{
  "id": "2010.04105",
  "version": "v1",
  "published": "2020-10-08T16:41:12.000Z",
  "updated": "2020-10-08T16:41:12.000Z",
  "title": "Estimation of the continuity constants for Bogovskiĭ and regularized Poincaré integral operators",
  "authors": [
    "Johnny Guzman",
    "Abner J. Salgado"
  ],
  "categories": [
    "math.AP",
    "cs.NA",
    "math.CA",
    "math.NA"
  ],
  "abstract": "We study the dependence of the continuity constants for the regularized Poincar\\'e and Bogovski\\u{\\i} integral operators acting on differential forms defined on a domain $\\Omega$ of $\\mathbb{R}^n$. We, in particular, study the dependence of such constants on certain geometric characteristics of the domain when these operators are considered as mappings from (a subset of) $L^2(\\Omega,\\Lambda^\\ell)$ to $H^1(\\Omega,\\Lambda^{\\ell-1})$, $\\ell \\in \\{1, \\ldots, n\\}$. For domains $\\Omega$ that are star shaped with respect to a ball $B$ we study the dependence of the constants on the ratio $diam(\\Omega)/diam(B)$. A program on how to develop estimates for higher order Sobolev norms is presented. The results are extended to certain classes of unions of star shaped domains.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-08T16:41:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20",
      "42B37",
      "35S05",
      "35C15",
      "53A45",
      "58J10",
      "65M60"
    ],
    "keywords": [
      "integral operators",
      "continuity constants",
      "estimation",
      "higher order sobolev norms",
      "dependence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}