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arXiv · 2307.02911

Sharp spectral gap estimates for higher-order operators on Cartan-Hadamard manifolds

Csaba Farkas·Sándor Kajántó·Alexandru Kristály
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Abstract

The goal of this paper is to provide sharp spectral gap estimates for problems involving higher-order operators (including both the clamped and buckling plate problems) on Cartan-Hadamard manifolds. The proofs are symmetrization-free -- thus no sharp isoperimetric inequality is needed -- based on two general, yet elementary functional inequalities. The spectral gap estimate for clamped plates solves a sharp asymptotic problem from Cheng and Yang [Proc. Amer. Math. Soc., 2011] concerning the behavior of higher-order eigenvalues on hyperbolic spaces, and answers a question raised in Krist\'aly [Adv. Math., 2020] on the validity of such sharp estimates in high-dimensional Cartan-Hadamard manifolds. As a byproduct of the general functional inequalities, various Rellich inequalities are established in the same geometric setting.

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Csaba Farkas, Sándor Kajántó, Alexandru Kristály. 2023-07-06. Sharp spectral gap estimates for higher-order operators on Cartan-Hadamard manifolds. https://arxiv.org/abs/2307.02911

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