{
  "id": "1801.04230",
  "version": "v1",
  "published": "2018-01-12T16:59:04.000Z",
  "updated": "2018-01-12T16:59:04.000Z",
  "title": "Asymptotics for the resolvent equation associated to the game-theoretic $p$-laplacian",
  "authors": [
    "Diego Berti",
    "Rolando Magnanini"
  ],
  "comment": "17 pages, 1 figure",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the (viscosity) solution $u^\\varepsilon$ of the elliptic equation $\\varepsilon^2\\Delta_p^G u= u$ in a domain (not necessarily bounded), satisfying $u=1$ on its boundary. Here, $\\Delta_p^G$ is the {\\it game-theoretic or normalized $p$-laplacian}. We derive asymptotic formulas for $\\varepsilon\\to 0^+$ involving the values of $u^\\varepsilon$, in the spirit of Varadhan's work \\cite{Va}, and its $q$-mean on balls touching the boundary, thus generalizing that obtained in \\cite{MS-AM} for $p=q=2$. As in a related parabolic problem, investigated in \\cite{BM}, we link the relevant asymptotic behavior to the geometry of the domain.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-12T16:59:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J92",
      "35J25",
      "35J40",
      "35Q91"
    ],
    "keywords": [
      "resolvent equation",
      "game-theoretic",
      "relevant asymptotic behavior",
      "varadhans work",
      "elliptic equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}