Improved lower bounds for Dirichlet eigenvalues of the Laplacian and poly-Laplacian on bounded Euclidean domains

TL;DR

This paper derives improved, optimal lower bounds for sums of Dirichlet eigenvalues of the Laplacian and poly-Laplacian on bounded Euclidean domains, advancing previous bounds by capturing all positive terms in polynomial expansions.

Contribution

The authors develop new polynomial expansions that lead to sharper, optimal lower bounds for eigenvalue sums, surpassing prior results in the literature.

Findings

Established Brezin-Li-Yau type lower bounds for eigenvalues.

Derived polynomial expansions capturing all positive terms.

Improved existing lower bounds, making them optimal.

Abstract

In this paper, we establish Brezin-Li-Yau type lower bounds for averaged sums of Dirichlet eigenvalues of the Laplacian and poly-Laplacian on bounded domains in Euclidean spaces. By deriving expansions of two binary polynomials which may be of independent interest, we improve several existing lower bounds of this kind in the literature. Furthermore, our lower bounds are optimal in the sense that our expansions capture all positive terms, whereas previous works only provided certain lower bounds for these two binary polynomials, effectively capturing only a subset of the positive terms identified in our expansions.