Deep estimates for higher eigenvalues of the poly-Laplacian

TL;DR

This paper derives improved lower bounds for higher eigenvalues of the poly-Laplacian, extending classical inequalities and addressing conjectures across various dimensions and boundary conditions.

Contribution

It introduces new sharp lower bounds for poly-Laplacian eigenvalues in arbitrary dimensions without restrictive conditions, improving existing inequalities and related conjectures.

Findings

Enhanced lower bounds for poly-Laplacian eigenvalues

Improved inequalities for Stokes eigenvalue problems

Progress on the Pólya conjecture in specific cases

Abstract

We investigate the lower bound for higher eigenvalues $\lambda_i$ of the poly-Laplace operator on a bounded domain and improve the famous Li-Yau inequality and its related results. Firstly, we consider the low dimensional cases for the P\'{o}lya conjecture, the clamped plate problem and the eigenvalue problem of the poly-Laplacian and deliver a series of deep eigenvalue inequalities for these problems respectively. Secondly, we establish a sharp lower bound for the eigenvalues of the poly-Laplacia in arbitrary dimension under some certain restrictive conditions. Finally, we provide an improved inequality for $\lambda_i$ in arbitrary dimension without any restrictive conditions. Our results also yield the improvement of the lower bounds for the Stokes eigenvalue problems and the Generalized P\'{o}lya conjecture.