{
  "id": "2101.02662",
  "version": "v1",
  "published": "2021-01-07T18:06:12.000Z",
  "updated": "2021-01-07T18:06:12.000Z",
  "title": "Variational $p$-harmonious functions: existence and convergence to $p$-harmonic functions",
  "authors": [
    "Evan W. Chandra",
    "Michinori Ishiwata",
    "Rolando Magnanini",
    "Hidemitsu Wadade"
  ],
  "comment": "17 pages, no figures, submitted paper",
  "categories": [
    "math.AP"
  ],
  "abstract": "In a recent paper, the last three authors showed that a game-theoretic $p$-harmonic function $v$ is characterized by an asymptotic mean value property with respect to a kind of mean value $\\nu_p^r[v](x)$ defined variationally on balls $B_r(x)$. In this paper, in a domain $\\Om\\subset\\RR^N$, $N\\ge 2$, we consider the operator $\\mu_p^\\ve$, acting on continuous functions on $\\ol{\\Om}$, defined by the formula $\\mu_p^\\ve[v](x)=\\nu^{r_\\ve(x)}_p[v](x)$, where $r_\\ve(x)=\\min[\\ve,\\dist(x,\\Ga)]$ and $\\Ga$ denotes the boundary of $\\Omega$. We first derive various properties of $\\mu^\\ve_p$ such as continuity and monotonicity. Then, we prove the existence and uniqueness of a function $u^\\ve\\in C(\\ol{\\Om})$ satisfying the Dirichlet-type problem: $$ u(x)=\\mu_p^\\ve[u](x) \\ \\mbox{ for every } \\ x\\in\\Om,\\quad u=g \\ \\mbox{ on } \\ \\Ga, $$ for any given function $g\\in C(\\Ga)$. This result holds, if we assume the existence of a suitable notion of barrier for all points in $\\Ga$. That $u^\\ve$ is what we call the \\textit{variational} $p$-harmonious function with Dirichlet boundary data $g$, and is obtained by means of a Perron-type method based on a comparison principle. \\par We then show that the family $\\{ u^\\ve\\}_{\\ve>0}$ gives an approximation scheme for the viscosity solution $u\\in C(\\ol{\\Om})$ of $$ \\De_p^G u=0 \\ \\mbox{ in }\\Om, \\quad u=g \\ \\mbox{ on } \\ \\Ga, $$ where $\\De_p^G$ is the so-called game-theoretic (or homogeneous) $p$-Laplace operator. In fact, we prove that $u^\\ve$ converges to $u$, uniformly on $\\ol{\\Om}$ as $\\ve\\to 0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-07T18:06:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J60",
      "35J92",
      "35K55",
      "35K92"
    ],
    "keywords": [
      "harmonic function",
      "harmonious function",
      "variational",
      "asymptotic mean value property",
      "convergence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}