{
  "id": "2412.20196",
  "version": "v1",
  "published": "2024-12-28T16:09:28.000Z",
  "updated": "2024-12-28T16:09:28.000Z",
  "title": "Optimal domains for the Cheeger inequality",
  "authors": [
    "Giuseppe Buttazzo"
  ],
  "comment": "10 pages, 0 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we prove the existence of an optimal domain $\\Omega_{opt}$ for the shape optimization problem $$\\max\\Big\\{\\lambda_q(\\Omega)\\ :\\ \\Omega\\subset D,\\ \\lambda_p(\\Omega)=1\\Big\\},$$ where $q<p$ and $D$ is a prescribed bounded subset of ${\\bf R}^d$. Here $\\lambda_p(\\Omega)$ (respectively $\\lambda_q(\\Omega)$) is the first eigenvalue of the $p$-Laplacian $-\\Delta_p$ (respectively $-\\Delta_q$) with Dirichlet boundary condition on $\\partial\\Omega$. This is related to the existence of optimal sets that minimize the generalized Cheeger ratio $${\\mathcal F}_{p,q}(\\Omega)=\\frac{\\lambda_p^{1/p}(\\Omega)}{\\lambda_q^{1/q}(\\Omega)}.$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-28T16:09:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49Q10",
      "49J45",
      "49R05",
      "35P15",
      "35J25"
    ],
    "keywords": [
      "optimal domain",
      "cheeger inequality",
      "shape optimization problem",
      "dirichlet boundary condition",
      "first eigenvalue"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}