{
  "id": "1406.6824",
  "version": "v1",
  "published": "2014-06-26T09:44:17.000Z",
  "updated": "2014-06-26T09:44:17.000Z",
  "title": "Existence of minimizers for eigenvalues of the Dirichlet-Laplacian with a drift",
  "authors": [
    "Barbara Brandolini",
    "Francesco Chiacchio",
    "Antoine Henrot",
    "Cristina Trombetti"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper deals with the eigenvalue problem for the operator $L=-\\Delta -x\\cdot \\nabla $ with Dirichlet boundary conditions. We are interested in proving the existence of a set minimizing any eigenvalue $\\lambda_k$ of $L$ under a suitable measure constraint suggested by the structure of the operator. More precisely we prove that for any $c>0$ and $k\\in \\mathbb{N} $ the following minimization problem $$ \\min\\left\\{\\lambda_k(\\Omega): \\> \\Omega \\>\\mbox{quasi-open} \\>\\mbox{set}, \\> \\int_\\Omega e^{|x|^2/2}dx\\le c\\right\\} $$ has a solution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-26T09:44:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J45",
      "35P05",
      "49G05"
    ],
    "keywords": [
      "dirichlet-laplacian",
      "minimizers",
      "dirichlet boundary conditions",
      "eigenvalue problem",
      "paper deals"
    ],
    "publication": {
      "doi": "10.1016/j.jde.2015.02.028",
      "journal": "Journal of Differential Equations",
      "year": 2015,
      "month": "Jul",
      "volume": 259,
      "number": 2,
      "pages": 708
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015JDE...259..708B"
    }
  }
}