New eigenvalue estimates involving Bessel functions
- 1 July 2021
- journal article
- research article
- Published by Universitat Autonoma de Barcelona in Publicacions Matemàtiques
- Vol. 65 (2), 681-726
- https://doi.org/10.5565/publmat6522109
Abstract
Given a compact Riemannian manifold (M-n, g) with boundary partial derivative M, we give an estimate for the quotient integral(partial derivative M) f d mu(g)/integral(M) f d mu(g), where f is a smooth positive function defined on M that satisfies some inequality involving the scalar Laplacian. By the mean value lemma established in [39], we provide a differential inequality for f which, under some curvature assumptions, can be interpreted in terms of Bessel functions. As an application of our main result, a new inequality is given for Dirichlet and Robin Laplacian. Also, a new estimate is established for the eigenvalues of the Dirac operator that involves a positive root of Bessel function besides the scalar curvature. Independently, we extend the Robin Laplacian on functions to differential forms. We prove that this natural extension defines a self-adjoint and elliptic operator whose spectrum is discrete and consists of positive real eigenvalues. In particular, we characterize its first eigenvalue and provide a lower bound of it in terms of Bessel functions.Keywords
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