Bruno, Oscar P. and Galkowski, Jeffrey (2020) Domains without dense Steklov nodal sets. Journal of Fourier Analysis and Applications, 26 (3). Art. No. 45. ISSN 1069-5869. doi:10.1007/s00041-020-09753-7. https://resolver.caltech.edu/CaltechAUTHORS:20190906-093833395
![]() |
PDF
- Published Version
Creative Commons Attribution. 1MB |
![]() |
PDF
- Submitted Version
See Usage Policy. 1MB |
Use this Persistent URL to link to this item: https://resolver.caltech.edu/CaltechAUTHORS:20190906-093833395
Abstract
This article concerns the asymptotic geometric character of the nodal set of the eigenfunctions of the Steklov eigenvalue problem −Δϕ_(σj) = 0, on Ω, ∂_νϕ_(σj) = σ_jϕ_(σj) on ∂Ω in two-dimensional domains Ω. In particular, this paper presents a dense family A of simply-connected two-dimensional domains with analytic boundaries such that, for each Ω∈A, the nodal set of the eigenfunction ϕ_(σj) “is not dense at scale σ_j⁻¹”. This result addresses a question put forth under “Open Problem 10” in Girouard and Polterovich (J Spectr Theory 7(2):321–359, 2017). In fact, the results in the present paper establish that, for domains Ω∈A, the nodal sets of the eigenfunctions ϕ_(σj) associated with the eigenvalue σ_j have starkly different character than anticipated: they are not dense at any shrinking scale. More precisely, for each Ω∈A there is a value r₁ > 0 such that for each j there is x_j ∈ Ω such that ϕ_(σj) does not vanish on the ball of radius r₁ around x_j.
Item Type: | Article | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Related URLs: |
| ||||||||||||||||
ORCID: |
| ||||||||||||||||
Additional Information: | © 2020 The Author(s). 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/. Received 18 September 2019; Revised 23 April 2020; Published 11 June 2020. Thanks to Agustin Fernandez Lado for writing the code numerical Steklov-eigenfunction solver and for providing the derivation presented in Appendix A. Thanks also to Jared Wunsch for suggesting part of the proof of Lemma 5.2 The authors are grateful to the American Institute of Mathematics where this research began. Thanks also to the anonymous referees for many helpful comments. J.G. is grateful to the National Science Foundation for support under the Mathematical Sciences Postdoctoral Research Fellowship DMS-1502661 and under DMS-1900434. O.B. gratefully acknowledges support by NSF, AFOSR and DARPA through contracts DMS-1714169, FA9550-15-1-0043 and HR00111720035, and the NSSEFF Vannevar Bush Fellowship under contract number N00014-16-1-2808. | ||||||||||||||||
Funders: |
| ||||||||||||||||
Subject Keywords: | Steklov; High frequency; Nodal set | ||||||||||||||||
Issue or Number: | 3 | ||||||||||||||||
Classification Code: | MSC: 35P20 | ||||||||||||||||
DOI: | 10.1007/s00041-020-09753-7 | ||||||||||||||||
Record Number: | CaltechAUTHORS:20190906-093833395 | ||||||||||||||||
Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20190906-093833395 | ||||||||||||||||
Official Citation: | Bruno, O.P., Galkowski, J. Domains Without Dense Steklov Nodal Sets. J Fourier Anal Appl 26, 45 (2020). https://doi.org/10.1007/s00041-020-09753-7 | ||||||||||||||||
Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||||||||||
ID Code: | 98474 | ||||||||||||||||
Collection: | CaltechAUTHORS | ||||||||||||||||
Deposited By: | George Porter | ||||||||||||||||
Deposited On: | 06 Sep 2019 20:24 | ||||||||||||||||
Last Modified: | 16 Nov 2021 17:39 |
Repository Staff Only: item control page