{
  "id": "1111.2801",
  "version": "v1",
  "published": "2011-11-11T17:11:21.000Z",
  "updated": "2011-11-11T17:11:21.000Z",
  "title": "Geometric-type Sobolev inequalities and applications to the regularity of minimizers",
  "authors": [
    "Xavier Cabre",
    "Manel Sanchon"
  ],
  "comment": "20 pages; 1 figure",
  "categories": [
    "math.AP"
  ],
  "abstract": "The purpose of this paper is twofold. We first prove a weighted Sobolev inequality and part of a weighted Morrey's inequality, where the weights are a power of the mean curvature of the level sets of the function appearing in the inequalities. Then, as main application of our inequalities, we establish new $L^q$ and $W^{1,q}$ estimates for semi-stable solutions of $-\\Delta u=g(u)$ in a bounded domain $\\Omega$ of $\\mathbb{R}^n$. These estimates lead to an $L^{2n/(n-4)}(\\Omega)$ bound for the extremal solution of $-\\Delta u=\\lambda f(u)$ when $n\\geq 5$ and the domain is convex. We recall that extremal solutions are known to be bounded in convex domains if $n\\leq 4$, and that their boundedness is expected ---but still unkwown--- for $n\\leq 9$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-11-11T17:11:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K57",
      "35B65"
    ],
    "keywords": [
      "geometric-type sobolev inequalities",
      "minimizers",
      "regularity",
      "extremal solution",
      "main application"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.2801C"
    }
  }
}