Hausdorff measure bounds for nodal sets of Steklov eigenfunctions

We study nodal sets of Steklov eigenfunctions in a bounded domain with ${\mathcal{C}}^2$ boundary. Our first result is a lower bound for the Hausdorff measure of the nodal set: we show that for $u_{\lambda }$ a Steklov eigenfunction, with eigenvalue $\lambda \mathrm{\ne }0$, ${\mathcal{H}}^{d-1}(\{u_{\lambda }=0\})\ge c_{\mathrm{\Omega }}$, where $c_{\mathrm{\Omega }}$ is independent of $\lambda$. We also prove an almost sharp upper bound, namely ${\mathcal{H}}^{d-1}(\{u_{\lambda }=0\})\le C_{\mathrm{\Omega }}\lambda \log (\lambda +e)$.