{
  "id": "1609.01231",
  "version": "v1",
  "published": "2016-09-05T17:35:51.000Z",
  "updated": "2016-09-05T17:35:51.000Z",
  "title": "Regularity of the optimal sets for some spectral functionals",
  "authors": [
    "Dario Mazzoleni",
    "Susanna Terracini",
    "Bozhidar Velichkov"
  ],
  "comment": "31 pages",
  "categories": [
    "math.AP",
    "math.FA",
    "math.OC"
  ],
  "abstract": "In this paper we study the regularity of the optimal sets for the shape optimization problem \\[ \\min\\Big\\{\\lambda_1(\\Omega)+\\dots+\\lambda_k(\\Omega)\\ :\\ \\Omega\\subset\\mathbb{R}^d,\\ \\text{open}\\ ,\\ |\\Omega|=1\\Big\\}, \\] where $\\lambda_1(\\cdot),\\dots,\\lambda_k(\\cdot)$ denote the eigenvalues of the Dirichlet Laplacian and $|\\cdot|$ the $d$-dimensional Lebesgue measure. We prove that the topological boundary of a minimizer $\\Omega_k^*$ is composed of a relatively open regular part which is locally a graph of a $C^{1,\\alpha}$ function and a closed singular part, which is empty if $d<d^*$, contains at most a finite number of isolated points if $d=d^*$ and has Hausdorff dimension smaller than $(d-d^*)$ if $d>d^*$, where the natural number $d^*\\in[5,7]$ is the smallest dimension at which minimizing one-phase free boundaries admit singularities. To achieve our goal, as an auxiliary result, we shall extend for the first time the known regularity theory for the one-phase free boundary problem to the vector-valued case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-05T17:35:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49Q10",
      "35R35",
      "47A75",
      "49R05"
    ],
    "keywords": [
      "optimal sets",
      "spectral functionals",
      "regularity",
      "one-phase free boundaries admit singularities",
      "one-phase free boundary problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}