{
  "id": "1710.00451",
  "version": "v1",
  "published": "2017-10-02T01:42:58.000Z",
  "updated": "2017-10-02T01:42:58.000Z",
  "title": "Regularity for Shape Optimizers: The Degenerate Case",
  "authors": [
    "Dennis Kriventsov",
    "Fanghua Lin"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider minimizers of \\[ F(\\lambda_1(\\Omega),\\ldots,\\lambda_N(\\Omega)) + |\\Omega|, \\] where $F$ is a function nondecreasing in each parameter, and $\\lambda_k(\\Omega)$ is the $k$-th Dirichlet eigenvalue of $\\Omega$. This includes, in particular, functions $F$ which depend on just some of the first $N$ eigenvalues, such as the often studied $F=\\lambda_N$. The existence of a minimizer, which is also a bounded set of finite perimeter, was shown recently. Here we show that the reduced boundary of the minimizers $\\Omega$ is made up of smooth graphs, and examine the difficulties in classifying the singular points. Our approach is based on an approximation (\"vanishing viscosity\") argument, which--counterintuitively--allows us to recover an Euler-Lagrange equation for the minimizers which is not otherwise available.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-02T01:42:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49R05",
      "49N60",
      "35R35"
    ],
    "keywords": [
      "degenerate case",
      "shape optimizers",
      "regularity",
      "th dirichlet eigenvalue",
      "finite perimeter"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}