Hausdorff measure bounds for nodal sets of Steklov eigenfunctions

TL;DR

This paper establishes lower and upper bounds for the Hausdorff measure of nodal sets of Steklov eigenfunctions in bounded domains, advancing understanding of their geometric properties as eigenvalues grow.

Contribution

It provides the first uniform lower bound and an almost sharp upper bound for the Hausdorff measure of Steklov eigenfunction nodal sets, independent of eigenvalue.

Findings

Lower bound: measure is at least a constant independent of eigenvalue.

Upper bound: measure grows at most like eigenvalue times log(eigenvalue).

Results are valid for domains with 2 boundary.

Abstract

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 $\lambda\neq 0$, $\mathcal{H}^{d-1}(\{u_{\lambda}=0\})\geq c_{\Omega}$, where $c_{\Omega}$ is independent of $\lambda$. We also prove an almost sharp upper bound, namely $\mathcal{H}^{d-1}(\{u_{\lambda}=0\})\leq C_{\Omega}\lambda\log(\lambda+e)$.