{
  "id": "2101.02886",
  "version": "v1",
  "published": "2021-01-08T07:34:59.000Z",
  "updated": "2021-01-08T07:34:59.000Z",
  "title": "A shape optimization problem on planar sets with prescribed topology",
  "authors": [
    "L. Briani",
    "G. Buttazzo",
    "F. Prinari"
  ],
  "categories": [
    "math.OC"
  ],
  "abstract": "We consider shape optimization problems involving functionals depending on perimeter, torsional rigidity and Lebesgue measure. The scaling free cost functionals are of the form $P(\\Omega)T^q(\\Omega)|\\Omega|^{-2q-1/2}$ and the class of admissible domains consists of two-dimensional open sets $\\Omega$ satisfying the topological constraints of having a prescribed number $k$ of bounded connected components of the complementary set. A relaxed procedure is needed to have a well-posed problem and we show that when $q<1/2$ an optimal relaxed domain exists. When $q>1/2$ the problem is ill-posed and for $q=1/2$ the explicit value of the infimum is provided in the cases $k=0$ and $k=1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-08T07:34:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49Q10",
      "49J45",
      "49R05",
      "35P15",
      "35J25"
    ],
    "keywords": [
      "shape optimization problem",
      "planar sets",
      "prescribed topology",
      "scaling free cost functionals",
      "two-dimensional open sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}