On the remainder term of the Berezin inequality on a convex domain

TL;DR

This paper refines bounds on the eigenvalues of the Laplacian on convex domains by improving the Berezin inequality's remainder term, leading to better eigenvalue estimates and extending previous results to higher dimensions.

Contribution

It generalizes and improves upper bounds for Riesz means of eigenvalues on convex domains by refining the Berezin inequality's remainder term, applicable in higher dimensions.

Findings

Improved upper bounds for Riesz means of eigenvalues.

Derived lower bounds for individual eigenvalues.

Extended previous planar results to higher-dimensional convex domains.

Abstract

We study the Dirichlet eigenvalues of the Laplacian on a convex domain in $\mathbb{R}^n$, with $n\geq 2$. In particular, we generalize and improve upper bounds for the Riesz means of order $\sigma\geq 3/2$ established in an article by Geisinger, Laptev and Weidl. This is achieved by refining estimates for a negative second term in the Berezin inequality. The obtained remainder term reflects the correct order of growth in the semi-classical limit and depends only on the measure of the boundary of the domain. We emphasize that such an improvement is for general $\Omega\subset\mathbb{R}^n$ not possible and was previously known to hold only for planar convex domains satisfying certain geometric conditions. As a corollary we obtain lower bounds for the individual eigenvalues $\lambda_k$, which for a certain range of $k$ improves the Li--Yau inequality for convex domains. However, for convex domains one can use different methods to obtain even stronger such lower bounds.