{
  "id": "2006.13403",
  "version": "v1",
  "published": "2020-06-24T00:38:34.000Z",
  "updated": "2020-06-24T00:38:34.000Z",
  "title": "The weighted Hardy inequality and 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$${\\!,}$ $d_\\Gamma$ the Euclidean distance to the boundary and $H=-\\divv(C\\,\\nabla)$ an elliptic operator with $C=(\\,c_{kl}\\,)>0$ where $c_{kl}=c_{lk}$ are real, bounded, Lipschitz functions. We assume that $C\\sim c\\,d_\\Gamma^{\\,\\delta}$ as $d_\\Gamma\\to0$ in the sense of asymptotic analysis where $c$ is a strictly positive, bounded, Lipschitz function and $\\delta\\geq0$. We also assume that there is an $r>0$ and a $ b_{\\delta,r}>0$ such that the weighted Hardy inequality \\[ \\int_{\\Gamma_{\\!\\!r}} d_\\Gamma^{\\,\\delta}\\,|\\nabla \\psi|^2\\geq b_{\\delta,r}^{\\,2}\\int_{\\Gamma_{\\!\\!r}} d_\\Gamma^{\\,\\delta-2}\\,| \\psi|^2 \\] is valid for all $\\psi\\in C_c^\\infty(\\Gamma_{\\!\\!r})$ where $\\Gamma_{\\!\\!r}=\\{x\\in\\Omega: d_\\Gamma(x)<r\\}$. We then prove that the condition $(2-\\delta)/2<b_\\delta$ is sufficient for the essential self-adjointness of $H$ on $C_c^\\infty(\\Omega)$ with $b_\\delta$ the supremum over $r$ of all possible $b_{\\delta,r}$ in the Hardy inequality. This result extends all known results for domains with smooth boundaries and also gives information on self-adjointness for a large family of domains with rough, e.g.\\ fractal, boundaries.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-24T00:38:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "31C25",
      "47D07"
    ],
    "keywords": [
      "symmetric diffusion operators",
      "weighted hardy inequality",
      "self-adjointness",
      "result extends",
      "smooth boundaries"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}