{
  "id": "math/0503025",
  "version": "v1",
  "published": "2005-03-01T22:27:00.000Z",
  "updated": "2005-03-01T22:27:00.000Z",
  "title": "The Effect of Curvature on the Best Constatnt in the Hardy-Sobolev Inequalities",
  "authors": [
    "N. Ghoussoub",
    "F. Robert"
  ],
  "comment": "39 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We address the question of attainability of the best constant in the following Hardy-Sobolev inequality on a smooth domain $\\Omega$ of \\mathbb{R}^n: $$ \\mu_s (\\Omega) := \\inf \\{\\int_{\\Omega}| \\nabla u|^2 dx; u \\in {H_{1,0}^2(\\Omega)} \\hbox{and} \\int_{\\Omega} \\frac {|u|^{2^{\\star}}}{|x|^s} dx =1\\}$$ when 0<s<2, 2^*:=2^*(s)=\\frac{2(n-s)}{n-2}, and when 0 is on the boundary $\\partial \\Omega$. This question is closely related to the geometry of $\\partial\\Omega$, as we extend here the main result obtained in [15] by proving that at least in dimension n >= 4, the negativity of the mean curvature of $\\partial \\Omega $ at 0 is sufficient to ensure the attainability of $\\mu_{s}(\\Omega)$. Key ingredients in our proof are the identification of symmetries enjoyed by the extremal functions correrresponding to the best constant in half-space, as well as a fine analysis of the asymptotic behaviour of appropriate minimizing sequences. The result holds true also in dimension 3 but the more involved proof will be dealt with in a forthcoming paper [17].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-03-01T22:27:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hardy-sobolev inequality",
      "best constatnt",
      "best constant",
      "result holds true",
      "mean curvature"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......3025G"
    }
  }
}