Eigenvalue estimates for a class of elliptic differential operators in divergence form

TL;DR

This paper provides eigenvalue estimates for elliptic operators in divergence form on Riemannian manifolds, extending known inequalities and applying Weyl's asymptotic formula to derive bounds for the drifting Laplacian.

Contribution

It introduces new eigenvalue bounds for a class of elliptic operators, extending previous results to the drifting Laplacian and generalizing inequalities on curved spaces.

Findings

Derived lower bounds for the mean of the first k eigenvalues.

Extended partial solutions to Pólya's conjecture for the drifting Laplacian.

Generalized inequalities for domains in spheres and projective spaces.

Abstract

We compute estimates for eigenvalues of a class of linear second-order elliptic differential operators in divergence form (with Dirichlet boundary condition) on a bounded domain in a complete Riemannian manifold. Our estimates are based upon the Weyl's asymptotic formula. As an application, we find a lower bound for the mean of the first k eigenvalues of the drifting Laplacian. In particular, we have extended for this operator a partial solution given by Cheng and Yang for the generalized conjecture of P\'olya. We also derive a second-Yang type inequality due to Chen and Cheng, and other two inequalities which generalize results by Cheng and Yang obtained for a domain in the unit sphere and for a domain in the projective space.