Universal Bounds for Eigenvalues of the Polyharmonic Operators

TL;DR

This paper establishes universal bounds for eigenvalues of polyharmonic operators on various compact domains, improving existing inequalities and covering a broad class of operators and geometries.

Contribution

It introduces sharper universal eigenvalue inequalities for polyharmonic operators, extending results to spherical domains and improving upon classical bounds.

Findings

Universal eigenvalue inequalities for polyharmonic operators.

Sharper bounds than Payne-Pólya-Weinberg and Yang inequalities.

Extensions to spherical domains and higher-order operators.

Abstract

We study eigenvalues of polyharmonic operators on compact Riemannian manifolds with boundary (possibly empty). In particular, we prove a universal inequality for the eigenvalues of the polyharmonic operators on compact domains in a Euclidean space. This inequality controls the $k$th eigenvalue by the lower eigenvalues, independently of the particular geometry of the domain. Our inequality is sharper than the known Payne-P\'olya-Weinberg type inequality and also covers the important Yang inequality on eigenvalues of the Dirichlet Laplacian. We also prove universal inequalities for the lower order eigenvalues of the polyharmonic operator on compact domains in a Euclidean space which in the case of the biharmonic operator and the buckling problem strengthen the estimates obtained by Ashbaugh. Finally, we prove universal inequalities for eigenvalues of polyharmonic operators of any order on compact domains in the sphere.