Kristaly, A. (2020) Advances in Mathematics
Ovidiu Ioan Moisescu
10 Noiembrie 2020
Kristaly, A. (2020) Fundamental tones of clamped plates in nonpositively curved spaces. Advances in Mathematics, 367, 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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