{
  "id": "2307.07345",
  "version": "v1",
  "published": "2023-07-14T13:49:45.000Z",
  "updated": "2023-07-14T13:49:45.000Z",
  "title": "Energy stability for a class of semilinear elliptic problems",
  "authors": [
    "Danilo Gregorin Afonso",
    "Alessandro Iacopetti",
    "Filomena Pacella"
  ],
  "comment": "33 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we consider semilinear elliptic problems in a bounded domain $\\Omega$ contained in a given unbounded Lipschitz domain $\\mathcal C \\subset \\mathbb R^N$. Our aim is to study how the energy of a solution behaves with respect to volume-preserving variations of the domain $\\Omega$ inside $\\mathcal C$. Once a rigorous variational approach to this question is set, we focus on the cases when $\\mathcal C$ is a cone or a cylinder and we consider spherical sectors and radial solutions or bounded cylinders and special one-dimensional solutions, respectively. In these cases, we show both stability and instability results, which have connections with related overdetermined problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-14T13:49:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J61",
      "35B35",
      "35B38",
      "49Q10"
    ],
    "keywords": [
      "semilinear elliptic problems",
      "energy stability",
      "special one-dimensional solutions",
      "solution behaves",
      "rigorous variational approach"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}