{
  "id": "1605.06294",
  "version": "v1",
  "published": "2016-05-20T11:29:18.000Z",
  "updated": "2016-05-20T11:29:18.000Z",
  "title": "Regularity of Minimizers of Shape Optimization Problems involving Perimeter",
  "authors": [
    "Guido De Philippis",
    "Jimmy Lamboley",
    "Michel Pierre",
    "Bozhidar Velichkov"
  ],
  "categories": [
    "math.OC"
  ],
  "abstract": "We prove existence and regularity of optimal shapes for the problem$$\\min\\Big\\{P(\\Omega)+\\mathcal{G}(\\Omega):\\ \\Omega\\subset D,\\ |\\Omega|=m\\Big\\},$$where $P$ denotes the perimeter, $|\\cdot|$ is the volume, and the functional $\\mathcal{G}$ is either one of the following:\\textless{}ul\\textgreater{}\\textless{}li\\textgreater{} the Dirichlet energy $E\\_f$, with respect to a (possibly sign-changing) function $f\\in L^p$;\\textless{}/li\\textgreater{}\\textless{}li\\textgreater{}a spectral functional of the form $F(\\lambda\\_{1},\\dots,\\lambda\\_{k})$, where $\\lambda\\_k$ is the $k$th eigenvalue of the Dirichlet Laplacian and $F:\\mathbb{R}^k\\to\\mathbb{R}$ is Lipschitz continuous and increasing in each variable.\\textless{}/li\\textgreater{}\\textless{}/ul\\textgreater{}The domain $D$ is the whole space $\\mathbb{R}^d$ or a bounded domain. We also give general assumptions on the functional $\\mathcal{G}$ so that the result remains valid.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-20T11:29:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "shape optimization problems",
      "regularity",
      "minimizers",
      "result remains valid",
      "dirichlet energy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}