Boundary Hölder continuity of stable solutions to semilinear elliptic problems in $C^{1,1}$ domains

Abstract

This article establishes the boundary Hölder continuity of stable solutions to semilinear elliptic problems in the optimal range of dimensions $n\le 9$, for $C^{1,1}$ domains. We consider equations $-Lu=f(u)$ 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 $-L-f^{\prime }(u)$. Our result is new even for the Laplacian, for which [Cabré, Figalli, Ros-Oton, and Serra, Acta Math. 224 (2020)] proved the Hölder continuity in $C^3$ domains.

Improved the presentation and removed Appendix C (now obsolete). To appear in J. Reine Angew. Math