Interior nodal sets of Steklov eigenfunctions on surfaces

We investigate the interior nodal sets $\mathcal{N}_λ$ of Steklov eigenfunctions on connected and compact surfaces with boundary. The optimal vanishing order of Steklov eigenfunctions is shown be $Cλ$. The singular sets $\mathcal{S}_λ$ are finite points on the nodal sets. We are able to prove that the Hausdorff measure $H^0(\mathcal{S}_λ)\leq Cλ^2$. Furthermore, we obtain an upper bound for the measure of interior nodal sets $H^1(\mathcal{N}_λ)\leq Cλ^{\frac{3}{2}}$. Here those positive constants $C$ depend only on the surfaces.

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