Boundary Hölder continuity of stable solutions to semilinear elliptic problems in $C^{1,1}$ domains
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916410082459648 |
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| author | Erneta, Iñigo U. |
| author_facet | Erneta, Iñigo U. |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_07062 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Boundary Hölder continuity of stable solutions to semilinear elliptic problems in $C^{1,1}$ domains Erneta, Iñigo U. Analysis of PDEs 35B35, 35B45, 35B65, 35J15, 35J61 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. |
| title | Boundary Hölder continuity of stable solutions to semilinear elliptic problems in $C^{1,1}$ domains |
| topic | Analysis of PDEs 35B35, 35B45, 35B65, 35J15, 35J61 |
| url | https://arxiv.org/abs/2305.07062 |