New eigenvalue estimates involving Bessel functions

Abstract

Given a compact Riemannian manifold $(M^n,g)$ with boundary $\partial M$, we give an estimate for the quotient $\frac{\int_{\partial M} f\,d\mu_g}{\int_M f\,d\mu_g}$, where $f$ is a smooth positive function defined on $M$ that satisfies some inequality involving the scalar Laplacian. By the mean value lemma established [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.