{
  "id": "1606.02195",
  "version": "v1",
  "published": "2016-06-07T16:09:10.000Z",
  "updated": "2016-06-07T16:09:10.000Z",
  "title": "Some isoperimetric inequalities on $\\mathbb{R} ^N$ with respect to weights $|x|^α$",
  "authors": [
    "A. Alvino",
    "F. Brock",
    "F. Chiacchio",
    "A. Mercaldo",
    "M. R. Posteraro"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "We solve a class of isoperimetric problems on $\\mathbb{R}^N $ with respect to weights that are powers of the distance to the origin. For instance we show that if $k\\in [0,1]$, then among all smooth sets $\\Omega$ in $\\mathbb{R} ^N$ with fixed Lebesgue measure, $\\int_{\\partial \\Omega } |x|^k \\, \\mathscr{H}_{N-1} (dx)$ achieves its minimum for a ball centered at the origin. Our results also imply a weighted Polya-Sz\\\"ego principle. In turn, we establish radiality of optimizers in some Caffarelli-Kohn-Nirenberg inequalities, and we obtain sharp bounds for eigenvalues of some nonlinear problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-07T16:09:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "51M16",
      "46E35",
      "46E30",
      "35P15"
    ],
    "keywords": [
      "isoperimetric inequalities",
      "sharp bounds",
      "isoperimetric problems",
      "caffarelli-kohn-nirenberg inequalities",
      "nonlinear problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}