Inequalities for eigenvalues of fourth-order elliptic operators in divergence form on complete Riemannian manifolds

TL;DR

This paper establishes new inequalities for eigenvalues of fourth-order elliptic operators on complete Riemannian manifolds, extending classical results and providing estimates for specific operators like the bi-drifted Cheng-Yau operator.

Contribution

It generalizes Payne-Pólya-Weinberger-Yang type inequalities to weighted divergence form operators on manifolds, including estimates for lower order eigenvalues and specific applications.

Findings

Derived inequalities for eigenvalues of fourth-order elliptic operators.

Extended classical eigenvalue bounds to weighted divergence form operators.

Provided eigenvalue estimates for the bi-drifted Cheng-Yau operator.

Abstract

We prove some inequalities of Payne-P\'olya-Weinberger-Yang type for eigenvalues of fourth-order elliptic operators in weighted divergence form on complete Riemannian manifolds which generalizes the corresponding result for the clamped plate problem. We also prove estimates for lower order eigenvalues that contain some of the estimates from the literature. As an application of our results, we obtain eigenvalues estimates for the bi-drifted Cheng-Yau operator.