{
  "id": "1310.2511",
  "version": "v1",
  "published": "2013-10-09T14:57:52.000Z",
  "updated": "2013-10-09T14:57:52.000Z",
  "title": "Optimal Regularity for The Signorini Problem and its Free Boundary",
  "authors": [
    "John Andersson"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We will show optimal regularity for minimizers of the Signorini problem for the Lame system. In particular if $\\u=(u^1,u^2,u^3)\\in W^{1,2}(B_1^+:\\R^3)$ minimizes $$ J(\\u)=\\int_{B_1^+}|\\nabla \\u+\\nabla^\\bot \\u|^2+\\lambda\\div(\\u)^2 $$ in the convex set $$ K=\\big\\{\\u=(u^1,u^2,u^3)\\in W^{1,2}(B_1^+:\\R^3);\\; u^3\\ge 0 \\textrm{on}\\Pi, $$ $$ \\u=f\\in C^\\infty(\\partial B_1) \\textrm{on}(\\partial B_1)^+ \\big\\}, $$ where $\\lambda\\ge 0$ say. Then $\\u\\in C^{1,1/2}(B_{1/2}^+)$. Moreover the free boundary, given by $$ \\Gamma_\\u=\\partial \\{x;\\;u^3(x)=0,\\; x_3=0\\}\\cap B_{1}, $$ will be a $C^{1,\\alpha}$ graph close to points where $\\u$ is not degenerate. Similar results have been know before for scalar partial differential equations (see for instance \\cite{AC} and \\cite{ACS}). The novelty of this approach is that it does not rely on maximum principle methods and is therefore applicable to systems of equations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-10-09T14:57:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "optimal regularity",
      "signorini problem",
      "free boundary",
      "scalar partial differential equations",
      "maximum principle methods"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.2511A"
    }
  }
}