Interior nodal sets of Steklov eigenfunctions on surfaces

TL;DR

This paper studies the properties of Steklov eigenfunctions on surfaces, establishing bounds on their vanishing order, singular points, and interior nodal set measures, advancing understanding of their geometric behavior.

Contribution

It provides new bounds on the vanishing order, singular set size, and interior nodal set measure of Steklov eigenfunctions on surfaces.

Findings

Vanishing order of eigenfunctions is bounded by Cλ.

Number of singular points is at most Cλ^2.

Interior nodal set measure is at most Cλ^{3/2}.

Abstract

We investigate the interior nodal sets $\mathcal{N}_\lambda$ of Steklov eigenfunctions on connected and compact surfaces with boundary. The optimal vanishing order of Steklov eigenfunctions is shown be $C\lambda$. The singular sets $\mathcal{S}_\lambda$ are finite points on the nodal sets. We are able to prove that the Hausdorff measure $H^0(\mathcal{S}_\lambda)\leq C\lambda^2$. Furthermore, we obtain an upper bound for the measure of interior nodal sets $H^1(\mathcal{N}_\lambda)\leq C\lambda^{\frac{3}{2}}$. Here those positive constants $C$ depend only on the surfaces.