{
  "id": "1305.2137",
  "version": "v1",
  "published": "2013-05-09T16:25:19.000Z",
  "updated": "2013-05-09T16:25:19.000Z",
  "title": "On the torsion function with Robin or Dirichlet boundary conditions",
  "authors": [
    "M. van den Berg",
    "D. Bucur"
  ],
  "comment": "19 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "For $p\\in (1,+\\infty)$ and $b \\in (0, +\\infty]$ the $p$-torsion function with Robin boundary conditions associated to an arbitrary open set $\\Om \\subset \\R^m$ satisfies formally the equation $-\\Delta_p =1$ in $\\Om$ and $|\\nabla u|^{p-2} \\frac{\\partial u}{\\partial n} + b|u|^{p-2} u =0$ on $\\partial \\Om$. We obtain bounds of the $L^\\infty$ norm of $u$ {\\it only} in terms of the bottom of the spectrum (of the Robin $p$-Laplacian), $b$ and the dimension of the space in the following two extremal cases: the linear framework (corresponding to $p=2$) and arbitrary $b>0$, and the non-linear framework (corresponding to arbitrary $p>1$) and Dirichlet boundary conditions ($b=+\\infty$). In the general case, $p\\not=2, p \\in (1, +\\infty)$ and $b>0$ our bounds involve also the Lebesgue measure of $\\Om$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-05-09T16:25:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J25",
      "35P99",
      "58J35"
    ],
    "keywords": [
      "dirichlet boundary conditions",
      "torsion function",
      "arbitrary open set",
      "extremal cases",
      "robin boundary conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.2137V"
    }
  }
}