{
  "id": "0706.4267",
  "version": "v3",
  "published": "2007-06-28T16:32:43.000Z",
  "updated": "2009-07-06T14:02:09.000Z",
  "title": "A mixed problem for the infinity laplacian via Tug-of-War games",
  "authors": [
    "Fernando Charro",
    "Jesus Garcia Azorero",
    "Julio D. Rossi"
  ],
  "comment": "13 pages. Final version",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we prove that a function $ u\\in\\mathcal{C}(\\bar{\\Omega})$ is the continuous value of the Tug-of-War game described in \\cite{PSSW} if and only if it is the unique viscosity solution to the infinity laplacian with mixed boundary conditions {-\\Delta_{\\infty}u(x)=0\\quad & \\text{in} \\Omega, \\frac{\\partial u}{\\partial n}(x)=0\\quad & \\text{on} \\Gamma_N, u(x)=F(x)\\quad & \\text{on} \\Gamma_D. By using the results in \\cite{PSSW}, it follows that this viscous PDE problem has a unique solution, which is the unique {\\it absolutely minimizing Lipschitz extension} to the whole $\\bar{\\Omega}$ (in the sense of \\cite{Aronsson} and \\cite{PSSW}) of the boundary data $ F:\\Gamma_D\\to\\R $.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2009-07-06T14:02:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J60",
      "91A05",
      "49L25",
      "35J25"
    ],
    "keywords": [
      "infinity laplacian",
      "tug-of-war game",
      "mixed problem",
      "unique viscosity solution",
      "mixed boundary conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0706.4267C"
    }
  }
}