Domains Without Dense Steklov Nodal Sets

• Published in: The Journal of fourier analysis and applications 2020-06, Vol.26 (3), Article 45
• Main Authors: Bruno, Oscar P., Galkowski, Jeffrey
• Format: Article
• Language: English
• Subjects: Abstract Harmonic Analysis; Approximations and Expansions; Domains; Eigenvalues; Eigenvectors; Fourier Analysis; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Partial Differential Equations; Signal,Image and Speech Processing; Two dimensional analysis
• ISSN: 1069-5869; 1531-5851
• DOI: 10.1007/s00041-020-09753-7

This article concerns the asymptotic geometric character of the nodal set of the eigenfunctions of the Steklov eigenvalue problem - Δ ϕ σ j = 0 , on Ω , ∂ ν ϕ σ j = σ j ϕ σ j 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 σ j “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 σ j 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 r 1 > 0 such that for each j there is x j ∈ Ω such that ϕ σ j does not vanish on the ball of radius r 1 around x j .