Nodal Sets of Steklov Eigenfunctions

Abstract

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 $Ω$ 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 $u|_{\partial Ω}$ in terms of the eigenvalue $λ$ as $λ$ grows to infinity. In case that the domain $Ω$ is analytic, we prove a polynomial bound O($λ^6$). 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].