Bruno, O; Galkowski, J; (2020) Domains without dense Steklov nodal sets. Journal of Fourier Analysis and Applications , 26 , Article 45. 10.1007/s00041-020-09753-7.

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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 .

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