Smoothness and stability in the Alt–Phillips problem

TL;DR

This paper investigates the Alt–Phillips free boundary problem for negative exponents, proving smoothness and deriving new stability conditions.

Contribution

A unified proof of smoothness and a new stability condition for the Alt–Phillips problem with negative exponents.

Findings

Smoothness of $C^{1,\alpha}$-regular free boundaries is proven using Schauder estimates for degenerate quasilinear PDEs.

A new stability condition is derived, ruling out nontrivial axially symmetric stable cones in low dimensions.

A variational criterion for stability recovers the one for minimal surfaces as $\gamma \rightarrow -2$.

Abstract

We study the one-phase Alt–Phillips free boundary problem, focusing on the case of negative exponents \documentclass[12pt]{minimal}
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\begin{document}$$\gamma \in (-2,0)$$\end{document}γ∈(-2,0). The goal of this paper is twofold. On the one hand, we prove smoothness of \documentclass[12pt]{minimal}
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\begin{document}$$C^{1,\alpha }$$\end{document}C1,α-regular free boundaries by reducing the problem to a class of degenerate quasilinear PDEs, for which we establish Schauder estimates. Such a method provides a unified proof of the smoothness for general exponents. On the other hand, by exploiting the higher regularity of solutions, we derive a new stability condition for the Alt–Phillips problem in the negative exponent regime, ruling out the existence of nontrivial axially symmetric stable cones in low dimensions. Finally, we provide a variational criterion for the stability of cones in the Alt–Phillips problem, which recovers the one for minimal surfaces in the singular limit as \documentclass[12pt]{minimal}
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\begin{document}$$\gamma \rightarrow -2$$\end{document}γ→-2.

Full text

Introduction

The one-phase Alt–Phillips free boundary problem concerns the study of non-negative minimizers of the functional

$$[eqn]$$

among functions with non-negative boundary datum on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial B_1$$\end{document} . The functional was first introduced in the 80 s by Phillips in [41], and later further investigated by Alt and Phillips in [3]. More broadly, variational problems involving the potential $[eqn]$ serve as a prototypical framework for studying general semilinear and quasilinear free boundary problems. When the potential lacks \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{1,1}$$\end{document} -regularity (e.g.,  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (-2,2)$$\end{document} ), it is natural to expect that minimizers may exhibit regions where the solution remains constant, thereby giving rise to a free boundary.

Beyond its original applications in modeling population dynamics [28], and porous catalysts [4], the Alt–Phillips problem is especially notable for its role in “interpolating” between several classical and well-studied free boundary problems. Specifically:

• (i) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =1$$\end{document} , the obstacle problem;
• (ii) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =0$$\end{document} , the Alt–Caffarelli problem (i.e., the Bernoulli problem);
• (iii) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \rightarrow -2$$\end{document} , the functional “approximates” the problem of minimal surfaces (see [19, 20]). Although the regularity theory for both the obstacle problem, the Alt–Caffarelli problem, and the minimal surfaces is fairly well understood (e.g., [8, 25, 37, 40, 43, 50]), it is only in recent years that several research groups have developed a fine regularity theory for minimizers of the Alt–Phillips functional [1, 18–21, 26, 33, 42, 51, 52].

The focus of this work is the case of negative exponents \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (-2,0)$$\end{document} , which has been introduced by De Silva and Savin in [19]. Precisely, as in the case of positive exponents, they show that minimizers exist by direct method and they are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} -Hölder continuous, with

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \beta {:}{=} \frac{2}{2-\gamma }. \end{aligned}$$\end{document}

This regularity is optimal, indeed the exponent \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} is the scaling parameter of the Alt–Phillips functional for every exponent. Then, if we set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _u{:}{=}\{u>0\}$$\end{document} , they proved that the free boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega _u$$\end{document} can be decomposed as the disjoint union

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \partial \Omega _u \cap B_1 = \textrm{Reg}(\partial \Omega _u)\cup \textrm{Sing}(\partial \Omega _u), $$\end{document}

where the regular part \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Reg}(\partial \Omega _u)$$\end{document} is locally the graph of a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{1,\alpha }$$\end{document} function and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Sing}(\partial \Omega _u)$$\end{document} is a closed singular set of Hausdorff dimension at most \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d-d^*(\gamma )$$\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}$$d^*(\gamma )\ge 3$$\end{document} . In [19], they observed one of the first remarkable differences between the cases of negative and positive exponents: indeed, the change of sign in the exponent makes the problem more degenerate and the free boundary condition is derived in terms of a second order expansion of the solution.

Furthermore, using a monotonicity formula and a dimension reduction argument, they showed that the dimensional threshold \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d^*(\gamma )$$\end{document} is the first dimension in which a minimizing cone for the Alt–Phillips problem exhibits singularities. Notice that, by the works of Caffarelli, Jerison and Kenig [7], Jerison and Savin [31], De Silva and Jerison [15], we know that for the Alt–Caffarelli problem, i.e.,  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =0$$\end{document} , minimizing cones are flat 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} for every dimension smaller than \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d^*(0)\in \{5,6,7\}$$\end{document} (we stress that the proof works also for stable cones).

Afterwards, in [20] they improved 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}$$d^*(\gamma )$$\end{document} by exploiting a distinctive phenomenon that arises as \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} approaches the limiting values \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-2$$\end{document} and 0. Indeed, by a fine compactness argument, they proved that the functionals \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} -converge to a Dirichlet-perimeter functional introduced by Athanasopoulos, Caffarelli, Kenig and Salsa in [5]. Therefore, by classical results concerning the dimension of the singular set for area-minimizing hypersurfaces (see [37, 43]), they proved that asymptotically

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \text {if } \,\gamma \sim -2\, \text { then } \,d^*(\gamma )\ge 8,\qquad \text {if } \,\gamma \sim 0\, \text { then } \,d^*(\gamma )\ge 5. $$\end{document}

Two equivalent formulations

Before presenting the main results, it is worth introducing an equivalent formulation of the Alt–Phillips problem, originally proposed in [3, Section 4]. Precisely, for every exponent \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (-2,2)$$\end{document} , given a minimizer u of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {J}_\gamma $$\end{document} and letting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} be as in (1.1), we set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w{:}{=}\beta u^{1/\beta }$$\end{document} . Then w is a minimizer of the functional

$$[eqn]$$

The auxiliary function w has been exploited both for the case of positive [3, 18, 33] and negative exponents [19, 20]. In these last contributions, the authors showed the convenience of this formulation for the study of regular points. Nevertheless, the variables u and w are interchangeable, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _u\equiv \Omega _w$$\end{document} , and thus our analysis of the regularity of the free boundary can equivalently be formulated in terms of w. In the case of negative exponents \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (-2,0)$$\end{document} , as initially highlighted by [19], minimizers of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {J}_\gamma $$\end{document} are solutions of the Euler–Lagrange equations

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\begin{aligned} \Delta u = \frac{\gamma }{2}u^{\gamma -1}\quad \text{ in } \Omega _{u}\cap B_{1},\qquad u^{\gamma }\left( \frac{|\nabla u|^2}{u^{\gamma }} - 1\right) =0\quad \text{ on } \partial \Omega _{u}\cap B_1. \end{aligned}\end{aligned}$$\end{document}

Equivalently, the free boundary condition can be rewritten as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\begin{aligned} \lim _{t \rightarrow 0^+} t^{2(\beta -1)}\left( \frac{u(x_{0}-t\nu _{x_0})}{t^\beta } - c_{\beta } \right) =0, \quad \text{ where } c_\beta {:}{=} \frac{1}{\beta ^{\beta }} = \left( \frac{2-\gamma }{2}\right) ^{\frac{2}{2-\gamma }}, \end{aligned}\end{aligned}$$\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}$$x_0 \in \partial \Omega _u \cap B_1$$\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}$$\nu _{x_0}$$\end{document} is the outer normal to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega _u$$\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} . We stress that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_\beta $$\end{document} is the constant associated to the one-dimensional solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_0(t){:}{=}c_\beta (t^+)^\beta $$\end{document} .

On the other hand, a direct computation shows that the Alt–Phillips problem can be viewed as a degenerate one-phase type free boundary problem, where w satisfies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\begin{aligned} \Delta w = \frac{s}{2}\frac{1-|\nabla w|^2}{w}\quad \text{ in } \Omega _w\cap B_1,\qquad w^s(|\nabla w|^2 -1) = 0\quad \text{ on } \partial \Omega _w\cap B_1. \end{aligned}\end{aligned}$$\end{document}

Equivalently, the free boundary condition for w is

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{t \rightarrow 0^+} t^{s}\left( \frac{w(x_0-t\nu _{x_0})}{t} - 1 \right) =0 \quad \text{ at } x_0\in \partial \Omega _w\cap B_1. \end{aligned}$$\end{document}

As noted in [19], unlike in the case of positive exponents, the free boundary condition cannot be easily derived through integration by parts or inner variations. We refer to [19, Proposition 4.4], where such a condition is obtained for minimizers via a calibration argument. Ultimately, by proving an improvement-of-flatness lemma for viscosity solutions of the auxiliary formulation, they deduce \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{1,\alpha }$$\end{document} -regularity for both the free boundary and the auxiliary function, in a neighborhood of regular points. We postpone the discussion of the natural free boundary condition for the Alt–Phillips problem to Remark 2.6.

The principle intention of this paper is to establish higher regularity of the free boundary at regular points and define the notion of stable singular cones for the Alt–Phillips functional in the negative regime, highlighting the connection with the stability inequalities for the Alt–Caffarelli functional [7] and for area-minimizing hypersurfaces [44].

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			\begin{document}$$C^\infty $$\end{document}C∞ regularity of regular points

Higher regularity of the free boundary in the Alt–Phillips problem has been recently established only in the range of positive exponents, with different methodologies. Precisely, at regular points, the smoothness was first proved for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (1,2)$$\end{document} in [26], then extended to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (2/3,2)$$\end{document} in [1], and more generally to all positive exponents \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,2)$$\end{document} in [42] (see also [14]).

In the following theorem, we conclude the study of higher regularity for the Alt–Phillips problem, by proving smoothness of the free boundary at regular points in the case of negative exponents \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (-2,0)$$\end{document} . Since our approach naturally extends to the full range of exponents \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (-2,2)$$\end{document} , we provide an alternative and unified proof for the smoothness of the regular part of the free boundary.

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}$$\gamma \in (-2,2)$$\end{document} and u be a local minimizer of the Alt–Phillips functional \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {J}_\gamma $$\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}$$B_1$$\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}$$x_0 \in \partial \Omega _u\cap B_1$$\end{document} is a regular free boundary point. Then:

• (i)the free boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega _u$$\end{document} is the graph of a smooth function in a neighborhood of \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} ;
• (ii) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w{:}{=} \beta u^{1/\beta }$$\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}$$u/{\textrm{dist}(x,\partial \Omega _u)^\beta }$$\end{document} are smooth in a neighborhood of \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} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\Omega }_u$$\end{document} .

Although Theorem 1.1 is stated in terms of local minimizers (see Definition 3.2), the proof actually holds for regular solutions of the corresponding Euler–Lagrange equations, in the sense of Definition 2.1. See Proposition 2.2 and Lemma 2.3 for the precise statements.

Our approach to prove Theorem 1.1 is to apply an hodograph transformation (see the seminal work of Kinderlehrer and Nirenberg [34]) in a neighborhood of regular points, which reformulates the higher regularity problem as a Schauder-type estimates for degenerate quasilinear PDEs with Neumann boundary condition. More precisely, we consider the change of coordinates \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi (x',x_d){:}{=}(x',w(x',x_d))$$\end{document} induced by the function w, and we show the existence of a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{1,\alpha }$$\end{document} -regular function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h:B_\delta \cap \{x_d\ge 0\} \rightarrow \mathbb {R}$$\end{document} satisfying

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w(x',h(x',x_d))=\beta u^{1/\beta }(x',h(x',x_d))= x_d, $$\end{document}

so that the free boundary is given by the graph of the trace of h over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{x_d=0\}$$\end{document} . If we assume that w is a local minimizer of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {E}_s$$\end{document} (resp. u a local minimizer of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {J}_\gamma $$\end{document} ), then the hodograph transform h is a local minimizer of

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal {F}(\varphi ){:}{=} \int _{B_\delta ^+}x_d^{s}\, F(\nabla \varphi )\,dx,\quad \text{ where } F(p){:}{=}\frac{|p|^2+1}{p_d}, $$\end{document}

and the minimality is formulated in terms of outer variations. Therefore, the hodograph transform h satisfies the following degenerate quasilinear PDE

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\begin{aligned} \textrm{div}\left( x_d^s\, DF(\nabla h)\right) =0 \quad \text{ in } B_\delta ^+,\qquad \lim _{x_{d}\rightarrow 0^+}x_d^s\, DF(\nabla h)\cdot e_d = 0 \quad \text{ on } B_\delta '. \end{aligned}\end{aligned}$$\end{document}

Here we denoted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_\delta ^+{:}{=}B_\delta \cap \{x_d>0\}$$\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}$$B_\delta '{:}{=}B_\delta \cap \{x_d=0\}$$\end{document} . Notice that:

• (i)for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (-2, 2/3)$$\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}$$s \in (-1,1)$$\end{document} , the weight \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_d^s$$\end{document} lies in the \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} -Muckenhoupt;
• (ii)for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in [2/3, 2)$$\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}$$s\ge 1$$\end{document} , the weight is merely integrable. Finally, being \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x'\mapsto h(x',0)$$\end{document} a local parametrization of the free boundary, we deduce the claimed result from the validity of Schauder-type estimates for the function h (see Theorem 1.2 below).

We emphasize that in [42], Restrepo and Ros-Oton adopt a different strategy, focusing on the PDE satisfied by the ratio of partial derivatives of a solution (see also [16, 17] in which this method was originally introduced in the context of the Alt–Caffarelli problem and the obstacle problem).

Degenerate quasilinear PDEs

The regularity theory for degenerate divergence-form PDEs has been extensively developed over the past decades, both in the setting of \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} -Muckenhoupt weights [22, 35] and in the so-called superdegenerate regime [47].

Our approach builds upon the analysis carried out by Terracini, Vita, and the second author in [47], where a Schauder-type theory is established for linear equations with integrable weights. However, to the best of our knowledge, a corresponding theory for quasilinear operators in the same setting is currently unknown. For this reason, our regularity result is formulated for general quasilinear PDEs, as we believe this result is of independent interest.

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}$$s>-1$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\in \mathbb {N}_{\ge 2}$$\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}$$\alpha \in (0,1)$$\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}$$v\in C^{1,\alpha }(\overline{B_1^+})$$\end{document} be a solution of

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\begin{aligned} \textrm{div}(x_d^s\,DF(\nabla v))=0\quad \text {in } B_1^+, \qquad \lim _{x_d\rightarrow 0^+}x_d^{s}\, D F(\nabla v)\cdot e_d =0\quad \text {on } B_1', \end{aligned}\end{aligned}$$\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}$$F:E \subset \mathbb {R}^d\rightarrow \mathbb {R}$$\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}$$(\nabla v)(\overline{B_1^+})\subset E$$\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}$$E\subset \mathbb {R}^d$$\end{document} is a connected bounded open set. Assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F\in C^{k,\alpha }(E)$$\end{document} is uniformly convex in E, i.e., there exist \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\lambda \le \Lambda <+\infty $$\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}$$ \lambda \textrm{Id}\le D^2 F(p)\le \Lambda \textrm{Id}\quad \text {for every }p\in E. $$\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}$$v\in C^{k,\alpha }(\overline{B_{1/2}^+})$$\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}$$ \Vert v\Vert _{C^{k,\alpha }(\overline{B^+_{1/2}})}\le C, $$\end{document}

for some constant \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} 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}$$d, s, k, \alpha , \lambda , \Lambda , \Vert F\Vert _{C^{k,\alpha }(E)}, \Vert v\Vert _{C^{1,\alpha }(\overline{ B_1^+})}$$\end{document} .

Stable solutions of the Alt–Phillips problem for \documentclass[12pt]{minimal}

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			\begin{document}$$\gamma \in (-2,0)$$\end{document}γ∈(-2,0)

In the Alt–Phillips problem, the classification of minimizing cones, i.e., homogeneous minimizers with isolated singularity, is a challenging open problem, at least in dimension \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge 3$$\end{document} . Even for the Alt–Caffarelli problem ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =0$$\end{document} ) the classification is only partially understood (see [7, 15, 31]). A key tool in addressing these questions is the concept of stability, which plays a central role in ruling out singularities by establishing the rigidity of singular cones in low dimensions (see [7]). In this direction, in [33] Karakhanyan and Sanz-Perela recently computed the second variation for the Alt–Phillips problem with positive exponent.

The following is the main result concerning the stability condition under inner variations for minimizing cones in the case of negative exponents. The smoothness of the regular part of the free boundary proved in Theorem 1.1 allows for second-order expansions of local minimizers, which are essential for computing the second variation.

Theorem 1.3

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (-2,0)$$\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}$$u\in C^{0,\beta }(\mathbb {R}^d)$$\end{document} be a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} -homogeneous global minimizer of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {J}_\gamma $$\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}$$\mathbb {R}^d$$\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}$$\partial \Omega _u$$\end{document} is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{1,\alpha }$$\end{document} -regular outside the origin, then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{ \Omega _u} |\nabla u|^2 \left( |\nabla f|^2 - \mathcal {A}^2_u\,f^2\right) \,dx\ge 0,\quad \text{ where } \mathcal {A}^2_u{:}{=} \frac{|\nabla ^2 u|^2}{|\nabla u|^2} - \frac{|\nabla ^2 u\nabla u|^2}{|\nabla u|^4}, \end{aligned}$$\end{document}

for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f \in C^\infty _c(\mathbb {R}^d\setminus \{0\})$$\end{document} . Equivalently, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w{:}{=}\beta u^{1/\beta }\in C^{0,1}(\mathbb {R}^d)$$\end{document} is 1-homogeneous global minimizer of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {E}_s$$\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}$$\mathbb {R}^d$$\end{document} , and if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega _w$$\end{document} is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{1,\alpha }$$\end{document} -regular outside the origin, then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{ \Omega _w} w^s|\nabla w|^2\left( |\nabla f|^2 - \mathcal {A}^2_w\, f^2\right) \,dx \ge 0,\quad \text{ where } \mathcal {A}^2_w{:}{=} \frac{|\nabla ^2 w|^2}{|\nabla w|^2} - \frac{|\nabla ^2 w\nabla w|^2}{|\nabla w|^4}, \end{aligned}$$\end{document}

for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f \in C^\infty _c(\mathbb {R}^d\setminus \{0\})$$\end{document} .

We stress that Theorem 1.3 is stated under the assumptions of homogeneity and global minimality, but the proof actually applies in a more general setting. In fact, in Definition 3.1 we introduce the notion of stable solutions, and in Proposition 3.3 we prove a local stability condition without requiring homogeneity. Finally, in Remark 3.7, we observe that Theorem 1.3 holds even for global stable solutions, regular outside the origin, in the sense of Definition 3.2 and Definition 2.1.

For the computation of the stability condition, it is more convenient to rely on the variational formulation associated to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {E}_s$$\end{document} , even though (1.3) and (1.4) are equivalent. Indeed, given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f \in C^\infty _c(\mathbb {R}^d \setminus \{0\})$$\end{document} , the condition (1.4) is deduced by computing the second derivative of the map

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} t \mapsto \mathcal {E}_s\left( w\circ (\textrm{Id}+t \xi )^{-1}\right) ,\quad \text{ where } \xi {:}{=}\frac{\nabla w}{|\nabla w|}f. \end{aligned}$$\end{document}

The choice of these variations is dictated by the strong degeneracy of the problem close to regular points. Indeed, while in the case of the Alt–Caffarelli problem ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = 0$$\end{document} ) and for positive exponents \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,2)$$\end{document} it is natural to consider variations of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi {:}{=} \frac{\nabla w}{|\nabla w|^2}f$$\end{document} and integrate by parts on the free boundary to reveal the role of mean curvature, in the case of negative exponents the presence of the weight \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w^s$$\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}$$s \in (-1, 0)$$\end{document} , obstructs such an approach and masks the contribution of the mean curvature. In this sense, the variations in (1.5) are more closely used in the study of minimal surfaces, where the vector field is typically chosen of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi =\nu f$$\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}$$\nu $$\end{document} denoting the normal to the surface (see [37]). We postpone the main differences between our stability condition and those in [7, 33] for non-negative exponents to Remark 3.8.

Notice that the quantity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}_u^2$$\end{document} in Theorem 1.3 controls the second fundamental form of the level sets of the solution at a given point (the same applies to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}_w$$\end{document} ). Precisely, as pointed out by Sternberg and Zumbrun in [46, Lemma 2.1],

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {A}^2_u = |A|^2 + \frac{|\nabla _T |\nabla u||^2}{|\nabla u|^2} \end{aligned}$$\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}$$\nabla _T$$\end{document} denotes the tangential gradient along the level set of u passing through \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in \Omega _u$$\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}$$|A|^2$$\end{document} is the squared norm of the second fundamental form of the same level set.

Axially symmetric cones

Using the stability condition in Theorem 1.3, we provide an initial result concerning the existence of minimizers with an isolated singularity. Since in the regime of negative exponents the problem interpolates between the Alt–Caffarelli problem and the theory of minimal surfaces, it is natural to construct examples of singular free boundaries that are either axially symmetric (see [15] for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = 0$$\end{document} ) or of Lawson’s type (see [12, 44] for minimal surfaces).

We emphasize that recently different groups of authors have focused on the study of singular points in the case of positive exponent. Indeed, in [33] they ruled out the existence of singular cones that are axially symmetric in low dimensions, when the exponent \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0,2/3)$$\end{document} . On the other hand, in [51, 52] the authors construct both radial and axially symmetric cones and show that these solutions are minimizers for \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} sufficiently close to 1.

In the following theorem, we show that for negative exponents, minimizing axially symmetric cones for the Alt–Phillips functional are trivial in low dimensions.

Theorem 1.4

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (-2,0)$$\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}$$u \in C^{0,\beta }(\mathbb {R}^d)$$\end{document} be a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} -homogeneous global minimizer of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {J}_\gamma $$\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}$$\mathbb {R}^d$$\end{document} . Assume that u is axially symmetric and

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \text {either}\quad d\le 6\quad \text {or}\quad d=7\ \text {and}\ \gamma <\frac{10-8\sqrt{5}}{11}\approx -0.7171. $$\end{document}

Then u is one-dimensional.

Although Theorem 1.4 is stated under the minimality assumption, the proof actually applies to the case of global stable solutions, regular outside the origin, in the sense of Definition 3.2 and Definition 2.1 (see Proposition 4.1). We also point out that the homogeneity assumption is not necessary (see Remark 4.2). Finally, we observe that Theorem 1.4 also holds for minimizers or stable solutions of the functional \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {E}_s$$\end{document} , under the corresponding assumptions.

Asymptotic results as \documentclass[12pt]{minimal}

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			\begin{document}$$\gamma \rightarrow -2$$\end{document}γ→-2

As already mentioned above, in [19, Theorem 2.4], the authors establish a \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} -convergence result of the (normalized) Alt–Phillips energies to the perimeter functional, as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \rightarrow -2$$\end{document} . This result provide a connection between minimizing free boundaries for the Alt–Phillips problem with area-minimizing hypersurfaces, in the same spirit of the theory of phase transition for the Allen–Cahn equation [2, 9, 38].

It is natural to ask whether this connection between the problems persists at the level of stable solutions. For instance, the singular perturbation problem associated with stable solutions of the Allen–Cahn equation has been studied only recently (see [27, 29, 36, 48, 49]), where the limit has been characterized in terms of stable minimal hypersurfaces. On the other hand, we refer to [11, 32] for a recent literature concerning stable solutions of free boundary problems.

In this paper, we also investigate the limiting behavior of stable solutions of the Alt–Phillips problem as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \rightarrow -2$$\end{document} , by showing that under \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{2,\alpha }$$\end{document} -regularity, uniform in \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} (see (5.6)), the limiting interface is a stable minimal hypersurface.

First, in Proposition 5.1, we provide a variational characterization of the stability condition of Theorem 1.3 in terms of an eigenvalue problem involving a weighted 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}$$\mathbb {S}^{d-1}$$\end{document} . Precisely, given a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} -homogeneous solution u and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma _u{:}{=}\Omega _u\cap \mathbb {S}^{d-1}$$\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}$$\begin{aligned} \lambda _s(\Sigma _u){:}{=} \min _{\begin{array}{c} \varphi \in C^\infty (\mathbb {S}^{d-1})\\ \varphi \not \equiv 0 \end{array}}\frac{\displaystyle \int _{\Sigma _u} |\nabla u|^2\big (|\nabla _S \varphi |^2 - \mathcal {A}_u^2 \varphi ^2\big )\,d\mathcal {H}^{d-1}}{\displaystyle \int _{\Sigma _u} |\nabla u|^2 \varphi ^2 \,d\mathcal {H}^{d-1}}\,, \end{aligned}$$\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}$$\nabla _S$$\end{document} is the tangential gradient on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}^{d-1}$$\end{document} . Then, we show that the stability condition is equivalent to require that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lambda _s(\Sigma _u)\ge -\left( \frac{d+s-2}{2}\right) ^2. \end{aligned}$$\end{document}

Finally, we address the asymptotic behavior of this criterion as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \rightarrow -2$$\end{document} , recovering the analogous result for stable minimal surfaces, thus obtaining that the limit interface is a stable minimal surface. For a discussion on the general limit and the role of the associated quadratic forms of the stability condition in Theorem 1.3, we refer to Remark 5.2.

A valuable direction for future research, though beyond the scope of the present work, would be to expand on this analysis and extend the asymptotic result in [48, 49] to the case of stable solutions for the Alt–Phillips problem with negative exponents.

Structure of the paper

In Sect. 2, we apply the hodograph map to regular solutions and we rewrite the problem in terms of a degenerate quasilinear elliptic PDE. Then we establish corresponding Schauder estimates in Theorem 1.2, which implies the \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} regularity of the free boundary in Theorem 1.1. In Sect. 3, we compute both the first and second variations of the functional \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {J}_\gamma $$\end{document} , leading to the stability condition in Theorem 1.3. In Sect. 4, we use the stability condition to prove that axially symmetric cones are trivial in low dimensions, i.e., Theorem 1.4. Finally, in Sect. 5 we prove the variational characterization of the stability condition and we discuss the asymptotic of stable cones as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \rightarrow -2$$\end{document} .

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			\begin{document}$$C^\infty $$\end{document}C∞ regularity of the free boundary

In this section, we show that the free boundary conditions are satisfied in a pointwise sense at regular points and then we rephrase the problem by an hodograph transformation. Therefore, the higher regularity of the free boundary follows by Schauder-type estimates for degenerate quasilinear PDEs with Neumann boundary conditions.

For the sake of clarity, we start by introducing the notion of regular solutions. For simplicity, the definition is given in terms of w, though an equivalent one holds for u.

Definition 2.1

(Regular solutions) Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (-2,2)$$\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}$$s{:}{=}\beta \gamma >-1$$\end{document} . We say that w is a regular solution in an open set D (unless otherwise specified, we take \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D{:}{=}B_1$$\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}$$\begin{aligned} \Delta w = \frac{s}{2}\frac{1-|\nabla w|^2}{w}\quad \text{ in } \Omega _w\cap D, \qquad {\left\{ \begin{array}{ll} |\nabla w|^2=1 & \text{ on } \partial \Omega _w \cap D, \text { if } s\ge 0\\ w^{s}(|\nabla w|^2-1)=0 & \text{ on } \partial \Omega _w \cap D, \text { if } s < 0 \end{array}\right. } \end{aligned}$$\end{document}

if, for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >-s$$\end{document} ,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w\in C^{1,\alpha }(\overline{\Omega }_w\cap D) \text { and the free boundary } \partial \Omega _w \text { is } C^{1,\alpha } \text {-regular in } D $$\end{document}

and (2.1) is satisfied in a pointwise sense. Precisely, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in (-1,0)$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\begin{aligned} \lim _{t \rightarrow 0^+} w^s(x_0-t\nu _{x_0})\Big (|\nabla w(x_0-t\nu _{x_0})|^2-1\Big )=0,\quad \text{ for } \text{ every } x_0 \in \partial \Omega _w \cap D, \end{aligned}\end{aligned}$$\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}$$\nu _{x_0}$$\end{document} is the outer normal to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega _w$$\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} .

The following is the main regularity result, which ultimately implies Theorem 1.1.

Proposition 2.2

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (-2,2)$$\end{document} and w be a regular solution in the sense of Definition 2.1. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\in \partial \Omega _w$$\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}$$r>0$$\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}$$ w\in C^{\infty }(\overline{\Omega }_w\cap B_r) \text { and the free boundary } \partial \Omega _w \text { is smooth in } B_r. $$\end{document}

Before addressing the problem of higher regularity, we first point out that local minimizers of the Alt–Phillips functional are regular solutions, close to regular free boundary points. This observation was essentialy contained in the \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} -regularity theorem proved in [18, 19].

Lemma 2.3

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (-2,2)$$\end{document} and w be a local minimizer to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {E}_s$$\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}$$s{:}{=}\beta \gamma $$\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}$$x_0\in \partial \Omega _w\cap B_1$$\end{document} is a regular free boundary point, then w is a regular solution in a ball \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_r(x_0)$$\end{document} , in the sense of Definition 2.1, for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r>0$$\end{document} .

Proof

It is not restrictive to assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0=0$$\end{document} . Then, up to rescaling the problem, by [19, Theorem 2.3] there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta \in (0,1)$$\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}$$ w\in C^{1,\delta }(\overline{\Omega }_w\cap B_r)$$\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}$$\partial \Omega _w$$\end{document} is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{1,\delta }$$\end{document} -regular in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_r$$\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}$$r>0$$\end{document} . Since in the case of positive exponents \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in [0,2)$$\end{document} the conclusion is immediate, we address the case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (-2,0)$$\end{document} . Notice that, for negative \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} , the result follows once we show that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w\in C^{1,\alpha }(\overline{\Omega }_w\cap B_r) \text { and the free boundary } \partial \Omega _w \text { is } C^{1,\alpha } \text {-regular in } B_r, $$\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}$$\alpha >-s$$\end{document} . In fact, being \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\nabla w|=1$$\end{document} on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega _w\cap B_r$$\end{document} , it implies that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \lim _{t\rightarrow 0^+}\left| w^s(x_0-t\nu _{x_0})\Big (|\nabla w(x_0-t\nu _{x_0})|^2-1\Big )\right| \le C \lim _{t \rightarrow 0^+}t^{s+\alpha }=0. $$\end{document}

As already observed in the proof of [19, Proposition 7.2], the linearization procedure allows to rewrite the regularity problem for \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} -flat free boundaries in terms of Schauder-type estimates for the degenerate linear PDE

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textrm{div}\left( x_d^s\, \nabla v\right) =0 \quad \text {in }B_1^+,\qquad \lim _{x_d\rightarrow 0^+}x_d^s\, \nabla v\cdot e_d = 0 \quad \text {on }B_1'. $$\end{document}

Since solutions of such linearized system are indeed smooth up to the boundary (see [45, Theorem 1.1]), we can show that for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,1)$$\end{document} there exists a dimensional radius \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho >0$$\end{document} , small enough, such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left( x\cdot \nu -\rho ^{1+\alpha }\varepsilon \right) _+\le w(x)\le \left( x\cdot \nu +\rho ^{1+\alpha }\varepsilon \right) _+\quad \text {for every } x\in B_\rho . \end{aligned}$$\end{document}

Then, fixed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha >-s$$\end{document} , the result follows by a standard iteration argument (see for instance [10], where such strategy is exploited for obtaining \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{2,\alpha }$$\end{document} -estimates in a one-phase type problem). \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 hodograph transform

In this section we write the hodograph transformation of a solution of the Alt–Phillips problem (see also [1, 26] in which this map was already exploited in the case of positive exponent \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (2/3,2)$$\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}$$x \in \{x_d\ge 0\}$$\end{document} it is convenient to write \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=(x',x_d)\in \mathbb {R}^{d-1}\times \mathbb {R}$$\end{document} . Without loss of generality, we can assume that w is a regular solution in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_1$$\end{document} , in the sense of Definition 2.1, and that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 \in \partial \Omega _w$$\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}$$\nabla w(0)= e_d$$\end{document} . Therefore, there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho >0$$\end{document} sufficiently small, such 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}$$ \Phi :\overline{\Omega }_w\cap B_\rho \rightarrow \mathbb {R}^d\cap \{y_d\ge 0\},\quad \Phi (x',x_d){:}{=}(x',w(x',x_d)),\quad $$\end{document}

is bijective from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\Omega }_w\cap B_\rho $$\end{document} onto an open neighborhood of the origin in the upper half-space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{y_d\ge 0\}$$\end{document} . Clearly, it is not restrictive to replace \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi (\overline{\Omega }_w\cap B_\rho )$$\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}$$\{y_d\ge 0\}\cap B_\delta $$\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}$$\delta >0$$\end{document} . Thus, \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} maps the free boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega _w\cap B_\rho $$\end{document} into \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B'_\delta $$\end{document} . On the other hand, the inverse function

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Phi ^{-1}:\overline{B_\delta ^+}\rightarrow \overline{\Omega }_w\cap B_\rho ,\quad \Phi ^{-1}(y',y_d){:}{=}(y',h(y',y_d)),\quad \end{aligned}$$\end{document}

is well-defined. Being \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w\in C^{1,\alpha }(\overline{\Omega }_w\cap B_\rho )\cap C^\infty _{\textrm{loc}}(\Omega _w\cap B_\rho )$$\end{document} , 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}$$h: \overline{B_\delta ^+} \rightarrow \mathbb {R}$$\end{document} is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{1,\alpha }$$\end{document} -regular in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{B_\delta ^+}$$\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}$$C^\infty $$\end{document} -regular in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_\delta ^+$$\end{document} . Throughout the paper, we refer to such h as the hodograph transform of w. By differentiating the identity

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ h(x',w(x',x_d))=x_d, \quad \text{ for } \text{ every } x=(x',x_d) \in \overline{\Omega }_w \cap B_\delta , $$\end{document}

we can derive the system satisfied by the hodograph transform h. For the sake of simplicity, in the computations that follow, the derivatives of w are evaluated at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\in \Omega _w\cap B_\delta $$\end{document} , and those of h at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y{:}{=}(x',w(x',x_d))\in B_\delta ^+$$\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}$$i,j=1,\dots ,d-1$$\end{document} , we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&w_d =\frac{1}{h_d},\qquad w_i=-\frac{h_i}{h_d},\\&w_{ij}= -\frac{h_{ij}}{h_d} +\frac{ h_j h_{id}}{h_d^2}+\frac{h_{jd}h_i}{h_d^2} -\frac{h_{dd} h_i h_j}{h_d^3},\\&w_{id}=-\frac{h_{id}}{h_d^2}+ \frac{h_{dd}\, h_i}{h_d^3},\qquad w_{dd}=- \frac{h_{dd}}{h_d^3}. \end{aligned} \end{aligned}$$\end{document}

Although this is a classical fact (see e.g., [13, Remark 3.1]), we emphasize that the hodograph transform h contains all the information of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega _w$$\end{document} , indeed h is a regular function that coincides on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_\rho '$$\end{document} with the parametrization of the free boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega _w$$\end{document} . Thus, the derivatives of 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}$$\partial \Omega _w$$\end{document} coincide with the partial derivatives \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_i$$\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}$$i = 1, \dots , d-1$$\end{document} .

More precisely, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega _w \cap B_\rho $$\end{document} is the graph 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}$$g: B_\rho '\rightarrow \mathbb {R}$$\end{document} in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e_d$$\end{document} -direction, then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w(x',g(x'))=0\quad \text{ for } \text{ every } (x',0)\in B_{\rho }'. $$\end{document}

Then, by differentiating the identity above, we obtain for \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}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} g_i(x') =-\frac{w_i(x',g(x'))}{w_d(x',g(x'))}= h_i(x',w(x',g(x'))) = h_i(x',0)\quad \text{ for } \text{ every } (x',0) \in B_{\rho }', \end{aligned}$$\end{document}

where the second equality follows from (2.3) and the last from the definition of g. Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e_d$$\end{document} is the interior normal to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega _w$$\end{document} at the origin, we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla h(0)=e_d$$\end{document} . By exploiting (2.3), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{s}{2}\frac{1-|\nabla w|^2}{w}= \frac{s}{2}\frac{1}{y_d}\left( 1-\frac{1+\sum _{i=1}^{d-1} h_i^2}{h_d^2}\right) . $$\end{document}

Hence, the interior condition (2.1) become

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} - \frac{h_{dd}}{h_{dd}^3}\left( 1 + \sum _{i= 1}^{d-1} h_i^2\right) + 2\sum _{i= 1}^{d-1} \frac{h_{i}h_{id}}{h_d^2} - \sum _{i=1}^{d-1}\frac{h_{ii}}{h_d} =\frac{s}{2y_d}\left( 1-\frac{1+\sum _{i=1}^{d-1}h_i^2}{h_d^2}\right) \quad \text {in }B_\delta ^+. \end{aligned}$$\end{document}

Similarly, the free boundary condition in (2.1) can be rewritten as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lim _{y_d \rightarrow 0^+} y_d^s\left( 1-\frac{1+\sum _{i=1}^{d-1}h_i^2}{h_d^2}\right) =0\quad \text {on }B_\delta '. \end{aligned}$$\end{document}

We now collect all the computations and we show that the hodograph transform satisfies a degenerate quasilinear elliptic PDE with a Neumann boundary condition. For the sake of completeness, we refer to [22, 35, 47] for a complete discussion on weighted Sobolev spaces and the correct notion of weak solutions.

Lemma 2.4

Let w be a regular solution, in the sense of Definition 2.1. Then its hodograph transform h, defined in (2.2), is a solution of

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textrm{div}\left( x_d^s\, DF(\nabla h)\right) =0 \quad \text {in }B_\delta ^+,\qquad \lim _{x_d\rightarrow 0^+}x_d^s\, DF(\nabla h)\cdot e_d = 0 \quad \text {on }B_\delta ', \end{aligned}$$\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}$$\begin{aligned} F(p){:}{=}\frac{|p|^2 + 1 }{p_d}. \end{aligned}$$\end{document}

Moreover, if w is a local minimizer of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {E}_s$$\end{document} , then the hodograph transform h minimizes

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {F}(\varphi ){:}{=} \int _{B_\delta ^+}x_d^{s}\, F(\nabla \varphi )\,dx, \end{aligned}$$\end{document}

among competitors in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1(B_\delta ^+,x_d^s\,dx)$$\end{document} with same trace on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\partial B_\delta )^+$$\end{document} .

Proof

We divide the proof into two cases.

Case 1: regular solutions of (2.1). First, by direct computation, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&D F(p) = \frac{2}{p_d}p -\frac{(1+|p|^2)}{p_d^2}e_d,\\&D^2 F(p)=\frac{2}{p_d}\textrm{Id}-2\frac{e_d\otimes p + p\otimes e_d}{p_d^2}+\frac{2(1+|p|^2)}{p_d^3}e_d\otimes e_d. \end{aligned}$$\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}$$\begin{aligned} \begin{aligned}&DF(\nabla h)\cdot e_i = 2\frac{h_i}{h_d}\quad \text{ for } \text{ every } i=1,\dots ,d-1,\\&DF(\nabla h)\cdot e_d = 2-\frac{1+|\nabla h|^2}{h_d^2}=1-\frac{1+\sum _{i=1}^{d-1}h_i^2}{h_d^2} \end{aligned} \end{aligned}$$\end{document}

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}$$\begin{aligned}&y_d^{-s}\textrm{div}\left( y_d^{s} DF(\nabla h)\right) =\sum _{i=1}^{d}\partial _i(DF(\nabla h)\cdot e_i)+s\frac{DF(\nabla h)\cdot e_d}{y_d}\\&\quad =2\sum _{i=1}^{d-1}\left( \frac{h_{ii}}{h_d}-\frac{h_{id}h_{i}}{h_d^2}\right) +\partial _d\left( -\frac{1+\sum _{i=1}^{d-1}h_i^2}{h_d^2}\right) +\frac{s}{y_d}\left( 1-\frac{1+\sum _{i=1}^{d-1}h_i^2}{h_d^2}\right) \\&\quad = 2\sum _{i=1}^{d-1}\left( \frac{h_{ii}}{h_d}-\frac{h_{id}h_{i}}{h_d^2}\right) +2\frac{h_{dd}}{h_d^3}-2\left( \sum _{i=1}^{d-1} \frac{h_i h_{id}}{h_d^2}-\frac{h_{dd}h_i^2}{h_d^3}\right) +\frac{s}{y_d}\left( 1-\frac{1+\sum _{i=1}^{d-1}h_i^2}{h_d^2}\right) \\&\quad =2\sum _{i=1}^{d-1}\left( \frac{h_{ii}}{h_d}-2\frac{h_{id}h_{i}}{h_d^2}\right) +2\frac{h_{dd}}{h_d^3}\left( 1+\sum _{i=1}^{d-1}h_i^2\right) +\frac{s}{y_d}\left( 1-\frac{1+\sum _{i=1}^{d-1}h_i^2}{h_d^2}\right) . \end{aligned}$$\end{document}

Finally, the weighted quasilinear PDE follows by combining the last line with (2.5) in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_\delta ^+$$\end{document} . Similarly, the boundary condition follows by comparing (2.6) with the second line in (2.9).

Case 2: local minimizers of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {E}_s$$\end{document} . First, by (2.3), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w^s(x)|\nabla w(x)|^2= y_d^{s}\,\frac{|\nabla _{x'} h(y)|^2 + 1 }{h_d^2(y)}\quad \text{ and } \quad \textrm{det}\nabla \Phi (x) = w_d(x) = \frac{1}{h_d (y)} $$\end{document}

where in both the equality we set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y{:}{=}\Phi (x)$$\end{document} . By applying the change of coordinate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y{:}{=}\Phi (x)$$\end{document} to the functional \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {E}_s$$\end{document} , we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \int _{B_\rho \cap \Omega _w}w^s(|\nabla w|^2 + 1)\, dx = \int _{B_\delta ^+} y_d^{s}\left( \frac{|\nabla _{x'} h|^2 + 1 }{h^2_d} + 1\right) h_d\,dy $$\end{document}

where the last factor comes from the first identity in (2.3). Therefore,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \int _{B_\rho \cap \Omega _w}w^s(|\nabla w|^2 +1)\, dx = \int _{B_\delta ^+} y_d^{s}\,F(\nabla h)\,dy $$\end{document}

and the Euler–Lagrange equations coincides with (2.7). \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 2.5

(Degenerate quasilinear problems and convexity) In the context of degenerate quasilinear operators, it is natural to say that (2.4) is uniformly elliptic if the function F is strictly convex (see Theorem 1.2). Precisely, when F is given as in (2.8), and thus is not convex, we localize the problem so that (2.4) is uniformly elliptic in a possibly smaller neighborhood of the origin (see the proof of Proposition 2.2).

Remark 2.6

(The free boundary condition) In [18, 19], the authors derived the free boundary condition associated to the Alt–Phillips problem, both for positive and negative exponents. Precisely, the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w{:}{=}\beta u^{1/\beta }$$\end{document} satisfies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\left\{ \begin{array}{ll} |\nabla w|^2=1 & \quad \text{ on } \partial \Omega _w \cap B_1, \text { if } \gamma \ge 0\\ w^{s}(|\nabla w|^2-1)=0 & \quad \text{ on } \partial \Omega _w \cap B_1, \text { if } \gamma < 0 \end{array}\right. } $$\end{document}

in a viscosity sense. The understanding of the free boundary condition is fundamental for the proof of an improvement-of-flatness lemma by a linearization argument. Nevertheless, the boundary condition associated to the hodograph transform of w (see the Neumann condition in (2.7)) and the linearized problem arising from the linearization in [19, Section 7] suggest that is natural to consider viscosity solutions of

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w^{s}(|\nabla w|^2-1)=0 \quad \text{ on } \partial \Omega _w \cap B_1, $$\end{document}

independently on the sign of the exponent \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (-2,2)$$\end{document} . Moreover, although for positive powers this condition is a priori weaker, the analysis in [47] suggests that a posteriori the two conditions are indeed equivalent.

Schauder estimates for degenerate quasilinear PDEs

In this section, we present a first result in the setting of Schauder estimates for degenerate quasilinear PDEs.

Before stating the main Schauder-type estimates, we recall the following lemma from [47, Lemma 2.4, Remark 2.5].

Lemma 2.7

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s>-1$$\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}$$\alpha \in (0,1)$$\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}$$f\in C^{k,\alpha }(\overline{B_1^+})$$\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}$$\begin{aligned} \varphi (x){:}{=}\frac{1}{x_d^s}\int _0^{x_d}t^sf(x',t)\,dt. \end{aligned}$$\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}$$\varphi _d\in C^{k,\alpha }(\overline{B_1^+})$$\end{document} and the following estimate holds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \varphi _d\Vert _{C^{k,\alpha }(\overline{B_{1/2}^+})}\le C\Vert f\Vert _{C^{k,\alpha }(\overline{B_{1/2}^+})}, \end{aligned}$$\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}$$C>0$$\end{document} depending only on d, s, k, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} .

Ultimately, Theorem 1.2 is a consequence of the following proposition.

Proposition 2.8

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s>-1$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\in \mathbb {N}_{\ge 1}, \alpha \in (0,1)$$\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}$$v\in C^{k,\alpha }(\overline{B_1^+})$$\end{document} be a solution of

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textrm{div}(x_d^s\,DF(\nabla v))=0\quad \text {in } B_1^+, \qquad \lim _{x_d\rightarrow 0^+}x_d^{s} D F(\nabla v)\cdot e_d =0\quad \text {on } B_1', \end{aligned}$$\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}$$F: E\subset \mathbb {R}^d\rightarrow \mathbb {R}$$\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}$$(\nabla v)(\overline{B_1^+})\subset E$$\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}$$E\subset \mathbb {R}^d$$\end{document} is a connected bounded open set. Assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F\in C^{k+1,\alpha }(E)$$\end{document} is uniformly convex in E, i.e., there exist \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\lambda \le \Lambda <+\infty $$\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}$$ \lambda \textrm{Id}\le D^2 F(p)\le \Lambda \textrm{Id}\quad \text {for every }p\in E. $$\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}$$v\in C^{k+1,\alpha }(\overline{B_{1/2}^+})$$\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}$$ \Vert v\Vert _{C^{k+1,\alpha }(\overline{B^+_{1/2}})}\le C, $$\end{document}

for some constant \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} 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}$$d, s, k, \alpha , \lambda , \Lambda , \Vert F\Vert _{C^{k+1,\alpha }(E)}, \Vert v\Vert _{C^{k,\alpha }(\overline{ B_1^+})}$$\end{document} .

Proof

The proof builds on the argument developed in the linear case in [47]. As already noted in [47], the bootstrap argument must account for the differing behavior of the operator along the tangential directions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e_i$$\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}$$i = 1, \dots , d-1$$\end{document} , and the vertical direction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e_d$$\end{document} . We mention that all the positive constants C in the proof depend only on d, s, k, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} , \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} , \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} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert F\Vert _{C^{k+1,\alpha }(E)}, \Vert v\Vert _{C^{k,\alpha }(\overline{ B_1^+})}$$\end{document} .

First, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A(p){:}{=}D^2 F(p): E \rightarrow \mathbb {R}^{d\times d}$$\end{document} be a matrix-valued function, uniformly elliptic in E, then we observe that for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,\ldots ,d-1$$\end{document} , the tangential partial derivative \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_i$$\end{document} solves the following linear degenerate elliptic problem

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textrm{div}(x_d^s\, A(\nabla v)\nabla v_i )=0\quad \text {in } B_1^+,\qquad \lim _{x_d\rightarrow 0^+} x_d^{s}\, A(\nabla v)\nabla v_i\cdot e_d =0\quad \text {on } B_1'. \end{aligned}$$\end{document}

Since the operator is differentiable in the tangential directions, (2.11) can be obtained using the difference quotient method. Then, being \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A(\nabla v(\cdot ))\in C^{k-1,\alpha }(\overline{B_1^+}),$$\end{document} by [47], we deduce that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_i\in C^{k,\alpha }(\overline{B_{1/2}^+})$$\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}$$\begin{aligned} \Vert v_i\Vert _{C^{k,\alpha }(\overline{B^+_{1/2}})}\le C \end{aligned}$$\end{document}

for some constant \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} .

Regarding the normal derivative, set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi {:}{=} DF(\nabla v)\cdot e_d$$\end{document} . Then, by the previous part of the proof, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$DF\in C^{k,\alpha }(E;\mathbb {R}^{d\times d})$$\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}$$v_i\in C^{k,\alpha }(\overline{B_{1/2}^+})$$\end{document} for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,\ldots ,d-1$$\end{document} , we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\begin{aligned} \varphi _i\in C^{k-1,\alpha }(\overline{B_{1/2}^+})\quad \text {for every}\quad i=1,\ldots ,d-1, \end{aligned}\end{aligned}$$\end{document}

and, by (2.12),

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert \varphi _i\Vert _{C^{k-1,\alpha }(\overline{ B_{1/2}^+})}\le C.\end{aligned}$$\end{document}

Then, let us prove that also the normal derivative \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi _d$$\end{document} is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{k-1,\alpha }$$\end{document} -regular. First, we notice that the interior condition in (2.10) can be rewritten as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varphi _d+\frac{s}{x_d}\varphi =-\sum _{i=1}^{d-1}\partial _i(DF(\nabla v)\cdot e_i)=:f\qquad \text{ in } B_1^+. \end{aligned}$$\end{document}

Notice, that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in C^{k-1,\alpha }(\overline{ B_{1/2}^+})$$\end{document} since both DF and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_i$$\end{document} are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{k,\alpha }$$\end{document} -regular, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,\ldots ,d-1$$\end{document} . Now, fixed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x',0)\in B_{1/2}'$$\end{document} the above identity for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\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}$$ x_d^{-s}\,\partial _d(x_d^s\, \varphi (x',x_d))= f(x',x_d)\quad \text{ for } x_d \in (1/2-|x'|^2)^{1/2}. $$\end{document}

Moreover, by solving the ODE with boundary condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{x_d\rightarrow 0^+}x_d^s \, \varphi (x',x_d)=0$$\end{document} , we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi (x',x_d)=\frac{1}{x_d^s}\int _0^{x_d} t^s f(x',t)\,dt\quad \text{ in } B_{1/2}^+. $$\end{document}

By Lemma 2.7, 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}$$\begin{aligned} \varphi _d\in C^{k-1,\alpha }(\overline{ B_{1/2}^+}) \end{aligned}$$\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}$$\begin{aligned} \Vert \varphi _d\Vert _{C^{k-1,\alpha }(\overline{ B_{1/2}^+})}\le C\Vert f\Vert _{C^{k-1,\alpha }(\overline{ B_{1/2}^+})}\le C \end{aligned}$$\end{document}

where in the last inequality we exploit the improved regularity along tangential derivatives in (2.12). Since we obtained that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{k,\alpha }$$\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}$$\overline{B_{1/2}^+}$$\end{document} , it remains to transfer this regularity to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_d$$\end{document} . Although \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} is closely related to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_d$$\end{document} , a separate argument is required to conclude the same regularity for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_d$$\end{document} .

By the uniform convexity of F, 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}$$A(p)e_d\cdot e_d\ge \lambda >0$$\end{document} for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in E$$\end{document} . Then, since DF is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{k,\alpha }$$\end{document} -regular, being that E is a connected bounded open set, by the global implicit function theorem there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H\in C^{k,\alpha }(E)$$\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}$$ t=D F(p',p_d)\cdot e_d\iff p_d=H(p',t),\,\quad \text{ for } \text{ every } p \in E,\ t\in DF(E). $$\end{document}

Notice that the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{k,\alpha }$$\end{document} norm of H depends only on the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{k+1,\alpha }$$\end{document} norm of F. Now, being \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi =DF(\nabla v)\cdot e_d \in C^{k,\alpha }(\overline{B_{1/2}^+})$$\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}$$\nabla _{x'}v \in C^{k,\alpha }(\overline{B_{1/2}^+};\mathbb {R}^{d-1})$$\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}$$\begin{aligned} v_d=H(\nabla _{x'}v,\varphi )\in C^{k,\alpha }(\overline{B_{1/2}^+}). \end{aligned}$$\end{document}

Moreover, by (2.12), (2.13) and (2.14), the following estimate holds

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert v_d\Vert _{C^{k,\alpha }(\overline{B_{1/2}^+})}\le C\sum _{i=1}^{d-1}\Vert v_i\Vert _{C^{k,\alpha }(\overline{B_{1/2}^+})}+\Vert \varphi \Vert _{C^{k,\alpha }(\overline{B_{1/2}^+})}\le C, \end{aligned}$$\end{document}

concluding the proof. \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 Theorem 1.2

It follows immediately from 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}$$\square $$\end{document}

Proof of \documentclass[12pt]{minimal}

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			\begin{document}$$C^\infty $$\end{document}C∞ regularity

Now we prove the main results of this section.

Proof of Proposition 2.2

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (-2,2)$$\end{document} and w be a regular solution, in the sense of Definition 2.1. Up to translation and rescaling, assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\in \partial \Omega _u$$\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}$$\nabla w(0)=e_d$$\end{document} . Therefore, as in Subsection 2.1, we can infer the existence of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho >0$$\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}$$\delta >0$$\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}$$ \Phi :\overline{\Omega }_w\cap B_\rho \rightarrow \mathbb {R}^d\cap \{x_d\ge 0\},\quad \Phi (x',x_d){:}{=}(x', w(x',x_d)),\quad $$\end{document}

is bijective from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\Omega }_w\cap B_\rho $$\end{document} into \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_\delta ^+$$\end{document} . Hence, there exists the hodograph transform \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\in C^{1,\alpha }(\overline{B_\delta ^+})$$\end{document} of w and, by Lemma 2.4, it is a solution of (2.7), with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla h(0)=e_d$$\end{document} .

We proceed by showing that the regularity of h can be improved. First, we observe that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ A(p){:}{=}D^2 F(p)=\frac{2}{p_d}\textrm{Id}-2\frac{e_d\otimes p+ p\otimes e_d}{p_d^2}+\frac{2(1+|p|^2)}{p_d^3}e_d\otimes e_d, $$\end{document}

is uniformly elliptic in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_\eta (e_d)$$\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}$$\eta >0$$\end{document} . In fact, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A(e_d)=2\textrm{Id}$$\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}$$\eta >0$$\end{document} sufficiently small, 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}$$ -\textrm{Id}\le A(p)-2\textrm{Id}\le \textrm{Id},\quad \text{ for } \text{ every } p \in B_\eta (e_d). $$\end{document}

Moreover, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla h(0)=e_d$$\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}$$A(\nabla h(0))=2\textrm{Id}$$\end{document} , by the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{1,\alpha }$$\end{document} -regularity of h in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{B_\delta ^+}$$\end{document} we deduce the existence of a radius \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma >0$$\end{document} , possibly smaller then \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} , such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (\nabla h)(\overline{B_\sigma ^+})\subset B_\eta (e_d). \end{aligned}$$\end{document}

Therefore, F is uniformly convex in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E{:}{=}B_\eta (e_d)$$\end{document} and, up to rescaling, the hodograph map h fulfills the hypotheses of Theorem 1.2. Since F is \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} regular in E, then h is \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} in a neighborhood of the origin. Finally, by exploiting the inverse of the hodograph transformation we deduce the existence of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r>0$$\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}$$w\in C^{\infty }(\overline{\Omega }_w\cap B_{r})$$\end{document} . Similarly, since the derivatives of 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}$$\partial \Omega _w$$\end{document} are given by the tangential derivatives of h (see (2.4)), the improved regularity of the hodograph transform implies that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega _w$$\end{document} is smooth in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{r}$$\end{document} . \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

The following result is a direct consequence of Proposition 2.2.

Proof of Theorem 1.1

The proof directly follows by Proposition 2.2. Indeed, let u be a local minimizer of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {J}_\gamma $$\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}$$B_1$$\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}$$x_0\in \partial \Omega _u$$\end{document} be a regular free boundary point. Without loss of generality, suppose that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0=0$$\end{document} and consider the auxiliary function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w{:}{=}\beta u^{1/\beta }$$\end{document} . Then w is a local minimizer to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {E}_s$$\end{document} , and, up to rescaling, by Lemma 2.3 it is a regular solution. Thus, by Proposition 2.2, there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r>0$$\end{document} small such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w\in C^{\infty }(\overline{\Omega }_w\cap B_r) \text { and the free boundary } \partial \Omega _w \text { is smooth in } B_r. $$\end{document}

Finally, the last part of the result follows by exploiting Hadamard’s Lemma. Indeed, up to localizing the problem in a smaller ball \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_r$$\end{document} in such a way the boundary normal coordinates are well-defined, we can deduce that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w(x)=\textrm{dist}(x,\partial \Omega _w)\psi (x)$$\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}$$\psi \in C^\infty (\overline{\Omega }_w\cap B_r)$$\end{document} . On the other hand, being \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\nabla w|=1$$\end{document} on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega _w$$\end{document} , we get \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi \equiv 1$$\end{document} on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega _w\cap B_r$$\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}$$\psi $$\end{document} is positive in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega _w \cap B_r$$\end{document} and thus

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{u}{\textrm{dist}(x,\partial \Omega _w)^\beta } = \left( \frac{1}{\beta }\frac{w}{\textrm{dist}(x,\partial \Omega _u)}\right) ^\beta =\left( \frac{1}{\beta }\psi (x)\right) ^\beta $$\end{document}

is smooth in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\Omega }_w\cap B_r$$\end{document} . \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

The stability condition

In this section we introduce the definitions of stable solutions for the Alt–Phillips problem. Then, we compute both the first and the second variations with respect to inner variations. Ultimately, we derive the rigidity condition of Theorem 1.3 for minimizing (and stable) cones.

Minimizers and stable solutions

In order to compute the stability condition, we can consider solutions of the problem whose free boundary is smooth around the points we want to deal with. Nevertheless, in light of the result of Sect. 2, it is not restrictive to work with regular solutions w in the sense of Definition 2.1, since their free boundary is smooth near the points of interest and the free boundary condition is pointwise satisfied.

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (-2,0)$$\end{document} , and B be a ball 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} , then we define

$$[eqn]$$

In the following we introduce the notion of stable solutions. In general, stability is defined for critical points of a variational functional with respect to a class of variations. However, for the Alt–Phillips problem with negative exponents, criticality must be replaced by a stronger assumption, as the validity of (2.1) cannot be directly deduced from domain variations (see Lemma 3.4 and the discussion in [19]). Throughout the paper, we address this by assuming that solutions are regular in the sense of Definition 2.1 near regular points.

Definition 3.1

(Stable solutions) Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (-2,0)$$\end{document} , and B be a ball 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} . 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}$$u: B \rightarrow \mathbb {R}$$\end{document} is a stable solution for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {J}_\gamma $$\end{document} in B, if

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\begin{aligned} \delta ^2 \mathcal {J}_\gamma (u,B)[\xi ]{:}{=}\frac{d^2}{d t^2}\bigg |_{t=0}\mathcal {J}_{\gamma }(u \circ \Phi _t^{-1},B)\ge 0,\quad \text{ where } \Phi _t(x){:}{=}x+t\xi (x), \end{aligned}\end{aligned}$$\end{document}

for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi \in C^{\infty }_c(B;\mathbb {R}^d)$$\end{document} . Equivalently, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w: B\rightarrow \mathbb {R}$$\end{document} is stable solution for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {E}_s$$\end{document} in B, if

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\begin{aligned} \delta ^2 \mathcal {E}_s(w,B)[\xi ]{:}{=}\frac{d^2}{d t^2}\bigg |_{t=0}\mathcal {E}_s(w \circ \Phi _t^{-1},B)\ge 0,\quad \text{ where } \Phi _t(x){:}{=}x+t\xi (x), \end{aligned}\end{aligned}$$\end{document}

for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi \in C^{\infty }_c(B;\mathbb {R}^d)$$\end{document} .

We also introduce the notion of local/global minimizers and global stable solutions.

Definition 3.2

Let B be a ball 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} , 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}$$u: B\rightarrow \mathbb {R}$$\end{document} is a local minimizer of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {J}_\gamma $$\end{document} in B, if

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal {J}_{\gamma }(u,B)\le \mathcal {J}_{\gamma }(v, B),\qquad \text {for all } v\in H^1(B) \text { such that } v-u \in H^1_0(B). $$\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}$$u: \mathbb {R}^d\rightarrow \mathbb {R}$$\end{document} is said to be a global minimizer of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {J}_\gamma $$\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}$$\mathbb {R}^d$$\end{document} (resp. global stable solution 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} ) if u is a local minimizer (resp. stable solution) in every ball \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B\subset \mathbb {R}^d$$\end{document} . The previous definitions can be naturally extended to the function w.

Second inner variation

The following result, is our first characterization of the stability condition, which ultimately implies Theorem 1.3.

Proposition 3.3

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (-2,0)$$\end{document} , and B be a ball 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} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f \in C^\infty _c(B)$$\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}$$u:B\rightarrow \mathbb {R}$$\end{document} be a local minimizer of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {J}_\gamma $$\end{document} in B; or a stable solution in B, in the sense of Definition 3.1. Assume that u is regular in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {supp}(f)$$\end{document} , in the sense of Definition 2.1. Then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \int _{ \Omega _u} |\nabla u|^2 \left( |\nabla f|^2 - \mathcal {A}^2_u\, f^2\right) \,dx\ge 0,\quad \text {where }\mathcal {A}^2_u{:}{=} \frac{|\nabla ^2 u|^2}{|\nabla u|^2} - \frac{|\nabla ^2 u\nabla u|^2}{|\nabla u|^4}. $$\end{document}

Equivalently, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w:B\rightarrow \mathbb {R}$$\end{document} be a local minimizer of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {E}_s$$\end{document} in B; or a stable solution in B in the sense of Definition 3.1. Assume that u is regular in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {supp}(f)$$\end{document} , in the sense of Definition 2.1. Then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \int _{ \Omega _w} w^s|\nabla w|^2 \left( |\nabla f|^2 - \mathcal {A}^2_w\, f^2\right) \,dx\ge 0,\quad \text {where }\mathcal {A}^2_w{:}{=} \frac{|\nabla ^2 w|^2}{|\nabla w|^2} - \frac{|\nabla ^2 w\nabla w|^2}{|\nabla w|^4}. $$\end{document}

In order to compute a second order expansion close to regular free boundaries, we work directly with local minimizers w of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {E}_s$$\end{document} (resp. stable solutions) being smooth close to regular points. Throughout the section, given a vector field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi :\mathbb {R}^d \rightarrow \mathbb {R}^d$$\end{document} we denote with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D\xi \in \mathbb {R}^{d\times d}$$\end{document} the matrix with entries \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(D\xi )_{ij}{:}{=}\partial _i \xi _j$$\end{document} , for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le i,j\le d$$\end{document} .

In the following result we compute the first variation of the functional \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {E}_s$$\end{document} along inner variations. Ultimately, we observe that regular solutions are indeed stationary along non-tangential variations. We stress that such computation is not necessary for the proof of Proposition 3.3.

Lemma 3.4

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (-2,0)$$\end{document} , and B be a ball 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} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w \in H^1(B)$$\end{document} , then for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi \in C^\infty _c(B;\mathbb {R}^d)$$\end{document} we set as 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}$$\mathcal {E}_s$$\end{document} at w 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}$$\xi $$\end{document} the quantity

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \delta \mathcal {E}_s(w,B)[\xi ]{:}{=}\frac{d}{d t}\bigg |_{t=0}\mathcal {E}_s(w \circ \Phi _t^{-1},B),\quad \text{ where } \Phi _t(x){:}{=}x+t\xi (x). $$\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}$$\begin{aligned} \delta \mathcal {E}_s(w,B)[\xi ]=\int _{\Omega _w}w^s\Big ((|\nabla w|^2+1){{\,\textrm{div}\,}}\xi -2D\xi \nabla w\cdot \nabla w\Big )\,dx. \end{aligned}$$\end{document}

Moreover, for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in C^\infty _c(B)$$\end{document} , if w is a regular solution in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {supp}(f)$$\end{document} , in the sense of Definition 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}$$\begin{aligned} \delta \mathcal {E}_s(w,B)\left[ \frac{\nabla w}{|\nabla w|} f\right] = 0. \end{aligned}$$\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}$$\xi \in C^\infty _c(B;\mathbb {R}^d)$$\end{document} be a smooth vector field with compact support in B and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi _t{:}{=}\textrm{Id}+t\xi $$\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}$$w_t:B\rightarrow \mathbb {R}$$\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}$$w_t(x){:}{=}w(\Phi ^{-1}_t(x))$$\end{document} . Then, 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}$$y=\Phi _t(x)$$\end{document} , we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\mathcal {E}_s(w \circ \Phi _t^{-1},B) =\int _{\Omega _{w_t}}w_t^s(|\nabla w_t|^2+1)\,dx\\&\quad =\int _{\Omega _w} w^s(\Phi _t^{-1}(x))\Big (\big |D\Phi _t^{-1}(x)\nabla w_t(\Phi _t^{-1}(x))\big |^2+1\Big )\,dx\\&\quad =\int _{\Omega _w}w^s(y)\Big (D\Phi _t^{-1}(\Phi _t(y))^T D\Phi _t^{-1}(\Phi _t(y))\nabla w(y)\cdot \nabla w(y)+1\Big )|\text {det}D\Phi _t(y)|\,dy\\&\quad =\int _{\Omega _w}w^s(y)\Big ([D\Phi _t(y)]^{-T}[D\Phi _t(y)]^{-1}\nabla w(y)\cdot \nabla w(y)+1\Big )|\text {det}D\Phi _t(y)|\,dy. \end{aligned}\end{aligned}$$\end{document}

Using the facts that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ D\Phi _t^{-1}=\textrm{Id}-tD\xi + o(t)\qquad \text {and}\qquad |\text {det}D\Phi _t|=1+t{{\,\textrm{div}\,}}\xi + o(t), $$\end{document}

we can expand the previous identity to first order in t, obtaining (3.1), as we claimed.

Now, suppose that w is a regular solution in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {supp}(f)$$\end{document} , in the sense of Definition 2.1. By exploiting the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{1,\alpha }$$\end{document} -regularity of both w and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega _w$$\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}$$\alpha >-s$$\end{document} ) we can rewrite (3.1) by integrating by parts. By the Rellich-type identity

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\,\textrm{div}\,}}\Big (|\nabla w|^2\xi -2(\xi \cdot \nabla w)\nabla w\Big )=|\nabla w|^2{{\,\textrm{div}\,}}\xi -2D\xi \nabla w\cdot \nabla w-2\Delta w(\xi \cdot \nabla w), \end{aligned}$$\end{document}

we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\begin{aligned}&\delta \mathcal {E}_s(w,B)[\xi ]= \int _{\Omega _w}\left[ w^s{{\,\textrm{div}\,}}\Big ((|\nabla w|^2+1)\xi -2(\xi \cdot \nabla w)\nabla w\Big )+2w^s\Delta w(\xi \cdot \nabla w)\right] \,dx \\&\quad = \int _{\Omega _w} \textrm{div}\Big ( w^s\,(|\nabla w|^2+1)\xi -2w^s(\xi \cdot \nabla w)\nabla w\Big )\, dx \\&\quad \quad +2\int _{\Omega _w} w^s\Delta w(\xi \cdot \nabla w) - \frac{s}{2}w^{s-1}\nabla w\cdot \Big ((|\nabla w|^2+1)\xi -2(\xi \cdot \nabla w)\nabla w\Big )\,dx \\&\quad = \int _{\Omega _w} \textrm{div}\Big ( w^s\,(1-|\nabla w|^2)\xi +2w^s|\nabla w|^2\xi -2w^s(\xi \cdot \nabla w)\nabla w\Big )\, dx \\&\quad \quad +2\int _{\Omega _w} w^s\left( \Delta w - \frac{s}{2}\frac{1-|\nabla w|^2}{w} \right) (\xi \cdot \nabla w) \,dx. \end{aligned}\end{aligned}$$\end{document}

By substituting (2.1), we already know that the last integral is null for every smooth vector field with compact support. On the other hand, for non-tangential variations, the previous computation can be further improved. Thus, given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in C^\infty _c(B)$$\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}$$ \xi := \frac{\nabla w}{|\nabla w|} f. $$\end{document}

Indeed, since in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega _w \cap B$$\end{document} the interior normal coincides with the gradient of w, the previous variation is perpendicular to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega _w$$\end{document} as well. Then, by direct substitution, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\delta \mathcal {E}_s(w,B)\left[ \frac{\nabla w}{|\nabla w|}f\right] = \int _{\Omega _w} \textrm{div}\bigg ( w^s\,(1-|\nabla w|^2)\frac{\nabla w}{|\nabla w|}f\bigg )\, dx \\&\quad =\int _{\partial \Omega _w} w^s\,(|\nabla w|^2-1)f \, d\mathcal {H}^{d-1} \end{aligned}$$\end{document}

where in the last equality we exploit the Gauss–Green formula as in [30, Proposition 2.7]. Precisely, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w \in C^{1,\alpha }(\overline{\Omega }_w\cap \text {supp}(f))$$\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}$$\alpha >-s$$\end{document} , it implies that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w^s(|\nabla w|^2-1)$$\end{document} extends continuously to zero on the free boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega _w$$\end{document} . Finally, (3.2) follows by the free boundary condition in (2.1). \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}

We proceed now to the study of the second variation.

Lemma 3.5

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (-2,0)$$\end{document} , and B be a ball 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} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in C^\infty _c(B)$$\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}$$w:B\rightarrow \mathbb {R}$$\end{document} be a regular solution in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {supp}(f)$$\end{document} , in the sense of Definition 2.1. Then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \delta ^2\mathcal {E}_s(w,B)\left[ \frac{\nabla w}{|\nabla w|}f\right] =\int _{\Omega _w}(f_1+f_2+f_3)\,dx, \end{aligned}$$\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}$$f_1:=\frac{1}{2}w^s\Big (|\nabla w|^2+1\Big ){{\,\textrm{div}\,}}\left[ \left( \frac{\Delta w\nabla w}{|\nabla w|^2}-\frac{1}{2}\frac{\nabla (|\nabla w|^2)}{|\nabla w|^2}\right) f^2 \right] , $$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\begin{aligned} f_2&:=w^s|\nabla f|^2|\nabla w|^2, \end{aligned}\end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f_3:=w^s\Big ((\Delta w)^2-|\nabla ^2w|^2\Big )f^2+w^s{{\,\textrm{div}\,}}\left[ \left( \frac{1}{2}\nabla (|\nabla w|^2)-\Delta w\nabla w\right) f^2\right] . \end{aligned}$$\end{document}

Proof

Given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi \in C^\infty _c(B;\mathbb {R}^d)$$\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}$$\Phi _t:=\textrm{Id} + t \xi $$\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}$$w_t:B\rightarrow \mathbb {R}$$\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}$$w_t(x):=w(\Phi ^{-1}_t(x))$$\end{document} . By (3.3) and the following second order expansions

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} D\Phi _t^{-1}&=\textrm{Id}-tD\xi +t^2[D\xi ]^2+ o(t^2),\\|\text {det}D\Phi _t|&=1+t{{\,\textrm{div}\,}}\xi + \frac{t^2}{2}\Big (({{\,\textrm{div}\,}}\xi )^2-\text {Tr}(D\xi )^2\Big )+o(t^2), \end{aligned}$$\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}$$\begin{aligned}\begin{aligned} \delta ^2 \mathcal {E}_s(w)[\xi ]&=\int _{\Omega _w} w^s\bigg (\Big (|\nabla w|^2+1\Big )\frac{({{\,\textrm{div}\,}}\xi )^2-\text {Tr}(D\xi )^2}{2}+|D\xi \nabla w|^2\\ &\quad +2\Big ([D\xi ]^2\nabla w\cdot \nabla w-D\xi \nabla w\cdot \nabla w {{\,\textrm{div}\,}}\xi \Big )\bigg )\,dx.\end{aligned}\end{aligned}$$\end{document}

Now, set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f_1:= w^s\Big (|\nabla w|^2+1\Big )\frac{({{\,\textrm{div}\,}}\xi )^2-\text {Tr}(D\xi )^2}{2}, \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_2:=w^s|D\xi \nabla w|^2, $$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f_3:=2w^s\Big ([D\xi ]^2\nabla w\cdot \nabla w-D\xi \nabla w\cdot \nabla w {{\,\textrm{div}\,}}\xi \Big ). \end{aligned}$$\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}$$f\in C^{\infty }_c(B)$$\end{document} , we take \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi :=\frac{\nabla w}{|\nabla w|}f$$\end{document} . For what it concerns \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_1$$\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}$$f_3$$\end{document} , we proceed by using some general differential identities, already exploited in [33, Subsection 4.2] for the study of the Alt–Phillips functional for positive exponents. In fact, by following their notations, the new formulations of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_1$$\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}$$f_3$$\end{document} follow by substituting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi =f|\nabla w|$$\end{document} in to the proof of [33, Theorem 4.1]. Regarding \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_2$$\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}$$\begin{aligned}\begin{aligned} \partial _{i}\xi _j&=\frac{\partial _{ij}w}{|\nabla w|}f+\frac{\partial _if\partial _jw}{|\nabla w|}-\frac{1}{|\nabla w|^2}\partial _j w \frac{(\nabla \partial _iw\cdot \nabla w)}{|\nabla w|}f. \end{aligned}\end{aligned}$$\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}$$\begin{aligned}\begin{aligned} D\xi \nabla w =\frac{\nabla ^2w\nabla w}{|\nabla w|}f+|\nabla w|\nabla f-\frac{1}{|\nabla w|^2}|\nabla w|^2\frac{\nabla ^2w\nabla w}{|\nabla w|} f=|\nabla w|\nabla f, \end{aligned}\end{aligned}$$\end{document}

which implies that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|D\xi \nabla w|^2=|\nabla w|^2|\nabla f|^2,$$\end{document} concluding the proof. \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 3.6

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (-2,0)$$\end{document} , and B be a ball 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} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in C^\infty _c(B)$$\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}$$w:B\rightarrow \mathbb {R}$$\end{document} be a local minimizer of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {E}_s$$\end{document} in B; or a stable solution in B, in the sense of Definition 3.1. Assume that w is regular in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {supp}(f)$$\end{document} , in the sense of Definition 2.1. If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_1$$\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}$$f_3$$\end{document} are as in Lemma 3.5, then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _{\Omega _w}(f_1+f_3)\,dx=-\int _{\Omega _w}w^s|\nabla w|^2\mathcal {A}_w^2f^2\,dx,\quad \text{ where } \mathcal {A}_w^2=\frac{|\nabla ^2w|^2}{|\nabla w|^2}-\frac{|\nabla ^2w\nabla w|^2}{|\nabla w|^4}. $$\end{document}

Proof

Since w is a regular solution in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {supp}(f)$$\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}$$C^{1,\alpha }$$\end{document} -regular free boundary in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {supp}(f)$$\end{document} , then, by Proposition 2.2,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w\in C^{\infty }(\overline{\Omega }_w\cap \text {supp}(f)) \text { and the free boundary } \partial \Omega _w \text { is smooth in } \text {supp}(f). $$\end{document}

Expanding the divergence of the second term in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_3$$\end{document} , we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\begin{aligned} f_3&:=w^s\Big ((\Delta w)^2-|\nabla ^2w|^2\Big )f^2+w^s{{\,\textrm{div}\,}}\left[ \left( \frac{\nabla (|\nabla w|^2)}{2}-\Delta w\nabla w\right) f^2\right] \\&=w^s\Big ((\Delta w)^2-|\nabla ^2w|^2\Big )f^2+w^s \frac{\Delta (|\nabla w|^2)}{2}f^2+w^s\frac{\nabla (|\nabla w|^2)}{2}\cdot \nabla f^2\\&\quad - w^s\textrm{div}\Big ((\Delta w \nabla w)f^2\Big )\\ &=w^s\Big ((\Delta w)^2-|\nabla ^2w|^2\Big )f^2+w^s \frac{\Delta (|\nabla w|^2)}{2}f^2+{{\,\textrm{div}\,}}\left( w^s\frac{\nabla (|\nabla w|^2)}{2}f^2\right) \\ &\quad -{{\,\textrm{div}\,}}\left( w^s\frac{\nabla (|\nabla w|^2)}{2}\right) f^2- w^s\textrm{div}\Big ((\Delta w \nabla w)f^2\Big ). \end{aligned}\end{aligned}$$\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}$$\begin{aligned} {{\,\textrm{div}\,}}\left( w^s\frac{\nabla (|\nabla w|^2)}{2}\right) =\frac{1}{2}\nabla w^s\cdot \nabla (|\nabla w|^2) +\frac{1}{2}w^s \Delta (|\nabla w|^2), \end{aligned}$$\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}$$\begin{aligned}\begin{aligned} f_3&=w^s\Big ((\Delta w)^2-|\nabla ^2w|^2\Big )f^2+{{\,\textrm{div}\,}}\left( w^s\frac{\nabla (|\nabla w|^2)}{2}f^2\right) -\frac{1}{2}\nabla w^s\cdot \nabla (|\nabla w|^2)f^2\\ &\qquad - w^s\textrm{div}\Big ((\Delta w \nabla w)f^2\Big ). \end{aligned}\end{aligned}$$\end{document}

Then we can rewrite \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_1+f_3=f_0+f_{\text {parts}},$$\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}$$\begin{aligned}\begin{aligned} f_0:=w^s\Big ((\Delta w)^2-|\nabla ^2w|^2\Big )f^2-\frac{1}{2}\nabla w^s\cdot \nabla (|\nabla w|^2)f^2 \end{aligned}\end{aligned}$$\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}$$\begin{aligned}\begin{aligned} f_{\text {parts}}&:=\frac{1}{2} w^s \big (|\nabla w|^2 +1\big )\textrm{div}\left[ \left( \frac{\Delta w\nabla w}{|\nabla w|^2} - \frac{1}{2} \frac{\nabla (|\nabla w|^2)}{|\nabla w|^2}\right) f^2\right] \\ &\quad +\textrm{div}\left( w^s \frac{\nabla (|\nabla w|^2)}{2}f^2\right) - w^s\textrm{div}\Big ((\Delta w \nabla w)f^2\Big ). \end{aligned}\end{aligned}$$\end{document}

By rewriting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{\text {parts}}$$\end{document} , we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\begin{aligned} f_{\text {parts}}&={{\,\textrm{div}\,}}\left[ w^s\left( \big (|\nabla w|^2+1\big )\left( \frac{1}{2}\frac{\Delta w\nabla w}{|\nabla w|^2}-\frac{1}{4} \frac{\nabla (|\nabla w|^2)}{|\nabla w|^2}\right) -\Delta w\nabla w+\frac{\nabla (|\nabla w|^2)}{2}\right) f^2\right] \\ &\quad + \int _{\Omega _w} \left[ - \frac{1}{2} \left( \frac{\Delta w \nabla w}{|\nabla w|^2} - \frac{1}{2} \frac{\nabla (|\nabla w|^2)}{|\nabla w|^2}\right) \nabla \big (w^s \big (1+|\nabla w|^2\big )\big )+\Delta w \nabla w \cdot \nabla w^s\right] f^2 \\ &= \textrm{div}(F) \\&\quad + \left[ - \frac{1}{2} \left( \frac{\Delta w \nabla w}{|\nabla w|^2} - \frac{1}{2} \frac{\nabla (|\nabla w|^2)}{|\nabla w|^2}\right) \nabla \big (w^s \big (1+|\nabla w|^2\big )\big )+\Delta w \nabla w \cdot \nabla w^s\right] f^2 \end{aligned}\end{aligned}$$\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}$$ F:= \frac{1}{2} w^s(1-|\nabla w|^2)\left( \frac{\Delta w \nabla w}{|\nabla w|^2} - \frac{\nabla ^2 w \nabla w}{|\nabla w|^2}\right) f^2. $$\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}$$\begin{aligned}\begin{aligned} f_{\text {parts}}-{{\,\textrm{div}\,}}(F)&= \bigg [- \frac{1}{2} \left( \frac{\Delta w \nabla w}{|\nabla w|^2} - \frac{1}{2} \frac{\nabla (|\nabla w|^2)}{|\nabla w|^2}\right) \Big (\nabla w^s \big (1+|\nabla w|^2\big )+w^s\nabla (|\nabla w|^2)\Big )\\ &\quad +\Delta w \nabla w \cdot \nabla w^s\bigg ]f^2\\ &= \frac{1}{2} \frac{\Delta w \nabla w\cdot \nabla w^s}{|\nabla w|^2}\big (|\nabla w|^2-1\big )f^2-\frac{1}{2}w^s\frac{\Delta w \nabla w\cdot \nabla (|\nabla w|^2)}{|\nabla w|^2}f^2\\ &\quad +\frac{1}{4}\frac{\nabla (|\nabla w|^2)\cdot \nabla w^s}{|\nabla w|^2}\big (1+|\nabla w|^2\big )f^2 +\frac{1}{4}w^s \frac{|\nabla (|\nabla w|^2)|^2}{|\nabla w|^2}f^2\\ &= -w^s (\Delta w)^2 f^2 -\frac{1}{2}w^s\frac{\Delta w \nabla w\cdot \nabla (|\nabla w|^2)}{|\nabla w|^2}f^2\\ &\quad +\frac{1}{4}\frac{\nabla (|\nabla w|^2)\cdot \nabla w^s}{|\nabla w|^2}\big (1+|\nabla w|^2\big )f^2 +w^s \frac{|\nabla ^2w\nabla w|^2}{|\nabla w|^2}f^2 \end{aligned}\end{aligned}$$\end{document}

where in the last equality we used the equation for w (see (2.1)). Combining the above identities, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\begin{aligned} f_0+f_{\text {parts}}&=-w^s\left( |\nabla ^2w|^2-\frac{|\nabla ^2w\nabla w|^2}{|\nabla w|^2}\right) f^2-\frac{1}{2}\nabla w^s\cdot \nabla (|\nabla w|^2)f^2\\ &\quad -\frac{1}{2}w^s\frac{\Delta w \nabla w\cdot \nabla (|\nabla w|^2)}{|\nabla w|^2}f^2\\&\quad +\frac{1}{4}\frac{\nabla (|\nabla w|^2)\cdot \nabla w^s}{|\nabla w|^2}\big (1+|\nabla w|^2\big )f^2+{{\,\textrm{div}\,}}(F)\\ &=-w^s\left( |\nabla ^2w|^2-\frac{|\nabla ^2w\nabla w|^2}{|\nabla w|^2}\right) f^2 -\frac{1}{2}w^s\frac{\Delta w \nabla w\cdot \nabla (|\nabla w|^2)}{|\nabla w|^2}f^2\\ &\quad +\frac{1}{4}\frac{\nabla (|\nabla w|^2)\cdot \nabla w^s}{|\nabla w|^2}\big (1-|\nabla w|^2\big )f^2+{{\,\textrm{div}\,}}(F) \end{aligned}\end{aligned}$$\end{document}

and so

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ f_0+f_{\text {parts}}=-w^s\left( |\nabla ^2w|^2-\frac{|\nabla ^2w\nabla w|^2}{|\nabla w|^2}\right) f^2+{{\,\textrm{div}\,}}(F), $$\end{document}

where in the last equality we used the PDE satisfied by w (see (2.1)). Hence, as in the proof of Lemma 3.4 we can apply the Gauss–Green formula as in [30, Proposition 2.7]. In fact, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w\in C^{\infty }(\overline{\Omega }_w\cap \text {supp}(f))$$\end{document} , the vector field F can be extended continuously to zero on the free boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega _w$$\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}$$\text {supp}(f)$$\end{document} and so F is divergence free in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _w\cap \text {supp}(f)$$\end{document} . Then the result follows integrating the last identity and using 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}$$\mathcal {A}_w^2$$\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}

We conclude with the proof of the stability condition in Theorem 1.3 and Proposition 3.3.

Proof of Proposition 3.3

It follows immediately by combining Lemma 3.5 and Lemma 3.6. \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 Theorem 1.3

Since the free boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega _w$$\end{document} is a smooth cone outside the origin, for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f \in C^\infty _c(\mathbb {R}^d\setminus \{0\})$$\end{document} we can apply Proposition 3.3 to conclude. \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 3.7

(Stability condition for stable solutions) Using Proposition 3.3, we observe that Theorem 1.3 holds even for global stable solutions u of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {J}_\gamma $$\end{document} , regular outside the origin, in the sense of Definition 3.2 and Definition 2.1. Similarly, Theorem 1.3 holds for w under the corresponding assumptions.

Remark 3.8

(Stability conditions for general exponents) Recently in [33], the authors obtained a stability condition for the Alt–Phillips problem with positive exponents, by computing the second variations of the form

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ t \mapsto \mathcal {E}_s\left( w\circ (\textrm{Id}+t \xi )^{-1}\right) ,\quad \text{ where } \xi :=\frac{\nabla w}{|\nabla w|^2}\varphi , $$\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}$$\varphi \in C^\infty _c(\mathbb {R}^d \setminus \{0\})$$\end{document} . Their condition has a different form from our (1.5) and 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}$$\begin{aligned} \int _{\Omega _w}w^s|\nabla \varphi |^2\,dx\ge \int _{\Omega _w}w^s\frac{\Delta w}{w}\varphi ^2\,dx,\quad \text{ for } \text{ every } \varphi \in C^\infty _c(\mathbb {R}^d {\setminus } \{0\}).\end{aligned}$$\end{document}

The choice of such variations is natural and it coincides with the one considered by Caffarelli, Jerison and Kenig in [7], for the Alt–Caffarelli problem ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =0$$\end{document} ), where they show that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \int _{\Omega _w}|\nabla \varphi |^2\,dx\ge \int _{\partial \Omega _w}H\varphi ^2\,d\mathcal {H}^{d-1},\quad \text{ for } \text{ every } \varphi \in C^\infty _c(\mathbb {R}^d {\setminus } \{0\}). $$\end{document}

Nevertheless, it is well-known that this latter inequality, can be also written in a Sternberg–Zumbrun form [46]. Indeed, by choosing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi :=|\nabla w|f$$\end{document} and integrating by parts, we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \int _{ \Omega _w} |\nabla w|^2\left( |\nabla f|^2 - \mathcal {A}^2_w\, f^2\right) \,dx \ge 0,\quad \text{ where } \mathcal {A}^2_w:= \frac{|\nabla ^2 w|^2}{|\nabla w|^2} - \frac{|\nabla ^2 w\nabla w|^2}{|\nabla w|^4}, $$\end{document}

which is exactly Theorem 1.3 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}$$s=0$$\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}$$\gamma =0$$\end{document} ). Naturally, a similar formulation holds for the case of positive exponents.

Surprisingly, 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}$$\gamma \in (-2,0)$$\end{document} , the presence of the singular weight \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w^s$$\end{document} prevents to write both the conditions. Heuristically, the stability condition can be computed by considering only variations of the form (1.5), which are more close to the one exploited in the study of stable minimal surfaces. Indeed, by comparing the stability condition (4.3) below with the one for positive exponents (3.4), we observe that in the negative regime, it is not possible to split the integrals and integrate by parts the divergence term in (4.3), since the weight \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w^s$$\end{document} is singular on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega _w$$\end{document} . Moreover, for negative exponents, such integration can be computed if and only if 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}$$\partial \Omega _w$$\end{document} is identically zero.

Axially symmetric cones

In this section, we consider the specific case of minimizing axially symmetric cones, namely homogeneous minimizers u that are invariant under rotations around a fixed axis. Precisely, we say that u is axially symmetric if, up to a rotation, we have that for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x =(x',x_d)\in \mathbb {R}^d$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u(x',x_d)=u(\tau ,x_d),\quad \text {where } \tau :=|x'|. \end{aligned}$$\end{document}

The main result of the section is Theorem 1.4, namely that minimizing axially symmetric cones are one-dimensional in low dimensions.

Before presenting the proof of the main result, we begin by rephrasing the stability condition under axial symmetry as an Hardy-type inequality. For simplicity of notation, we set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u_\tau :=\nabla u\cdot \frac{(x',0)}{\tau }. \end{aligned}$$\end{document}

The same notations apply to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w:=\beta u^{1/\beta }$$\end{document} . The following is the main result towards the proof of Theorem 1.4.

Proposition 4.1

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (-2,0)$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge 3$$\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}$$u\in C^{0,\beta }(\mathbb {R}^d)$$\end{document} be a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} -homogeneous global minimizer of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {J}_{\gamma }$$\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}$$\mathbb {R}^d$$\end{document} ; or a global stable solution in the sense of Definition 3.2. Assume that u is axially symmetric and that is regular outside the origin, in the sense of Definition 2.1. Then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{\Omega _u} u_\tau ^2 \left( |\nabla \eta |^2-(d-2)\frac{\eta ^2}{\tau ^2}\right) \,dx\ge 0,\end{aligned}$$\end{document}

for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta \in C^\infty _c(\mathbb {R}^d)$$\end{document} .

Equivalently, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w\in C^{0,1}(\mathbb {R}^d)$$\end{document} be a 1-homogeneous global minimizer of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {E}_{s}$$\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}$$\mathbb {R}^d$$\end{document} ; or a global stable solution, in the sense of Definition 3.2. Assume that w is axially symmetric and that is regular outside the origin, in the sense of Definition 2.1. Then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{\Omega _w}w^s w_\tau ^2 \left( |\nabla \eta |^2-(d-2)\frac{\eta ^2}{\tau ^2}\right) \,dx\ge 0,\end{aligned}$$\end{document}

for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta \in C^\infty _c(\mathbb {R}^d)$$\end{document} .

The proof of Proposition 4.1 is carried out by working directly with w and proving (4.2), from which (4.1) immediately follows. The key point is to use \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:=\frac{1}{|\nabla w|}w_\tau \eta $$\end{document} as a test function in the stability condition of Theorem 1.3, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} is a cut-off function 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}$$\mathbb {R}^d$$\end{document} . We stress, in relation to (1.5), that such a test function corresponds to testing the second variation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {E}_s$$\end{document} with the vector field \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi =\frac{\nabla w}{|\nabla w|^2}w_\tau \eta . $$\end{document}

We proceed directly with the proof of Proposition 4.1 (see also [33] for a similar result in the context of positive exponents).

Proof of Proposition 4.1

Since w is a regular solution in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^d\setminus \{0\}$$\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}$$C^{1,\alpha }$$\end{document} -regular free boundary in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^d\setminus \{0\}$$\end{document} , then, by Proposition 2.2,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w\in C^{\infty }(\overline{\Omega }_w\cap (\mathbb {R}^d\setminus \{0\})) \text { and the free boundary } \partial \Omega _w \text { is smooth in } \mathbb {R}^d\setminus \{0\}. $$\end{document}

The proof is divided in three steps.

Step 1. We start by proving that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{\Omega _w}w^s|\nabla \varphi |^2\,dx-\int _{\Omega _w}\bigg [{{\,\textrm{div}\,}}\left( w^s\frac{\nabla (|\nabla w|^2)}{2|\nabla w|^2}\varphi ^2\right) +w^s\frac{\Delta w}{w}\varphi ^2\bigg ]\,dx\ge 0, \end{aligned}$$\end{document}

for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi \in C^\infty _c(\mathbb {R}^d\setminus \{0\})$$\end{document} , computing the following second variation

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \delta ^2 \mathcal {E}_s(w,\mathbb {R}^d)\left[ \frac{\nabla w}{|\nabla w|^2}\varphi \right] \ge 0. $$\end{document}

Precisely, by substituting \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:=\frac{\varphi }{|\nabla w|}$$\end{document} in the stability condition for w in Theorem 1.3, we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\int _{\Omega _w}w^s|\nabla w|^2\left| \nabla \left( \frac{\varphi }{|\nabla w|}\right) \right| ^2\,dx\\&\quad -\int _{\Omega _w}w^s|\nabla w|^2\left( \frac{|\nabla ^2w|^2}{|\nabla w|^2}-\frac{|\nabla ^2w\nabla w|^2}{|\nabla w|^4}\right) \frac{\varphi ^2}{|\nabla w|^2}\,dx\ge 0. \end{aligned}$$\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}$$\begin{aligned}\begin{aligned} |\nabla w|^2\left| \nabla \left( \frac{\varphi }{|\nabla w|}\right) \right| ^2&=|\nabla w|^2\left| \frac{\nabla \varphi }{|\nabla w|}-\frac{\varphi \nabla |\nabla w|}{|\nabla w|^2}\right| ^2 \\ &=|\nabla w|^2\left( \frac{|\nabla \varphi |^2}{|\nabla w|^2}+\varphi ^2\frac{|\nabla |\nabla w||^2}{|\nabla w|^4}-\frac{\nabla (|\nabla w|)\cdot \nabla \varphi ^2}{|\nabla w|^3}\right) \\ &=|\nabla \varphi |^2+\varphi ^2\frac{|\nabla ^2w\nabla w|^2}{|\nabla w|^4}-\frac{\nabla |\nabla w|}{|\nabla w|}\cdot \nabla \varphi ^2, \end{aligned}\end{aligned}$$\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}$$\begin{aligned}\begin{aligned} \int _{\Omega _w}w^s|\nabla \varphi |^2\,dx- \int _{\Omega _w}\left[ w^s\frac{\nabla |\nabla w|}{|\nabla w|}\cdot \nabla \varphi ^2+w^s\frac{|\nabla ^2w|^2}{|\nabla w|^2}\varphi ^2-2w^s\frac{|\nabla ^2w\nabla w|^2}{|\nabla w|^4}\varphi ^2 \right] \,dx\ge 0.\end{aligned}\end{aligned}$$\end{document}

By applying the Leibnitz rule to the second term above, we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\begin{aligned}&\int _{\Omega _w}w^s|\nabla \varphi |^2\,dx-\int _{\Omega _w}\bigg [{{\,\textrm{div}\,}}\left( w^s \frac{\nabla |\nabla w|}{|\nabla w|}\varphi ^2\right) -{{\,\textrm{div}\,}}\left( w^s\frac{\nabla |\nabla w|}{|\nabla w|}\right) \varphi ^2+w^s\frac{|\nabla ^2w|^2}{|\nabla w|^2}\varphi ^2\\ &\quad -2w^s\frac{|\nabla ^2w\nabla w|^2}{|\nabla w|^4}\varphi ^2 \bigg ]\,dx\ge 0. \end{aligned}\end{aligned}$$\end{document}

Expanding the third term above using that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\,\textrm{div}\,}}\left( w^s\frac{\nabla |\nabla w|}{|\nabla w|}\right) ={{\,\textrm{div}\,}}\left( \frac{\nabla (|\nabla w|^2)}{2|\nabla w|^2}\right) , \end{aligned}$$\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}$$\begin{aligned}\begin{aligned}&\int _{\Omega _w}w^s|\nabla \varphi |^2\,dx-\int _{\Omega _w}\bigg [{{\,\textrm{div}\,}}\left( w^s\frac{\nabla |\nabla w|}{|\nabla w|}\varphi ^2\right) -\frac{\nabla w^s\cdot \nabla (|\nabla w|^2)}{2|\nabla w|^2}\varphi ^2-w^s\frac{\Delta (|\nabla w|^2)}{2|\nabla w|^2}\varphi ^2\\ &\quad +w^s\frac{|\nabla (|\nabla w|^2)|^2}{2|\nabla w|^4}\varphi ^2+w^s\frac{|\nabla ^2w|^2}{|\nabla w|^2}\varphi ^2-2w^s\frac{|\nabla ^2w\nabla w|^2}{|\nabla w|^4}\varphi ^2\bigg ]\,dx\ge 0.\end{aligned}\end{aligned}$$\end{document}

Using the Bochner’s identity and the PDE satisfied by w (see (2.1)), we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{1}{2}\Delta (|\nabla w|^2)&=|\nabla ^2w|^2+\nabla (\Delta w)\cdot \nabla w\\&=|\nabla ^2w|^2- \frac{\Delta w}{w}|\nabla w|^2-\frac{s}{2}\frac{\nabla (|\nabla w|^2)\cdot \nabla w }{w}. \end{aligned}$$\end{document}

Moreover, since

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{|\nabla (|\nabla w|^2)|^2}{2|\nabla w|^4} =2\frac{|\nabla ^2w\nabla w|^2}{|\nabla w|^4},$$\end{document}

then we obtain (4.3), concluding Step 1.

Step 2. In the second step, we prove (4.2) for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta \in C^\infty _c(\mathbb {R}^d)$$\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}$$\text {supp}(\eta )\subset \{\tau >0\}$$\end{document} . We choose \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi :=c\eta $$\end{document} in the above inequality (4.3), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c:=w_\tau $$\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}$$\begin{aligned}\begin{aligned}&\int _{\Omega _w}w^s|\nabla (c\eta )|^2\,dx=\int _{\Omega _w}\bigg [w^s|\nabla \eta |^2c^2+w^s|\nabla c|^2\eta ^2+w^sc\nabla c\cdot \nabla \eta ^2\bigg ]\,dx\\&\quad =\int _{\Omega _w}\bigg [w^s|\nabla \eta |^2c^2+w^s|\nabla c|^2\eta ^2+{{\,\textrm{div}\,}}\big (w^s(c\nabla c) \eta ^2\big )-{{\,\textrm{div}\,}}(w^sc\nabla c)\eta ^2\bigg ]\,dx\\&\quad =\int _{\Omega _w}\bigg [w^s|\nabla \eta |^2c^2+{{\,\textrm{div}\,}}\big (w^s(c\nabla c) \eta ^2\big )-w^sc\Delta c\eta ^2\\&\qquad -sw^{s-1}c\nabla w\cdot \nabla c\eta ^2\bigg ]\,dx, \end{aligned}\end{aligned}$$\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}$$\begin{aligned}\begin{aligned}\begin{aligned} 0&\le \int _{\Omega _w}\bigg [w^s|\nabla (c\eta )|^2\,dx-{{\,\textrm{div}\,}}\left( w^s\frac{\nabla (|\nabla w|^2)}{2|\nabla w|^2}c^2\eta ^2\right) -w^s\frac{\Delta w}{w}c^2\eta ^2\bigg ]\,dx\\ &=\int _{\Omega _w}\bigg [w^s|\nabla \eta |^2c^2-w^s\bigg (\Delta c+s\frac{\nabla w\cdot \nabla c}{w}+\frac{\Delta w}{w}c\bigg )c\eta ^2 \\ &\quad +{{\,\textrm{div}\,}}\left( w^s(c\nabla c) \eta ^2-w^s\frac{\nabla (|\nabla w|^2)}{2|\nabla w|^2}c^2\eta ^2\right) \bigg ]\,dx. \end{aligned}\end{aligned}\end{aligned}$$\end{document}

By direct differentiation (see for instance [33, Equation 6.4]) 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}$$\begin{aligned} \Delta c+s\frac{\nabla w\cdot \nabla c}{w}+\frac{\Delta w}{w}c=(d-2)\frac{c}{\tau ^2}\quad \text {in } \Omega _w\cap \{\tau >0\}, \end{aligned}$$\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}$$\begin{aligned}\begin{aligned}\begin{aligned} 0&\le \int _{\Omega _w}\bigg [w^s|\nabla \eta |^2c^2-(d-2)w^s\frac{c^2}{\tau ^2}\eta ^2+{{\,\textrm{div}\,}}\left( w^s(c\nabla c) \eta ^2-w^s\frac{\nabla (|\nabla w|^2)}{2|\nabla w|^2}c^2\eta ^2\right) \bigg ]\,dx. \end{aligned} \end{aligned}\end{aligned}$$\end{document}

The step is concluded once we prove that the third term above is zero. Hence, set F to be the vector field defined as

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} F:=(c\nabla c) \eta ^2-\frac{\nabla (|\nabla w|^2)}{2|\nabla w|^2}c^2\eta ^2, \end{aligned}$$\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}$$\begin{aligned} \int _{\Omega _w}{{\,\textrm{div}\,}}(w^sF)\,dx=\lim _{\varepsilon \rightarrow 0^+}\int _{\{w>\varepsilon \}}{{\,\textrm{div}\,}}(w^sF)\,dx=\lim _{\varepsilon \rightarrow 0}\int _{\partial \{w>\varepsilon \}}w^s (F\cdot \nu )\,d\mathcal {H}^{d-1}, \end{aligned}$$\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}$$\nu =-\frac{\nabla w}{|\nabla w|}$$\end{document} . We observe that the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F\cdot \nu $$\end{document} is smooth in a neighborhood of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega _w \cap \text {supp}(\eta )$$\end{document} . Moreover, if H 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}$$\partial \Omega _w$$\end{document} pointing towards the complement of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _w$$\end{document} , by following the same argument in [24, Equation 4.4], we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} c_\nu =Hc\quad \text {on }\partial \Omega _w,\qquad \text {and}\qquad H=\frac{\nabla (|\nabla w|^2)}{2|\nabla w|^2}\cdot \nu \quad \text {on }\partial \Omega _w. \end{aligned}$$\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}$$F\cdot \nu =0$$\end{document} on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega _w$$\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}$$\varepsilon >0$$\end{document} small enough, so that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |(F\cdot \nu )| \le C\varepsilon \quad \text {in a neighborhood of } \partial \Omega _w\cap \text {supp}(\eta ), \end{aligned}$$\end{document}

for some constant \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} depending on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} but 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, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1+s>0$$\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}$$\begin{aligned} \lim _{\varepsilon \rightarrow 0^+}\left| \int _{\partial \{w>\varepsilon \}}w^s (F\cdot \nu )\,d\mathcal {H}^{d-1}\right| \le C\lim _{\varepsilon \rightarrow 0^+}\varepsilon ^{1+s}=0, \end{aligned}$$\end{document}

as we claimed.

Step 3. The proof is concluded once we show that (4.2) holds for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta \in C^\infty _c(\mathbb {R}^d)$$\end{document} . This extension follows by standard arguments (see e.g., [24, 33]). Precisely, given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta \in C^\infty _c(\mathbb {R}^d)$$\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}$$\eta _\varepsilon :=\eta \zeta _\varepsilon $$\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}$$\zeta _\varepsilon $$\end{document} is first chosen as an axially symmetric cut-off function, and subsequently replaced by a radial one. Precisely, if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\zeta \in C^\infty _c(\mathbb {R})$$\end{document} is the one dimensional cut-off function such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\zeta \equiv 0$$\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}$$(-\infty ,1/2)$$\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}$$\zeta \equiv 1$$\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}$$(1,+\infty )$$\end{document} , then we set first \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\zeta _\varepsilon =\zeta (\tau /\varepsilon )$$\end{document} and subsequently \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\zeta _\varepsilon =\zeta (|x|/\varepsilon )$$\end{document} . Since by regularity of w and the axially symmetric property, 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}$$w^sw_\tau ^2\rightarrow 0$$\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}$$\tau \rightarrow 0^+$$\end{document} , then sending \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} (here we are using the assumption that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge 3$$\end{document} ), we conclude the stability inequality (4.2) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\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}

Now we are ready to prove Theorem 1.4.

Proof of Theorem 1.4

The case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=2$$\end{document} follows by standard arguments (see e.g., [23]). Indeed, we can directly use the logarithmic cut-off function

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x):={\left\{ \begin{array}{ll} 1& \quad \text {if }|x|\le 1, \\ \frac{\log R-\log |x|}{\log R}& \quad \text {if }1\le |x|\le R, \\ 0& \quad \text {if }|x|\ge R, \\ \end{array}\right. }$$\end{document}

in the stability condition (1.3) of Theorem 1.3. The conclusion follwos by applying the Sternberg–Zumbrun formula (1.6) and sending \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R\rightarrow +\infty $$\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}$$d\ge 3$$\end{document} , which allows us to apply Proposition 4.1. We proceed by proving that w is one-dimensional, which implies the same result on u. Observe that, by a standard approximation argument, the inequalities in (4.1) holds even for test functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} which are just Lipschitz and not \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} . Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta >0$$\end{document} a small constant to be chosen later. For every \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}$$R\ge 1$$\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}$$\eta _{\varepsilon ,R}$$\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}$$\eta _{\varepsilon ,R}:={\left\{ \begin{array}{ll} \tau ^{-\theta }\zeta _R& \quad \text {if }\tau >\varepsilon , \\ \varepsilon ^{-\theta }\zeta _R& \quad \text {if }\tau \le \varepsilon , \\ \end{array}\right. }$$\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}$$\zeta _R\in C^\infty (\mathbb {R}^d)$$\end{document} is a cut-off function such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\zeta _R\ge 0$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\zeta _R\equiv 1$$\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}$$B_R$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\zeta _R\equiv 0$$\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}$$B_{2R}$$\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}$$\begin{aligned} |\nabla \zeta _R|\le \frac{C}{R}.\end{aligned}$$\end{document}

Since w is 1-homogeneous, we notice

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} w^sw_\tau ^2\le CR^s\quad \text {in }B_{2R}. \end{aligned}$$\end{document}

We compute first the two terms of (4.2) in Proposition 4.1. First, we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\begin{aligned}\int _{\mathbb {R}^d}w^s\frac{w_\tau ^2}{\tau ^2}\eta _{\varepsilon ,R}\,dx=\int _{B_{2R}\cap \{\tau >\varepsilon \}}w^sw_\tau ^2\tau ^{-2\theta -2}\zeta _R^2\,dx+\int _{B_{2R}\cap \{\tau \le \varepsilon \}}w^sw_\tau ^2 \tau ^{-2}\varepsilon ^{-2\theta }\zeta _R^2\,dx.\end{aligned}\end{aligned}$$\end{document}

Secondly, since

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\nabla \eta _{\varepsilon ,R}|^2\le {\left\{ \begin{array}{ll} \theta ^2\tau ^{-2\theta -2}\zeta _R& \quad \text {in } B_R\cap \{\tau>\varepsilon \}, \\ \theta ^2\tau ^{-2\theta -2}\zeta _R+\tau ^{-2\theta }|\nabla \zeta _R|^2& \quad \text {in } \left( B_{2R}{\setminus } B_R\right) \cap \{\tau >\varepsilon \}, \\ \varepsilon ^{-2\theta }|\nabla \zeta _R|^2& \quad \text {in } B_{2R}\cap \{\tau \le \varepsilon \}, \\ \end{array}\right. }$$\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}$$\begin{aligned}\begin{aligned} \int _{\mathbb {R}^d}w^s w_\tau ^2|\nabla \eta _{\varepsilon ,R}|^2\,dx&\le \theta ^2\int _{B_{2R}\cap \{\tau>\varepsilon \}}w^sw_\tau ^2\tau ^{-2\theta -2}\zeta _R^2\,dx\\ &\quad +\int _{\left( B_{2R}{\setminus } B_R\right) \cap \{\tau >\varepsilon \}}w^sw_\tau ^2\tau ^{-2\theta }|\nabla \zeta _R|^2\,dx\\ &\quad +\int _{B_{2R}\cap \{\tau \le \varepsilon \}}w^sw_\tau ^2\varepsilon ^{-2\theta }|\nabla \zeta _R|^2\,dx.\end{aligned}\end{aligned}$$\end{document}

Therefore, by Proposition 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}$$\begin{aligned} \begin{aligned} (d-2)\int _{B_{2R}\cap \{\tau>\varepsilon \}}w^sw_\tau ^2\tau ^{-2\theta -2}\zeta _R^2\,dx&\le (d-2)\int _{B_{2R}\cap \{\tau>\varepsilon \}}w^sw_\tau ^2\tau ^{-2\theta -2}\zeta _R^2\,dx\\ &\quad +(d-2)\int _{B_{2R}\cap \{\tau \le \varepsilon \}}w^sw_\tau ^2 \tau ^{-2}\varepsilon ^{-2\theta }\zeta _R^2\,dx \\ &\le \theta ^2\int _{B_{2R}\cap \{\tau>\varepsilon \}}w^sw_\tau ^2\tau ^{-2\theta -2}\zeta _R^2\,dx \\ &\quad \int _{\left( B_{2R}{\setminus } B_R\right) \cap \{\tau >\varepsilon \}}w^sw_\tau ^2\tau ^{-2\theta }|\nabla \zeta _R|^2\,dx\\ &\quad \qquad +\int _{B_{2R}\cap \{\tau \le \varepsilon \}}w^sw_\tau ^2\varepsilon ^{-2\theta }|\nabla \zeta _R|^2\,dx, \end{aligned}\end{aligned}$$\end{document}

leading to

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\begin{aligned}\begin{aligned} (d-2-\theta ^2)\int _{B_{2R}\cap \{\tau>\varepsilon \}}w^sw_\tau ^2\tau ^{-2\theta -2}\zeta _R^2\,dx&\le \int _{\left( B_{2R}{\setminus } B_R\right) \cap \{\tau >\varepsilon \}}w^sw_\tau ^2\tau ^{-2\theta }|\nabla \zeta _R|^2\,dx\\ &\quad +\int _{B_{2R}\cap \{\tau \le \varepsilon \}}w^sw_\tau ^2\varepsilon ^{-2\theta }|\nabla \zeta _R|^2\,dx.\end{aligned} \end{aligned}\end{aligned}$$\end{document}

An estimate of the right-hand side is obtained by considering the two integrals separately.

• (i)For the first integral, we change the coordinate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\mapsto (\tau ,t)$$\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}$$\tau >\varepsilon $$\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}$$t\in (-2R,2R)$$\end{document} , so that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$dx=\tau ^{d-2}\,d\tau \,dt$$\end{document} . Then, using (4.4) and (4.5), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\begin{aligned}\begin{aligned}\int _{\left( B_{2R}{\setminus } B_R\right) \cap \{\tau>\varepsilon \}}w^sw_\tau ^2\tau ^{-2\theta }|\nabla \zeta _R|^2\,dx&\le CR^{s-2}\int _{\left( B_{2R}{\setminus } B_R\right) \cap \{\tau >\varepsilon \}}\tau ^{-2\theta }\,dx\\ &\le CR^{s-1}\int _{\varepsilon }^{2R}\tau ^{d-2-2\theta }\,d\tau \\ &=CR^{s-1}\left( R^{d-1-2\theta }-\varepsilon ^{d-1-2\theta }\right) . \end{aligned}\end{aligned}\end{aligned}$$\end{document}

• (ii)For the second integral, by (4.4) and (4.5), we have

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}\begin{aligned}\int _{B_{2R}\cap \{\tau \le \varepsilon \}}w^sw_\tau ^2\varepsilon ^{-2\theta }|\nabla \zeta _R|^2\,dx\le CR^{s-2}\varepsilon ^{-2\theta }\int _{B_{2R}\cap \{\tau \le \varepsilon \}}\,dx= CR^{s-1}\varepsilon ^{d-1-2\theta }.\end{aligned}\end{aligned}$$\end{document}

Combining (4.6) with the last two estimates above, 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}$$\begin{aligned} (d-2-\theta ^2)\int _{B_{2R}\cap \{\tau >\varepsilon \}}w^sw_\tau ^2\tau ^{-2\theta -2}\zeta _R^2\,dx\le CR^{d+s-2-2\theta }+CR^{s-1}\varepsilon ^{d-1-2\theta }.\end{aligned}$$\end{document}

Finally, the conclusion follows by choosing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \in \mathbb {R}$$\end{document} so that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} d-2-\theta ^2>0\quad \text {and}\quad d+s-2-2\theta <0.\end{aligned}$$\end{document}

Indeed, noting that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d-1-2\theta >\theta ^2+1-2\theta \ge 0$$\end{document} , we may first let \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} and then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R\rightarrow +\infty $$\end{document} in (4.7), which, in view of (4.8), yields \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\tau \equiv 0$$\end{document} . That is w is one-dimensional. Such a choice of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \in \mathbb {R}$$\end{document} is possible provided that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sqrt{d-2}>\frac{d+s-2}{2}. \end{aligned}$$\end{document}

Equivalently, this condition holds if

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 2+(1-\sqrt{1-s})^2<d<2+(1+\sqrt{1-s})^2. \end{aligned}$$\end{document}

The desired conclusion follows by rewriting the above inequality in terms of \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}$$\square $$\end{document}

Remark 4.2

(On the homogeneity assumption) In the proof of Theorem 1.4, the homogeneity assumption is in fact unnecessary. It was only used to derive the estimate (4.5). However, this bound actually follows from the regularity of the solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u\in C^{0,\beta }(\mathbb {R}^d)$$\end{document} , together with the identity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_\tau ^2=\beta ^{-s}w^sw_\tau ^2$$\end{document} .

Asymptotic of stable cones as \documentclass[12pt]{minimal}

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			\begin{document}$$\gamma \rightarrow -2$$\end{document}γ→-2

In this section, we begin by providing a variational characterization of the stability condition for homogeneous global solutions, formulated as an eigenvalue problem for a weighted Laplace-Beltrami operator. Ultimately, we provide some insight into the limit as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \rightarrow -2$$\end{document} , by showing the the convergence of this stability criterion to that associated with stable minimal cones.

A stability criterion for cones

In the context of homogeneous solutions with isolated singularities, a stability condition can be reformulated as a lower bound for an eigenvalue problem associated with a 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}$$\mathbb {S}^{d-1}$$\end{document} . As already observed in [31] for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = 0$$\end{document} (i.e., the Alt–Caffarelli problem), and in [6, 44] for the limiting case \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \rightarrow -2$$\end{document} (i.e., the minimal surfaces), this criterion serves as a key tool in ruling out the presence of singularities for homogeneous free boundaries (see also [39] for the analogue in the capillary context).

The following result is the analogue of these criteria for the Alt–Phillips problem, where the eigenvalue problem involves a degenerate 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}$$\mathbb {S}^{d-1}$$\end{document} . For clarity, it is stated in terms of the auxiliary function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w:= \beta u^{1/\beta }$$\end{document} , although it can be equivalently formulated in terms of u. In the following, 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}$$\nabla _S$$\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}$$ \textrm{div}_S$$\end{document} respectively the tangential gradient and the tangential divergence on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}^{d-1}$$\end{document} .

Proposition 5.1

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (-2,0)$$\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}$$w \in C^{0,1}(\mathbb {R}^d)$$\end{document} be a 1-homogeneous global regular solution outside the origin, in the sense of Definition 2.1. Given the regular section \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma _w:=\Omega _w \cap \mathbb {S}^{d-1}$$\end{document} , consider the degenerate eigenvalue problem on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}^{d-1}$$\end{document}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lambda _s(\Sigma _w):= \min _{\begin{array}{c} \varphi \in C^\infty (\mathbb {S}^{d-1})\\ \varphi \not \equiv 0 \end{array}}\frac{\displaystyle \int _{\Sigma _w} w^s|\nabla w|^2\big (|\nabla _S \varphi |^2 - \mathcal {A}_w^2 \varphi ^2\big )\,d\mathcal {H}^{d-1}}{\displaystyle \int _{\Sigma _w} w^s|\nabla w|^2 \varphi ^2 \,d\mathcal {H}^{d-1}}\,, \end{aligned}$$\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}$$\mathcal {A}^2_w$$\end{document} as in (1.4). Then, u is stable 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} 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}$$\begin{aligned} \lambda _s(\Sigma _w)\ge -\left( \frac{d+s-2}{2}\right) ^2 \end{aligned}$$\end{document}

Proof

Throughout the proof, we often identity a point \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} with its spherical coordinates \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(r,\theta )$$\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}$$r>0$$\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}$$\theta \in \mathbb {S}^{d-1}$$\end{document} . Notice also that, since w is a 1-homogeneous function, 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}$$w^s$$\end{document} is s-homogeneous, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\nabla w|$$\end{document} is 0-homogeneous and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}_w^2$$\end{document} is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-2)$$\end{document} -homogeneous.

Thus, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(r,\theta ):=g(r)\varphi (\theta )$$\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}$$\varphi \in C^\infty (\mathbb {S}^{d-1})$$\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}$$g \in C^\infty _c((0,+\infty ))$$\end{document} . Using the coarea formula and the homogeneity of w, we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{\Omega _w} w^s |\nabla w|^2 \left( |\nabla f|^2 -\mathcal {A}^2_w f^2\right) \,dx&=\int _0^{\infty } r^{s+d-3}g^2(r)\,dr \int _{\Sigma _w} w^s|\nabla w|^2\\&\quad \left( |\nabla _S \varphi |^2 - \mathcal {A}_w^2\varphi ^2\right) \,d\mathcal {H}^{d-1}\\&\quad + \int _0^{\infty } r^{s+d-1}(g'(r))^2\,dr \int _{\Sigma _w} w^s|\nabla w|^2 \varphi ^2 \,d\mathcal {H}^{d-1}. \end{aligned}$$\end{document}

On the other hand, since the Hardy’s inequality implies that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \inf \left\{ \frac{\int _0^\infty r^{s+d-1}(g'(r))^2\,dr}{\int _0^\infty r^{s+d-3}g(r)^2\,dr} : g \in C^\infty _c((0,+\infty ))\right\} = \left( \frac{d+s-2}{2}\right) ^2, $$\end{document}

we deduce that, for 1-homogeneous solutions, the stability condition (1.4) holds 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}$$\begin{aligned} \int _{\Sigma _w} w^s|\nabla w|^2\left( |\nabla _S \varphi |^2 - \mathcal {A}_w^2\varphi ^2\right) \,d\mathcal {H}^{d-1} + \left( \frac{d+s-2}{2}\right) ^2 \int _{\Sigma _w} w^s|\nabla w|^2\varphi ^2 \,d\mathcal {H}^{d-1}\ge 0 \end{aligned}$$\end{document}

for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi \in C^\infty (\mathbb {S}^{d-1})$$\end{document} . Finally, by 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}$$\lambda _s(\Sigma _w)$$\end{document} , it is immediate to see that (5.3) is equivalent to the claimed lower bound (5.2). \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}

In light of the stability condition for positive exponents [33, Theorem 4.1], the previous result can be extended to the whole range \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (-2,2)$$\end{document} . Naturally, in the case of non-negative exponents, the weighted eigenvalue \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _s(\Sigma _w)$$\end{document} admits alternative formulations depending on the notion of stability exploited in the proof (see the discussion in Remark 3.8).

For the case of minimal surfaces, the stability criterion for cones can be stated as follows (see [44]). Let C be a minimal cone 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} , smooth outside the origin, so that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M:= C\cap \mathbb {S}^{d-1}$$\end{document} is smooth in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}^{d-1}$$\end{document} . Consider the eigenvalue problem

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Lambda (M):= \min _{\begin{array}{c} \phi \in C^\infty (\mathbb {S}^{d-1})\\ \phi \not \equiv 0 \end{array}}\frac{\displaystyle \int _{M} \left( |\nabla _M \phi |^2 -|A_M|^2 \phi ^2\right) \,d\mathcal {H}^{d-2}}{\displaystyle \int _{M} \phi ^2 \,d\mathcal {H}^{d-2}}\,, \end{aligned}$$\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}$$|A_M|^2$$\end{document} is the squared norm of the second fundamental form of M and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla _M$$\end{document} is the tangential gradient on M. Then, C is stable 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} 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}$$\begin{aligned} \Lambda (M)\ge -\left( \frac{d-3}{2}\right) ^2. \end{aligned}$$\end{document}

The eigenvalue \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda (M)$$\end{document} is the first eigenvalue of Jacobi operator associated to M. Indeed, if we set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_M:=\Delta _M + |A_M|^2$$\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}$$\Delta _M$$\end{document} is the Laplace-Beltrami operator on M, 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}$$\phi :\mathbb {S}^{d-1}\rightarrow \mathbb {R}$$\end{document} satisfying

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ -L_M \phi = \Lambda (M) \phi \quad \text{ on } M,\qquad \int _M \phi ^2 \,d\mathcal {H}^{d-2}=1. $$\end{document}

Exploring the limit as \documentclass[12pt]{minimal}

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			\begin{document}$$\gamma \rightarrow -2$$\end{document}γ→-2

In order to provide some insight into the limit of the stability criterion as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \rightarrow -2$$\end{document} , we begin by sketching the phenomena that arise in this singular regime. The key point in this discussion is the behavior of the weighted measure associated with (5.1) as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \rightarrow -1$$\end{document} (see (5.7) below).

Given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_k \rightarrow -1$$\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}$$w_k$$\end{document} be a sequence of 1-homogeneous global stable solutions, regular outside the origin, in the sense of Definition 3.2 and Definition 2.1, such that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} w_k|_{\mathbb {S}^{d-1}} \in C^{2,\alpha }(\overline{\Sigma }_{w_k}) \text { and } \Sigma _{w_k}:=\Omega _{w_k}\cap \mathbb {S}^{d-1} \text { is } C^{2,\alpha }\text { -regular, uniformly in } k. \end{aligned}$$\end{document}

The results in Sect. 2 do not provide estimates that are uniform in k. As far as we know, uniform \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{1,\alpha }$$\end{document} estimates have been established only in [20], for minimizers of a normalized Alt–Phillips functional. The proof of assumption (5.6) is rather involved and would require a more detailed refinement of our regularity theory, which goes beyond the scope of this paper.

Under such compactness, we proceed by studying the asymptotic behavior of both the Euler–Lagrange equation and the Rayleigh-type quotient corresponding to (5.1), 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} . Roughly speaking, the proof consists into showing that

$$[eqn]$$

where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M:=\lim _{k\rightarrow \infty }\partial \Sigma _{w_k}$$\end{document} . The presence of the coefficient \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1+s_k)$$\end{document} is coherent with the normalization introduced in [19, Theorem 2.4], where they shown that the perimeter functional coincides with \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}$$\mathcal {E}_s$$\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}$$s\rightarrow -1$$\end{document} . Precisely, in the context of minimizers, they show that $[eqn]$ 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} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_\infty :=\lim _{k\rightarrow +\infty }w_k$$\end{document} . Then (5.7) follows by noticing

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ |\nabla w_k^{1+s_k}| = (1+s_k)w_k^{s_k} |\nabla w_k|,\qquad \text{ and }\qquad |\nabla w_k|=1\quad \text {on }\partial \Sigma _{w_k}. $$\end{document}

We also notice that (5.6) allows to deduce that M is minimal, i.e.,  \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{M}=0$$\end{document} . Indeed, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_k$$\end{document} be 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}$$\partial \Sigma _{w_k}$$\end{document} oriented towards the complement \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _{w_k}$$\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}$$\nu _k$$\end{document} be the outer normal to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Sigma _{w_k}$$\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}$$ w_k(x-t\nu _k(x)) = t - \frac{H_k}{2(1+s_k)}t^2 + O(t^{2+\alpha }),\quad \text{ for } x\in \partial \Sigma _{w_k}. $$\end{document}

In light of the higher regularity result of Sect. 2, the computation of the second normal derivative is essentially equivalent to one in [33, Lemma 5.1]. Hence, by substituting such expansion in the free boundary condition (2.1) (alternatively (1.2)), we deduce that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{t \rightarrow 0^+} t^{1+s_k}H_k = 0$$\end{document} for every \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} . Therefore, the claim follows by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{2,\alpha }$$\end{document} -compactness.

In the present section we proceed by underlying the effect of such phenomenon by showing the convergence of the stability criterion Proposition 5.1, under the uniform regularity assumptions.

Step 1: the asymptotic of the first eigenfunction. We begin by introducing some notations. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w:=w_k$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s:=s_k$$\end{document} and set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Sigma :=\Omega _w \cap \mathbb {S}^{d-1},\qquad M_t:=\partial \{w>t\}\cap \mathbb {S}^{d-1}. $$\end{document}

Notice that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_0=\partial \Sigma $$\end{document} . Since w has an isolated singularity at the origin, both \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}$$M_t$$\end{document} are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{2,\alpha }$$\end{document} -regular in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}^{d-1}$$\end{document} , uniformly in k. Now, let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi :=\varphi _{k}:\mathbb {S}^{d-1} \rightarrow \mathbb {R}$$\end{document} be an eigenfunction corresponding to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _s:=\lambda _{s_k}(\Sigma )$$\end{document} . By rewriting the Euler–Lagrange equations associated to (5.1), we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} -\left( \textrm{div}_S(w^s|\nabla w|^2 \nabla _S \varphi ) +\mathcal {A}_w^2 w^s|\nabla w|^2 \varphi \right) = \lambda _s w^s|\nabla w|^2 \varphi & \quad \text{ in } \Sigma , \\ w^s |\nabla w|^2 \nabla _S \varphi \cdot \nabla _S w = 0 & \quad \text{ on } \partial \Sigma . \end{array}\right. }\end{aligned}$$\end{document}

We stress that the boundary condition is understood in a limiting sense, as we approach \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Sigma $$\end{document} . Moreover, we assume that the following \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} -type normalization holds true

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (1+s)\int _{\mathbb {S}^{d-1}}w^s|\nabla w|^2 \varphi ^2\,d\mathcal {H}^{d-1}=1, \end{aligned}$$\end{document}

for every \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} . In light of the uniform \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{2,\alpha }$$\end{document} -regularity of w 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} , by [47, Theorem 1.1] we deduce that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi \in C^{2,\alpha }(\overline{\Sigma })$$\end{document} , uniformly in k.

Now, for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\le t \le \Vert {w} \Vert _{L^\infty (\mathbb {S}^{d-1})}$$\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}$$H_{M_t}$$\end{document} 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}$$M_t\subset \mathbb {S}^{d-1}$$\end{document} , pointing towards the complement of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{w>t\}$$\end{document} , and with

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \nu := -\frac{\nabla _S w}{|\nabla _S w|} $$\end{document}

the outer normal vector to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_t$$\end{document} , tangent to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}^{d-1}$$\end{document} . Thus, the decomposition

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Delta _S \varphi = \Delta _{M_t} \varphi - H_{M_t} \varphi _{\nu } + \varphi _{\nu \nu }\quad \text{ on } M_t, $$\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}$$\begin{aligned}&\textrm{div}_S(w^s|\nabla w|^2 \nabla _S \varphi ) + \mathcal {A}_w^2 w^s|\nabla w|^2 \varphi = w^s|\nabla w|^2 (\Delta _{M_t} \varphi + \mathcal {A}_w^2 \varphi - H_{M_t}\varphi _{\nu })\\&\quad + w^s|\nabla w|^2\varphi _{\nu \nu } + \nabla _S (w^s|\nabla w|^2)\cdot \nabla _S \varphi \quad \text{ on } M_t. \end{aligned}$$\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}$$\nu $$\end{document} is orthogonal to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_t$$\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}$$\begin{aligned} \partial _{\nu } (w^s|\nabla w|^2 \varphi _{\nu })&= w^s |\nabla w|^2 \varphi _{{\nu } {\nu }} + \partial _{\nu }(w^s |\nabla w|^2)\varphi _{\nu }\\&= w^s |\nabla w|^2 \varphi _{{\nu } {\nu }} + \nabla _S(w^s |\nabla w|^2)\cdot \nabla _S \varphi -\nabla _{M_t}(w^s |\nabla w|^2)\cdot \nabla _{M_t}\varphi \\&= w^s |\nabla w|^2 \varphi _{{\nu } {\nu }} + \nabla _S(w^s |\nabla w|^2)\cdot \nabla _S \varphi - w^s\nabla _{M_t}(|\nabla w|^2)\cdot \nabla _{M_t}\varphi , \end{aligned}$$\end{document}

where in the last equality we use that w is constant on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_t$$\end{document} . Finally, 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}$$\begin{aligned}&\textrm{div}_S(w^s|\nabla w|^2 \nabla _S \varphi ) +\mathcal {A}_w^2 w^s|\nabla w|^2 \varphi \\&\quad = w^s|\nabla w|^2 \left( \Delta _{M_t} \varphi + \mathcal {A}_w^2\varphi - H_{M_t}\varphi _{\nu } + \frac{\nabla _{M_t}(|\nabla w|^2)}{|\nabla w|^2}\cdot \nabla _{M_t}\varphi \right) + \partial _{\nu } (w^s |\nabla w|^2\varphi _{\nu }) \end{aligned}$$\end{document}

on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_t$$\end{document} . Hence, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} is a an eigenfunction associated to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _s$$\end{document} , by (5.8) we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ - \partial _{\nu } (w^s |\nabla w|^2\varphi _{\nu }) = w^s |\nabla w|^2 F_t\qquad \text{ on } M_t, $$\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}$$ F_t:= \Delta _{M_t} \varphi + \mathcal {A}^2_w \varphi + \lambda _s \varphi - H_{M_t}\varphi _{\nu } +\frac{\nabla _{M_t}(|\nabla w|^2)}{|\nabla w|^2}\cdot \nabla _{M_t}\varphi . $$\end{document}

By combining (1.6) with the boundary condition from (5.8) 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}$$|\nabla w| = 1$$\end{document} on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Sigma $$\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}$$ F_0 \equiv L_{\partial \Sigma } \varphi +\lambda _s \varphi \quad \text{ on } \partial \Sigma , $$\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}$$L_{\partial \Sigma }$$\end{document} is the Jacobi operator associated to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Sigma $$\end{document} defined above.

Let us denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_\delta (\partial \Sigma )$$\end{document} a \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} -neighborhood of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Sigma $$\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}$$\Sigma $$\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}$$\partial \Sigma $$\end{document} is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{2,\alpha }$$\end{document} -regular in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}^{d-1}$$\end{document} , uniformly in k, there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho _0>0$$\end{document} for which in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{\rho _0}(\partial \Sigma )$$\end{document} we can exploit a Fermi-type coordinate system. Indeed, we set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x(\theta ,\rho ):=\chi _{z(\theta )}(\rho )$$\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}$$z: U \rightarrow \mathbb {S}^{d-1}$$\end{document} is a local parametrization of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Sigma $$\end{document} and, fixed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z \in \partial \Sigma $$\end{document} , the map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho \mapsto \chi _{z}(\rho )$$\end{document} is a solution of

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\left\{ \begin{array}{ll} \chi _z'(\rho ) = -\nu (\chi _z(\rho ))& \quad \text{ for } \rho \in (0,\rho _0) \\ \chi _z(0)=z. \end{array}\right. } $$\end{document}

For the sake of readability, we omit the dependence of the new coordinates from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \in U$$\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}$$w \in C^{2,\alpha }(\overline{\Sigma })$$\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}$$|\nabla w|=1$$\end{document} on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Sigma $$\end{document} , then in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{\rho _0}(\partial \Sigma )$$\end{document} we can rewrite the condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w(x)=t$$\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}$$ w(\chi _z(\tau ))=t \quad \iff \quad w(z -\tau \nu + O(\tau )) = t \quad \iff \quad t=\tau (1+O(\tau )). $$\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}$$\begin{aligned} \begin{aligned} (w^s|\nabla w|^2\nabla _S\varphi \cdot \nu )(\chi _{z}(\rho ))&= \int _0^\rho (\nabla _S(w^s|\nabla w|^2\nabla _S\varphi \cdot \nu )(\chi _z(\tau ))\cdot \chi '_z(\tau )\,d\tau \\&= \int _0^\rho (\nabla _S(w^s|\nabla w|^2\nabla _S\varphi \cdot \nu )(\chi _z(\tau ))\cdot (-\nu (\chi _z(\tau )))\,d\tau \\&= \int _0^\rho (w^s|\nabla w|^2 F_{w(\chi _z(\tau ))})(\chi _z(\tau ))\,d\tau \\ &= \int _0^\rho (w^s|\nabla w|^2 F_{\tau (1+O(\tau ))})(\chi _z(\tau ))\,d\tau . \end{aligned} \end{aligned}$$\end{document}

In the following computations we restrict to the values of t for which \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_t\subset N_{\rho _0}(\partial \Sigma )$$\end{document} . Again, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w \in C^{2,\alpha }(\overline{\Sigma })$$\end{document} , uniformly in k, there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C>0$$\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}$$\begin{aligned} |(w^s|\nabla w|^2)(\chi _z(\tau )) - \tau ^s| \le C\tau ^{s+1},\quad \text{ in } N_{\rho _0}(\partial \Sigma ). \end{aligned}$$\end{document}

Fixed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z\in \partial \Sigma $$\end{document} , if we set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(\tau ):=F_{\tau (1+O(\tau ))}(\chi _z(\tau ))$$\end{document} , we can exploit the one-dimensional nature of the construction, and get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\bigg |(1+s)\int _0^\rho (w^s|\nabla w|^2)(\chi _z(\tau )) f(\tau )\,d\tau - f(0)\bigg | \\&\quad \le (1+s)\int _0^\rho \tau ^s |f(\tau )-f(0)|\,d\tau + C(1+s) \int _0^\rho \tau ^{s+1}|f(\tau )|\,d\tau ,\\&\quad \le (1+s) C \left( [f]_{C^{0,\alpha }}+\Vert {f} \Vert _{L^\infty } \right) \rho ^{s+1+\alpha }, \end{aligned} \end{aligned}$$\end{document}

where the right-hand side converges to zero 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} , being \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f \in C^{0,\alpha }([0,\rho _0])$$\end{document} , uniformly in k. Notice also that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(0)=F_0(z)$$\end{document} , for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z \in \partial \Sigma $$\end{document} . Therefore, by exploiting the asymptotic of w and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\nabla w|$$\end{document} close to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Sigma $$\end{document} , and combining (5.10), (5.11), (5.12), we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ |(1+s)\rho ^s(\nabla _S\varphi \cdot \nu )(\chi _z(\rho )) - F_0(z)| \le (1+s)C \rho ^{s+1}, $$\end{document}

for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho \in (0,\rho _0)$$\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}$$z \in \partial \Sigma $$\end{document} . Similarly, by applying the coarea formula, the normalization (5.9) implies

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \int _{\partial \Sigma }\varphi ^2\,d\mathcal {H}^{d-2}-1 \right| \le (1+s)C\rho ^{s+1}. \end{aligned}$$\end{document}

By rephrasing the last estimates with the original functions, we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\left| L_{\partial \Sigma _{w_k}}\varphi _k + \lambda _{s_k}(\Sigma _{w_k})\varphi _k\right| \le (1+s_k)C\rho _0^{s+1} \,\,\,\text{ on } \partial \Sigma _{w_k},\\&\left| \int _{\partial \Sigma _{w_k}}\varphi _k^2\,d\mathcal {H}^{d-2}-1 \right| \le (1+s_k)C\rho _0^{s+1}. \end{aligned}$$\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}$$w_k,\varphi _k \in C^{2,\alpha }(\overline{\Sigma }_{w_k})$$\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}$$\partial \Sigma _{w_k}$$\end{document} is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{2,\alpha }$$\end{document} -regular, uniformly in k, we can proceed by passing to the limit 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} . If we set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ w_\infty :=\lim _{k\rightarrow \infty }w_{k},\quad \varphi _\infty :=\lim _{k\rightarrow \infty }\varphi _{k},\quad M:=\lim _{k\rightarrow \infty }\partial \Sigma _{w_{k}},\quad \lambda _\infty :=\lim _{k\rightarrow +\infty }\lambda _{s_k}(\Sigma _{w_k}) $$\end{document}

we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ -L_{M}\varphi _\infty =\lambda _{\infty }\varphi _\infty \quad \text{ on } M,\qquad \int _{M} \varphi _\infty \,d\mathcal {H}^{d-2}=1,\qquad \lambda _\infty \ge -\left( \frac{d-3}{2}\right) ^2. $$\end{document}

Step 2: the asymptotic of the Rayleigh quotient. We want to conclude by showing that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _\infty =\Lambda (M)$$\end{document} , so that M is a stable minimal cone. A priori, the claim would follow by showing that the Rayleigh-type quotient in (5.1) converges to the one in (5.4) 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} . Nevertheless, as pointed out in Remark 5.2 below, such convergence does not hold in general. Our approach is therefore to prove this convergence for a particular family of test functions.

Thus, let us omit the dependence on the index k and consider the same notations introduced above. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi \in C^\infty (\mathbb {S}^{d-1})$$\end{document} satisfying the normalization (5.9). We set

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \widetilde{\varphi }(\chi _{z(\theta )}(\tau ))=\varphi (z(\theta )) \end{aligned}$$\end{document}

so that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{\varphi }\in C^\infty (\mathbb {S}^{d-1})$$\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}$$ \widetilde{\varphi }=\varphi \quad \text {on }\partial \Sigma ,\qquad \nabla _S\widetilde{\varphi }\cdot \nu =0\quad \text {on }\partial \Sigma . $$\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}$$\widetilde{\varphi }$$\end{document} is regular, we also have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w^s|\nabla w|^2 \nabla _S\widetilde{\varphi }\cdot \nu =0$$\end{document} on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Sigma $$\end{document} . In particular, since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla _S = \nabla _{\partial \Sigma } + (\nabla _S \cdot \nu )\nu $$\end{document} , it implies that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla _S\widetilde{\varphi }=\nabla _{\partial \Sigma }\varphi $$\end{document} . If we define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g(x):=|\nabla _S \widetilde{\varphi }|^2 - \mathcal {A}^2_w \widetilde{\varphi }^2$$\end{document} , then, by (1.6) and the definition on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{\varphi }$$\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}$$\begin{aligned} g\equiv |\nabla _{\partial \Sigma } \varphi |^2 - |A_{\partial \Sigma }|^2 \varphi ^2 \quad \text{ on } \partial \Sigma . \end{aligned}$$\end{document}

It is not restrictive to assume that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{supp}\varphi \subset N_{\rho _0}(\partial \Sigma )$$\end{document} . In this neighborhood, the coordinate system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x(\theta ,\tau ):=\chi _{z(\theta )}(\tau )$$\end{document} induce the following expansion of the volume element

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ d\mathcal {H}^{n-1}(x)= (1+O(\tau ))d\mathcal {H}^{d-2}(\theta )d\tau , $$\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}$$d\mathcal {H}^{d-2}(\theta )$$\end{document} is the \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 surface element on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Sigma $$\end{document} (see [7, Remark 3]). Then,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \int _{\Sigma } w^s|\nabla w|^2\,g\,d\mathcal {H}^{d-1} = \int _{\partial \Sigma }\left( \int _0^{\rho _0}(w^s|\nabla w|^2\,g)(\chi _{z(\theta )}(\tau ))(1+O(\tau ))\,d\tau \right) \,d\mathcal {H}^{d-2}(\theta ). $$\end{document}

By combining that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau \mapsto g(\chi _{z(\theta )}(\tau )) \in C^{0,\alpha }([0,\rho ])$$\end{document} and the asymptotic in (5.12), we get

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \lim _{k\rightarrow +\infty }(1+s_k)\int _{\Sigma _{w_k}} w_k^{s_k}|\nabla w_k|^2\,g\,d\mathcal {H}^{d-1}&=\int _{M}g(\chi _{z(\theta )}(0))\,d\mathcal {H}^{d-2} \\ &=\int _{M}(|\nabla _{M}\varphi |^2-|A_{M}|^2 \varphi ^2)\,d\mathcal {H}^{d-2}, \end{aligned} \end{aligned}$$\end{document}

where the last identity follows by (5.15). For what it concerns the asymptotic of the denominator in the Rayleigh quotient, by combining

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| \int _{\partial \Sigma _{w_k}}\widetilde{\varphi }^2\,d\mathcal {H}^{d-2}-(1+s_k)\int _{\Sigma _{w_k}}w_k^{s_k}|\nabla w_k|^2 \widetilde{\varphi }^2\,d\mathcal {H}^{d-1} \right| \le (1+s_k)C\rho _0^{s_k+1}$$\end{document}

with (5.13), we finally get that

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \lambda _\infty =\lim _{k\rightarrow +\infty }\lambda _{s_k}(\Sigma _{w_k})\le \frac{\displaystyle \int _{M} \left( |\nabla _M \varphi |^2 -|A_M|^2 \varphi ^2\right) \,d\mathcal {H}^{d-2}}{\displaystyle \int _{M} \varphi ^2 \,d\mathcal {H}^{d-2}},\quad \text{ for } \text{ every } \varphi \in C^\infty (\mathbb {S}^{d-1}), $$\end{document}

which implies that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _\infty =\Lambda (M)$$\end{document} .

Remark 5.2

(The quadratic form associated to the stability) In the previous steps, we focused on the asymptotic behavior of homogeneous stable solutions of the Alt–Phillips problem as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \rightarrow -2$$\end{document} , motivated by their relevance in the classification of singular points. However, under the regularity assumption (5.6), this asymptotic analysis can be readily extended to stable solutions regular outside the origin, in the sense of Definition 3.1 and Definition 2.1, by showing that the limit of the associated free boundaries is indeed a stable minimal surface.

Our analysis highlights a deep difference between the asymptotic of stable regular solutions and the one of the quadratic form associated to the stability condition, i.e.,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ Q_s(f):=\int _{ \Omega _w} w^s|\nabla w|^2\left( |\nabla f|^2 - \mathcal {A}^2_w\, f^2\right) \,dx. $$\end{document}

An additional term appears in 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}$$Q_s$$\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}$$s \rightarrow -1$$\end{document} , corresponding to the derivative of the test functions in the direction orthogonal to the limiting interface \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M:= \lim _k \partial \Omega _{w_k}$$\end{document} . Nevertheless, the stability of this limit interface can be obtained by choosing test functions of type (5.14). This behavior is not unexpected, as it mirrors phenomena observed for the singular limit of stable solutions to the Allen–Cahn equation (see [29, 36]).