{
  "id": "0902.3216",
  "version": "v1",
  "published": "2009-02-18T18:42:32.000Z",
  "updated": "2009-02-18T18:42:32.000Z",
  "title": "A Free boundary problem for the $p(x)$- Laplacian",
  "authors": [
    "Julián Fernández Bonder",
    "Sandra Martínez",
    "Noemi Wolanski"
  ],
  "comment": "35 pages, submitted",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the optimization problem of minimizing $\\int_{\\Omega}|\\nabla u|^{p(x)}+ \\lambda \\chi_{\\{u>0\\}} dx$ in the class of functions $W^{1,p(\\cdot)}(\\Omega)$ with $u-\\phi_0\\in W_0^{1,p(\\cdot)}(\\Omega)$, for a given $\\phi_0\\geq 0$ and bounded. $W^{1,p(\\cdot)}(\\Omega)$ is the class of weakly differentiable functions with $\\int_\\Omega |\\nabla u|^{p(x)} dx<\\infty$. We prove that every solution $u$ is locally Lipschitz continuous, that it is a solution to a free boundary problem and that the free boundary, $\\Omega\\cap\\partial\\{u>0\\}$, is a regular surface.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-02-18T18:42:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R35",
      "35B65"
    ],
    "keywords": [
      "free boundary problem",
      "regular surface",
      "optimization problem",
      "weakly differentiable functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0902.3216F"
    }
  }
}