{
  "id": "1604.00520",
  "version": "v1",
  "published": "2016-04-02T16:09:02.000Z",
  "updated": "2016-04-02T16:09:02.000Z",
  "title": "A natural approach to the asymptotic mean value property for the $p$-Laplacian",
  "authors": [
    "Michinori Ishiwata",
    "Rolando Magnanini",
    "Hidemitsu Wadade"
  ],
  "comment": "19 pages, submitted",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $1\\le p\\le\\infty$. We show that a function $u\\in C(\\mathbb R^N)$ is a viscosity solution to the normalized $p$-Laplace equation $\\Delta_p^n u(x)=0$ if and only if the asymptotic formula $$ u(x)=\\mu_p(\\ve,u)(x)+o(\\ve^2) $$ holds as $\\ve\\to 0$ in the viscosity sense. Here, $\\mu_p(\\ve,u)(x)$ is the $p$-mean value of $u$ on $B_\\ve(x)$ characterized as a unique minimizer of $$ \\inf_{\\la\\in\\RR}\\nr u-\\la\\nr_{L^p(B_\\ve(x))}. $$ This kind of asymptotic mean value property (AMVP) extends to the case $p=1$ previous (AMVP)'s obtained when $\\mu_p(\\ve,u)(x)$ is replaced by other kinds of mean values. The natural definition of $\\mu_p(\\ve,u)(x)$ makes sure that this is a monotonic and continuous (in the appropriate topology) functional of $u$. These two properties help to establish a fairly general proof of (AMVP), that can also be extended to the (normalized) parabolic $p$-Laplace equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-02T16:09:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J60",
      "35K55",
      "35J92",
      "35K92"
    ],
    "keywords": [
      "asymptotic mean value property",
      "natural approach",
      "laplace equation",
      "viscosity solution",
      "fairly general proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160400520I"
    }
  }
}