{
  "id": "2210.14806",
  "version": "v1",
  "published": "2022-10-26T15:53:06.000Z",
  "updated": "2022-10-26T15:53:06.000Z",
  "title": "On the first eigenvalue of the Laplacian for polygons",
  "authors": [
    "Emanuel Indrei"
  ],
  "comment": "61 pages, 14 figures",
  "categories": [
    "math.AP",
    "math-ph",
    "math.DG",
    "math.FA",
    "math.MG",
    "math.MP"
  ],
  "abstract": "In 1947, P\\'olya proved that if $n=3,4$ the regular polygon $P_n$ minimizes the principal frequency of an n-gon with given area $\\alpha>0$ and suggested that the same holds when $n \\ge 5$. In $1951,$ P\\'olya & Szeg\\\"o discussed the possibility of counterexamples in the book \"Isoperimetric Inequalities In Mathematical Physics.\" This paper constructs explicit $(2n-4)$--dimensional polygonal manifolds $\\mathcal{M}(n, \\alpha)$ and proves the existence of a computable $N \\ge 5$ such that for all $n \\ge N$, the admissible $n$-gons are given via $\\mathcal{M}(n, \\alpha)$ and there exists an explicit set $ \\mathcal{A}_{n}(\\alpha) \\subset \\mathcal{M}(n,\\alpha)$ such that $P_n$ has the smallest principal frequency among $n$-gons in $\\mathcal{A}_{n}(\\alpha)$. Inter-alia when $n \\ge 3$, a formula is proved for the principal frequency of a convex $P \\in \\mathcal{M}(n,\\alpha)$ in terms of an equilateral $n$-gon with the same area; and, the set of equilateral polygons is proved to be an $(n-3)$--dimensional submanifold of the $(2n-4)$--dimensional manifold $\\mathcal{M}(n,\\alpha)$ near $P_n$. If $n=3$, the formula completely addresses a 2006 conjecture of Antunes and Freitas and another problem mentioned in \"Isoperimetric Inequalities In Mathematical Physics.\" Moreover, a solution to the sharp polygonal Faber-Krahn stability problem for triangles is given and with an explicit constant. The techniques involve a partial symmetrization, tensor calculus, the spectral theory of circulant matrices, and $W^{2,p}/BMO$ estimates. Last, an application is given in the context of electron bubbles.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-26T15:53:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "first eigenvalue",
      "sharp polygonal faber-krahn stability problem",
      "isoperimetric inequalities",
      "smallest principal frequency",
      "mathematical physics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 61,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}