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Domains without dense Steklov nodal sets

Bruno, O; Galkowski, J; Domains without dense Steklov nodal sets. (In press).

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Abstract

This article concerns the asymptotic geometric character of the nodal set of the eigenfunctions of the Steklov eigenvalue problem −∆φσj = 0, on Ω, ∂νφσj = σjφσj on ∂Ω in two-dimensional domains Ω. In particular, this paper presents a dense family A of simplyconnected two-dimensional domains with analytic boundaries such that, for each Ω ∈ A, the nodal set of the eigenfunction φσj “is not dense at scale σ −1 j ”. This result addresses a question put forth under “Open Problem 10” in Girouard and Polterovich, J. Spectr. Theory, 321-359 (2017). In fact, the results in the present paper establish that, for domains Ω ∈ A, the nodal sets of the eigenfunctions φσj associated with the eigenvalue σj 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 xj ∈ Ω such that φσj does not vanish on the ball of radius r1 around xj .

Type: Article
Title: Domains without dense Steklov nodal sets
Publisher version: https://arxiv.org/abs/1908.03307
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10096035
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