \frac{\delta }{2\Lambda } + {\bar{C}} \frac{\delta ^2}{4\Lambda ^2} \right\} \le \frac{\delta }{\Lambda }. \end{aligned}$$\end{document}

Lastly, to prove that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{{\delta }/{10 \Lambda }} (x_0) \subset \Phi (\mathcal {B}_{{\delta }/{\Lambda }}(z_0))$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} sufficiently large, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \in [0,1]$$\end{document} we define the map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{\Phi }}_t :B^{d-1}_{{\delta }/{2\Lambda }} \times \left( z_0 - {\delta }/{2\Lambda }, z_0 + {\delta }/{2\Lambda } \right) \rightarrow \mathbb {R}^d$$\end{document} as follows

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\widehat{\Phi }}_t(y,z):= t (\Phi (y,z) - x_0) + (1-t) (y,z-z_0). \end{aligned}$$\end{document}

Hence, using (2.22) we shall write

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\widehat{\Phi }}_t(y,z) = (y, z-z_0) - t z_0 (k_1 y_1, \dots , k_{d-1} y_{d-1}, 0) + O(\left| y\right| ^2 + \left| z-z_0\right| ^2). \end{aligned}$$\end{document}

The previous formula implies that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} is sufficiently large, then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |{\widehat{\Phi }}_t(y,z)| > {\delta }/{10 \Lambda }, \qquad \forall t \in [0,1], \forall (y,z) \in \partial \left( B_{{\delta }/{2 \Lambda }}^{d-1} \times (z_0 - {\delta }/{ 2 \Lambda }, z_0 + {\delta }/{2 \Lambda }) \right) .\nonumber \\ \end{aligned}$$\end{document}

Therefore, for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \in B_{{\delta }/{ 10 \Lambda }}(0)$$\end{document} , the standard properties of the degree imply that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textrm{deg}\Big ({\widehat{\Phi }}_1,p,B_{{\delta }/{2 \Lambda }}^{d-1} \times (z_0 - {\delta }/{ 2 \Lambda }, z_0 + {\delta }/{2 \Lambda })\Big )\\ = \textrm{deg}\Big ({\widehat{\Phi }}_0,p,B_{{\delta }/{2 \Lambda }}^{d-1} \times (z_0 - {\delta }/{ 2 \Lambda }, z_0 + {\delta }/{2 \Lambda })\Big ) = 1,\nonumber \\ \end{aligned}$$\end{document}

and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{{\delta }/{10 \Lambda }} (x_0) \subset \Phi (\mathcal {B}_{{\delta }/{\Lambda }}(z_0))$$\end{document} . In particular, (2.29), (2.31)–(2.33) are satisfied if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document} is large enough. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Approximation of sets

The proof of Theorem 1.1 is more direct when the set E is smooth and intersects the boundary of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} transversely (in a measure theoretic sense). To handle the general case, we approximate E with smooth bounded open sets such that both the Perimeter and the Willmore energy of the approximating sets in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} converge to those of E. When \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega = \mathbb {R}^d$$\end{document} there are several ways to construct such an approximation, see e.g. [6] and the references therein. In the following lemma we show that the same conclusion holds when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} is a bounded open set of class  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^1$$\end{document} . Moreover, the approximating sets that we consider intersect the boundary of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} transversely.

Lemma 2.18

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E \subset \mathbb {R}^d$$\end{document} be a bounded open set with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial E \in C^2$$\end{document} . For any bounded open set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}^d$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega \in C^1$$\end{document} , there exists a sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ E_j \}_{j \in \mathbb {N}}$$\end{document} of smooth bounded open sets of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^d$$\end{document} such that (A1) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaystyle \lim _{j \rightarrow \infty } \left| E_j \Delta E\right| = 0$$\end{document} ,(A2) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaystyle \lim _{j \rightarrow \infty } \textrm{Per}(E_j,\Omega ) = \textrm{Per}(E,\Omega )$$\end{document} ,(A3) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaystyle \lim _{j \rightarrow \infty } \mathcal {W}(\partial E_j, \Omega ) = \mathcal {W}(\partial E, \Omega )$$\end{document} ,(A4) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaystyle \lim _{j \rightarrow \infty } \mathcal {H}^{d-1}(\partial E_j \cap \partial \Omega ) = 0$$\end{document} ,(A5) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaystyle \sup _{j \in \mathbb {N}} \Vert {H_{\partial E_j}} \Vert _{C^0(\partial E_j)} \le \Vert {H_{\partial E}} \Vert _{C^0(\partial E)} +1.$$\end{document}

Proof

To begin, we show that there exists a sequence of sets of class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2$$\end{document} with the above properties. Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega $$\end{document} is of class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^1$$\end{document} , then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{\partial \Omega } \in C(\partial \Omega ; \mathbb {S}^{d-1})$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{\partial \Omega }$$\end{document} denotes the inner unit normal to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} . Hence, we find a smooth vector field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X \in C^{\infty }_c(\mathbb {R}^d; \mathbb {R}^d)$$\end{document} such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \langle X(x), N_{\partial \Omega }(x) \rangle \le - \frac{1}{2}, \qquad \forall x \in \partial \Omega . \end{aligned}$$\end{document}

For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t >0$$\end{document} , we define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_t(x):=x+tX(x)$$\end{document} . In particular, since X has compact support, there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_0 > 0$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_t$$\end{document} is a diffeomorphism of class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^\infty $$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \in (0,t_0]$$\end{document} . We consider the set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} E_t:=f_t(E), \end{aligned}$$\end{document}

and we claim that there exists a sequence of positive real numbers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ t_j \}_{j \in \mathbb {N}}$$\end{document} converging to zero for which the corresponding sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_j:=E_{t_j}$$\end{document} fulfill (A1), (A2), (A3), (A4) and (A5).

Letting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_t:= f_t^{-1}$$\end{document} , we prove that for t small enough \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_t$$\end{document} pushes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} inside, that is there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_1 \in (0,t_0]$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_t(\Omega ) \subset \Omega $$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \in (0,t_1]$$\end{document} . Suppose by contradiction that there exist a sequence of positive real numbers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ t_n \}_{n \in \mathbb {N}}$$\end{document} converging to zero and a sequence of points \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ y_n \}_{n \in \mathbb {N}} \subset \Omega $$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_n:=g_{t_n}(y_n) \in \mathbb {R}^{d} \setminus \Omega $$\end{document} . Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} is bounded, we can assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_n$$\end{document} converges to a limit point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_{\infty } \in {\overline{\Omega }}$$\end{document} . On the other hand, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_t$$\end{document} converges uniformly to the identity map as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \rightarrow 0$$\end{document} , therefore \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_n$$\end{document} converges to the same limit point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_{\infty }$$\end{document} . Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_n \in \mathbb {R}^{d} \setminus \Omega $$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \in \mathbb {N}$$\end{document} , we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_{\infty } \in \partial \Omega $$\end{document} . Using a local chart near \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_{\infty }$$\end{document} , we reduce to the case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_\infty = 0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega = \{ x \in \mathbb {R}^d :x_d < 0 \}$$\end{document} . Hence, it is clear that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(y_n - x_n)/t_n$$\end{document} is pointing inside \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} . On the other hand, by (2.34) and the definition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_t$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{n \rightarrow \infty } \Big \langle \frac{y_n-x_n}{t_n}, N_{\partial \Omega }(z_{\infty }) \Big \rangle & = \lim _{n \rightarrow \infty } \Big \langle \frac{f_{t_n}(x_n)-x_n}{t_n}, N_{\partial \Omega }(z_{\infty }) \Big \rangle \\ & = \langle X(z_{\infty }), N_{\partial \Omega }(z_{\infty }) \rangle \le - \frac{1}{2}, \end{aligned}$$\end{document}

which is a contradiction because \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{\partial \Omega }(z_{\infty })$$\end{document} is pointing inside \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} .

Proof of (A1). For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \in (0,t_0]$$\end{document} , we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial E_t = f_t(\partial E) \in C^2$$\end{document} since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_t$$\end{document} is a smooth diffeomorphism and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial E \in C^2$$\end{document} . Moreover, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in \mathbb {R}^d \setminus \partial E$$\end{document} , it holds that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbbm {1}_{E_t}(x) \rightarrow \mathbbm {1}_{E}(x)$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\rightarrow 0$$\end{document} . Indeed, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in \mathbb {R}^d, t \in (0,t_0]$$\end{document} we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathbbm {1}_{f_t(E)}(x) = \mathbbm {1}_{f_t(E)} (f_t(g_t(x)) = \mathbbm {1}_{E}(g_t(x)). \end{aligned}$$\end{document}

Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_t$$\end{document} converges to the identity map uniformly as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \rightarrow 0$$\end{document} and E is open, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in E$$\end{document} then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_t(x) \in E$$\end{document} we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_t(x) \in E$$\end{document} for t small enough. The same argument works for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in \mathbb {R}^d \setminus {\overline{E}}$$\end{document} , thus proving (A1).

Proof of (A2). Since the Perimeter is lower semicontinuous with respect to the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document} -convergence of sets (see e.g. [39, Proposition 12.15]), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \liminf _{t \rightarrow 0} \textrm{Per}(E_t, \Omega ) \ge \textrm{Per}(E, \Omega ). \end{aligned}$$\end{document}

We prove the opposite inequality. We set for convenience \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{E_t}:=\mathcal {H}^{d-1} \llcorner \partial E_t$$\end{document} . It is known (see e.g [39, (17.6), (17.29), (17.30)]) that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (g_t)_{\#} \mu _{E_t} = p_t \mu _E, \quad \text {with} \quad \Vert {p_t} \Vert _{C^0(\partial E)} = 1 + O(t). \end{aligned}$$\end{document}

Combining this fact with the property that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset g_t^{-1}(\Omega )$$\end{document} we obtain

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mu _{E_t} (\Omega ) \le \mu _{E_t} (g_t^{-1}(\Omega )) = (g_t)_{\#} \mu _{E_t} (\Omega ) = \mu _{E} (\Omega ) + O(t). \end{aligned}$$\end{document}

Then, (A2) follows by taking the limsup as t goes to zero in the previous inequality.

Proof of (A3) and (A5). It was proved by Schätzle (see [50]) that the Willmore functional is lower semicontinuous with respect to the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document} -convergence of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2$$\end{document} sets, that is

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \liminf _{t \rightarrow 0} \mathcal {W}(\partial E_t, \Omega ) \ge \mathcal {W}(\partial E, \Omega ). \end{aligned}$$\end{document}

In our case, there are simpler and more direct ways to show the lower semicontinuity property above. For example, taking into account (A2), then (2.36) follows by an application of Reshetnyak’s continuity theorem, see [4, Remark 2] and also [38, Lemma 2] for a particular case. At this point, we claim that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} H_{\partial E_t}(y) = H_{\partial E}(g_t(y)) + O(t), \qquad \forall y \in \partial E_t, \end{aligned}$$\end{document}

where the reminder term is uniform with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y \in \partial E_t$$\end{document} . Thus, the sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{E_t\}_{t \in (0, t_1] }$$\end{document} satisfy (A5) provided that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_1$$\end{document} is small enough. By (2.35), (2.37) and since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset g_t^{-1}(\Omega )$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {W}(\partial E_t, \Omega ) \le \mathcal {W}(\partial E_t, \Omega ) + O(t) \end{aligned}$$\end{document}

and the conclusion follows taking the limsup as t goes to zero in the inequality above. To check (2.37), let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi \in C^2(\mathbb {R}^d)$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E = \{ x :\psi (x) > 0 \}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla \psi (x) \ne 0$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in \partial E$$\end{document} . It is well known that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} N_{\partial E} = \frac{\nabla \psi }{\left| \nabla \psi \right| } \quad \hbox { on}\ \{\psi = 0\}, \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} H_{\partial E} = - \textrm{div} \bigg ( \frac{\nabla \psi }{\left| \nabla \psi \right| } \bigg ) = - \frac{\Delta \psi }{\left| \nabla \psi \right| } + \frac{\nabla ^2 \psi [\nabla \psi , \nabla \psi ]}{\left| \nabla \psi \right| ^3} \quad \hbox { on}\ \{\psi = 0\}, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{\partial E}$$\end{document} denotes the inner unit normal to E. In particular, the right-hand sides in the previous identities do not depend on the particular choice of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document} . For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \in (0,t_1]$$\end{document} , we define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi _t \in C^2(\mathbb {R}^d)$$\end{document} by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi _t:= \psi \circ g_t$$\end{document} . It is clear that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_t = \{ y :\psi _t(y) > 0 \}$$\end{document} and that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla \psi _t (y) \ne 0$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y \in \partial E_t$$\end{document} . We set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_t(y):= \nabla g_t(y)$$\end{document} and we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_t^*(y)$$\end{document} its transpose matrix. A direct computation shows

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \nabla \psi _t(y) = G_t^*(y) \nabla \psi (g_t(y)), \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \nabla ^2 \psi _t(g_t(y)) = G_t^*(y) \nabla ^2 \psi (g_t(y)) G_t(y) + \langle \nabla \psi (g_t(y)), \nabla ^2 g_t (y) \rangle , \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla ^2 g_t$$\end{document} is the vector valued Hessian of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_t$$\end{document} . Combining (2.39)–(2.41) and the fact that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert {g_t - \textrm{Id}} \Vert _{C^2(\mathbb {R}^d)} \rightarrow 0$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \rightarrow 0$$\end{document} we deduce (2.37).

Proof of (A4). For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_2 \in (0,t_1]$$\end{document} , we define the set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {N}(t_2):=\{ t \in (0,t_2] :\mathcal {H}^{d-1}(\partial E_t \cap \partial \Omega ) > 0 \}, \end{aligned}$$\end{document}

and we claim that there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_2 \in (0,t_1]$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {N}(t_2)$$\end{document} is at most countable. If this is the case, then we find a sequence of positive real numbers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ t_j \}_{j \in \mathbb {N}}$$\end{document} converging to zero for which the corresponding sets satisfy a stronger property than (A4), namely \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}^{d-1}(\partial E_{t_j} \cap \partial \Omega ) = 0$$\end{document} .

We write \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial E_t \cap \partial \Omega = A_t \cup B_t$$\end{document} , where

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A_t&:= \{ z \in \partial E_t \cap \partial \Omega :T_z \partial E_t = T_z \partial \Omega \} = \{ z \in \partial E_t \cap \partial \Omega :\left| \langle N_{\partial E_t}(z) , N_{\partial \Omega }(z) \rangle \right| = 1 \}, \\ B_t&:= \{ z \in \partial E_t \cap \partial \Omega :T_z \partial E_t \ne T_z \partial \Omega \} = \{ z \in \partial E_t \cap \partial \Omega :\left| \langle N_{\partial E_t}(z) , N_{\partial \Omega }(z) \rangle \right| < 1 \}, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{\partial E_t}$$\end{document} denotes the inner unit normal to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_t$$\end{document} . It is well known (see e.g. [30, Section 1, Theorem 3.3]) that the transverse intersection of two submanifolds of codimensions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_2$$\end{document} is either empty or a submanifold of codimension \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_1+k_2$$\end{document} . Therefore, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \in (0,t_1]$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_t$$\end{document} is either empty or a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(d-2)$$\end{document} -dimensional submanifold of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^d$$\end{document} . In both cases, we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}^{d-1}(B_t)=0$$\end{document} . We prove that there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_2 \in (0,t_1]$$\end{document} such that for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t,s \in (0, t_2]$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \ne s$$\end{document} we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_t \cap A_s = \emptyset $$\end{document} . Suppose by contradiction that there exist two sequences of positive real numbers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0< s_n < t_n$$\end{document} , with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_n \rightarrow 0$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \rightarrow \infty $$\end{document} , and two sequences of points \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ x_n \}, \{ y_n \} \subset \partial E$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{t_n}(x_n) = f_{s_n}(y_n) \in \partial \Omega $$\end{document} . Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial E$$\end{document} is compact, up to subsequences, we may assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_n \rightarrow z_{\infty } \in \partial E$$\end{document} . Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_n = f_{t_n}(x_n) - t_n X(x_n)$$\end{document} , it follows that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_n:=f_{t_n}(x_n) \rightarrow z_{\infty }$$\end{document} and we infer that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_{\infty } \in \partial \Omega $$\end{document} . Moreover, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| x_n - y_n\right| = \left| t_n X(x_n) - s_n X(y_n)\right| \le 2 t_n \max \{ \left| X(x)\right| :x \in \partial E \}, \end{aligned}$$\end{document}

therefore \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_n \rightarrow z_\infty $$\end{document} . We claim that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_{\infty } \in A_0$$\end{document} . Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_n=f_{t_n}(x_n) \in A_{t_n}$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \in \mathbb {N}$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \langle N_{\partial E_t}(z_n), N_{\partial \Omega }(z_n) \rangle \right| = 1. \end{aligned}$$\end{document}

On the other hand, from (2.38) and (2.40) we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} N_{\partial E_t}(z) = \frac{G_t^*(z)N_{\partial E}(g_t(z))}{\left| G_t^*(z)N_{\partial E}(g_t(z))\right| }, \qquad \forall z \in \partial E_t. \end{aligned}$$\end{document}

Therefore, we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{\partial E_t}(z_n) \rightarrow N_{\partial E}(z_{\infty })$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \rightarrow \infty $$\end{document} , since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_n \rightarrow z_{\infty }$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_t$$\end{document} converges in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^1$$\end{document} to the identity map as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \rightarrow 0$$\end{document} . Then, by (2.42) we derive

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \langle N_{\partial E}(z_{\infty }), N_{\partial \Omega }(z_{\infty }) \rangle \right| = \lim _{n \rightarrow \infty } \left| \langle N_{\partial E_t}(z_n), N_{\partial \Omega }(z_n) \rangle \right| = 1. \end{aligned}$$\end{document}

This proves that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_{\infty } \in A_0 = \{ z \in \partial E \cap \partial \Omega :\left| \langle N_{\partial E}(z), N_{\partial \Omega }(z) \rangle \right| = 1 \}$$\end{document} .

At this point, we claim that the map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F :\partial E \times \mathbb {R}\rightarrow \mathbb {R}^d$$\end{document} defined as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(x,t):=f_t(x)$$\end{document} is a local diffeomorphism around \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(z_{\infty },0)$$\end{document} . If this is the case, then we find a contradiction since, for n large, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(x_n, t_n)=F(y_n,s_n)$$\end{document} implies that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_n, t_n)=(y_n,s_n)$$\end{document} , but \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_n < t_n$$\end{document} . To prove the claim we have to check that the differential of F at the point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(z_{\infty },0)$$\end{document} is surjective. It is not difficult to see that the image of the differential at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(z_\infty , 0)$$\end{document} is given by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} V:=T_{z_{\infty }} \partial E \oplus \textrm{Span} (X(z_{\infty })). \end{aligned}$$\end{document}

Now, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V=\mathbb {R}^d$$\end{document} if and only if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle X(z_{\infty }), N_{\partial E}(z_{\infty }) \rangle \ne 0$$\end{document} . Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_{\infty } \in A_0$$\end{document} we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| \langle X(z_{\infty }), N_{\partial E}(z_{\infty }) \rangle \right| = \left| \langle X(z_{\infty }), N_{\partial \Omega }(z_{\infty }) \rangle \right| $$\end{document} and the latter is different from zero because of (2.34).

Building smooth sets. To summarize, we have constructed a sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{ E_j \}_{j \in \mathbb {N}}$$\end{document} of bounded open sets of class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2$$\end{document} satisfying (A1), (A2), (A3), (A5) and the additional property \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}^{d-1}(\partial E_j \cap \partial \Omega ) = 0$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \in \mathbb {N}$$\end{document} . To conclude, we want to pass from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2$$\end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^\infty $$\end{document} sets. For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \in \mathbb {N}$$\end{document} , there exists a sequence of smooth bounded open sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{k,j}$$\end{document} converging to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_j$$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1(\mathbb {R}^d)$$\end{document} , as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \rightarrow \infty $$\end{document} , and such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{k \rightarrow \infty } \textrm{Per}(E_{k,j}, \mathbb {R}^d) = \textrm{Per}(E_j, \mathbb {R}^d) \quad \text {and} \quad \lim _{k \rightarrow \infty } \mathcal {W}(\partial E_{k,j}, \mathbb {R}^d) = \mathcal {W}(\partial E_j, \mathbb {R}^d). \end{aligned}$$\end{document}

Moreover, the sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{E_{j,k}\}_{k\in \mathbb {N}}$$\end{document} can be chosen such that (A5) is satisfied, with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\partial E_j}$$\end{document} at the right-hand side. We refer e.g. to [6] for a rigorous proof of this fact and more general results about the approximation by smooth sets on the whole Euclidean space. By \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}^{d-1}(\partial E_j \cap \partial \Omega ) = 0$$\end{document} , (2.43) and the lower semicontinuity of the Perimeter and the Willmore functional on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} and on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^d \setminus {\overline{\Omega }}$$\end{document} , we infer

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{k \rightarrow \infty } \textrm{Per}(E_{k,j}, \Omega ) = \textrm{Per}(E_j, \Omega ) \quad \text {and} \quad \lim _{k \rightarrow \infty } \mathcal {W}(\partial E_{k,j}, \Omega ) = \mathcal {W}(\partial E_j, \Omega ). \end{aligned}$$\end{document}

Moreover, by the upper semicontinuity of the evaluation on closed sets with respect to the weak convergence of measures we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \limsup _{k \rightarrow \infty } \mathcal {H}^{d-1}(\partial E_{k,j} \cap \partial \Omega ) \le \mathcal {H}^{d-1}(\partial E_j \cap \partial \Omega ) = 0. \end{aligned}$$\end{document}

The conclusion follows from (2.44), (2.45) and a diagonal argument. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

On the decay of optimal profile

In this section, we discuss the proof of Theorem 1.2. Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w'$$\end{document} solves the fractional Allen–Cahn equation, then for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda > 0$$\end{document} we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} ((-\Delta )^s + \lambda ) w' = (\lambda - W''(w)) w', \end{aligned}$$\end{document}

The proof of Theorem 1.2 relies on the decay properties of the fundamental solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{s,\lambda }$$\end{document} of the operator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda + (-\Delta )^s$$\end{document} , whose symbol is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda + \left| 2\pi \xi \right| ^{2s}$$\end{document} . Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{s,\lambda }$$\end{document} is formally defined by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} G_{s,\lambda }(x) = {\mathscr {F}}_1^{-1} \left( \left( \lambda + \left| 2 \pi \xi \right| ^{2s} \right) ^{-1}\right) (x) = \int _{\mathbb {R}} \frac{e^{2\pi ix \xi }}{ \lambda + \left| 2\pi \xi \right| ^{2s} } \, d\xi , \end{aligned}$$\end{document}

and for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi \in \mathbb {R}$$\end{document} we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1}{ \lambda + \left| 2\pi \xi \right| ^{2s} } = \int _0^{+\infty } e^{-\lambda t} \exp \left( - t \left| 2 \pi \xi \right| ^{2s} \right) \, dt, \end{aligned}$$\end{document}

then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{s,\lambda }$$\end{document} formally satisfies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} G_{s,\lambda }(x) = \int _{\mathbb {R}} e^{2\pi i x \xi } \int _0^{+\infty } e^{-\lambda t}\exp \left( - t \left| 2\pi \xi \right| ^{2s}\right) \, dt \, d \xi = \int _{0}^{+\infty } e^{-\lambda t} P^{(s)}_1 (t, x) \, dt. \end{aligned}$$\end{document}

The above computation can be made rigorous.

Proposition 3.1

Fix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in (0,1)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda >0$$\end{document} . For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in \mathbb {R}$$\end{document} , set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} G_{s,\lambda }(x): = \int _{0}^{+\infty } e^{-\lambda t} P^{(s)}_1(t,x) \, dt. \end{aligned}$$\end{document}

Then, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{s,\lambda } \in L^1(\mathbb {R})$$\end{document} and it holds that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathscr {F}}_1(G_{s,\lambda })(\xi ) = \frac{1}{ \lambda + \left| 2\pi \xi \right| ^{2s} } \qquad \forall \xi \in \mathbb {R}. \end{aligned}$$\end{document}

Moreover, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{s,\lambda } \in C^{\infty }(\mathbb {R}{\setminus } \{0\})$$\end{document} and for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \ge 0 $$\end{document} it holds that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \partial _x^k G_{s,\lambda } (x)\right| \le C(s,\lambda , k) \left| x\right| ^{-k-1-2s} \qquad \forall x\ne 0. \end{aligned}$$\end{document}

Proof

To begin, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P^{(s)}_1(t, x) >0$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(t,x) \in (0, +\infty ) \times \mathbb {R}$$\end{document} , we notice that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{s,\lambda }(x)$$\end{document} is always well defined with values in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0, +\infty ]$$\end{document} . Then, by (2.6) and Fubini’s theorem, it results that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{\mathbb {R}} G_{s,\lambda }(x)&= \int _0^{+\infty } \int _{\mathbb {R}} e^{-\lambda t} t^{-\frac{1}{2s}} P^{(s)}_1\left( 1, t^{-\frac{1}{2s}}x\right) \, dx \, dt \\&= \int _0^{+\infty } e^{-\lambda t} \int _{\mathbb {R}} P^{(s)}_{1} (1, y)\, dy \, dt < +\infty . \end{aligned}$$\end{document}

From now on, we neglect constants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(k,s,\lambda )>0$$\end{document} . Moreover, by Proposition 2.4, we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} G_{s,\lambda }(x) \lesssim \int _{0}^{+\infty } e^{-\lambda t} t^{-\frac{1}{2s}} \frac{1}{1+ \left| x\right| ^{2s+1} t^{-\frac{2s+1}{2s}} }\, dt \lesssim \left| x\right| ^{-1-2s}\int _{0}^{+\infty } e^{-\lambda t} t\, dt \lesssim \left| x\right| ^{-1-2s}, \end{aligned}$$\end{document}

thus proving (3.3) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k = 0$$\end{document} . To compute the Fourier transform, by Fubini’s theorem and the inversion formula in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} , we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathscr {F}}_1( G_{s,\lambda })(\xi )&= \int _0^{+\infty } e^{-\lambda t} {\mathscr {F}}_1 \left( P_1^{(s)}(t, \cdot )\right) (\xi ) \, dt \\ &= \int _{0}^{+\infty } e^{-\lambda t} \exp \left( - \left| 2\pi \xi \right| ^{2s} t \right) \, dt = \frac{1}{\lambda + \left| 2 \pi \xi \right| ^{2s} }, \end{aligned}$$\end{document}

thus proving (3.2). To prove that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{s,\lambda }$$\end{document} is smooth away from the origin, we check that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _0^{+\infty } e^{-\lambda t} \left| \partial _x^k P^{(s)}_1(t,x)\right| \, dt \lesssim \left| x\right| ^{-k-1-2s} \qquad \forall k\in \mathbb {N}\qquad \forall x \ne 0. \end{aligned}$$\end{document}

Indeed, by (2.6) and Proposition 2.5 we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _0^{+\infty } e^{-\lambda t} \left| \partial _x^k P^{(s)}_1(t,x)\right| \, dt&= \int _0^{+\infty } e^{-\lambda t} t^{-\frac{k+1}{2s}} \left| \partial _x^k P^{(s)}_1(1, t^{-\frac{1}{2s}} x) \right| \, dt \\ &\lesssim \int _0^{+\infty } e^{-\lambda t} t^{-\frac{k+1}{2s}} \frac{1}{1+ t^{-\frac{k+1+2s}{2s}} \left| x\right| ^{-k-1-2s}} \, dt \\ &\lesssim \left| x\right| ^{-k-1-2s} \int _0^{+\infty } e^{-\lambda t} t \, dt, \end{aligned}$$\end{document}

thus proving (3.4). Then, it is easy to check that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{s,\lambda }$$\end{document} is smooth away from the origin and it holds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial ^k_x G_{s,\lambda }(x) = \int _0^{+\infty } e^{-\lambda t} \partial _x^k P^{(1)}_s(t,x) \, dt. \end{aligned}$$\end{document}

Thus, (3.3) follows by (3.4). \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

The proof of Theorem 1.2 follows from Proposition 3.1.

Proof of Theorem 1.2

We perform the proof by induction. We start with the case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=2$$\end{document} . We neglect constants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(k,s, W)>0$$\end{document} . Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w'$$\end{document} solves (3.5), iterating the estimates in [53, Proposition 2.8 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-$$\end{document} 2.9], it is readily checked that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w \in W^{2, \infty }(\mathbb {R})$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert {w} \Vert _{W^{2,\infty }(\mathbb {R})} \lesssim 1$$\end{document} . Letting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda = W''(\pm 1)$$\end{document} , since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w' \in L^\infty (\mathbb {R})$$\end{document} solves

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (\lambda + (-\Delta )^s) w' = (\lambda - W''(w)) w', \end{aligned}$$\end{document}

then by (3.2) we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} w'(x) = \int _{\mathbb {R}} G_{s,\lambda }(x-z) \Psi (z) w'(z) \, dz, \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{s,\lambda }$$\end{document} is given by (3.1) and we set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi (z) = \lambda - W''(w(z))$$\end{document} . Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{s,\lambda }$$\end{document} is smooth away from 0 (see Proposition 3.1), for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x > 1$$\end{document} we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} w''(x)&= \int _{\left| x-z\right| \ge \frac{x}{2}} G_{s,\lambda } (x-z) \partial _z [ \Psi (z) w'(z) ] \, dz + \int _{\left| x-z\right| < \frac{x}{2}} G_{s,\lambda } (x-z) \partial _z [ \Psi (z) w'(z) ] \, dz \\ &= \int _{|x-z|\ge \frac{x}{2}} \partial _x G_{s,\lambda } (x-z) \Psi (z) w'(z)\,dz -\Bigg [ G_{s,\lambda }(x-z)\Psi (z) w'(z) \Bigg ]_{\frac{x}{2}}^{\frac{3x}{2}} \\ &\qquad +\int _{\frac{x}{2}}^{\frac{3x}{2}} G_{s,\lambda } (x-z) \left[ -W'''(w(z))(w'(z))^2 + (\lambda -W''(w(z)))w''(z) \right] \, dz. \end{aligned}$$\end{document}

By the decay properties of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{s,\lambda }, \partial _x G_{s,\lambda }$$\end{document} (see Proposition 3.1) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w \in W^{2,\infty }(\mathbb {R})$$\end{document} , we infer that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| w''(x)\right| \lesssim \left| x\right| ^{-2-2s} + \left| x\right| ^{-2s} \Vert {w''} \Vert _{L^\infty ([{x}/{2},{3x}/{2}])}. \end{aligned}$$\end{document}

Then, letting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h > 1+ {1}/{s}$$\end{document} be an integer and iterating h times the above estimate, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x> 2^{h+1}$$\end{document} we find that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| w''(x)\right| \lesssim \left| x\right| ^{-2-2s} + \left| x\right| ^{-2s h} \Vert {w''} \Vert _{L^\infty ([2^{-h}x,2^{h}x])} \lesssim \left| x\right| ^{-2-2s} \left( 1 + \Vert {w''} \Vert _{L^\infty ([1, +\infty ))}\right) , \end{aligned}$$\end{document}

thus proving (1.11) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x > 2^{h+1}$$\end{document} . The estimate for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \le -2^{h+1}$$\end{document} is analogous, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w''$$\end{document} is odd, and the case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in [-2^{h+1},2^{h+1}]$$\end{document} is trivial, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w'' \in L^\infty (\mathbb {R})$$\end{document} . Then, (1.11) is proved for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=2$$\end{document} .

Fix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \ge 2$$\end{document} and assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W \in C^{k+2}(\mathbb {R})$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w \in C^{k}(\mathbb {R})$$\end{document} and (1.11) is proved for any derivative of order smaller than or equal to k. We prove that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w\in C^{k+1}(\mathbb {R})$$\end{document} and (1.11) holds for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _x^{k+1} w$$\end{document} . Differentiating k times (1.7), we find that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (-\Delta )^s \partial _x^{k} w = -\partial _x^{k} W'(w), \end{aligned}$$\end{document}

where the right-hand side satisfies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert {\partial _x^{k} W'(w) } \Vert _{L^\infty (\mathbb {R})} \lesssim 1$$\end{document} . Hence, iterating [53, Proposition 2.8 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-$$\end{document} 2.9], it is readily checked that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _x^k w \in C^{1}(\mathbb {R})$$\end{document} and it holds \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert {\partial _x^{k+1} w} \Vert _{L^\infty (\mathbb {R})} \lesssim 1$$\end{document} . Then, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{s,\lambda }$$\end{document} is smooth away from the origin (see Proposition 3.1), by (3.6) and integrating by parts k times, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x>1$$\end{document} we find that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \partial _x^{k+1} w(x)&= \int _{\left| x-z\right| \ge \frac{x}{2}} G_{s,\lambda }(x-z) \partial _z^k[\Psi (z) w'(z)] \, dz \\&+ \int _{\left| x-z\right|< \frac{x}{2}} G_{s,\lambda }(x-z) \partial _z^k[\Psi (z) w'(z)] \, dz \\ &= \int _{\left| x-z\right| \ge \frac{x}{2}} \partial _x^k G_s(x-z) \Psi (z) w'(z)\, dz \\&- \sum _{i=0}^{k-1} \bigg [ \partial _x^{i} G_{s,\lambda }(x-z) \partial ^{k-1-i}_z[\Psi (z) w'(z)] \bigg ]_{z=\frac{x}{2}}^{z= \frac{3x}{2}} \\ &\qquad + \int _{\left| x-z\right| < \frac{x}{2}} G_{s,\lambda }(x-z) \partial _z^k[\Psi (z) w'(z)] \, dz = A+ \sum _{i=0}^{k-1} B_i + C. \end{aligned}$$\end{document}

We estimate separately each term. To begin, by Proposition 3.1, we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| A\right| \lesssim \left| x\right| ^{-k-1-2s} \Vert {\Psi w'} \Vert _{L^1(\mathbb {R})} \lesssim \left| x\right| ^{-k-1-2s}. \end{aligned}$$\end{document}

For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j = 0, \dots , k$$\end{document} , by the chain rule and since (1.11) holds up to the order k, it is easy to estimate

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \partial _z^j \Psi (z)\right| \lesssim \left| z\right| ^{-j-2s}. \end{aligned}$$\end{document}

Then, fix an index \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i = 0, \dots , k-1$$\end{document} . By Leibniz rule, we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \partial ^i_z [\Psi (z) w'(z)]\right|&\le \sum _{j=0}^i \left( {\begin{array}{c}i\\ j\end{array}}\right) \left| \partial _z^j\Psi (z)\right| \left| \partial _z^{i+1-j} w(z)\right| \lesssim \left| z\right| ^{-i-1-4s}. \end{aligned}$$\end{document}

Therefore, by Proposition 3.1 and (3.8), we infer that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{i=0}^{k-1} \left| B_i\right| \lesssim \left| x\right| ^{-k-1-6s}. \end{aligned}$$\end{document}

Lastly, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{s,\lambda } \in L^1(\mathbb {R})$$\end{document} , using (1.11) up to order k and by (3.7), we estimate

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| C\right|&\lesssim \sum _{j=1}^{k-1} \Vert {(\partial _z^j \Psi ) ( \partial _z^{k-j+1} w ) } \Vert _{L^\infty ([{x}/{2}, {3x}/{2}])} + \Vert { \Psi \partial _z^{k+1} w } \Vert _{L^\infty ([{x}/{2}, {3x}/{2}])} \\ &\lesssim \left| x\right| ^{-k-1-4s} + \left| x\right| ^{-2s} \Vert { \partial _z^{k+1} w } \Vert _{L^\infty ([{x}/{2}, {3x}/{2}])}. \end{aligned}$$\end{document}

To summarize, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x>1$$\end{document} we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \partial _x^{k+1} w(x)\right| \lesssim \left| x\right| ^{-k-1-2s} + \left| x\right| ^{-2s} \Vert { \partial _z^{k+1} w } \Vert _{L^\infty ([{x}/{2}, {3x}/{2}])}. \end{aligned}$$\end{document}

Then, letting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h > 1+ {(k+1)}/{2s}$$\end{document} be an integer and iterating h times the above estimate, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x> 2^{h+1}$$\end{document} we find that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \partial _x^{k+1 }w(x)\right| \lesssim \left| x\right| ^{-k-1-2s} + \left| x\right| ^{-2s h} \Vert {\partial _x^{k+1 }w} \Vert _{L^\infty ([2^{-h}x,2^{h}x])}\\ \lesssim \left| x\right| ^{-k-1-2s} \left( 1 + \Vert {\partial _x^{k+1 }w} \Vert _{L^\infty ([1, +\infty ))}\right) , \end{aligned}$$\end{document}

thus proving (1.11) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x > 2^{h+1}$$\end{document} for the derivative of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k+1$$\end{document} . The estimate for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \le -2^{h+1}$$\end{document} is analogous, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _x^{k+1} w$$\end{document} is odd or even, and the case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in [-2^{h+1},2^{h+1}]$$\end{document} is trivial, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _x^{k+1} w$$\end{document} is uniformly bounded. Then, the proof is concluded. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Remark 3.2

Using [53, Proposition 2.8 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-$$\end{document} 2.9] as in the proof of Theorem 1.2, assuming \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W \in C^{k,\alpha }_{\textrm{loc}}$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha + 2\,s >1$$\end{document} would still suffice to prove that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w \in C^{k, \beta }(\mathbb {R})$$\end{document} for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta >0$$\end{document} . However, the main purpose of Theorem 1.2 is to study the decay rate of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _x^k w$$\end{document} . Since an integration by part is needed in our argument to prove (1.11), we have to assume \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W \in C^{k+1}$$\end{document} .

As a corollary, we obtain the following result.

Corollary 3.3

Fix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in (0,1)$$\end{document} . Let E be a bounded open set of class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2$$\end{document} according to Definition 2.6 and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta >0$$\end{document} be given by Lemma 2.9. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_\varepsilon $$\end{document} be defined by (4.1). Then, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \ge 1$$\end{document} it holds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\varepsilon \in (0,1)} \Vert {(-\Delta )^s u_\varepsilon } \Vert _{L^\infty (\mathbb {R}^d \setminus \Sigma _{{\delta }/{\Lambda }})} \le C(d,s,W,\delta , \Lambda ). \end{aligned}$$\end{document}

Proof

We neglect multiplicative constants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(d,s,\delta ,W, \Lambda )>0$$\end{document} . By Lemma 2.1, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\varepsilon \in (0,1)} \Vert {(-\Delta )^s u_\varepsilon } \Vert _{L^\infty (\mathbb {R}^d \setminus \Sigma _{{\delta }/{\Lambda }})} \lesssim \Vert {u_\varepsilon } \Vert _{L^\infty (\mathbb {R}^d)} + \Vert {u_\varepsilon } \Vert _{C^2( \mathbb {R}^d \setminus \Sigma _{{\delta }/{2\Lambda }} )}. \end{aligned}$$\end{document}

Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert {u_\varepsilon } \Vert _{L^\infty (\mathbb {R}^d)} = 1$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} , it remains to estimate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_\varepsilon $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2(\mathbb {R}^d \setminus \Sigma _{{\delta }/{2\Lambda }} )$$\end{document} . Recalling that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{\Sigma } \in C^2(\mathbb {R}^d)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| \beta _\Sigma (x)\right| \ge {\delta }/{2\Lambda }$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in \mathbb {R}^d \setminus \Sigma _{{\delta }/{2\Lambda }}$$\end{document} (see Definition 2.10), we need to estimate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon (t) = w\left( {t}/{\varepsilon }\right) $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2( \{ \left| t\right| > {\delta }/{2\Lambda } \})$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \in (0,1)$$\end{document} . Indeed, by Theorem 1.2 we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| w'_\varepsilon (t)\right| + \left| w''_{\varepsilon }(t)\right| = \left| \frac{1}{\varepsilon } w'\left( \frac{t}{\varepsilon } \right) \right| + \left| \frac{1}{\varepsilon ^2} w''\left( \frac{t}{\varepsilon }\right) \right| \lesssim \varepsilon ^{2s} \qquad \forall \left| t\right| \ge \frac{\delta }{2\Lambda } \qquad \forall \varepsilon \in (0,1). \end{aligned}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Expansion of the fractional Laplacian around the boundary

As we explained in the introduction, the expansion of the fractional Laplacian in Fermi coordinates for the function defined by (4.1) is crucial for proving our main result. In this section, we provide a complete proof of this expansion. We emphasize that our approach closely follows some computations from [14, Section 3], which deal with a three-dimensional setting. However, we adapt these computations to our framework, carefully keeping track of constants and error terms.

Theorem 4.1

Let W be a double-well potential satisfying (W1), (W2), (W3). Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w: \mathbb {R}\rightarrow (-1,1)$$\end{document} be the one-dimensional optimal profile and for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} let us set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon (z) = w \left( {z}/{\varepsilon }\right) $$\end{document} . Assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in \left( {1}/{2},1\right) $$\end{document} . Let E be a bounded open set of class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^3$$\end{document} and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma = \partial E$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta >0$$\end{document} be such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _\Sigma $$\end{document} is well defined according to Definition 2.10 and let us set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u_\varepsilon (x) = w_\varepsilon (\beta _\Sigma (x)). \end{aligned}$$\end{document}

There exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _0, C \ge 1$$\end{document} depending on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} with the following property. For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \ge \Lambda _0$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \in (0,1)$$\end{document} there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_{\varepsilon ,\Lambda }: \Sigma _{{\delta }/{10 \Lambda }} \rightarrow \mathbb {R}$$\end{document} such that for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0 \in \Sigma _{{\delta }/{10 \Lambda }}$$\end{document} it holds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (-\Delta )^s u_\varepsilon (x_0)&= (-\partial _{zz})^s w_\varepsilon (z_0) + \frac{\gamma _{1,s}}{2} \frac{H_{\Sigma }(x_0')}{(2s-1)} \int _{-{\delta }/{\Lambda }}^{{\delta }/{\Lambda }} \frac{w_\varepsilon '(z_0+{\bar{z}})}{\left| {\bar{z}}\right| ^{2s-1}} \, d{\bar{z}} + \mathcal {R}_{\varepsilon , \Lambda }(x_0), \end{aligned}$$\end{document}

where we set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_0 = \textrm{dist}_{\Sigma }(x_0)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0' = \pi _{\Sigma } (x_0)$$\end{document} . Moreover, it holds that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert {\mathcal {R}_{\varepsilon ,\Lambda }} \Vert _{L^\infty (\Sigma _{{\delta }/{10 \Lambda }})} \le C \Lambda ^{2s}. \end{aligned}$$\end{document}

Proof

For the reader convenience, we split the proof in several steps. To begin, we summarize the strategy adopted. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _0$$\end{document} be the geometric constant given by Lemma 2.17 and fix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \ge \Lambda _0$$\end{document} . Then, we take \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0 \in \Sigma _{{\delta }/{10 \Lambda }}$$\end{document} and we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0' = \pi _{\Sigma }(x_0), z_0 = \textrm{dist}_\Sigma (x_0)$$\end{document} . Then, with the same notation as Lemma 2.17 we take Fermi coordinates around \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0'$$\end{document} , i.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi : B_\delta ^{d-1} \times (-\delta , \delta ) \rightarrow \mathbb {R}^d$$\end{document} . We recall that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0 = \Phi (0, z_0)$$\end{document} with respect to this coordinate system. For convenience, we set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}_{{\delta }/{\Lambda }}(x_0): = \Phi (\mathcal {B}_{{\delta }/{\Lambda }}(z_0))$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}_{{\delta }/{\Lambda }}(z_0) \subset B_\delta ^{d-1} \times (-\delta , \delta )$$\end{document} is defined by (2.27). Since the fractional Laplacian is a singular integral (see (2.1)) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}_{{\delta }/{\Lambda }}(x_0)$$\end{document} is an open set around \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0$$\end{document} (see Lemma 2.17), we analyse separately the contribution in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}_{{\delta }/{\Lambda }}(x_0)$$\end{document} and in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^d \setminus \mathcal {T}_{{\delta }/{\Lambda }}(x_0)$$\end{document} . More precisely, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (-\Delta )^s u_\varepsilon (x_0)&= \gamma _{d,s} \int _{\mathbb {R}^d \setminus \mathcal {T}_{{\delta }/{\Lambda }}(x_0)} \frac{u_\varepsilon (x_0) - u_\varepsilon (x)}{\left| x-x_0\right| ^{d+2s}} \, dx \\&+ \lim _{\nu \rightarrow 0} \gamma _{d,s} \int _{\mathcal {T}_{{\delta }/{\Lambda }}(x_0) \setminus B_\nu (x_0)} \frac{u_\varepsilon (x_0) - u_\varepsilon (x)}{\left| x-x_0\right| ^{d+2s}} \, dx. \end{aligned}$$\end{document}

Unless otherwise specified, the reminders involved in the following computations satisfy bounds depending on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} . In particular, they are independent on the choice of the point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0$$\end{document} . This fact follows essentially by Remark 2.12, Lemmas 2.15 and 2.17.

Step 1: Estimating the outer contribution. Up to an implicit constant depending only on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d, \delta , \Sigma $$\end{document} , by Lemma 2.17 we estimate the integral in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^d \setminus \mathcal {T}_{{\delta }/{\Lambda }}(x_0)$$\end{document} as follows

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \int _{\mathbb {R}^d \setminus \mathcal {T}_{{\delta }/{\Lambda }}(x_0)} \frac{u_\varepsilon (x_0)-u_\varepsilon (x)}{\left| x-x_0\right| ^{d+2s}} \, dx \right|&\lesssim \int _{\mathbb {R}^d \setminus B_{{\delta }/{10 \Lambda }}(x_0) } \frac{1}{\left| x-x_0\right| ^{d+2s}} \, dx \lesssim \Lambda ^{2s}. \end{aligned}$$\end{document}

Step 2: Rewriting the inner contribution. The estimate of the singular term is much more delicate and it requires a careful analysis. To ease the notation, given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu >0$$\end{document} , we set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta ^s_{\nu } u_\varepsilon (x_0):= \int _{\mathcal {T}_{{\delta }/{\Lambda }} (x_0) \setminus B_\nu (x_0)} \frac{u_\varepsilon (x) - u_\varepsilon (x_0)}{\left| x-x_0\right| ^{d+2s}} \, dx. \end{aligned}$$\end{document}

We changed sign to avoid many negative terms in the computations. We aim to write the integral in (4.5) as an integral with respect to the variables z, y. Furthermore, writing the Euclidean distance as in Remark 2.16, by definition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T}_{{\delta }/{\Lambda } } (x_0)$$\end{document} and recalling that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi $$\end{document} is a diffeomorphism by Lemma 2.17, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {T}_{{\delta }/{\Lambda }}(x_0) \setminus B_{\nu }(x_0) = \Phi (\mathcal {U}^1_{\nu }), \end{aligned}$$\end{document}

where we set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {U}^1_{\nu }:= \left\{ (y,z) \in \mathcal {B}_{{\delta }/{\Lambda }} (z_0) :\begin{aligned} \left| z-z_0 + Y(y)\cdot N(y) + z_0(1-N_d(y))\right| ^2 + \left| (Y(y) - z_0 e_d)_\tau \right| ^2 \ge \nu ^2 \end{aligned} \right\} . \end{aligned}$$\end{document}

Here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}_{{\delta }/{\Lambda }}(z_0)$$\end{document} is defined by (2.27). Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _\Sigma $$\end{document} is the proper distance from the boundary (see Definition 2.10 and Lemma 2.17) in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi (\mathcal {U}^1_{\nu })$$\end{document} , changing variables in (4.5) and recalling that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_\varepsilon $$\end{document} is given by (4.1), we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta ^s_{\nu } u_\varepsilon (x_0) = \int _{\mathcal {U}^1_{\nu }} \frac{(w_\varepsilon (z)-w_\varepsilon (z_0)) \left| \det (\nabla \Phi (y,z))\right| }{\left( \left| z-z_0 + Y(y)\cdot N(y) + z_0(1-N_d(y))\right| ^2 + \left| (Y(y) - z_0 e_d)_\tau \right| ^2\right) ^{\frac{d+2s}{2}}} \, dz \, dy. \end{aligned}$$\end{document}

Next, with the notation of Lemma 2.17, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y\in {\mathscr {B}}_{{\delta }/{\Lambda }}^{d-1}(z_0)$$\end{document} , we set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\bar{z}}(y,z) = z-z_0 + Y(y)\cdot N(y) + z_0 (1- N_d(y)). \end{aligned}$$\end{document}

Therefore, changing again variables, we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta ^s_{\nu } u_\varepsilon (x_0) = \int _{\mathcal {U}^2_{\nu }} \frac{(w_\varepsilon (z(y, {\bar{z}}))-w_\varepsilon (z_0)) \left| \det (\nabla \Phi (y,z(y, {\bar{z}})))\right| }{\left( \left| {\bar{z}}\right| ^2 + \left| (Y(y) - z_0 e_d)_\tau \right| ^2\right) ^{\frac{d+2s}{2}}} \, d{\bar{z}} \, dy, \end{aligned}$$\end{document}

where we set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {U}^2_{\nu }:= \left\{ (y,{\bar{z}}) \in {\mathscr {B}}_{{\delta }/{\Lambda }}^{d-1}(z_0) \times \left( -{\delta }/{\Lambda }, {\delta }/{\Lambda }\right) :\left| {\bar{z}}\right| ^2 + \left| (Y(y) - z_0 e_d)_\tau \right| ^2 \ge \nu ^2 \right\} . \end{aligned}$$\end{document}

Hence, setting

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\bar{y}}(y) = (\textrm{Id} - \nabla ^2 g(0) z_0) y, \end{aligned}$$\end{document}

and changing again variables, we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta ^s_{\nu } u_\varepsilon (x_0) = \int _{\mathcal {U}^3_{\nu }} \frac{(w_\varepsilon (z(y({\bar{y}}), {\bar{z}}))-w_\varepsilon (z_0)) }{\left( \left| {\bar{z}}\right| ^2 + \left| (Y(y({\bar{y}})) - z_0 e_d)_\tau \right| ^2\right) ^{\frac{d+2s}{2}}} \frac{\left| \det (\nabla \Phi (y({\bar{y}}),z(y({\bar{y}}), {\bar{z}})))\right| }{\left| \det (\textrm{Id} - z_0 \nabla ^2 g (0))\right| } \, d{\bar{z}} \, d{\bar{y}}, \end{aligned}$$\end{document}

where we set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {U}^3_{\nu }:= \left\{ ({\bar{y}},{\bar{z}}) \in B_{{\delta }/{\Lambda }}^{d-1} \times \left( -{\delta }/{\Lambda }, {\delta }/{\Lambda } \right) :\left| {\bar{z}}\right| ^2 + \left| (Y(y({\bar{y}})) - z_0 e_d)_\tau \right| ^2 \ge \nu ^2 \right\} . \end{aligned}$$\end{document}

Step 3: Computing the leading order terms. In order to estimate the integral in (4.8), we compute the leading orders of the terms involved. By Definition 2.14, (4.7) and (2.24), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| (Y(y({\bar{y}})) - z_0 e_d)_\tau \right| ^2 = \left| {\bar{y}}\right| ^2 + O(\left| z_0\right| \left| {\bar{y}}\right| ^3) + O(\left| {\bar{y}}\right| ^4). \end{aligned}$$\end{document}

By Lemma 2.15 and (4.7), it is clear that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} Y(y({\bar{y}}))\cdot N(y({\bar{y}})) = -\frac{1}{2} \sum _{i=1}^{d-1} {\bar{y}}_i^2 k_i + O( \left| z_0\right| \left| {\bar{y}}\right| ^2 ), \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 1- N_d(y({\bar{y}})) = O(\left| {\bar{y}}\right| ^2), \end{aligned}$$\end{document}

Thus, by (4.6), (4.7), (4.11) and (4.12), we infer that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} z(y, {\bar{z}}) = z_0 + {\bar{z}} + \frac{1}{2} \sum _{i=1}^{d-1} y_i^2 k_i + O(\left| y\right| ^3) + O( \left| z_0\right| \left| y\right| ^2). \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} z(y({\bar{y}}), {\bar{z}}) = z_0 + {\bar{z}} + f(z_0, {\bar{y}}), \end{aligned}$$\end{document}

where we set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f(z_0, {\bar{y}}) = \sum _{i=1}^{d-1} \frac{1}{2} k_i {\bar{y}}_i^2 + O( \left| z_0\right| \left| {\bar{y}}\right| ^2) + O( \left| {\bar{y}}\right| ^3). \end{aligned}$$\end{document}

By Lemma 2.15, (4.13) and (4.6), we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\det \nabla \Phi (y, z(y, {\bar{z}})) \nonumber \\&= \prod _{i=1}^{d-1} (1- ({\bar{z}}+z_0 + O(\left| y\right| ^2)) k_i ) + O((\left| {\bar{z}}\right| + \left| z_0\right| + \left| y\right| ^2 ) \left| y\right| ) + O(\left| y\right| ^2) \nonumber \\ &= \prod _{i=1}^{d-1} (1- k_i (z_0 + {\bar{z}})) + O(\left| z_0\right| \left| y\right| ) + O(\left| {\bar{z}}\right| \left| y\right| ) + O (\left| y\right| ^2) . \end{aligned}$$\end{document}

By (4.7) and (4.16), we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\frac{\det \nabla \Phi (y({\bar{y}}), z(y({\bar{y}}) , {\bar{z}}))}{\det (\textrm{Id} - z_0 \nabla ^2 g (0))} \nonumber \\&= \left( \prod _{i=1}^{d-1} (1- z_0 k_i - {\bar{z}} k_i ) + O((\left| z_0\right| + \left| {\bar{z}}\right| )\left| {\bar{y}}\right| + \left| {\bar{y}}\right| ^2) \right) \left( \prod _{i=1}^{d-1} (1- z_0 k_i )\right) ^{-1} \nonumber \\ &= \prod _{i=1}^{d-1} \left( 1- {\bar{z}} \frac{k_i }{ 1- z_0 k_i } \right) + O((\left| z_0\right| + \left| {\bar{z}}\right| )\left| {\bar{y}}\right| + \left| {\bar{y}}\right| ^2) \nonumber \\ &= 1- {\bar{z}} H_{\Sigma }(x_0') + O(\left| z_0\right| \left| {\bar{y}}\right| ) + O(\left| {\bar{y}}\right| ^2) + O ( \left| {\bar{z}}\right| ^2). \end{aligned}$$\end{document}

From now on, we denote by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \rho ^2 = \left| {\bar{z}}\right| ^2 + \left| {\bar{y}}\right| ^2. \end{aligned}$$\end{document}

Hence, by (4.10), (4.14), (4.15), (4.17), we write (4.8) as follows

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta ^s_{\nu } u_\varepsilon (x_0) = \int _{\mathcal {U}^3_{\nu }} \frac{\left[ w_\varepsilon ( z_0 + {\bar{z}} + f(z_0, {\bar{y}}))-w_\varepsilon (z_0)\right] \left[ 1- {\bar{z}} H_{\Sigma }(x_0') + O(\left| z_0\right| \left| {\bar{y}}\right| ) + O( \rho ^2) \right] }{\left( \rho ^2 + O(\left| z_0\right| \left| {\bar{y}}\right| ^3) + O (\left| {\bar{y}}\right| ^4) ) \right) ^{\frac{d+2s}{2}}} \, d{\bar{z}} \, d{\bar{y}}. \end{aligned}$$\end{document}

By standard manipulations, we write

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\frac{1- {\bar{z}} H_\Sigma (x_0') + O(\left| z_0\right| \left| {\bar{y}}\right| ) + O(\rho ^2) }{\left( \rho ^2 + O(\left| z_0\right| \left| {\bar{y}}\right| ^3) + O (\left| {\bar{y}}\right| ^4) ) \right) ^{\frac{d+2s}{2}}}\\&= \frac{1- {\bar{z}} H_\Sigma (x_0') + O(\left| z_0\right| \left| {\bar{y}} \right| ) + O(\rho ^2) }{ \rho ^{d+2s} } \left( 1 + O(\left| z_0\right| \left| {\bar{y}}\right| ) + O(\left| {\bar{y}}\right| ^2) \right) \\&= \frac{1- {\bar{z}} H_\Sigma (x_0') + O(\left| z_0\right| \left| {\bar{y}} \right| ) + O(\rho ^2) }{ \rho ^{d+2s} }. \end{aligned}$$\end{document}

Hence, by (4.18) we write

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta ^s_{\nu } u_\varepsilon (x_0) = \int _{\mathcal {U}^3_{\nu }} \frac{ g_\varepsilon ({\bar{y}}, {\bar{z}}, z_0) }{\rho ^{d+2s}} \, d{\bar{z}} \, d{\bar{y}}, \end{aligned}$$\end{document}

where we denote by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} g_\varepsilon ({\bar{y}}, {\bar{z}}, z_0):= \left[ w_\varepsilon ( z_0 + {\bar{z}} + f(z_0, {\bar{y}}))-w_\varepsilon (z_0)\right] \left[ 1- {\bar{z}} H_{\Sigma }(x_0') + O(\left| z_0\right| \left| {\bar{y}}\right| ) + O(\rho ^2) \right] \nonumber \\ \end{aligned}$$\end{document}

We claim that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{\nu \rightarrow 0} \int _{\mathcal {U}^3_{\nu }} \frac{g_\varepsilon ({\bar{y}}, {\bar{z}}, z_0)}{\rho ^{d+2s}} \, d{\bar{z}} \, d{\bar{y}} = \lim _{\nu \rightarrow 0} \int _{\mathcal {C}_{\nu }} \frac{g_\varepsilon ({\bar{y}}, {\bar{z}}, z_0)}{\rho ^{d+2s}} \, d{\bar{z}} \, d{\bar{y}}, \nonumber \\ \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}_\nu $$\end{document} is the complement of a ball in a cylinder

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {C}_{\nu }:= \left\{ ({\bar{y}},{\bar{z}}) \in B_{{\delta }/{\Lambda }}^{d-1} \times \left( -{\delta }/{\Lambda }, {\delta }/{\Lambda }\right) :\rho \ge \nu \right\} . \end{aligned}$$\end{document}

To begin, we estimate the symmetric difference between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {U}_\nu ^3$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {C}_\nu $$\end{document} . By (4.9), (4.10) and (4.21), if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\bar{y}}, {\bar{z}}) \in \mathcal {U}_\nu ^3 \setminus \mathcal {C}_\nu $$\end{document} we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \nu ^2 - O(\left| {\bar{y}}\right| ^3) \le \rho ^2 \le \nu ^2$$\end{document} . Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| {\bar{y}}\right| \le \rho \le \nu $$\end{document} , we find a purely geometric constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{c}}>0$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho ^2 \in ( \nu ^2- {\bar{c}} \nu ^{3}, \nu ^2 )$$\end{document} . Similarly, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\bar{y}}, {\bar{z}}) \in \mathcal {C}_\nu \setminus \mathcal {U}_\nu ^3 $$\end{document} , we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho ^2 \in ( \nu ^2, \nu ^2 + {\bar{c}} \nu ^{3})$$\end{document} . To summarize, we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {U}^3_{\nu } \Delta \mathcal {C}_{\nu } \subset \{ \nu - {\bar{c}} \nu ^2 \le \rho \le \nu + {\bar{c}} \nu ^2 \}. \end{aligned}$$\end{document}

Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon $$\end{document} is Lipschitz, f satisfies (4.15) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in (0,1)$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{\nu \rightarrow 0} \left| \int _{\mathcal {U}^3_{\nu }} \frac{g_\varepsilon ({\bar{y}}, {\bar{z}}, z_0) }{\rho ^{d+2s}} \, d {\bar{y}}\, d{\bar{z}} - \int _{\mathcal {C}_\nu } \frac{g_\varepsilon ({\bar{y}}, {\bar{z}}, z_0) }{\rho ^{d+2s}} \, d {\bar{y}}\, d{\bar{z}} \right|&\lesssim \lim _{\nu \rightarrow 0} \int _{\nu - {\bar{c}} \nu ^2}^{\nu + {\bar{c}} \nu ^2} \frac{\rho + \rho ^2 }{\rho ^{d+2s}} \rho ^{d-1} \, d \rho = 0. \end{aligned}$$\end{document}

Here the implicit constant depends on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} , but it is independent of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} . Hence, (4.20) is proved.

Step 4: Collecting the estimates. To summarize, by (4.20) and letting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f,g_\varepsilon $$\end{document} be as in (4.15), (4.19), we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{\nu \rightarrow 0} \Delta _\nu ^s u_\varepsilon (x_0)&= \lim _{\nu \rightarrow 0} \int _{\mathcal {C}_\nu } \frac{w_\varepsilon (z_0 + {\bar{z}}) - w_\varepsilon (z_0) }{\rho ^{d+2s}} (1- {\bar{z}} H_{\Sigma }(x_0')) \, d {\bar{z}} \, d {\bar{y}} \nonumber \\&\quad + \int _{\mathcal {C}_\nu } \frac{w_\varepsilon '(z_0 + {\bar{z}}) f(z_0, {\bar{y}})}{\rho ^{d+2s}} (1- {\bar{z}} H_{\Sigma }(x_0')) \, d{\bar{z}}\, d {\bar{y}}\nonumber \\&\quad + \int _{\mathcal {C}_\nu } \frac{w_\varepsilon (z_0 + {\bar{z}} + f(z_0, {\bar{y}}) ) - w_\varepsilon (z_0+ {\bar{z}}) - w_\varepsilon '(z_0+{\bar{z}}) f(z_0, {\bar{y}}) }{\rho ^{d+2s}} (1- {\bar{z}} H_{\Sigma }(x_0'))\, d{\bar{z}}\, d {\bar{y}}\nonumber \\&\quad + O \left( \int _{\mathcal {C}_\nu } \frac{\left| w_\varepsilon (z_0+ {\bar{z}} + f(z_0, {\bar{y}}) )- w_\varepsilon (z_0) \right| }{\rho ^{d+2s}} ( \left| z_0\right| \left| {\bar{y}}\right| + \rho ^2 ) \, d{\bar{z}} \, d {\bar{y}} \right) .\nonumber \\&= \lim _{\nu \rightarrow 0} I_1^\nu + I_2^\nu + I_3^\nu + O(I_4^\nu ). \end{aligned}$$\end{document}

To conclude the proof, we estimate separately the four terms. For the reader’s convenience, we postpone these computations to Sect. 4.1. We summarize the contribution of each term:

• (i) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_1^\nu $$\end{document} gives the fractional Laplacian of w and a contribution of the principal curvatures;
• (ii) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_2^\nu $$\end{document} gives the remaining part of the term involving the principal curvatures;
• (iii) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_3^\nu $$\end{document} is the nonlinear error;
• (iv) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_4^\nu $$\end{document} is the error when expanding the ambient distance and the Jacobian. To conclude, by (4.4), (4.22), Lemmas 4.3–4.6, we infer that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{\nu \rightarrow 0} \gamma _{d,s} \Delta ^s_\nu u_\varepsilon (x_0) = -(-\Delta )^s w_ \varepsilon (x_0) - \frac{\gamma _{1,s}}{2} \frac{H_{\Sigma } (x_0')}{2s-1} \int _{-{\delta }/{\Lambda }}^{{\delta }/{\Lambda }} \frac{w_\varepsilon '(z_0+{\bar{z}})}{\left| {\bar{z}}\right| ^{2s-1}} \, d{\bar{z}} + \mathcal {R}_{\Lambda , \varepsilon }(x_0), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_{\Lambda ,\varepsilon }: \Sigma _{{\delta }/{10 \Lambda }} \rightarrow \mathbb {R}$$\end{document} is a bounded function satisfying (4.3). Then, the proof is concluded. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Estimates of the four integrals

We estimate the terms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_1^\nu , I_2^\nu , I_3^\nu , I_4^\nu $$\end{document} in (4.22). Throughout this section, we implicitly assume \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \in (0,1)$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \ge \Lambda _0$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _0$$\end{document} is given by Lemma 2.17. We neglect constants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C(d,s,W,\delta , \Sigma )>0$$\end{document} , whereas it is crucial to keep the dependence of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon , \Lambda $$\end{document} explicit We recall that w is the optimal profile and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon (z) = w \left( {z}/{\varepsilon }\right) $$\end{document} . We need a preliminary lemma.

Lemma 4.2

Fix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in ({1}/{2},1)$$\end{document} . For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_0 \in \mathbb {R}, \ell , \varepsilon >0$$\end{document} it holds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & \int _{-\ell }^\ell \frac{w_\varepsilon (z_0+z)-w_\varepsilon (z_0)}{\left| z\right| ^{2s+1}} z \, dz \nonumber \\ & \quad = \frac{\ell ^{1-2s}}{1-2s} (w_\varepsilon (\ell +z_0) - w_\varepsilon (z_0-\ell )) + \frac{1}{2s-1} \int _{-\ell }^\ell \frac{w_\varepsilon '(z_0+z)}{\left| z\right| ^{2s-1}} \, dz. \end{aligned}$$\end{document}

Proof

We have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{-\ell }^\ell \frac{w_\varepsilon (z_0+z) - w_\varepsilon (z_0)}{\left| z\right| ^{2s+1}} z \, dz = \left( \int _0^\ell + \int _{-\ell }^0 \right) \frac{w_\varepsilon (z_0+z) - w_\varepsilon (z_0)}{\left| z\right| ^{2s+1}} z \, dz = I+II. \end{aligned}$$\end{document}

Then, carefully integrating by parts I, we get that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} I&= \lim _{\nu \rightarrow 0} \int _{\nu }^\ell \frac{w_\varepsilon (z_0+z)-w_\varepsilon (z_0)}{z^{2s}} \, dz \\ &= \lim _{\nu \rightarrow 0} \left[ \frac{z^{1-2s}}{1-2s} (w_\varepsilon (z_0+z)-w_\varepsilon (z_0)) \right] _{z=\nu }^{z=\ell } + \frac{1}{2s-1} \int _{\nu }^\ell \frac{w_\varepsilon '(z_0+z)}{z^{2s-1}} \, dz \\ &= \lim _{\nu \rightarrow 0} \frac{\ell ^{1-2s}}{1-2s} (w_\varepsilon (\ell +z_0)-w_\varepsilon (z_0)) - \frac{\nu ^{1-2s}}{1-2s} (w_\varepsilon (\nu +z_0) - w_\varepsilon (z_0)) \\&+ \frac{1}{2s-1} \int _{\nu }^\ell \frac{w_\varepsilon '(z_0+z)}{z^{2s-1}} \, dz. \end{aligned}$$\end{document}

Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon $$\end{document} is Lipschitz continuous and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in (0,1)$$\end{document} , we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{\nu \rightarrow 0} \left| \frac{\nu ^{1-2s }}{1-2s} (w_\varepsilon (\nu +z_0) - w_\varepsilon (z_0))\right| \lesssim \lim _{\nu \rightarrow 0} \nu ^{2-2s} = 0. \end{aligned}$$\end{document}

Recalling that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon $$\end{document} is increasing, by monotone convergence, we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&I -\frac{\ell ^{1-2s}}{1-2s} (w_\varepsilon (\ell +z_0)-w_\varepsilon (z_0)) \\&= \lim _{\nu \rightarrow 0} \frac{1}{2s-1} \int _{\nu }^\ell \frac{w_\varepsilon '(z_0+z)}{z^{2s-1}}\, dz \\&\quad = \frac{1}{2s-1} \int _0^{\ell } \frac{w_\varepsilon '(z_0+z)}{z^{2s-1}} \, dz. \end{aligned}$$\end{document}

Similarly, it is easy to check that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} II = \frac{\ell ^{1-2s}}{1-2s} (w_\varepsilon (z_0) - w_\varepsilon (z_0-\ell )) + \frac{1}{2s-1} \int _{-\ell }^0 \frac{w_\varepsilon '(z_0+z)}{\left| z\right| ^{2s-1}} \, dz. \end{aligned}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 4.3

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_1^\nu $$\end{document} be given by (4.22). It holds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{x_0 \in \Sigma _{{\delta }/{10 \Lambda }}} \left| \lim _{\nu \rightarrow 0} I_1^\nu + \frac{1}{\gamma _{d,s}}(-\Delta )^s w_\varepsilon (z_0) + \frac{\gamma _{1,s}}{\gamma _{d,s}} \cdot \frac{H_{\Sigma }(x_0') }{(2s-1)} \int _{-{\delta }/{\Lambda }}^{{\delta }/{\Lambda }} \frac{w_\varepsilon '(z_0+{\bar{z}})}{\left| {\bar{z}}\right| ^{2s-1}} \, d{\bar{z}} \right| \lesssim \Lambda ^{2s}. \end{aligned}$$\end{document}

Proof

For simplicity, we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell = {\delta }/{\Lambda }$$\end{document} . Using the second order difference to get rid of the principal value, we split

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{\nu \rightarrow 0} I_1^\nu&= \int _{B_\ell ^{d-1}} \int _{-\ell }^\ell \frac{w_\varepsilon (z_0 + {\bar{z}}) + w_\varepsilon (z_0-{\bar{z}}) -2 w_\varepsilon (z_0) }{2 \rho ^{d+2s}} \, d {\bar{z}} \, d {\bar{y}} \\ &\qquad - H_{\Sigma }(x_0') \int _{B_\ell ^{d-1}} \int _{-\ell }^\ell \frac{w_\varepsilon (z_0 + {\bar{z}}) - w_\varepsilon (z_0)}{\rho ^{d+2s}} {\bar{z}} \, d {\bar{z}} \, d {\bar{y}} \\ &= I_{1,1} + I_{1,2}. \end{aligned}$$\end{document}

Estimate of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{1,1}$$\end{document} . By Lemma 6.4 and recalling that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| w_\varepsilon \right| \le 1$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} I_{1,1}&= \frac{\gamma _{1,s}}{\gamma _{d,s}} \int _{-\ell }^{\ell } \frac{w_\varepsilon (z_0 +{\bar{z}}) + w_\varepsilon (z_0 - {\bar{z}}) - 2 w_\varepsilon (z_0)}{2 \left| {\bar{z}}\right| ^{1+2s}} \, d{\bar{z}} + O\left( \ell ^{-2s}\right) \\&= - \frac{1}{\gamma _{d,s}} (-\Delta )^s w_\varepsilon (z_0) + O\left( \ell ^{-2s}\right) . \end{aligned}$$\end{document}

Estimate of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{1,2}$$\end{document} . Similarly, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} I_{1,2}&= -H_{\Sigma }(x_0') \int _{-\ell }^{\ell } \left( \frac{\gamma _{1,s}}{\gamma _{d,s}} \frac{w_\varepsilon (z_0 + {\bar{z}})- w_\varepsilon (z_0)}{\left| {\bar{z}}\right| ^{1+2s}} + O\left( \ell ^{-1-2s} \right) \right) {\bar{z}} \, d {\bar{z}} \\ &= - H_{\Sigma } (x_0') \frac{\gamma _{1,s}}{\gamma _{d,s}} \int _{-\ell }^{\ell } \frac{w_\varepsilon (z_0 + {\bar{z}})- w_\varepsilon (z_0)}{\left| {\bar{z}}\right| ^{1+2s}} {\bar{z}} \, d {\bar{z}} + O\left( \ell ^{1-2s} \right) . \end{aligned}$$\end{document}

Then, using (4.23) and recalling that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| w_\varepsilon \right| \le 1$$\end{document} , we conclude

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| I_{1,2} + H_{\Sigma }(x_0') \frac{\gamma _{1,s}}{(2s-1) \gamma _{d,s}} \int _{-\ell }^\ell \frac{w_\varepsilon '(z_0+{\bar{z}})}{\left| {\bar{z}}\right| ^{2s-1}} \, d{\bar{z}}\right| \lesssim \ell ^{1-2s}. \end{aligned}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 4.4

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_2^\nu $$\end{document} be defined by (4.22). It holds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{x_0 \in \Sigma _{{\delta }/{10 \Lambda }}} \left| \lim _{\nu \rightarrow 0} I_2^\nu - \frac{H_{\Sigma }(x_0') }{2 (2s-1) } \cdot \frac{\gamma _{1,s}}{\gamma _{d,s}} \int _{-{\delta }/{\Lambda }}^{{\delta }/{\Lambda }} \frac{w'_\varepsilon (z_0+{\bar{z}})}{\left| {\bar{z}}\right| ^{2s-1}} \, d {\bar{z}} \right| \lesssim \Lambda ^{2s-1}. \end{aligned}$$\end{document}

Proof

For simplicity, we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell = {\delta }/{\Lambda }$$\end{document} . Then, we split

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{\nu \rightarrow 0} I_2^\nu&= \int _{B_\ell ^{d-1}} \int _{-\ell }^\ell \frac{w_\varepsilon '(z_0+ {\bar{z}}) \sum _{i=1}^{d-1} k_i {\bar{y}}_i^2 }{2 \rho ^{d+2s}} (1- {\bar{z}} H_{\Sigma }(x_0')) \, d {\bar{z}}\, d {\bar{y}} \\ &\quad + O \left( \left| z_0\right| \int _{B_\ell ^{d-1}} \int _{-\ell }^\ell \frac{w'_\varepsilon (z_0 + {\bar{z}})}{\rho ^{d+2s}} \left| {\bar{y}}\right| ^2 \, d {\bar{z}} \, d {\bar{y}} \right) \\&+ O \left( \int _{B_\ell ^{d-1}} \int _{-\ell }^\ell \frac{w'_\varepsilon (z_0 + {\bar{z}})}{\rho ^{d+2s}} \left| {\bar{y}}\right| ^3 \, d {\bar{z}} \, d {\bar{y}} \right) \\ &= I_{2,1} + O\left( I_{2,2}\right) + O\left( I_{2,3}\right) \end{aligned}$$\end{document}

and we estimate separately each term.

Estimate of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{2,1}$$\end{document} . By Lemma 6.4 and the fundamental theorem of calculus, we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} I_{2,1}&= \frac{H_{\Sigma }(x_0')}{2(2s-1)} \int _{-\ell }^\ell w_\varepsilon '(z_0 + {\bar{z}})(1- {\bar{z}} H_{\Sigma }(x_0')) \left( \frac{\gamma _{1,s}}{\gamma _{d,s}} \cdot \frac{1}{\left| {\bar{z}}\right| ^{2s-1}} + O\left( \ell ^{1-2s}\right) \right) \, d{\bar{z}} \\ &= \frac{H_{\Sigma }(x_0') }{2 (2s-1) }\cdot \frac{\gamma _{1,s}}{\gamma _{d,s}} \int _{-\ell }^\ell \frac{w'_\varepsilon (z_0+{\bar{z}})}{\left| {\bar{z}}\right| ^{2s-1}} \, d {\bar{z}} + O\left( \ell ^{1-2s}\right) . \end{aligned}$$\end{document}

Estimate of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{2,2}$$\end{document} . By Lemma 6.4 we compute

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} I_{2,2}&= \left| z_0\right| \int _{-\ell }^{\ell } w_\varepsilon '(z_0 + {\bar{z}}) \int _{B_\ell ^{d-1}} \frac{\left| {\bar{y}}\right| ^2}{ (\left| {\bar{z}}\right| ^2 + \left| {\bar{y}}\right| ^2)^{\frac{d+2s}{2}}}\, d {\bar{y}} \, d {\bar{z}} \\ &\lesssim \left| z_0\right| \int _{-\ell }^{\ell } \frac{ w_\varepsilon '(z_0 + {\bar{z}}) }{\left| {\bar{z}}\right| ^{2s-1}} \, d{\bar{z}} + \left| z_0\right| \ell ^{1-2s} \int _{-\ell }^{\ell } w_\varepsilon '(z_0 + {\bar{z}})\, d {\bar{z}} = A + B. \end{aligned}$$\end{document}

By the fundamental theorem of calculus, we compute

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} B \lesssim \left| z_0\right| \ell ^{1-2s} \int _{\mathbb {R}} w_\varepsilon '(z_0 + {\bar{z}})\, d {\bar{z}} \lesssim \ell ^{2-2s}. \end{aligned}$$\end{document}

To estimate A, we split

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A = \left| z_0\right| \left( \int _{\left| {\bar{z}}\right| \le {\left| z_0\right| }/{2}} + \int _{{\left| z_0\right| }/{2} \le \left| {\bar{z}}\right| \le \ell } \right) \frac{w_\varepsilon '(z_0+ {\bar{z}})}{\left| {\bar{z}}\right| ^{2s-1}}\, d{\bar{z}} = A_1 + A_2. \end{aligned}$$\end{document}

Then, by (2.4) we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A_1&\lesssim \left| z_0\right| \sup _{\left| \zeta \right| \ge {\left| z_0\right| }/{2} } w_\varepsilon '(\zeta ) \int _{\left| {\bar{z}}\right| \le {\left| z_0\right| }/{2}} \frac{1}{\left| {\bar{z}}\right| ^{2s-1}} \, d {\bar{z}} \lesssim \frac{\left| z_0\right| }{\varepsilon } \frac{1}{1+ \left| z_0\right| ^{1+2s} \varepsilon ^{-1-2s}} \left| z_0\right| ^{2-2s} \lesssim \ell ^{2-2s}, \end{aligned}$$\end{document}

since the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \mapsto \frac{t}{1+ \left| t\right| ^{1+2s}}$$\end{document} is bounded in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}$$\end{document} . To estimate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_2$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A_2 \lesssim \left| z_0\right| ^{2-2s} \int _{-\ell }^{\ell } w_\varepsilon '(z_0 + {\bar{z}})\, d {\bar{z}} \lesssim \ell ^{2-2s}. \end{aligned}$$\end{document}

Estimate of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{2,3}$$\end{document} . Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d+2s-3 < d-1$$\end{document} , we estimate

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| I_{2,3}\right|&\lesssim \int _{-\ell }^\ell w_\varepsilon '(z_0 + {\bar{z}}) \int _{B_\ell ^{d-1}} \frac{1}{\left| {\bar{y}}\right| ^{d+2s-3}} \, d {\bar{y}} \lesssim \ell ^{2-2s} \int _{-\ell }^{\ell } w_\varepsilon '(z_0 + {\bar{z}}) \, d {\bar{z}} \lesssim \ell ^{2-2s}. \end{aligned}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 4.5

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_3^\nu $$\end{document} be defined by (4.22). It holds that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{x_0 \in \Sigma _{{\delta }/{10 \Lambda }} } \limsup _{\nu \rightarrow 0} \left| I_3^\nu \right| \lesssim \varepsilon ^{2s} \Lambda ^{4s-2} + \Lambda ^{2s-2}. \end{aligned}$$\end{document}

Proof

For simplicity, we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell = {\delta }/{\Lambda }$$\end{document} . By Taylor’s expansion, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{\nu \rightarrow 0} I_3^\nu = \int _0^1 (1-t) \int _{B_{\ell }^{d-1}} \int _{-\ell }^{\ell } \frac{w_\varepsilon ''(z_0 + {\bar{z}} + t f(z_0, {\bar{y}}) ) f(z_0, {\bar{y}})^2 }{\rho ^{d+2s}} (1- {\bar{z}} H_{\Sigma }(x_0'))\, d{\bar{z}}\, d {\bar{y}} \end{aligned}$$\end{document}

Hence, integrating by parts twice, we get that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{\nu \rightarrow 0} I_3^\nu&= \int _0^1 \int _{B_{\ell }^{d-1}} \bigg [ \left[ w'_\varepsilon (z_0 + {\bar{z}} + t f({\bar{y}}, z_0)) \frac{1- {\bar{z}} H_{\Sigma }(x_0')}{\rho ^{d+2s}} \right] _{{\bar{z}} = -\ell }^{{\bar{z}}= \ell } \\ &\qquad - \left[ w_{\varepsilon }(z_0+ {\bar{z}} + t f({\bar{y}}, z_0) ) \frac{d}{d {\bar{z}}} \left( \frac{1- {\bar{z}} H_{\Sigma }(x_0')}{\rho ^{d+2s}} \right) \right] _{{\bar{z}}=-\ell }^{{\bar{z}} = \ell } \\ &\qquad + \int _{-\ell }^{\ell } w_\varepsilon (z_0 + {\bar{z}} + t f({\bar{y}}, z_0)) \frac{d^2}{d {\bar{z}}^2} \left( \frac{1- {\bar{z}} H_{\Sigma }(x_0')}{\rho ^{d+2s}} \right) \, d {\bar{z}} \bigg ] f({\bar{y}}, z_0)^2 \, d {\bar{y}} \, dt \\ &= I_{3,1} + I_{3,2} + I_{3,3}. \end{aligned}$$\end{document}

Estimate of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{3,1}$$\end{document} . To estimate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{3,1}$$\end{document} , by (2.4), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} w'_\varepsilon \left( z_0 + \ell + t f({\bar{y}}, z_0)\right)&\lesssim \frac{ \varepsilon ^{-1}}{1+ \varepsilon ^{-1-2s} \left| z_0 + \ell + t f({\bar{y}}, z_0)\right| ^{1+2s} }\\&\lesssim \frac{ \varepsilon ^{2s} }{\left| z_0 + \ell + t f({\bar{y}}, z_0)\right| ^{1+2s}}. \end{aligned}$$\end{document}

Moreover, by (4.15), it results that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| f({\bar{y}}, z_0)\right| \le C \ell ^2$$\end{document} for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{y}} \in B_{\ell }^{d-1}$$\end{document} . Here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C>0$$\end{document} is a purely geometric constant. Recall that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| z_0\right| \le {\ell }/{10}$$\end{document} . Hence, we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| z_0 + \ell + t f({\bar{y}}, z_0)\right| \ge \ell - \frac{\ell }{10} - C \ell ^2 \ge \frac{\ell }{2}, \end{aligned}$$\end{document}

provided that we take \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \ge \Lambda _0$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _0$$\end{document} in Lemma 2.17 is chosen large enough, depending on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma $$\end{document} . Thus, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f({\bar{y}}, z_0)^2 = O (\left| {\bar{y}}\right| ^4)$$\end{document} (see (4.15)), we estimate

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _0^1 \int _{B_{\ell }^{d-1}}&\left[ w'_\varepsilon (z_0 + \ell + t f({\bar{y}}, z_0)) \frac{1- \ell H_{\Sigma }(x_0')}{( \ell ^2 + \left| {\bar{y}}\right| ^2 )^{\frac{d+2s}{2}}} \right] f({\bar{y}}, z_0)^2 \, d {\bar{y}} \, dt \lesssim \varepsilon ^{2s} \ell ^{2-4s}. \end{aligned}$$\end{document}

Similarly, we estimate the evaluation in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\ell $$\end{document} , thus proving that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| I_{3,1}\right| \lesssim \varepsilon ^{2s} \ell ^{2-4s}$$\end{document} .

Estimate of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{3,2}$$\end{document} . By direct computation, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\bar{y}}, {\bar{z}}) \in B_{\ell }^{d-1} \times \left( -\ell ,\ell \right) $$\end{document} we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & \frac{d}{d {\bar{z}}} \left( \frac{1- {\bar{z}} H_{\Sigma }(x_0')}{\rho ^{d+2s}} \right) = -\frac{H_{\Sigma } (x_0') }{\rho ^{d+2s}} - (d+2s) \frac{{\bar{z}} - {\bar{z}}^2 H_{\Sigma }(x_0')}{\rho ^{d+2s+2}}, \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & \left| \frac{d^2}{d {\bar{z}}^2} \left( \frac{1- {\bar{z}} H_{\Sigma }(x_0')}{\rho ^{d+2s}} \right) \right| \lesssim \frac{1}{\rho ^{d+2s+2}}. \end{aligned}$$\end{document}

Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left| w_\varepsilon \right| \le 1$$\end{document} , by (4.24) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f({\bar{y}}, z_0)^2 = O (\left| {\bar{y}}\right| ^4)$$\end{document} (see (4.15)), we estimate

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| I_{3,2}\right| \lesssim \int _{B_\ell ^{d-1}} \frac{\left| {\bar{y}}\right| ^4}{ (\ell ^2 + \left| {\bar{y}}\right| ^2)^{\frac{d+2s}{2}} } \, d {\bar{y}} \lesssim \ell ^{3-2s} \end{aligned}$$\end{document}

Estimate of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{3,3}$$\end{document} . Using (4.25), Lemma 6.4 and recalling that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f({\bar{y}}, z_0)^2 = O (\left| {\bar{y}}\right| ^4)$$\end{document} (see (4.15)), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in \left( {1}/{2},1\right) $$\end{document} , it is readily checked that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| I_{3,3}\right|&\lesssim \int _{B_{\ell }^{d-1}} \int _{-\ell }^{\ell }\frac{\left| {\bar{y}}\right| ^4}{\rho ^{d+2s+2}} \, d{\bar{z}} \, d{\bar{y}} \lesssim \int _{-\ell }^{\ell } \left( \left| {\bar{z}}\right| ^{1-2s} + O\left( \ell ^{1-2s} \right) \right) \, d{\bar{z}} \lesssim \ell ^{2-2s}. \end{aligned}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Lemma 4.6

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_4^\nu $$\end{document} be defined by (4.22). It holds that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{x_0 \in \Sigma _{{\delta }/{10 \Lambda }}} \limsup _{\nu \rightarrow 0} \left| I_4^\nu \right| \lesssim \Lambda ^{2s-2}. \end{aligned}$$\end{document}

Proof

For simplicity, we denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell = {\delta }/{\Lambda }$$\end{document} . Then, we split

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} I_4&\le \left| z_0\right| \int _{B_\ell ^{d-1}} \int _{-\ell }^\ell \frac{\left| w_\varepsilon (z_0 + {\bar{z}} + f({\bar{y}}, z_0)) - w_\varepsilon (z_0+ {\bar{z}}) \right| }{\rho ^{d+2s-1}} \, d {\bar{z}} \, d {\bar{y}} \\ &\qquad + \left| z_0\right| \int _{B_\ell ^{d-1}} \int _{-\ell }^\ell \frac{\left| w_\varepsilon (z_0 + {\bar{z}}) - w_\varepsilon (z_0) \right| }{\rho ^{d+2s-1}} \, d {\bar{z}} \, d {\bar{y}} \\ &\qquad + \int _{B_\ell ^{d-1}} \int _{-\ell }^\ell \frac{\left| w_\varepsilon (z_0 + {\bar{z}} + f({\bar{y}}, z_0)) - w_\varepsilon (z_0) \right| }{\rho ^{d+2s-2}} \, d {\bar{z}} \, d {\bar{y}} \\ &= I_{4,1}+ I_{4,2}+ I_{4,3}. \end{aligned}$$\end{document}

Estimate of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{4,1}$$\end{document} . Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f({\bar{y}}, z_0) = O(\left| {\bar{y}}\right| ^2)$$\end{document} (see (4.15)) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| z_0\right| \le {\ell }/{10}$$\end{document} , by the fundamental theorem of calculus and integrating by parts, we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&I_{4,1} \le \ell \int _0^1 \int _{B_\ell ^{d-1}} \int _{-\ell }^\ell \frac{w'_\varepsilon (z_0 + {\bar{z}} + t f({\bar{y}}, z_0))}{\rho ^{d+2s-3}} \, d {\bar{z}} \, d {\bar{y}} \, dt\\&\quad = \ell \int _0^1 \int _{B_\ell ^{d-1}} \left( \left[ \frac{w_\varepsilon (z_0+ {\bar{z}} + t f({\bar{y}}, z_0))}{\rho ^{d+2s-3}} \right] _{{\bar{z}}= -\ell }^{{\bar{z}}= \ell } + (d+2s-3) \int _{-\ell }^\ell \frac{w_\varepsilon (z_0+ {\bar{z}} + t f({\bar{y}}, z_0)) {\bar{z}}}{\rho ^{d+2s-1}} \, d {\bar{z}} \right) \, d {\bar{y}} \, dt\\&\quad \lesssim \ell ^{3-2s} + \int _{B_\ell ^{d-1}} \int _{-\ell }^\ell \frac{1}{\rho ^{d+2s-2}} \, d {\bar{z}} \, d {\bar{y}} \lesssim \ell ^{2-2s}. \end{aligned}$$\end{document}

Estimate of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{4,2}$$\end{document} . The estimate of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{4,2}$$\end{document} is similar to that of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{2,2}$$\end{document} in Lemma 4.4. By the fundamental theorem of calculus and recalling that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d+2s-1 > d-1$$\end{document} , we compute

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| I_{4,2}\right|&\le \left| z_0\right| \int _0^1 \int _{-\ell }^\ell \left| {\bar{z}}\right| w_\varepsilon '(z_0 + t {\bar{z}}) \int _{B_\ell ^{d-1}} \frac{1}{(\left| {\bar{z}}\right| ^2 +\left| {\bar{y}}\right| ^2)^{\frac{d+2s-1}{2}}} \, d {\bar{y}} \, d {\bar{z}} \, dt \\ &= \left| z_0\right| \int _0^1 \int _{-\ell }^\ell \left| {\bar{z}}\right| ^{1-2s} w_\varepsilon '(z_0 + t {\bar{z}}) \int _{B_{{\ell }/{\left| {\bar{z}}\right| }}^{d-1}} \frac{1}{(1 +\left| {\bar{y}}\right| ^2)^{\frac{d+2s-1}{2}}} \, d {\bar{y}} \, d {\bar{z}} \, dt \\ &\le \left| z_0\right| \int _0^1 \frac{1}{t^{2-2s}} \int _{-\ell t}^{\ell t} \frac{w_\varepsilon '({\bar{z}} + z_0) }{\left| {\bar{z}}\right| ^{2s-1}}\, d {\bar{z}} \, dt. \end{aligned}$$\end{document}

Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2-2s <1 $$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| I_{4,2}\right| \lesssim \left| z_0\right| \int _{-\ell }^{\ell } \frac{w_\varepsilon '({\bar{z}} + z_0) }{\left| {\bar{z}}\right| ^{2s-1}}\, d {\bar{z}} \lesssim \ell ^{2-2s} \end{aligned}$$\end{document}

as for the term A in the estimate of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{2,2}$$\end{document} in Lemma 4.4.

Estimate of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{4,3}$$\end{document} . Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| w_\varepsilon \right| \le 1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d+2\,s-2 < d$$\end{document} , we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} I_{4,3}&\lesssim \int _{B_\ell ^{d-1}} \int _{-\ell }^\ell \frac{1}{\rho ^{d+2s-2}} \, d {\bar{z}} \, d {\bar{y}} \lesssim \ell ^{2-2s}. \end{aligned}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

On the finiteness of a constant

In this section, we show that the constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa _\star $$\end{document} in (1.9) is finite. Motivated by Theorem 4.1, we introduce the following notation.

Definition 5.1

Fix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in ({1}/{2},1)$$\end{document} and let w be the optimal profile of Theorem 2.2. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell , \varepsilon >0$$\end{document} , set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon (z) = w \left( {z}/{\varepsilon }\right) $$\end{document} and define

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \eta _{\varepsilon ,\ell }(z_0) = \int _{-\ell }^\ell \frac{w_\varepsilon '(z_0+z)}{\left| z\right| ^{2s-1}} \, dz. \end{aligned}$$\end{document}

Remark 5.2

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta _{\varepsilon ,\ell }$$\end{document} be given by (5.1). Since w is strictly increasing and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w'$$\end{document} is even and globally integrable, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta _{\varepsilon , \ell }$$\end{document} is nonnegative, even and globally bounded. Moreover, by the change of variable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z= {{\bar{z}}}/{\varepsilon }$$\end{document} it holds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \eta _{\varepsilon ,\ell }(z_0) = \varepsilon ^{1-2s} \eta _{1,{\ell }/{\varepsilon }}\left( {z_0 }/{\varepsilon } \right) . \end{aligned}$$\end{document}

Finally, we show the following result.

Proposition 5.3

Fix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell , \ell '>0$$\end{document} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta _{\varepsilon ,\ell }$$\end{document} be defined by (5.1). For any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in ({3}/{4},1)$$\end{document} , it holds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{\varepsilon \rightarrow 0^+} \int _{-{\ell '}/{\varepsilon }}^{{\ell '}/{\varepsilon }} \eta _{1, {\ell }/{\varepsilon }}(z)^2 \, dz =: \mu _w \in (0,\infty ). \end{aligned}$$\end{document}

The constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _w$$\end{document} depends only on w and it is independent of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell , \ell '$$\end{document} . For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s = {3}/{4}$$\end{document} , it holds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{\varepsilon \rightarrow 0^+} \frac{1}{\left| \log (\varepsilon )\right| } \int _{-{\ell '}/{\varepsilon }}^{{\ell '}/{\varepsilon }} \eta _{1, {\ell }/{\varepsilon }}(z)^2 \, dz = 8. \end{aligned}$$\end{document}

Proof

Fix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell , \ell ' >0$$\end{document} . Given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} we estimate the decay of the inner integral uniformly with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} . From now on, unless otherwise specified, we neglect constants depending on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s,W, \ell , \ell '$$\end{document} . By the decay of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w'$$\end{document} (see Theorem 2.2) we estimate

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\varepsilon >0} \sup _{\left| z\right| \le 1} \int _{-{\ell }/{\varepsilon }}^{{\ell }/{\varepsilon }} \frac{w'(z+t)}{\left| t\right| ^{2s-1}} \, dt \lesssim \int _{\left| t\right| \le 2} \frac{1}{\left| t\right| ^{2s-1}} \, dt + \sup _{\left| z\right| \le 1} \int _{ \left| t\right| \ge 2 } \frac{1}{1 + \left| t+z\right| ^{1+2s} } \, dt < +\infty . \nonumber \\ \end{aligned}$$\end{document}

To estimate the inner integral for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| z\right| \ge 1$$\end{document} , we fix a constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu >1$$\end{document} and we define

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} I_{z, \nu }:= \int _{\left| t\right| \le {\left| z\right| }/{\nu }} \frac{w'(t+z)}{\left| t\right| ^{2s-1}}\, dt, \quad II_{z, \nu } =: \int _{{\left| z\right| }/{\nu } \le \left| t\right| \le \nu \left| z\right| } \frac{w'(t+z)}{\left| t\right| ^{2s-1}} \, dt, \quad III_{z,\nu }:= \int _{ \left| t\right| \ge \nu \left| z\right| } \frac{w'(t+z)}{\left| t\right| ^{2s-1}} \, dt. \end{aligned}$$\end{document}

For the second term we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \nu ^{1-2s} \left| z\right| ^{1-2s} \int _{{\left| z\right| }/{\nu } \le \left| t\right| \le \nu \left| z\right| } w'(t+z) \, dt \le II_{z,\nu } \le \nu ^{2s-1} \left| z\right| ^{1-2s} \int _{{\left| z\right| }/{\nu } \le \left| t\right| \le \nu \left| z\right| } w'(t+z) \, dt. \end{aligned}$$\end{document}

Moreover, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ z\ge 1 $$\end{document} by direct computation and (2.5), we estimate

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\int _{{\left| z\right| }/{\nu } \le \left| t\right| \le \nu \left| z\right| } w'(t+z) \, dt \\&= w(z(1+\nu )) - w\left( z \left( 1+ \frac{1}{\nu }\right) \right) + w\left( z \left( 1- \frac{1}{\nu }\right) \right) - w(z(1-\nu )) \\ &= 2 + O(\left| z\right| ^{-2s}), \end{aligned}$$\end{document}

where the reminder depends also on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} . The same estimate can be proved for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z<-1$$\end{document} . Hence, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| z\right| \ge 1$$\end{document} we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 2 \nu ^{1-2s} \left| z\right| ^{1-2s} + O(\left| z\right| ^{1-4s}) \le II_{z,\nu } \le 2 \nu ^{2s-1} \left| z\right| ^{1-2s} + O(\left| z\right| ^{1-4s}). \end{aligned}$$\end{document}

By Theorem 2.2, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| z\right| \ge 1$$\end{document} we have the following estimate for the first term

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & I_{z,\nu } \lesssim \left| z\right| ^{-1-2s} \left( 1- \frac{1}{\nu }\right) ^{-1-2s} \int _{\left| t\right| \le {\left| z\right| }/{\nu }} \frac{1}{\left| t\right| ^{2s-1}}\, dt\nonumber \\ & \quad \le O(\left| z\right| ^{1-4s}), \end{aligned}$$\end{document}

where the reminder depends on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} . Similarly, for the third term, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| z\right| \ge 1$$\end{document} we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & III_{z, \nu } \le \nu ^{1-2s} \left| z\right| ^{1-2s} \int _{\left| t\right| \ge \nu \left| z\right| } w'(t+z) \, dt \le \nu ^{1-2s} \left| z\right| ^{1-2s} \int _{\left| t'\right| \ge \left| z\right| (\nu -1)} w'(t')\, dt' \nonumber \\ & \quad \le O(\left| z\right| ^{1-4s}), \end{aligned}$$\end{document}

where the reminder depends on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} . Then, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| z\right| \in \left( 1, {\ell }/{\varepsilon \nu }\right) $$\end{document} , we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} II_{z, \nu } \le \int _{-{\ell }/{\varepsilon }}^{{\ell }/{\varepsilon }} \frac{w'(t+z)}{\left| t\right| ^{2s-1}} \, dt \le I_{z,\nu } + II_{z,\nu } +III_{z,\nu } \end{aligned}$$\end{document}

and by (5.6)–(5.8) we estimate

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 2\nu ^{1-2s} \left| z\right| ^{1-2s} + O(\left| z\right| ^{1-4s}) \le \int _{-{\ell }/{\varepsilon }}^{{\ell }/{\varepsilon }} \frac{w'(z+t)}{\left| t\right| ^{2s-1}} \, dt \le 2 \nu ^{2s-1} \left| z\right| ^{1-2s} + O(\left| z\right| ^{1-4s}).\nonumber \\ \end{aligned}$$\end{document}

Here the reminders depend on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} , but they are independent of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} . With the same technique, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| z\right| \in \left( {\ell }/{\varepsilon \nu }, {\ell \nu }/{\varepsilon } \right) $$\end{document} we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{-{\ell }/{\varepsilon }}^{{\ell }/{\varepsilon }} \frac{w'(z+t)}{\left| t\right| ^{2s-1}} \, dt \le I_{z,\nu } + II_{z,\nu } \le O(\left| z\right| ^{1-2s}). \nonumber \\ \end{aligned}$$\end{document}

Similarly, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| z\right| \ge {\ell \nu }/{\varepsilon }$$\end{document} we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{-{\ell }/{\varepsilon }}^{{\ell }/{\varepsilon }} \frac{w'(z+t)}{\left| t\right| ^{2s-1}} \, dt \le I_{z,\nu } \le O(\left| z\right| ^{1-4s}). \end{aligned}$$\end{document}

Here the reminders depend on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} but not on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} . In conclusion, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in \left( {3}/{4}, 1\right) $$\end{document} it is trivial to see that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mu _w = \int _{\mathbb {R}} \left( \int _{\mathbb {R}} \frac{w'(t+z)}{\left| t\right| ^{2s-1}} \, dt \right) ^2 \, d z \in (0,+\infty ) \end{aligned}$$\end{document}

and it is independent of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell , \ell '$$\end{document} . To show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _w$$\end{document} is finite, using (5.5), (5.9)–(5.11) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu =2$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sup _{\varepsilon> 0} \int _{-{\ell '}/{\varepsilon }}^{{\ell '}/{\varepsilon }} \left( \int _{-{\ell }/{\varepsilon }}^{{\ell }/{\varepsilon } } \frac{w'(t+z)}{\left| t\right| ^{2s-1}} \, dt \right) ^2 \, dz&\lesssim 1+ \int _{ \left| z\right| >1 } \left| z\right| ^{2-4s}\, dz < +\infty . \end{aligned}$$\end{document}

To conclude, we consider the case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s = {3}/{4}$$\end{document} . Then, using (5.5), (5.9)–(5.11) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu >1$$\end{document} (to be chosen later), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\limsup _{\varepsilon \rightarrow 0^+} \frac{1}{\left| \log (\varepsilon )\right| }\int _{-{\ell '}/{\varepsilon }}^{{\ell '}/{\varepsilon }} \left( \int _{-{\ell }/{\varepsilon }}^{{\ell }/{\varepsilon } } \frac{w'(t+z)}{\left| t\right| ^{\frac{1}{2}}} \, dt \right) ^2 \, dz \le \limsup _{\varepsilon \rightarrow 0^+} \frac{1}{\left| \log (\varepsilon )\right| }\int _{\mathbb {R}} \left( \int _{-{\ell }/{\varepsilon }}^{{\ell }/{\varepsilon } } \frac{w'(t+z)}{\left| t\right| ^{\frac{1}{2}}} \, dt \right) ^2 \, dz \ \nonumber \\&\quad \le \limsup _{\varepsilon \rightarrow 0^+} \frac{1}{\left| \log (\varepsilon )\right| } \left( \int _{\left| z\right| \le 1} + \int _{1 \le \left| z\right| \le {\ell }/{\varepsilon \nu }} + \int _{{\ell }/{\varepsilon \nu } \le \left| z\right| \le {\ell \nu }/{\varepsilon }} + \int _{ \left| z\right| \ge {\ell \nu }/{\varepsilon } } \right) \left( \int _{-{\ell }/{\varepsilon }}^{{\ell }/{\varepsilon } } \frac{w'(t+z)}{\left| t\right| ^{\frac{1}{2}}} \, dt \right) ^2 \, dz\nonumber \\&\quad \le \limsup _{\varepsilon \rightarrow 0^+} \frac{1}{\left| \log (\varepsilon )\right| } \bigg ( O(1) + \int _{1 \le \left| z\right| \le {\ell }/{\varepsilon \nu } } \left( \frac{4\nu }{\left| z\right| } + O(\left| z\right| ^{-4}) \right) \, dz + \int _{ {\ell }/{\varepsilon \nu } \le \left| z\right| \le {\ell \nu }/{ \varepsilon } } O(\left| z\right| ^{-1}) \, dz\nonumber \\&\qquad + \int _{ \left| z\right| \ge {\ell \nu }/{\varepsilon } } O(\left| z\right| ^{-4}) \, dz \bigg ) = 8 \nu . \end{aligned}$$\end{document}

Using the lower bound in (5.9) and setting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{\ell }}:= \min \{\ell , \ell '\}$$\end{document} , we estimate

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \liminf _{\varepsilon \rightarrow 0^+} \frac{1}{\left| \log (\varepsilon )\right| }&\int _{-{\ell '}/{\varepsilon }}^{{\ell '}/{\varepsilon }} \left( \int _{-{\ell }/{\varepsilon }}^{{\ell }/{\varepsilon } } \frac{w'(t+z)}{\left| t\right| ^{\frac{1}{2}}} \, dt \right) ^2 \, dz\nonumber \\ &\ge \liminf _{\varepsilon \rightarrow 0^+} \frac{1}{\left| \log (\varepsilon )\right| }\int _{ 1 \le \left| z\right| \le {{\bar{\ell }}}/{\varepsilon \nu } } \left( \int _{-{\ell }/{\varepsilon }}^{{\ell }/{\varepsilon } } \frac{w'(t+z)}{\left| t\right| ^{\frac{1}{2}}} \, dt \right) ^2 \, dz \nonumber \\ &\ge \liminf _{\varepsilon \rightarrow 0^+} \frac{1}{\left| \log (\varepsilon )\right| } \int _{1 \le \left| z\right| \le {{\bar{\ell }}}/{\varepsilon \nu } } \left( \frac{4 \nu ^{-1}}{\left| z\right| } + O(\left| z\right| ^{-4}) \right) \, dz \ge 8 \nu ^{-1}. \end{aligned}$$\end{document}

Combining (5.12) and (5.13) and letting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu \rightarrow 1^+$$\end{document} , we prove (5.4). \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Proof of the main result

This section is entirely devoted to the proof of Theorem 1.1. Our analysis is based on the following heuristic argument which allows to guess the right scaling of the energy for different values of the parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in \left( {1}/{2},1\right) $$\end{document} . Given a smooth set E, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_\varepsilon (x) = w_\varepsilon (\beta _\Sigma (x))$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon $$\end{document} is the scaled one-dimensional optimal profile and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _\Sigma $$\end{document} is the modified signed distance function (see Definition 2.10). We split the energy in the contribution around the interface and the contribution far from the interface. By Theorem 4.1, we exploit the cancellations in the energy around the interface due to the structure of the recovery sequence and we use the decay property of the optimal profile to estimate separately the two terms in the integral away from the interface. More precisely, it is not difficult to see that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\int _{\Omega } \left( \varepsilon ^{2s-1} (-\Delta )^s u_\varepsilon + \frac{W'(u_\varepsilon )}{\varepsilon } \right) ^2 \, dx\\&\simeq \, \varepsilon ^{4s-2} \mathcal {W}(\Sigma , \Omega ) \int _{-\delta }^\delta \left( \int _{-\delta }^\delta \frac{w_\varepsilon '(z+t)}{\left| t\right| ^{2s-1}} \, dt \right) ^2 \, dz\\&\quad + \varepsilon ^{4s-2} \left( \Vert {R_\varepsilon } \Vert _{L^\infty (\Sigma _\delta )}^2 + \Vert {(-\Delta )^s u_\varepsilon } \Vert ^2_{L^\infty (\Omega \setminus \Sigma _\delta )}\right) \\&\quad + \varepsilon ^{-2} \Vert {W'(u_\varepsilon )} \Vert _{L^\infty (\Omega \setminus \Sigma _\delta )}^2\\&\simeq \, \varepsilon \mathcal {W}(\Sigma , \Omega ) \int _{-{\delta }/{\varepsilon }}^{{\delta }/{\varepsilon }} \left( \int _{-{\delta }/{\varepsilon }}^{{\delta }/{\varepsilon }} \frac{w'(z+t)}{\left| t\right| ^{2s-1}} \, dt \right) ^2 \, dz + O(\varepsilon ^{4s-2}). \end{aligned}$$\end{document}

Moreover, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{-{\delta }/{\varepsilon }}^{{\delta }/{\varepsilon }} \left( \int _{-{\delta }/{\varepsilon }}^{{\delta }/{\varepsilon }} \frac{w'(z+t)}{\left| t\right| ^{2s-1}} \, dt \right) ^2 \, dz \simeq {\left\{ \begin{array}{ll} 1 & \text { if } s \in \left( {3}/{4},1 \right) , \\[0.5ex] \left| \log (\varepsilon )\right| & \text { if } s = {3}/{4}, \\[0.5ex] \varepsilon ^{4s-3} & \text { if } s \in ({1}/{2},{3}/{4}), \end{array}\right. } \end{aligned}$$\end{document}

where we proved the first two estimates in Lemma 5.3, while the last one can be obtained with an simple modification of the argument. Therefore, we conclude that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{\Omega } \left( \varepsilon ^{2s-1} (-\Delta )^s u_\varepsilon + \frac{W'(u_\varepsilon )}{\varepsilon } \right) ^2 \, dx \simeq {\left\{ \begin{array}{ll} \varepsilon \mathcal {W}(\Sigma , \Omega ) + O(\varepsilon ^{4s-2}) & s \in \left( {3}/{4}, 1 \right) , \\[1ex] \varepsilon \left| \log \varepsilon \right| \mathcal {W}(\Sigma , \Omega ) + O(\varepsilon ^{4s-2}) & s = {3}/{4}, \\[1ex] \varepsilon ^{4s-2} \mathcal {W}(\Sigma , \Omega ) + O(\varepsilon ^{4s-2}) & s \in \left( {1}/{2}, {3}/{4} \right) . \end{array}\right. } \end{aligned}$$\end{document}

This argument shows different behaviours according to the value of the parameter s. Indeed, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s > {3}/{4}$$\end{document} then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \mathcal {W}(\Sigma , \Omega )$$\end{document} is the leading order term and, after dividing the energy by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} , we see the Willmore functional in the limit. The situation is similar for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s = {3}/{4}$$\end{document} , provided that we divide the energy by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \left| \log (\varepsilon )\right| $$\end{document} . Roughly speaking, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in \left[ {3}/{4}, 1\right) $$\end{document} , it turns out that the energy around the interface is much larger than the energy away from the interface and, after introducing an appropriate scaling, the limit has a purely local behaviour. In the proof of Theorem 1.1, we make this argument rigorous. This heuristic breaks down when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in \left( {1}/{2}, {3}/{4}\right) $$\end{document} . Our analysis suggests that all terms have the same order of magnitude and, after dividing the energy by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon ^{4s-2}$$\end{document} , a local/nonlocal behaviour might persist in the limit. To conclude, it seems that a finer analysis of both the error term in the expansion of the fractional Laplacian around the interface and the energy away from the interface would be needed to understand the limiting behaviour of our energy in the case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in \left( {1}/{2}, {3}/{4}\right) $$\end{document} . From now on, we focus on the case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in \left[ {3}/{4},1\right) $$\end{document} .

Proof of Theorem 1.1

We divide the proof in some steps. We recall that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\star $$\end{document} is the constant in (1.6) and we define

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \kappa _\star := {\left\{ \begin{array}{ll} \displaystyle \frac{\gamma _{1,s}^2}{4(2s-1)^2} \int _{\mathbb {R}} \left( \int _{\mathbb {R}} \frac{w'(t+z)}{\left| t\right| ^{2s-1}} \, dt \right) ^2 \, dz & s \in \left( {3}/{4}, 1\right) , \\[0.5ex] 8 \gamma _{1, {3}/{4}}^2 & s = {3}/{4}. \end{array}\right. } \end{aligned}$$\end{document}

Step 1. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in \left( {3}/{4}, 1 \right) $$\end{document} and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E \subset \mathbb {R}^d$$\end{document} be a bounded open set with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial E \in C^3$$\end{document} . Define

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u_\varepsilon (x) = w_\varepsilon (\beta _\Sigma (x)), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\varepsilon (z) = w \left( {z}/{\varepsilon } \right) $$\end{document} , w is the one-dimensional optimal profile and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _\Sigma $$\end{document} is the modified distance function according to Definition 2.10. We claim that for any bounded open set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}^d$$\end{document} we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \limsup _{\varepsilon \rightarrow 0^+} \left( \mathcal {F}_{s, \varepsilon } + \mathcal {G}_{s,\varepsilon } \right) (u_\varepsilon , \Omega ) \le c_\star \mathcal {H}^{d-1}(\partial E \cap {\overline{\Omega }}) + \kappa _{\star } \int _{\partial E \cap {\overline{\Omega }}} H_{\partial E}(y)^2 \, d \mathcal {H}^{d-1}(y). \end{aligned}$$\end{document}

The same conclusions holds for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s = {3}/{4}$$\end{document} adding an extra \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| \log (\varepsilon )\right| ^{-1}$$\end{document} in front of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}_{s,\varepsilon }$$\end{document} .

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in \left[ {3}/{4}, 1\right) $$\end{document} , by a simple modification of the proof of [49, Proposition 4.7] due to the fact that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _\Sigma $$\end{document} is the proper distance function in a tubular neighbourhood of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma = \partial E$$\end{document} , it is readily checked that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \limsup _{\varepsilon \rightarrow 0^+} \mathcal {F}_{\varepsilon ,s}(u_\varepsilon , \Omega ) \le c_{\star } \mathcal {H}^{d-1}( \Sigma \cap {\overline{\Omega }}). \end{aligned}$$\end{document}

We point out that no regularity on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega $$\end{document} is needed. Therefore, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in \left( {3}/{4}, 1 \right) $$\end{document} it remains to check that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \limsup _{\varepsilon \rightarrow 0^+} \mathcal {G}_{s,\varepsilon }(u_\varepsilon , \Omega ) \le \kappa _\star \int _{\Sigma \cap {\overline{\Omega }}} H_{\Sigma }(y)^2 \, d \mathcal {H}^{d-1}(y). \end{aligned}$$\end{document}

If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s = {3}/{4}$$\end{document} , (6.4) holds adding an extra \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| \log (\varepsilon )\right| ^{-1}$$\end{document} in front of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}_{s,\varepsilon }$$\end{document} .

The case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in \left( {3}/{4}, 1 \right) $$\end{document} . Given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta >0$$\end{document} as in Definition 2.10 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _0 \ge 1 $$\end{document} as in Theorem 4.1, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \ge \Lambda _0$$\end{document} we split

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {G}_{s,\varepsilon }(u_\varepsilon , \Omega )&= \left( \frac{1}{\varepsilon } \int _{\Omega \cap \Sigma _{{\delta }/{10 \Lambda }} } + \frac{1}{\varepsilon } \int _{\Omega \setminus \Sigma _{{\delta }/{10 \Lambda }}} \right) \left( \varepsilon ^{2s-1} (-\Delta )^s u_\varepsilon (x) + \frac{W'(u_\varepsilon (x))}{\varepsilon } \right) ^2 dx\,\\&= I_{\varepsilon ,\Lambda } + II_{\varepsilon ,\Lambda }. \end{aligned}$$\end{document}

We claim that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{\varepsilon \rightarrow 0^+} II_{\varepsilon ,\Lambda } = 0 \qquad \forall \Lambda \ge \Lambda _0. \end{aligned}$$\end{document}

To deal with the energy away from the interface, we neglect constants independent of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} . Thus, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \ge \Lambda _0$$\end{document} , we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} II_{\varepsilon ,\Lambda }&\lesssim \varepsilon ^{4s-3} \int _{\Omega \setminus \Sigma _{{\delta }/{10 \Lambda }}} \left( (-\Delta )^s u_\varepsilon (x) \right) ^2 \, dx \\&\quad + \varepsilon ^{-3} \int _{\Omega \setminus \Sigma _{{\delta }/{10 \Lambda }}} \left( W'(u_\varepsilon (x)) \right) ^2 \, dx = II_{1, \varepsilon ,\Lambda } + II_{2, \varepsilon , \Lambda }. \end{aligned}$$\end{document}

By Lemma 3.3, we conclude that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$II_{1, \varepsilon , \Lambda } \rightarrow 0$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0$$\end{document} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s>{3}/{4}$$\end{document} . Regarding \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_{2, \varepsilon , \Lambda }$$\end{document} , since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W'$$\end{document} is Lipschitz and odd, using (2.5), the monotonicity of w and the fact that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| \beta _{\Sigma }(x)\right| > {\delta }/{10 \Lambda }$$\end{document} in the domain of integration (see Definition 2.10), we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} II_{2, \varepsilon ,\Lambda }&= \varepsilon ^{-3} \int _{ E \setminus \Sigma _{{\delta }/{10 \Lambda }}} (W'(u_\varepsilon (x)) - W'(1))^2\, dx + \varepsilon ^{-3} \int _{(\Omega \setminus E) \setminus \Sigma _{{\delta }/{10 \Lambda }} } (W'(u_\varepsilon (x)) - W'(-1) )^2 \, dx \\ &\lesssim \varepsilon ^{-3} \int _{E \setminus \Sigma _{{\delta }/{10 \Lambda }}} \left| u_\varepsilon (x) - 1\right| ^2\, dx + \varepsilon ^{-3} \int _{(\Omega \setminus E) \setminus \Sigma _{{\delta }/{10 \Lambda }} } \left| u_\varepsilon (x) + 1 \right| ^2 \, dx \\ &\lesssim \varepsilon ^{-3} \int _{E \setminus \Sigma _{{\delta }/{10 \Lambda }} } \left| 1- w\left( \frac{\delta }{10 \Lambda \varepsilon }\right) \right| ^2\, dx + \varepsilon ^{-3} \int _{(\Omega \setminus E) \setminus \Sigma _{{\delta }/{10 \Lambda }} } \left| 1+ w\left( -\frac{\delta }{10 \Lambda \varepsilon }\right) \right| ^2 \, dx \lesssim \varepsilon ^{4s-3}. \end{aligned}$$\end{document}

Hence, we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$II_{2,\varepsilon ,\Lambda } \rightarrow 0$$\end{document} as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0^+$$\end{document} , since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s >{3}/{4}$$\end{document} .

Letting be \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa _\star $$\end{document} as in (6.1), it remains to check that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \inf _{\Lambda \ge \Lambda _0} \limsup _{\varepsilon \rightarrow 0^+} I_{\varepsilon ,\Lambda } \le \kappa _\star \int _{\Sigma \cap {\overline{\Omega }}} H_{\Sigma }(x) ^2 \, d \mathcal {H}^{d-1}(x). \end{aligned}$$\end{document}

Then, (6.4) follows by (6.5) and (6.6). The proof of (6.6) is based on the expansion of the fractional Laplacian (4.2). Fix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \ge \Lambda _0$$\end{document} . Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _\Sigma $$\end{document} is the proper signed distance in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma _{{\delta }/{10 \Lambda }}$$\end{document} (see Definition 2.10), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & I_{\varepsilon ,\Lambda } = \frac{1}{\varepsilon }\nonumber \\ & \quad \int _{\Sigma _{{\delta }/{10 \Lambda }} \cap \Omega } \left( \frac{W'(u_\varepsilon (x))}{\varepsilon } + \varepsilon ^{2s-1} \left( (-\partial _{zz})^s w_\varepsilon (z) + \frac{\gamma _{1,s}}{4s-2} H_{\Sigma }(x') \eta _{\varepsilon ,{\delta }/{10 \Lambda }} (z) + \mathcal {R}_{\varepsilon , \Lambda }(x) \right) \right) ^2 \, dx, \nonumber \\ \end{aligned}$$\end{document}

where we set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x'= \pi _{\Sigma }(x)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z = \textrm{dist}_{\Sigma }(x)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta _{\varepsilon , {\delta }/{10 \Lambda }}$$\end{document} is given by (5.1) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_{\varepsilon , \Lambda }$$\end{document} is uniformly bounded with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty (\Sigma _{{\delta }/{10 \Lambda }})$$\end{document} (see Theorem 4.1). By the scaling properties of the fractional Laplacian and the fact that w solves (2.3), it results that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (-\partial _{zz})^s w_\varepsilon (z) = \varepsilon ^{-2s} (-\partial _{zz})^s w\left( {z}/{\varepsilon }\right) = -\varepsilon ^{-2s} W'\left( w\left( {z}/{\varepsilon }\right) \right) = - \varepsilon ^{-2s} W'(u_\varepsilon (x)). \end{aligned}$$\end{document}

Therefore, (6.7) reads as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} I_{\varepsilon , \Lambda } = \varepsilon ^{4s-3} \int _{\Sigma _{{\delta }/{10 \Lambda }} \cap \Omega } \left( \frac{\gamma _{1,s}}{4s-2} H_{\Sigma }(x') \eta _{\varepsilon ,{\delta }/{10 \Lambda }}(z) + \mathcal {R}_{\varepsilon , \Lambda }(x) \right) ^2 \, dx. \end{aligned}$$\end{document}

Furthermore, using (5.2) and changing variables, we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varepsilon ^{4s-3} \int _{-{\delta }/{10 \Lambda }}^{{\delta }/{10 \Lambda }} \eta _{\varepsilon ,{\delta }/{10 \Lambda }} (z)^2 \, dz&= \varepsilon ^{-1} \int _{-{\delta }/{10 \Lambda }}^{{\delta }/{10 \Lambda }} \eta _{1,{\delta }/{10 \varepsilon \Lambda }}\left( \frac{z}{\varepsilon }\right) ^2 \, dz = \int _{-{\delta }/{10 \varepsilon \Lambda }}^{{\delta }/{10 \varepsilon \Lambda }} \eta _{1,{\delta }/{10 \varepsilon \Lambda }}(z)^2 \, dz. \end{aligned}$$\end{document}

Thus, using Lemma 5.3 we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{\varepsilon \rightarrow 0^+} \varepsilon ^{4s-3} \int _{-{\delta }/{10 \Lambda }}^{{\delta }/{10 \Lambda }} \eta _{\varepsilon , {\delta }/{10 \Lambda }}(z)^2 \, dz = \int _{\mathbb {R}} \left( \int _{\mathbb {R}} \frac{w'(t+z)}{\left| t\right| ^{2s-1}} \, dt \right) ^2 \, dz = \mu _w. \end{aligned}$$\end{document}

Thus, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s >{3}/{4}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_{\varepsilon , \Lambda }$$\end{document} is uniformly bounded with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\infty }(\Sigma _{{\delta }/{10 \Lambda }})$$\end{document} , we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \limsup _{\varepsilon \rightarrow 0^+} I_{\varepsilon ,\Lambda } = \frac{\gamma _{1,s}^2}{4(2s-1)^2} \limsup _{\varepsilon \rightarrow 0^+} \varepsilon ^{4s-3} \int _{\Omega \cap \Sigma _{{\delta }/{10 \Lambda }}} \left| H_{\Sigma }(x') \eta _{\varepsilon ,{\delta }/{10 \Lambda }}(z)\right| ^2 \, dx. \end{aligned}$$\end{document}

We remark that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Sigma _{{\delta }/{10 \Lambda }} \cap \Omega \subset \left\{ x'+z N_{\Sigma }(x') :x' \in \pi _{\Sigma }\left( \Sigma _{{\delta }/{10 \Lambda }} \cap \Omega \right) , \ z \in \left( -{\delta }/{10 \Lambda }, {\delta }/{10 \Lambda } \right) \right\} . \end{aligned}$$\end{document}

Thus, by (6.8), (6.9) and the Coarea formula, we write

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \limsup _{\varepsilon \rightarrow 0^+} I_{\varepsilon , \Lambda }&= \limsup _{\varepsilon \rightarrow 0^+} \varepsilon ^{4s-3} \int _{-{\delta }/{10 \Lambda }}^{{\delta }/{10 \Lambda }} \left| \eta _{\varepsilon ,{\delta }/{10 \Lambda }} (z)\right| ^2 \left( \int _{ \pi _{\Sigma } (\Sigma _{{\delta }/{10 \Lambda }} \cap \Omega ) } \left| H_{\Sigma }(x') \right| ^2 \, d \mathcal {H}^{d-1}(x') \right) \, dz \\ &= \mu _w \int _{ \pi _{\Sigma } (\Sigma _{{\delta }/{10 \Lambda }} \cap \Omega ) } \left| H_{\Sigma }(x') \right| ^2 \, d \mathcal {H}^{d-1}(x'), \end{aligned}$$\end{document}

Hence, (6.6) follows by the computation above and the fact \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bigcap _{\Lambda \ge \Lambda _0} \pi _{\Sigma } (\Sigma _{{\delta }/{10 \Lambda }} \cap \Omega ) \subset \Sigma \cap {\bar{\Omega }}$$\end{document} .

The case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s = {3}/{4}$$\end{document} . Here, we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4s-3 = 0$$\end{document} , and hence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon ^{4s-3} =1 $$\end{document} , for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon > 0$$\end{document} . However, there is an extra factor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| \log (\varepsilon )\right| ^{-1}$$\end{document} in the definition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}_{s, \varepsilon }$$\end{document} and Lemma 5.3 is modified accordingly. Therefore, the proof is similar to the previous case and we leave the straightforward modifications to the reader.

Step 2. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega , E \subset \mathbb {R}^d$$\end{document} be bounded open sets with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega \in C^1$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial E \in C^2$$\end{document} . By Lemma 2.18, we find a sequence of smooth sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{E_j\}_{j \in \mathbb {N}}$$\end{document} satisfying (A1), (A2), (A3), (A4), (A5). By the argument shown in the previous step, for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \in \mathbb {N}$$\end{document} we find a sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_\varepsilon ^j \rightarrow \chi _{E_j}$$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1_{\textrm{loc}}(\mathbb {R}^d)$$\end{document} such that (6.3) holds for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_j$$\end{document} . By (A2) and (A4), we have that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}^{d-1}(\partial E_j \cap {\overline{\Omega }}) \rightarrow \text {Per}(E, \Omega )$$\end{document} . Then, by (A3), (A4), (A5), it is immediate to check that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{j \rightarrow \infty } \int _{\partial E_j \cap {\overline{\Omega }}} H_{\partial E_j}(y)^2 \, d \mathcal {H}^{d-1}(y) = \mathcal {W}(\Sigma , \Omega ). \end{aligned}$$\end{document}

Then, by a diagonal argument we build a sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_\varepsilon \rightarrow \chi _E$$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1_{\textrm{loc}}(\mathbb {R}^d)$$\end{document} such that (6.3) holds. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Remark 6.1

We point out that by a simple modification of the proof of Theorem 1.1 it is possible to check that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{\varepsilon \rightarrow 0^+} \left( \mathcal {F}_{s, \varepsilon } + \mathcal {G}_{s,\varepsilon } \right) (u_\varepsilon , \Omega ) = c_\star \mathcal {H}^{d-1}(\partial E \cap \Omega ) + \kappa _{\star } \int _{\partial E \cap \Omega } H_{\partial E}(y)^2 \, d \mathcal {H}^{d-1}(y), \end{aligned}$$\end{document}

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in \left( {3}/{4},1 \right) $$\end{document} , E is a bounded open set of class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^3$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_\varepsilon $$\end{document} is defined by (6.2) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} is any bounded open set whose boundary intersects transversally \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial E$$\end{document} , that is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}^{d-1}(\partial E \cap \partial \Omega ) = 0$$\end{document} . Here, no regularity on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega $$\end{document} is needed. The same conclusions holds for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s = {3}/{4}$$\end{document} adding an extra \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| \log (\varepsilon )\right| ^{-1}$$\end{document} in front of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}_{s,\varepsilon }$$\end{document} . We leave the details to the interested reader.

Remark 6.2

Once Theorem 1.1 is established, we can prove (6.3) for bounded open sets E of class  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2$$\end{document} . Indeed, given any bounded open set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} (without further assumptions), we find a sequence of smooth bounded open sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _j$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bigcap _j \Omega _j = \Omega $$\end{document} . Then, using Theorem 1.1 for E in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _j$$\end{document} and recalling that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{j \rightarrow \infty } c_\star \textrm{Per}(E, \Omega _j) + \kappa _\star \mathcal {W}(E, \Omega _j) = c_\star \mathcal {H}^{d-1}(\partial E\cap {\overline{\Omega }}) + \kappa _\star \int _{\partial E \cap {\overline{\Omega }}} H_{\partial E}(y)^2 \, d \mathcal {H}^{d-1}(y), \end{aligned}$$\end{document}

we build the sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{u_\varepsilon \}$$\end{document} by a standard diagonal argument.

Appendix: The exact constants

We collect some useful results on the exact value of some constants involved in our computations. We recall that the Euler Gamma function and Beta function are defined respectively by

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Gamma (x) = \int _0^{+\infty } t^{x-1} e^{-t} \, dt, \qquad \textrm{B}(x,y) = \int _0^1(1-t)^{x-1} t^{y-1} \, dt \qquad x,y >0. \end{aligned}$$\end{document}

We recall that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & \Gamma (x+1) = x \Gamma (x), \qquad \Gamma ({1}/{2}) = \sqrt{\pi },\nonumber \\ & \qquad \mathcal {H}^{d-1} (\mathbb {S}^{d-1}) = \frac{2 \pi ^{\frac{d}{2}}}{\Gamma \left( \frac{d}{2}\right) }, \qquad \textrm{B}(x,y) = \frac{\Gamma (x) \Gamma (y)}{\Gamma (x+y)}. \end{aligned}$$\end{document}

Lemma 6.3

Given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a>-1, b > a+1$$\end{document} , it holds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 2\int _0^\infty \frac{ r^a }{ (r^2+1)^{\frac{b}{2}} }\,dr = \frac{ \Gamma \left( \frac{a+1}{2}\right) \Gamma \left( \frac{b-(a+1)}{2}\right) }{ \Gamma \left( \frac{b}{2}\right) }. \end{aligned}$$\end{document}

Proof

The computation is straightforward. After the change of variable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t= (r^2+1)^{-1}$$\end{document} , by standard manipulations, we find that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 2 \int _0^\infty \frac{ r^a }{ (r^2+1)^{\frac{b}{2}} } \, dr&= \int _0^1 t^{\frac{b}{2}} \left( \frac{1-t}{t} \right) ^{\frac{a-1}{2}} \frac{dt}{t^2} = \textrm{B}\left( \frac{b-a-1}{2}, \frac{a+1}{2} \right) . \end{aligned}$$\end{document}

Hence, (6.11) follows by the above computation and (6.10). \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

Corollary 6.4

Fix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta >0$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha , \beta \ge 0$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2s+\beta +1 >\alpha $$\end{document} . There exists an explicit constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M(d,s,\alpha , \beta ) >0$$\end{document} such that for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z \in (-\delta , \delta )$$\end{document} it holds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{B_\delta ^{d-1}} \frac{\left| y\right| ^\alpha }{(\left| z\right| ^2 + \left| y\right| ^2)^{\frac{d+2s + \beta }{2}}} \, d y = \mathcal {M}(d,s,\alpha ,\beta ) \left| z\right| ^{\alpha -1-\beta -2s} + O(\delta ^{\alpha -1 -\beta -2s }). \end{aligned}$$\end{document}

More precisely, it holds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & \int _{B_\delta ^{d-1}} \frac{1}{(\left| z\right| ^2 + \left| y\right| ^2)^{\frac{d+2s}{2}}} \, d y = \frac{\gamma _{1,s}}{\gamma _{d,s}} \left| z\right| ^{-1-2s} + O(\delta ^{-1-2s}), \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & \int _{B_\delta ^{d-1}} \frac{\left| y\right| ^2}{(\left| z\right| ^2 + \left| y\right| ^2)^{\frac{d+2s}{2}}} \, d y = \frac{(d-1) \gamma _{1,s}}{(2s-1) \gamma _{d,s} } \left| z\right| ^{1-2s} + O(\delta ^{1-2s}), \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} & \int _{B_\delta ^{d-1}} \frac{y_i^2}{(\left| z\right| ^2 + \left| y\right| ^2)^{\frac{d+2s}{2}}} \, d y = \frac{1}{2s-1} \frac{ \gamma _{1,s}}{ \gamma _{d,s} } \left| z\right| ^{1-2s} + O(\delta ^{1-2s}) \qquad i = 1, \dots , d-1.\nonumber \\ \end{aligned}$$\end{document}

The reminders are uniform with respect to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z \in (-\delta , \delta )$$\end{document} .

Proof

We set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {M}(d,s,\alpha , \beta ):= \int _{\mathbb {R}^{d-1}} \frac{\left| y\right| ^{\alpha }}{(1+\left| y\right| ^2)^{\frac{d+2s+\beta }{2}}} \, dy \end{aligned}$$\end{document}

and we remark that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M(d,s,\alpha , \beta )$$\end{document} is finite since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d+2s+\beta - \alpha > d-1$$\end{document} . Then, to check (6.12), we change variables \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y= \left| z\right| {\tilde{y}}$$\end{document} and we compute

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\int _{B_\delta ^{d-1}} \frac{\left| y\right| ^{\alpha }}{(\left| z\right| ^2 + \left| y\right| ^2)^{\frac{d+2s+\beta }{2}}} \, d y = \left| z\right| ^{\alpha -1-\beta -2s} \int _{B^{d-1}_{{\delta }/{\left| z\right| } }} \frac{\left| {\tilde{y}}\right| ^{\alpha }}{ (1+ \left| {\tilde{y}}\right| ^2) ^{\frac{d+2s +\beta }{2}}} \, d {\tilde{y}} \\ &= \left| z\right| ^{\alpha -1-\beta -2s} \left( \mathcal {M}(d,s,\alpha ,\beta ) - \int _{\left| {\tilde{y}}\right| \ge \frac{\delta }{\left| z\right| } } \frac{\left| {\tilde{y}}\right| ^{\alpha }}{ (1+ \left| {\tilde{y}}\right| ^2) ^{\frac{d+2s+\beta }{2}}} \, d {\tilde{y}} \right) \\ &= \left| z\right| ^{\alpha -1-\beta -2s} \left( \mathcal {M}(d,s,\alpha ,\beta ) - O \left( \left( \frac{\delta }{\left| z\right| } \right) ^{\alpha -1-\beta -2s} \right) \right) . \end{aligned}$$\end{document}

To prove (6.13) and (6.14), we compute the explicit value of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M}(d,s,\alpha , \beta )$$\end{document} . When \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\alpha , \beta ) = (0,0)$$\end{document} , by Lemmas 6.3, (6.10) and (2.2) we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {M}(d,s,0,0)&= \int _{\mathbb {R}^{d-1}} \frac{1}{(1+ \left| y\right| ^2)^{\frac{d+2s}{2}}} \, dy \\&= \mathcal {H}^{d-2}(\mathbb {S}^{d-2}) \int _0^{+\infty } \frac{r^{d-2}}{(1+r^2)^{\frac{d+2s}{2}}} \, dr = \pi ^{\frac{d-1}{2}} \frac{\Gamma (\frac{2s+1}{2})}{\Gamma (\frac{d+2s}{2})} = \frac{\gamma _{1,s}}{\gamma _{d,s}}, \end{aligned}$$\end{document}

Similarly, it is immediate to check that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {M}(d,s, 2,0) = \frac{d-1}{2s-1} \cdot \frac{\gamma _{1,s}}{\gamma _{d,s}}. \end{aligned}$$\end{document}

Lastly, we observe that (6.15) follows by (6.14), since by symmetry we have that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{B_\delta ^{d-1}} \frac{y_i^2}{(\left| z\right| ^2 + \left| y\right| ^2)^{\frac{d+2s}{2}}} \, d y = \frac{1}{d-1} \int _{B_\delta ^{d-1}} \frac{\left| y\right| ^2}{(\left| z\right| ^2 + \left| y\right| ^2)^{\frac{d+2s}{2}}} \, d y \qquad i =1, \dots , d-1. \end{aligned}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}