{
  "id": "2002.04064",
  "version": "v1",
  "published": "2020-02-10T19:55:57.000Z",
  "updated": "2020-02-10T19:55:57.000Z",
  "title": "Regularity of all minimizers of a class of spectral partition problems",
  "authors": [
    "Hugo Tavares",
    "Alessandro Zilio"
  ],
  "comment": "27 pages, 6 figures",
  "categories": [
    "math.AP",
    "math.OC"
  ],
  "abstract": "We study a rather broad class of optimal partition problems with respect to monotone and coercive functional costs that involve the Dirichlet eigenvalues of the partitions. We show a sharp regularity result for the entire set of minimizers for a natural relaxed version of the original problem, together with the regularity of eigenfunctions and a universal free boundary condition. Among others, our result covers the cases of the following functional costs \\[ (\\omega_1, \\dots, \\omega_m) \\mapsto \\sum_{i=1}^{m} \\left( \\sum_{j=1}^{k_i} \\lambda_{j}(\\omega_i)^{p_i}\\right)^{1/p_i}, \\quad \\prod_{i=1}^{m} \\left( \\prod_{j=1}^{k_i} \\lambda_{j}(\\omega_i)\\right), \\quad \\prod_{i=1}^{m} \\left( \\sum_{j=1}^{k_i} \\lambda_{j}(\\omega_i)\\right) \\] where $(\\omega_1, \\dots, \\omega_m)$ are the sets of the partition and $\\lambda_{j}(\\omega_i)$ is the $j$-th Laplace eigenvalue of the set $\\omega_i$ with zero Dirichlet boundary conditions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-10T19:55:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B25",
      "35J47",
      "35J50",
      "35R35",
      "49N60"
    ],
    "keywords": [
      "spectral partition problems",
      "minimizers",
      "zero dirichlet boundary conditions",
      "universal free boundary condition",
      "functional costs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}