Published August 7, 2025 | Version Published
Journal Article Open

Riesz means asymptotics for Dirichlet and Neumann Laplacians on Lipschitz domains

  • 1. ROR icon Ludwig-Maximilians-Universität München
  • 2. ROR icon California Institute of Technology
  • 3. ROR icon University of Gothenburg

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.

Copyright and License

© 2025, The Author(s). Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Acknowledgement

The authors are grateful for the anonymous referee’s careful reading and helpful suggestions. We would like to thank Jean Lagacé for stimulating discussions on asymptotics for polygons and Nikolay Filonov for making us aware of a number of inaccuracies in an earlier version of this paper.

Funding

Open Access funding enabled and organized by Projekt DEAL. Partial support through US National Science Foundation grant DMS-1954995 (R.L.F.), as well as through German Research Foundation grants EXC-2111-390814868 and TRR 352-Project-ID 470903074 (R.L.F.), the Knut and Alice Wallenberg foundation grant KAW 2017.0295 (S.L.), as well as the Swedish Research Council grant no. 2023-03985 (S.L.) is acknowledged.

Files

s00222-025-01352-x.pdf

Files (1.9 MB)

Name Size Download all
md5:06ae5cdede668f275829c843dca8617b
1.9 MB Preview Download

Additional details

Related works

Describes
Journal Article: https://rdcu.be/eAtMu (ReadCube)

Funding

Ludwig-Maximilians-Universität München

Caltech Custom Metadata

Caltech groups
Division of Physics, Mathematics and Astronomy (PMA)
Publication Status
Published