{
  "id": "1205.5052",
  "version": "v3",
  "published": "2012-05-22T20:36:39.000Z",
  "updated": "2013-09-23T13:24:19.000Z",
  "title": "Analysis of a free boundary at contact points with Lipschitz data",
  "authors": [
    "Aram Karakhanyan",
    "Henrik Shahgholian"
  ],
  "comment": "Accepted: Transactions of AMS, 2013",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we consider a minimization problem for the functional $$ J(u)=\\int_{B_1^+}|\\nabla u|\\sp 2+\\lambda_{+}^2\\chi_{\\{u>0\\}}+\\lambda_{-}^2\\chi_{\\{u\\leq0\\}}, $$ in the upper half ball $B_1^+\\subset\\R^n, n\\geq 2$ subject to a Lipschitz continuous Dirichlet data on $\\partial B_1^+$. More precisely we assume that $0\\in \\partial \\{u>0\\}$ and the derivative of the boundary data has a jump discontinuity. If $0\\in \\bar{\\partial(\\{u>0\\} \\cap B_1^+)}$ then (for $n=2$ or $n>3$ and one-phase case) we prove, among other things, that the free boundary $\\partial \\{u>0\\}$ approaches the origin along one of the two possible planes given by $$ \\gamma x_1 = \\pm x_2, $$ where $\\gamma$ is an explicit constant given by the boundary data and $\\lambda_\\pm$ the constants seen in the definition of $J(u)$. Moreover the speed of the approach to $\\gamma x_1=x_2$ is uniform.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-09-23T13:24:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R35"
    ],
    "keywords": [
      "free boundary",
      "contact points",
      "lipschitz data",
      "boundary data",
      "lipschitz continuous dirichlet data"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1205.5052K"
    }
  }
}