Nodal sets of Steklov eigenfunctions near the boundary: Inner radius estimates

TL;DR

This paper investigates the behavior of Steklov eigenfunctions near the boundary of Lipschitz domains, showing dense nodal sets close to the boundary and establishing bounds on nodal domains in two dimensions.

Contribution

It provides new boundary estimates for Steklov eigenfunctions and characterizes the size of nodal domains near the boundary in Lipschitz domains.

Findings

Nodal sets are wavelength dense near the boundary.

Nodal domains contain a half-ball of radius proportional to 1/λ near the boundary.

Contrasts behavior of eigenfunctions deep inside the domain.

Abstract

We show that Steklov eigenfunctions in a bounded Lipschitz domain have wavelength dense nodal sets near the boundary, in contrast to what can happen deep inside the domain. As a converse, in a two-dimensional Lipschitz domain $\Omega$, we prove that any nodal domain of a Steklov eigenfunction contains a half-ball centered at $\partial\Omega$ of radius $c_{\Omega}/\lambda$.