{
  "id": "1207.6344",
  "version": "v1",
  "published": "2012-07-26T17:38:28.000Z",
  "updated": "2012-07-26T17:38:28.000Z",
  "title": "A new symmetry criterion based on the distance function and applications to PDE's",
  "authors": [
    "Graziano Crasta",
    "Ilaria Fragalà"
  ],
  "comment": "17 pages",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "We prove that, if $\\Omega\\subset \\mathbb{R}^n$ is an open bounded starshaped domain of class $C^2$, the constancy over $\\partial \\Omega$ of the function $$\\varphi(y) = \\int_0^{\\lambda(y)} \\prod_{j=1}^{n-1}[1-t \\kappa_j(y)]\\, dt$$ implies that $\\Omega$ is a ball. Here $k_j(y)$ and $\\lambda(y)$ denote respectively the principal curvatures and the cut value of a boundary point $y \\in \\partial \\Omega$. We apply this geometric result to different symmetry questions for PDE's: an overdetermined system of Monge-Kantorovich type equations (which can be viewed as the limit as $p \\to + \\infty$ of Serrin's symmetry problem for the $p$-Laplacian), and equations in divergence form whose solutions depend only on the distance from the boundary in some subset of their domain.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-07-26T17:38:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35N25",
      "49K20",
      "35J70",
      "53A07"
    ],
    "keywords": [
      "distance function",
      "symmetry criterion",
      "applications",
      "monge-kantorovich type equations",
      "serrins symmetry problem"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1016/j.jde.2013.06.003",
      "journal": "Journal of Differential Equations",
      "year": 2013,
      "volume": 255,
      "number": 7,
      "pages": 2082
    },
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013JDE...255.2082C"
    }
  }
}