{
  "id": "1101.5224",
  "version": "v1",
  "published": "2011-01-27T08:56:06.000Z",
  "updated": "2011-01-27T08:56:06.000Z",
  "title": "An isoperimetric inequality for eigenvalues of the bi-harmonic operator",
  "authors": [
    "Q. Ding",
    "G. Feng",
    "Y. Zhang"
  ],
  "comment": "12 pages",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "} In this article, we put forward a Neumann eigenvalue problem for the bi-harmonic operator $\\Delta^2$ on a bounded smooth domain $\\Om$ in the Euclidean $n$-space ${\\bf R}^n$ ($n\\ge2$) and then prove that the corresponding first non-zero eigenvalue $\\Upsilon_1(\\Om)$ admits the isoperimetric inequality of Szeg\\\"o-Weinberger type: $\\Upsilon_1(\\Om)\\le \\Upsilon_1(B_{\\Om})$, where $B_{\\Om}$ is a ball in ${\\bf R}^n$ with the same volume of $\\Om$. The isoperimetric inequality of Szeg\\\"o-Weinberger type for the first nonzero Neumann eigenvalue of the even-multi-Laplacian operators $\\Delta^{2m}$ ($m\\ge1$) on $\\Om$ is also exploited.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-01-27T08:56:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "isoperimetric inequality",
      "bi-harmonic operator",
      "first nonzero neumann eigenvalue",
      "corresponding first non-zero eigenvalue",
      "neumann eigenvalue problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1101.5224D"
    }
  }
}