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. Conversely, in a 2D Lipschitz domain $\Omega $, we prove that any nodal domain of a Steklov eigenfunction contains a half-ball centered at $\partial \Omega $ of radius $c_{\Omega }/{\lambda }$.
Doubling inequalities and nodal sets in periodic elliptic homogenization
