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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2305.07062 |
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Table of Contents:
- This article establishes the boundary Hölder 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 $Ω\subset \mathbb{R}^n$, with $u = 0$ on $\partial Ω$, 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 [Cabré, Figalli, Ros-Oton, and Serra, Acta Math. 224 (2020)] proved the Hölder continuity in $C^3$ domains.