{
  "id": "1201.5328",
  "version": "v2",
  "published": "2012-01-25T17:17:39.000Z",
  "updated": "2012-01-30T11:25:50.000Z",
  "title": "An isoperimetric result for the fundamental frequency via domain derivative",
  "authors": [
    "Carlo Nitsch"
  ],
  "comment": "12 pages, minor corrections, the proof of Lemma 2.3 was shortened and clarified",
  "categories": [
    "math.OC"
  ],
  "abstract": "The Faber-Krahn deficit $\\delta\\lambda$ of an open bounded set $\\Omega$ is the normalized gap between the values that the first Dirichlet Laplacian eigenvalue achieves on $\\Omega$ and on the ball having same measure as $\\Omega$. For any given family of open bounded sets of $\\R^N$ ($N\\ge 2$) smoothly converging to a ball, it is well known that both $\\delta\\lambda$ and the isoperimetric deficit $\\delta P$ are vanishing quantities. It is known as well that, at least for convex sets, the ratio $\\frac{\\delta P}{\\delta \\lambda}$ is bounded by below by some positive constant (see \\cite{BNT,PW}), and in this note, using the technique of the shape derivative, we provide the explicit optimal lower bound of such a ratio as $\\delta P$ goes to zero.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-01-30T11:25:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P15",
      "49R05",
      "35J25"
    ],
    "keywords": [
      "fundamental frequency",
      "isoperimetric result",
      "domain derivative",
      "open bounded set",
      "first dirichlet laplacian eigenvalue achieves"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1201.5328N"
    }
  }
}