Complementary asymptotically sharp estimates for eigenvalue means of Laplacians

TL;DR

This paper derives sharp asymptotic bounds for Laplacian eigenvalues, improving classical inequalities by including second-order terms, using refined variational methods and boundary corrections.

Contribution

It introduces new asymptotically sharp inequalities for Laplacian eigenvalues that include second terms, complementing existing bounds like Berezin-Li-Yau and Kröger.

Findings

Derived inequalities contain second terms for eigenvalue means

Applicable to both Dirichlet and Neumann boundary conditions

Utilized boundary-corrected test functions in variational principles

Abstract

We present asymptotically sharp inequalities, containing a second term, for the Dirichlet and Neumann eigenvalues of the Laplacian on a domain, which are complementary to the familiar Berezin-Li-Yau and Kr\"oger inequalities in the limit as the eigenvalues tend to infinity. We accomplish this in the framework of the Riesz mean $R_1(z)$ of the eigenvalues by applying the averaged variational principle with families of test functions that have been corrected for boundary behaviour.