{
  "id": "1002.2770",
  "version": "v1",
  "published": "2010-02-14T12:04:17.000Z",
  "updated": "2010-02-14T12:04:17.000Z",
  "title": "Optimal Shape for Elliptic Problems with Random Perturbations",
  "authors": [
    "Giuseppe Buttazzo",
    "Faustino Maestre"
  ],
  "comment": "17 pages, 6 figures",
  "categories": [
    "math.OC",
    "math.AP"
  ],
  "abstract": "In this paper we analyze the relaxed form of a shape optimization problem with state equation $\\{{array}{ll} -div \\big(a(x)Du\\big)=f\\qquad\\hbox{in}D \\hbox{boundary conditions on}\\partial D. {array}.$ The new fact is that the term $f$ is only known up to a random perturbation $\\xi(x,\\omega)$. The goal is to find an optimal coefficient $a(x)$, fulfilling the usual constraints $\\alpha\\le a\\le\\beta$ and $\\displaystyle\\int_D a(x) dx\\le m$, which minimizes a cost function of the form $$\\int_\\Omega\\int_Dj\\big(x,\\omega,u_a(x,\\omega)\\big) dx dP(\\omega).$$ Some numerical examples are shown in the last section, to stress the difference with respect to the case with no perturbation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-02-14T12:04:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49Q10",
      "49J45",
      "46E35",
      "47A10",
      "74P05"
    ],
    "keywords": [
      "random perturbation",
      "elliptic problems",
      "optimal shape",
      "shape optimization problem",
      "state equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1002.2770B"
    }
  }
}