Global Schauder theory for minimizers of the $H^s(\Omega)$ energy

TL;DR

This paper establishes sharp boundary regularity results for minimizers of a fractional Sobolev energy functional, revealing optimal Hölder continuity up to the boundary for solutions of the regional fractional Laplacian.

Contribution

It provides the first sharp regularity estimates for solutions of the regional fractional Laplacian, including both Neumann and Dirichlet boundary conditions, and demonstrates their optimality.

Findings

Solutions are in $C^{2s+eta}(ar{ abla})$ for Neumann boundary conditions.

Solutions satisfy $u/ ext{dist}^{2s-1} o C^{1+eta}(ar{ abla})$ for Dirichlet conditions.

Regularity estimates are proven to be optimal and fail beyond certain Hölder exponents.

Abstract

We study the regularity of minimizers of the functional $\mathcal E(u):= [u]_{H^s(\Omega)}^2 +\int_\Omega fu$. This corresponds to understanding solutions for the regional fractional Laplacian in $\Omega\subset\mathbb R^N$. More precisely, we are interested on the global (up to the boundary) regularity of solutions, both in the case of free minimizers in $H^s(\Omega)$ (i.e., Neumann problem), or in the case of Dirichlet condition $u\in H^s_0(\Omega)$ when $s>\frac12$. Our main result establishes the sharp regularity of solutions in both cases: $u\in C^{2s+\alpha}(\overline\Omega)$ in the Neumann case, and $u/\delta^{2s-1}\in C^{1+\alpha}(\overline\Omega)$ in the Dirichlet case. Here, $\delta$ is the distance to $\partial\Omega$, and $\alpha<\alpha_s$, with $\alpha_s\in (0,1-s)$ and $2s+\alpha_s>1$. We also show the optimality of our result: these estimates fail for $\alpha>\alpha_s$, even when $f$ and $\partial\Omega$ are $C^\infty$.