{
  "id": "2305.07062",
  "version": "v1",
  "published": "2023-05-11T18:00:28.000Z",
  "updated": "2023-05-11T18:00:28.000Z",
  "title": "Boundary Hölder continuity of stable solutions to semilinear elliptic problems in $C^{1,1}$ domains",
  "authors": [
    "Iñigo U. Erneta"
  ],
  "comment": "32 pages",
  "categories": [
    "math.AP"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-11T18:00:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B35",
      "35B45",
      "35B65",
      "35J15",
      "35J61"
    ],
    "keywords": [
      "semilinear elliptic problems",
      "boundary hölder continuity",
      "stable solutions",
      "linear elliptic operator",
      "article establishes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}