Domains without dense Steklov nodal sets

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.