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Domains without dense Steklov nodal sets

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.

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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.

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Bruno, Oscar P.0000-0001-8369-3014
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 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.
Funding AgencyGrant Number
NSF Postdoctoral FellowshipDMS-1502661
Air Force Office of Scientific Research (AFOSR)FA9550-15-1-0043
Defense Advanced Research Projects Agency (DARPA)HR00111720035
National Security Science and Engineering Faculty FellowshipN00014-16-1-2808
Vannever Bush Faculty FellowshipUNSPECIFIED
Subject Keywords:Steklov; High frequency; Nodal set
Issue or Number:3
Classification Code:MSC: 35P20
Record Number:CaltechAUTHORS:20190906-093833395
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Official Citation:Bruno, O.P., Galkowski, J. Domains Without Dense Steklov Nodal Sets. J Fourier Anal Appl 26, 45 (2020).
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:98474
Deposited By: George Porter
Deposited On:06 Sep 2019 20:24
Last Modified:16 Nov 2021 17:39

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