{
  "id": "0807.2196",
  "version": "v1",
  "published": "2008-07-14T16:32:53.000Z",
  "updated": "2008-07-14T16:32:53.000Z",
  "title": "Regularity of the optimal shape for the first eigenvalue of the Laplacian with volume and inclusion constraints",
  "authors": [
    "Tanguy Briançon",
    "Jimmy Lamboley"
  ],
  "categories": [
    "math.OC"
  ],
  "abstract": "We consider the well-known following shape optimization problem: $$\\lambda_1(\\Omega^*)=\\min_{\\stackrel{|\\Omega|=a} {\\Omega\\subset{D}}} \\lambda_1(\\Omega), $$ where $\\lambda_1$ denotes the first eigenvalue of the Laplace operator with homogeneous Dirichlet boundary condition, and $D$ is an open bounded set (a box). It is well-known that the solution of this problem is the ball of volume $a$ if such a ball exists in the box $D$ (Faber-Krahn's theorem). In this paper, we prove regularity properties of the boundary of the optimal shapes $\\Omega^*$ in any case and in any dimension. Full regularity is obtained in dimension 2.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-07-14T16:32:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "first eigenvalue",
      "optimal shape",
      "inclusion constraints",
      "homogeneous dirichlet boundary condition",
      "shape optimization problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009AnIHP..26.1149B"
    }
  }
}