Fundamental tones of clamped plates in nonpositively curved spaces

Alexandru Kristály

doi:10.1016/j.aim.2020.107113

Abstract

We study Lord Rayleigh's problem for clamped plates on an arbitrary n-dimensional (n≥2) Cartan-Hadamard manifold (M,g) with sectional curvature K≤−κ2 for some κ≥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 (n−1)416κ4 whenever the κ-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 −κ2 provided that v≤cn/κn with c2≈21.031 and c3≈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. K≡κ=0). Sharp asymptotic estimates of the fundamental tone of small and large geodesic balls of low-dimensional hyperbolic spaces are also given. The sharpness of our results requires the validity of the κ-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.

Highlights

Introduction and main results LetΩ ⊂ Rn be a bounded domain (n ≥ 2), and consider the eigenvalue problemΔ2u = Γu in Ω, u = |∇u| = 0 on ∂Ω, (1.1)associated with the vibration of a clamped plate
We study Lord Rayleigh’s problem for clamped plates on an arbitrary n-dimensional (n ≥ 2) Cartan-Hadamard manifold (M, g) with sectional curvature K ≤ −κ2 for some κ ≥ 0
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 −κ2 provided that v ≤ cn/κn with c2 ≈ 21.031 and c3 ≈ 1.721, respectively

Summary

Space-forms

Let κ ≥ 0 and Nκn be the n-dimensional space-form with constant sectional curvature. −κ2. Let κ ≥ 0 and Nκn be the n-dimensional space-form with constant sectional curvature. (H−n κ2 , gκ) is a Cartan-Hadamard manifold with constant sectional curvature −κ2. Let (M, g) be an n-dimensional Cartan-Hadamard manifold with sectional curvature bounded above by −κ2 for some κ ≥ 0. The κ-Cartan-Hadamard conjecture on (M, g) (called as the generalized Cartan-Hadamard conjecture) states that the κ-sharp isoperimetric inequality holds on (M, g), i.e. for every open bounded Ω ⊂ M one has. The κ-Cartan-Hadamard conjecture holds for every κ ≥ 0 on space-forms with constant sectional curvature −κ2 (of any dimension), see Dinghas [18], and on CartanHadamard manifolds with sectional curvature bounded above by −κ2 of dimension 2, see Bol [5], and of dimension 3, see Kleiner [26]. In higher dimensions and for κ > 0, the conjecture is still open; for a detailed discussion, see Kloeckner and Kuperberg [28]

Gaussian hypergeometric function

Generic scheme

Fundamental tones in high-dimensions: nonoptimal estimates