Lower bounds for interior nodal sets of Steklov eigenfunctions

Abstract

We study the interior nodal sets, $Z_\lambda$ of Steklov eigenfunctions in an $n$-dimensional relatively compact manifolds $M$ with boundary and show that one has the lower bounds $|Z_\lambda|\ge c\lambda^{\frac{2-n}2}$ for the size of its $(n-1)$-dimensional Hausdorff measure. The proof is based on a Dong-type identity and estimates for the gradient of Steklov eigenfunctions, similar to those in \cite{SZ1} and \cite{SZ2}, respectively.