{
  "id": "2004.10134",
  "version": "v1",
  "published": "2020-04-21T16:26:34.000Z",
  "updated": "2020-04-21T16:26:34.000Z",
  "title": "Fractional-Order Operators on Nonsmooth Domains",
  "authors": [
    "Helmut Abels",
    "Gerd Grubb"
  ],
  "comment": "49 pages",
  "categories": [
    "math.AP",
    "math.FA"
  ],
  "abstract": "For the fractional Laplacian $(-\\Delta )^a$, $a\\in(0,1)$, and its nonsmooth pseudodifferential generalizations $P$, even of order $2a$, we establish regularity properties in $L_p$-Sobolev spaces of Bessel-potential type $H^s_p$ for the solutions $u$ of the homogeneous Dirichlet problem on nonsmooth domains $\\Omega \\subset \\mathbb R^n$: $Pu=f$ on $\\Omega $, $\\operatorname{supp} u\\subset\\overline\\Omega$. A main result is that for bounded $C^{1+\\tau }$-domains and operators $P$ with $C^\\tau $-dependence on $x$ (any $\\tau \\in(0,\\infty )$), the solutions with $f$ given in $H^s_p(\\Omega)$ ($s\\in [0,\\tau-2a)$, $p\\in(1,\\infty)$) have $H^{s+2a}_p$-regularity in the interior of the domain and a singular behavior with a factor $d^a$ close to the boundary, where $d(x)\\sim \\operatorname{dist}(x,\\partial\\Omega )$ (more precisely, $u$ belongs to a certain $a$-transmission space $H_p^{a(s+2a)}(\\overline\\Omega)$). This was previously only known for $\\tau =\\infty $. The result has required a new development of methods to handle nonsmooth coordinate changes for pseudodifferential operators, which have not been available before; this constitutes another main contribution of the paper.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-21T16:26:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35S15",
      "35R11",
      "35S05",
      "47G30",
      "60G52"
    ],
    "keywords": [
      "nonsmooth domains",
      "fractional-order operators",
      "handle nonsmooth coordinate changes",
      "nonsmooth pseudodifferential generalizations",
      "main contribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}