{
  "id": "1407.6233",
  "version": "v1",
  "published": "2014-07-23T14:25:45.000Z",
  "updated": "2014-07-23T14:25:45.000Z",
  "title": "A sharp inequality for Sobolev functions",
  "authors": [
    "Pedro M. Girão"
  ],
  "comment": "4 pages",
  "journal": "C. R. Math. Acad. Sci. Paris 334 (2002), 105-108",
  "doi": "10.1016/S1631-073X(02)02215-X",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $N\\geq 5$, $a>0$, $\\Omega$ be a smooth bounded domain in $\\mathbb{R}^{N}$, $2^*=\\frac{2N}{N-2}$, $2^\\#=\\frac{2(N-1)}{N-2}$ and $||u||^2=|\\nabla u|_{2}^2+a|u|_{2}^2$. We prove there exists an $\\alpha_{0}>0$ such that, for all $u\\in H^1(\\Omega)\\setminus\\{0\\}$, $$\\frac{S}{2^{\\frac 2N}}\\leq\\frac{||u||^2}{|u|_{2^*}^2}\\left(1+\\alpha_{0}\\frac{|u|_{2^\\#}^{2^\\#}}{||u||\\cdot|u|_{2^*}^{2^*/2}}\\right).$$ This inequality implies Cherrier's inequality.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-23T14:25:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35",
      "35J65"
    ],
    "keywords": [
      "sobolev functions",
      "sharp inequality",
      "inequality implies cherriers inequality",
      "smooth bounded domain"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.6233G"
    }
  }
}