{
  "id": "2305.07058",
  "version": "v1",
  "published": "2023-05-11T18:00:05.000Z",
  "updated": "2023-05-11T18:00:05.000Z",
  "title": "Energy estimate up to the boundary for stable solutions to semilinear elliptic problems",
  "authors": [
    "Iñigo U. Erneta"
  ],
  "comment": "25 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We obtain a universal energy estimate up to the boundary for stable solutions of semilinear equations with variable coefficients. Namely, we consider solutions to $- L u = f(u)$, where $L$ is a linear uniformly elliptic operator and $f$ is $C^1$, such that the linearized equation $-L - f'(u)$ has nonnegative principal eigenvalue. Our main result is an estimate for the $L^{2+\\gamma}$ norm of the gradient of stable solutions vanishing on the flat part of a half-ball, for any nonnegative and nondecreasing $f$. This bound only requires the elliptic coefficients to be Lipschitz. As a consequence, our estimate continues to hold in general $C^{1,1}$ domains if we further assume the nonlinearity $f$ to be convex. This result is new even for the Laplacian, for which a $C^3$ regularity assumption on the domain was needed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-11T18:00:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B35",
      "35B45",
      "35B65",
      "35J15",
      "35J61"
    ],
    "keywords": [
      "semilinear elliptic problems",
      "stable solutions",
      "universal energy estimate",
      "linear uniformly elliptic operator",
      "semilinear equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}