{
  "id": "1312.3909",
  "version": "v1",
  "published": "2013-12-13T19:15:01.000Z",
  "updated": "2013-12-13T19:15:01.000Z",
  "title": "Shape Optimization Problems for Metric Graphs",
  "authors": [
    "Giuseppe Buttazzo",
    "Berardo Ruffini",
    "Bozhidar Velichkov"
  ],
  "comment": "23 pages, 11 figures, ESAIM Control Optim. Calc. Var., (to appear)",
  "doi": "10.1051/cocv/2013050",
  "categories": [
    "math.OC",
    "math.CO"
  ],
  "abstract": "We consider the shape optimization problem $$\\min\\big\\{{\\mathcal E}(\\Gamma)\\ :\\ \\Gamma\\in{\\mathcal A},\\ {\\mathcal H}^1(\\Gamma)=l\\ \\big\\},$$ where ${\\mathcal H}^1$ is the one-dimensional Hausdorff measure and ${\\mathcal A}$ is an admissible class of one-dimensional sets connecting some prescribed set of points ${\\mathcal D}=\\{D_1,\\dots,D_k\\}\\subset{\\mathbb R}^d$. The cost functional ${\\mathcal E}(\\Gamma)$ is the Dirichlet energy of $\\Gamma$ defined through the Sobolev functions on $\\Gamma$ vanishing on the points $D_i$. We analyze the existence of a solution in both the families of connected sets and of metric graphs. At the end, several explicit examples are discussed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-12-13T19:15:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49R05",
      "49Q20",
      "49J45",
      "81Q35"
    ],
    "keywords": [
      "shape optimization problem",
      "metric graphs",
      "one-dimensional hausdorff measure",
      "one-dimensional sets",
      "cost functional"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.3909B"
    }
  }
}