Fundamental tones of clamped plates in nonpositively curved spaces

TL;DR

This paper investigates the fundamental frequencies of clamped plates in nonpositively curved spaces, establishing sharp bounds and isoperimetric inequalities in 2- and 3-dimensional Cartan-Hadamard manifolds, with implications for elliptic PDEs.

Contribution

It provides new sharp bounds and isoperimetric inequalities for the fundamental tone of clamped plates in nonpositively curved spaces, extending classical results to curved geometries.

Findings

Lower bounds for fundamental tones in nonpositively curved spaces.

Sharp isoperimetric inequalities for small plates in 2D and 3D.

Conditions for existence of solutions to biharmonic PDEs in hyperbolic disks.

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.