# Kristaly, A. (2020) Advances in Mathematics

*Autor:
Ovidiu Ioan Moisescu
*
*Publicat:
10 Noiembrie 2020*

**Kristaly, A.** (2020) Fundamental tones of clamped plates in nonpositively curved spaces. *Advances in Mathematics*, 367, 107113.

DOI: https://doi.org/10.1016/j.aim.2020.107113

✓ Publisher: *Elsevier*

✓ Web of Science Core Collection: *Science Citation Index Expanded*

✓ Categories: *Mathematics*

✓ Article Influence Score (AIS): **1.923** (2019) / **Q1**

**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 <= -kappa(2) for some kappa >= 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)(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 <= c(n)/kappa(n) with c(2) approximate to 21.031 and c(3) approximate to 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 equivalent to kappa = 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 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.

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