Boundary H\"{o}lder continuity of stable solutions to semilinear elliptic problems in $C^{1,1}$ domains

TL;DR

This paper proves boundary Hölder continuity of stable solutions to semilinear elliptic equations in $C^{1,1}$ domains for dimensions up to 9, extending previous results even for the Laplacian in less regular domains.

Contribution

It establishes the boundary Hölder regularity of stable solutions in $C^{1,1}$ domains for the first time, improving upon prior results that required smoother domains.

Findings

Hölder continuity holds for dimensions n ≤ 9

Results apply to equations with variable coefficients

Extends known regularity results to less smooth domains

Abstract

This article establishes the boundary H\"{o}lder continuity of stable solutions to semilinear elliptic problems in the optimal range of dimensions $n \leq 9$, for $C^{1,1}$ domains. We consider equations $- L u = f(u)$ in a bounded $C^{1,1}$ domain $\Omega \subset \mathbb{R}^n$, with $u = 0$ on $\partial \Omega$, where $L$ is a linear elliptic operator with variable coefficients and $f \in C^1$ is nonnegative, nondecreasing, and convex. The stability of $u$ amounts to the nonnegativity of the principal eigenvalue of the linearized equation $- L - f'(u)$. Our result is new even for the Laplacian, for which [Cabr\'{e}, Figalli, Ros-Oton, and Serra, Acta Math. 224 (2020)] proved the H\"{o}lder continuity in $C^3$ domains.