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

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 $c_{\Omega}/\lambda$.