{
  "id": "0909.4696",
  "version": "v2",
  "published": "2009-09-25T16:21:15.000Z",
  "updated": "2009-09-28T13:15:32.000Z",
  "title": "Regularity of minimizers of semilinear elliptic problems up to dimension four",
  "authors": [
    "Xavier Cabre"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the class of semi-stable solutions to semilinear equations $-\\Delta u=f(u)$ in a bounded smooth domain $\\Omega$ of $R^n$ (with $\\Omega$ convex in some results). This class includes all local minimizers, minimal, and extremal solutions. In dimensions $n \\leq 4$, we establish an priori $L^\\infty$ bound which holds for every positive semi-stable solution and every nonlinearity $f$. This estimate leads to the boundedness of all extremal solutions when $n=4$ and $\\Omega$ is convex. This result was previously known only in dimensions $n\\leq 3$ by a result of G. Nedev. In dimensions $5 \\leq n \\leq 9$ the boundedness of all extremal solutions remains an open question. It is only known to hold in the radial case $\\Omega=B_R$ by a result of A. Capella and the author.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-09-28T13:15:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B45",
      "35J61"
    ],
    "keywords": [
      "semilinear elliptic problems",
      "regularity",
      "semi-stable solution",
      "extremal solutions remains",
      "bounded smooth domain"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0909.4696C"
    }
  }
}