{
  "id": "1911.03018",
  "version": "v1",
  "published": "2019-11-08T03:35:41.000Z",
  "updated": "2019-11-08T03:35:41.000Z",
  "title": "On self-adjointness of symmetric diffusion operators",
  "authors": [
    "Derek W Robinson"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $\\Omega$ be a domain in $\\Ri^d$ with boundary $\\Gamma$ and let $d_\\Gamma$ denote the Euclidean distance to $\\Gamma$. Further let $H=-\\divv(C\\nabla)$ where $C=(\\,c_{kl}\\,)>0$ with $c_{kl}=c_{lk}$ are real, bounded, Lipschitz continuous functions and $D(H)=C_c^\\infty(\\Omega)$. Assume also that there is a $\\delta\\geq0$ such that $\\|C/d_\\Gamma^{\\,\\delta}-aI\\|\\to 0$ as $d_\\Gamma\\to0$ with $\\delta\\geq0$ where $a$ is a bounded Lipschitz continuous function with $a\\geq\\mu>0$ on a boundary layer $\\Gamma_{\\!\\!r}=\\{x\\in\\Omega: d_\\Gamma(x)<r\\}$. Finally we require $|(\\divv C).(\\nabla d_\\Gamma)|d_\\Gamma^{\\,-\\delta+1}$ to be bounded on~$\\Gamma_{\\!\\!r}$. Then we prove that if $\\Omega$ is a $C^2$-domain, or if $\\Omega=\\Ri^d\\backslash S$ where $S$ is a countable set of positively separated points, or if $\\Omega=\\Ri^d\\backslash \\overline \\Pi$ with $\\Pi$ a convex set whose boundary has Hausdorff dimension $d_H\\in \\{1,\\ldots, d-1\\}$ then the condition $\\delta>2-(d-d_H)/2$ is sufficient for $H$ to be essentially self-adjoint as an operator on $L_2(\\Omega)$. In particular $\\delta>3/2$ suffices for $C^2$-domains. Finally we prove that $\\delta\\geq 3/2$ is necessary in the $C^2$-case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-08T03:35:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "symmetric diffusion operators",
      "self-adjointness",
      "convex set",
      "boundary layer",
      "bounded lipschitz continuous function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}