Geometrical Versions of improved Berezin-Li-Yau Inequalities

TL;DR

This paper derives improved upper bounds for Riesz means of eigenvalues of the Dirichlet Laplacian on bounded domains, incorporating geometric boundary properties, and establishes new lower bounds on eigenvalues that enhance classical inequalities.

Contribution

It introduces geometric-dependent remainder terms in eigenvalue bounds, refining Berezin and Li-Yau inequalities for the Dirichlet Laplacian.

Findings

Improved upper bounds on Riesz means with geometric boundary terms

New lower bounds on individual eigenvalues under certain geometric conditions

Enhanced inequalities reflecting correct semi-classical growth

Abstract

We study the eigenvalues of the Dirichlet Laplace operator on an arbitrary bounded, open set in $\R^d$, $d \geq 2$. In particular, we derive upper bounds on Riesz means of order $\sigma \geq 3/2$, that improve the sharp Berezin inequality by a negative second term. This remainder term depends on geometric properties of the boundary of the set and reflects the correct order of growth in the semi-classical limit. Under certain geometric conditions these results imply new lower bounds on individual eigenvalues, which improve the Li-Yau inequality.