Riesz means asymptotics for Dirichlet and Neumann Laplacians on Lipschitz domains

TL;DR

This paper establishes two-term asymptotics for Riesz means of eigenvalues of Dirichlet and Neumann Laplacians on Lipschitz domains, with universal bounds for convex domains based on geometric features.

Contribution

It provides the first two-term asymptotic formulas for Riesz means on Lipschitz domains and introduces non-asymptotic bounds for convex domains depending only on geometric characteristics.

Findings

Two-term asymptotics for Riesz means of Laplacian eigenvalues.

Universal bounds for convex domains matching asymptotic terms.

Use of non-asymptotic Tauberian theorems in spectral analysis.

Abstract

We consider the eigenvalues of the Dirichlet and Neumann Laplacians on a bounded domain with Lipschitz boundary and prove two-term asymptotics for their Riesz means of arbitrary positive order. Moreover, when the underlying domain is convex, we obtain universal, non-asymptotic bounds that correctly reproduce the two leading terms in the asymptotics and depend on the domain only through simple geometric characteristics. Important ingredients in our proof are non-asymptotic versions of various Tauberian theorems.