MiS Preprint Repository

We study the nodal set of the Steklov eigenfunctions on the boundary of a smooth bounded domain in ${\mathbb{R}}^n$ - the eigenfunctions of the Dirichlet-to-Neumann map.

Under the assumption that the domain $\mathrm{\Omega }$ is $C^2$, we prove a doubling property for the eigenfunction $u$. We estimate the Hausdorff ${\mathcal{H}}^{n-2}$-measure of the nodal set of $u{|}_{\partial \mathrm{\Omega }}$ in terms of the eigenvalue $\lambda$ as $\lambda$ grows to infinity. In case that the domain $\mathrm{\Omega }$ is analytic, we prove a polynomial bound O(${\lambda }^6$).

Our arguments, which make heavy use of Almgren's frequency functions, are built on the previous works [Garofalo and Lin, CPAM 40 (1987), no. 3; Lin, CPAM 42 (1989), no. 6].