{
  "id": "1903.00881",
  "version": "v1",
  "published": "2019-03-03T11:16:51.000Z",
  "updated": "2019-03-03T11:16:51.000Z",
  "title": "The Soap Bubble Theorem and a $p$-Laplacian overdetermined problem",
  "authors": [
    "Francesca Colasuonno",
    "Fausto Ferrari"
  ],
  "comment": "18 pages, 0 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the $p$-Laplacian equation $-\\Delta_p u=1$ for $1<p<2$, on a regular bounded domain $\\Omega\\subset\\mathbb R^N$, with $N\\ge2$, under homogeneous Dirichlet boundary conditions. In the spirit of Alexandrov's Soap Bubble Theorem and of Serrin's symmetry result for the overdetermined problems, we prove that if the mean curvature $H$ of $\\partial\\Omega$ is constant, then $\\Omega$ is a ball and the unique solution of the Dirichlet $p$-Laplacian problem is radial. The main tools used are integral identities, the $P$-function, and the maximum principle.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-03T11:16:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J92",
      "35B06",
      "35N25",
      "53A10",
      "35A23"
    ],
    "keywords": [
      "laplacian overdetermined problem",
      "homogeneous dirichlet boundary conditions",
      "alexandrovs soap bubble theorem",
      "serrins symmetry result",
      "laplacian equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}