Lower bounds for interior nodal sets of Steklov eigenfunctions

TL;DR

This paper establishes lower bounds on the size of interior nodal sets of Steklov eigenfunctions on compact manifolds with boundary, using identities and gradient estimates.

Contribution

It provides the first known lower bounds for the measure of interior nodal sets of Steklov eigenfunctions in higher dimensions.

Findings

Lower bounds for nodal set measure: |Z_λ| ≥ c λ^{(2-n)/2}

Use of Dong-type identity and gradient estimates

Results applicable to n-dimensional manifolds with boundary

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.