Estimates for lower bounds of eigenvalues of the poly-Laplacian and quadratic polynomial operator of the Laplacian

TL;DR

This paper provides improved lower bound estimates for the sums of the first k eigenvalues of poly-Laplacian and quadratic polynomial Laplacian operators under Dirichlet boundary conditions.

Contribution

It introduces new bounds for eigenvalues of higher-order poly-Laplacian and quadratic polynomial Laplacian operators, enhancing previous estimates.

Findings

Derived new lower bounds for eigenvalue sums

Improved upon previous eigenvalue estimates

Applicable to poly-Laplacian and quadratic polynomial operators

Abstract

In this paper, we investigate the Dirchlet eigenvalue problems of poly-Laplacian with any order and quadratic polynomial operator of the Laplacian. We give some estimates for lower bounds of the sums of their first $k$ eigenvalues which improve the previous results.