{
  "id": "2010.00441",
  "version": "v1",
  "published": "2020-10-01T14:33:23.000Z",
  "updated": "2020-10-01T14:33:23.000Z",
  "title": "Regularity of the optimal sets for the second Dirichlet eigenvalue",
  "authors": [
    "Dario Mazzoleni",
    "Baptiste Trey",
    "Bozhidar Velichkov"
  ],
  "comment": "25 pages",
  "categories": [
    "math.AP",
    "math.OC",
    "math.SP"
  ],
  "abstract": "This paper is dedicated to the regularity of the optimal sets for the second eigenvalue of the Dirichlet Laplacian. Precisely, we prove that if the set $\\Omega$ minimizes the functional \\[ \\mathcal F_\\Lambda(\\Omega)=\\lambda_2(\\Omega)+\\Lambda |\\Omega|, \\] among all subsets of a smooth bounded open set $D\\subset \\mathbb{R}^d$, where $\\lambda_2(\\Omega)$ is the second eigenvalue of the Dirichlet Laplacian on $\\Omega$ and $\\Lambda>0$ is a fixed constant, then $\\Omega$ is equivalent to the union of two disjoint open sets $\\Omega_+$ and $\\Omega_-$, which are $C^{1,\\alpha}$-regular up to a (possibly empty) closed set of Hausdorff dimension at most $d-5$, contained in the one-phase free boundaries $D\\cap \\partial\\Omega_+\\setminus\\partial\\Omega_-$ and $D\\cap\\partial\\Omega_-\\setminus\\partial\\Omega_+$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-01T14:33:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R35",
      "49Q10",
      "47A75"
    ],
    "keywords": [
      "second dirichlet eigenvalue",
      "optimal sets",
      "regularity",
      "dirichlet laplacian",
      "second eigenvalue"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}