Upper Bound Estimates for Eigenvalues of the Poly-Laplacian

TL;DR

This paper derives sharp upper bounds for the sum of the first k eigenvalues of the poly-Laplacian with Dirichlet boundary conditions, extending previous results and improving bounds for large k when the order is two.

Contribution

It provides a new, sharper upper bound for eigenvalue sums of the poly-Laplacian, generalizing and improving upon prior work for any order and large k.

Findings

Established a sharp upper bound for eigenvalue sums.

Extended previous results to any order of the poly-Laplacian.

Improved bounds specifically for the case when order is two and k is large.

Abstract

In this paper, we obtain a sharp upper bound for the sum of the first $k$-th eigenvalues for this Dirichlet problem of poly-Laplacian with any order, which is viewed as an extension of the result due to Cheng and Wei (Journal of Differential Equations, 255 (2013), 220-233). In particular, if $l=2$ and $k$ is large enough, we give an important improvement of their result.