Weyl’s Law for the Steklov Problem on Surfaces with Rough Boundary

The validity of Weyl’s law for the Steklov problem on domains with Lipschitz boundary is a well-known open question in spectral geometry. We answer this question in two dimensions and show that Weyl’s law holds for an even larger class of surfaces with rough boundaries. This class includes domains w...

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Autores principales: Karpukhin, Mikhail, Lagacé, Jean, Polterovich, Iosif
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer Berlin Heidelberg 2023
Materias:
Article
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10415495/
https://www.ncbi.nlm.nih.gov/pubmed/37576945
http://dx.doi.org/10.1007/s00205-023-01912-6
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author Karpukhin, Mikhail
Lagacé, Jean
Polterovich, Iosif
author_facet Karpukhin, Mikhail
Lagacé, Jean
Polterovich, Iosif
author_sort Karpukhin, Mikhail
collection PubMed
description The validity of Weyl’s law for the Steklov problem on domains with Lipschitz boundary is a well-known open question in spectral geometry. We answer this question in two dimensions and show that Weyl’s law holds for an even larger class of surfaces with rough boundaries. This class includes domains with interior cusps as well as “slow” exterior cusps. Moreover, the condition on the speed of exterior cusps cannot be improved, which makes our result, in a sense optimal. The proof is based on the methods of Suslina and Agranovich combined with some observations about the boundary behaviour of conformal mappings.
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spelling pubmed-104154952023-08-12 Weyl’s Law for the Steklov Problem on Surfaces with Rough Boundary Karpukhin, Mikhail Lagacé, Jean Polterovich, Iosif Arch Ration Mech Anal Article The validity of Weyl’s law for the Steklov problem on domains with Lipschitz boundary is a well-known open question in spectral geometry. We answer this question in two dimensions and show that Weyl’s law holds for an even larger class of surfaces with rough boundaries. This class includes domains with interior cusps as well as “slow” exterior cusps. Moreover, the condition on the speed of exterior cusps cannot be improved, which makes our result, in a sense optimal. The proof is based on the methods of Suslina and Agranovich combined with some observations about the boundary behaviour of conformal mappings. Springer Berlin Heidelberg 2023-08-10 2023 /pmc/articles/PMC10415495/ /pubmed/37576945 http://dx.doi.org/10.1007/s00205-023-01912-6 Text en © The Author(s) 2023 https://creativecommons.org/licenses/by/4.0/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/ (https://creativecommons.org/licenses/by/4.0/) .
spellingShingle Article
Karpukhin, Mikhail
Lagacé, Jean
Polterovich, Iosif
Weyl’s Law for the Steklov Problem on Surfaces with Rough Boundary
title Weyl’s Law for the Steklov Problem on Surfaces with Rough Boundary
title_full Weyl’s Law for the Steklov Problem on Surfaces with Rough Boundary
title_fullStr Weyl’s Law for the Steklov Problem on Surfaces with Rough Boundary
title_full_unstemmed Weyl’s Law for the Steklov Problem on Surfaces with Rough Boundary
title_short Weyl’s Law for the Steklov Problem on Surfaces with Rough Boundary
title_sort weyl’s law for the steklov problem on surfaces with rough boundary
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10415495/
https://www.ncbi.nlm.nih.gov/pubmed/37576945
http://dx.doi.org/10.1007/s00205-023-01912-6
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