Interior nodal sets of Steklov eigenfunctions on surfaces

Abstract

We investigate the interior nodal sets ${\mathcal{N}}_{\lambda }$ of Steklov eigenfunctions on connected and compact surfaces with boundary. The optimal vanishing order of Steklov eigenfunctions is shown be $C\lambda$. The singular sets ${\mathcal{S}}_{\lambda }$ consist of finitely many points on the nodal sets. We are able to prove that the Hausdorff measure $H^0\left({\mathcal{S}}_{\lambda }\right)$ is at most $C{\lambda }^2$. Furthermore, we obtain an upper bound for the measure of interior nodal sets, $H^1\left({\mathcal{N}}_{\lambda }\right)\le C{\lambda }^{3∕2}$. Here the positive constants $C$ depend only on the surfaces.