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 $$-\Delta {\varphi }_{{\sigma }_j}=0,\text{ on }\Omega ,{\partial }_{\nu }{\varphi }_{{\sigma }_j}={\sigma }_j{\varphi }_{{\sigma }_j}\text{ on }\partial \Omega$$ in two-dimensional domains $\Omega$. In particular, this paper presents a dense family $\mathcal{A}$ of simply-connected two-dimensional domains with analytic boundaries such that, for each $\Omega \in \mathcal{A}$, the nodal set of the eigenfunction ${\varphi }_{{\sigma }_j}$ "is $not$ dense at scale ${\sigma }_j^{-1}$". 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 $\Omega \in \mathcal{A}$, the nodal sets of the eigenfunctions ${\varphi }_{{\sigma }_j}$ associated with the eigenvalue ${\sigma }_j$ have starkly different character than anticipated: they are not dense at any shrinking scale. More precisely, for each $\Omega \in \mathcal{A}$ there is a value $r_1>0$ such that for each $j$ there is $x_j\in \Omega$ such that ${\varphi }_{{\sigma }_j}$ does not vanish on the ball of radius $r_1$ around $x_j$.

21 Pages, 5 figures