{
  "id": "1909.02350",
  "version": "v1",
  "published": "2019-09-05T12:09:32.000Z",
  "updated": "2019-09-05T12:09:32.000Z",
  "title": "Fundamental tones of clamped plates in nonpositively curved spaces",
  "authors": [
    "Alexandru Kristály"
  ],
  "comment": "27 pages, 3 figures, 2 tables",
  "categories": [
    "math.AP",
    "math-ph",
    "math.DG",
    "math.MP"
  ],
  "abstract": "We study Lord Rayleigh's problem for clamped plates on an arbitrary $n$-dimensional $(n\\geq 2)$ Cartan-Hadamard manifold $(M,g)$ with sectional curvature $\\textbf{K}\\leq -\\kappa^2$ for some $\\kappa\\geq 0.$ We first prove a McKean-type spectral gap estimate, i.e. the fundamental tone of any domain in $(M,g)$ is universally bounded from below by $\\frac{(n-1)^4}{16}\\kappa^4$ whenever the $\\kappa$-Cartan-Hadamard conjecture holds on $(M,g)$, e.g. in 2- and 3-dimensions due to Bol (1941) and Kleiner (1992), respectively. In 2- and 3-dimensions we prove sharp isoperimetric inequalities for sufficiently small clamped plates, i.e. the fundamental tone of any domain in $(M,g)$ of volume $v>0$ is not less than the corresponding fundamental tone of a geodesic ball of the same volume $v$ in the space of constant curvature $-\\kappa^2$ provided that $v\\leq c_n/\\kappa^n$ with $c_2\\approx 21.031$ and $c_3\\approx 1.721$, respectively. In particular, Rayleigh's problem in Euclidean spaces resolved by Nadirashvili (1992) and Ashbaugh and Benguria (1995) appears as a limiting case in our setting (i.e. $\\textbf{K}\\equiv\\kappa=0$). The sharpness of our results requires the validity of the $\\kappa$-Cartan-Hadamard conjecture (i.e. sharp isoperimetric inequality on $(M,g)$) and peculiar properties of the Gaussian hypergeometric function, both valid only in dimensions 2 and 3; nevertheless, some nonoptimal estimates of the fundamental tone of arbitrary clamped plates are also provided in high-dimensions. As an application, by using the sharp isoperimetric inequality for small clamped hyperbolic discs, we give necessarily and sufficient conditions for the existence of a nontrivial solution to an elliptic PDE involving the biharmonic Laplace-Beltrami operator.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-05T12:09:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P15",
      "53C21",
      "35J35",
      "35J40"
    ],
    "keywords": [
      "fundamental tone",
      "clamped plates",
      "nonpositively curved spaces",
      "sharp isoperimetric inequality",
      "mckean-type spectral gap estimate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}