{
  "id": "1911.06686",
  "version": "v1",
  "published": "2019-11-15T15:17:48.000Z",
  "updated": "2019-11-15T15:17:48.000Z",
  "title": "Asymptotic behavior of $u$-capacities and singular perturbations for the Dirichlet-Laplacian",
  "authors": [
    "Laura Abatangelo",
    "Virginie Bonnaillie-Noël",
    "Corentin Léna",
    "Paolo Musolino"
  ],
  "comment": "46 pages, 13 figures",
  "categories": [
    "math.AP",
    "math.FA"
  ],
  "abstract": "In this paper we study the asymptotic behavior of $u$-capacities of small sets and its application to the analysis of the eigenvalues of the Dirichlet-Laplacian on a bounded planar domain with a small hole. More precisely, we consider two (sufficiently regular) bounded open connected sets $\\Omega$ and $\\omega$ of $\\mathbb{R}^2$, containing the origin. First, if $\\varepsilon$ is positive and small enough and if $u$ is a function defined on $\\Omega$, we compute an asymptotic expansion of the $u$-capacity $\\mathrm{Cap}_\\Omega(\\varepsilon \\omega, u)$ as $\\varepsilon \\to 0$. As a byproduct, we compute an asymptotic expansion for the $N$-th eigenvalues of the Dirichlet-Laplacian in the perforated set $\\Omega \\setminus (\\varepsilon \\overline{\\omega})$ for $\\varepsilon$ close to $0$. Such formula shows explicitly the dependence of the asymptotic expansion on the behavior of the corresponding eigenfunction near $0$ and on the shape $\\omega$ of the hole.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-15T15:17:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P20",
      "31C15",
      "31B10",
      "35B25",
      "35C20"
    ],
    "keywords": [
      "asymptotic behavior",
      "singular perturbations",
      "dirichlet-laplacian",
      "asymptotic expansion",
      "small hole"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}