Interior nodal sets of Steklov eigenfunctions on surfaces

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 $\mathcal{S}_\lambda$ are finite points on the nodal sets. We are able to prove that the Hausdorff measure $H^0(\mathcal{S}_\lambda)\leq C\lambda^2$. Furthermore, we obtain an upper bound for the measure of interior nodal sets $H^1(\mathcal{N}_\lambda)\leq C\lambda^{\frac{3}{2}}$. Here those positive constants $C$ depend only on the surfaces.