{
  "id": "1602.04618",
  "version": "v1",
  "published": "2016-02-15T10:46:05.000Z",
  "updated": "2016-02-15T10:46:05.000Z",
  "title": "On Polya's inequality for torsional rigidity and first Dirichlet eigenvalue",
  "authors": [
    "M. van den Berg",
    "V. Ferone",
    "C. Nitsch",
    "C. Trombetti"
  ],
  "comment": "18 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $\\Omega$ be an open set in Euclidean space with finite Lebesgue measure $|\\Omega|$. We obtain some properties of the set function $F:\\Omega\\mapsto \\R^+$ defined by $$ F(\\Omega)=\\frac{T(\\Omega)\\lambda_1(\\Omega)}{|\\Omega|} ,$$ where $T(\\Omega)$ and $\\lambda_1(\\Omega)$ are the torsional rigidity and first eigenvalue of the Dirichlet Laplacian respectively. We improve the classical P\\'olya bound $F(\\Omega)\\le 1,$ and show that $$F(\\Omega)\\le 1- \\nu_m T(\\Omega)|\\Omega|^{-1-\\frac2m},$$ where $\\nu_m$ depends on $m$ only. For any $\\epsilon\\in (0,1)$ we construct an open set $\\Omega_{\\epsilon}$ such that $F(\\Omega_{\\epsilon})\\ge 1-\\epsilon$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-15T10:46:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49J45",
      "49R05",
      "35P15",
      "47A75",
      "35J25"
    ],
    "keywords": [
      "first dirichlet eigenvalue",
      "torsional rigidity",
      "polyas inequality",
      "open set",
      "finite lebesgue measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160204618V"
    }
  }
}