{
  "id": "2407.01237",
  "version": "v1",
  "published": "2024-07-01T12:27:35.000Z",
  "updated": "2024-07-01T12:27:35.000Z",
  "title": "A note on the maximization of the first Dirichlet eigenvalue for perforated planar domains",
  "authors": [
    "Manuel Dias"
  ],
  "comment": "27 pages, 7 figures. Comments are welcomed",
  "categories": [
    "math.AP",
    "math.OC",
    "math.SP"
  ],
  "abstract": "In this work we prove that given an open bounded set $\\Omega \\subset \\mathbb{R}^2$ with a $C^2$ boundary, there exists $\\epsilon := \\epsilon(\\Omega)$ small enough such that for all $0 < \\delta < \\epsilon$ the maximum of $\\{\\lambda_1(\\Omega - B_{\\delta}(x)):B_{\\delta} \\subset \\Omega\\}$ is never attained when the ball is close enough to the boundary. In particular it is not obtained when $B_\\delta(x)$ is touching the boundary $\\partial \\Omega$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-01T12:27:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J25",
      "49R05",
      "49Q10",
      "35B65",
      "47J10"
    ],
    "keywords": [
      "first dirichlet eigenvalue",
      "perforated planar domains",
      "maximization",
      "open bounded set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}