Hausdorff measure bounds for nodal sets of Steklov eigenfunctions

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

28 pages, minor revisions. Version accepted for publication in Analysis & PDE

References in corpus (2)

• Local Asymmetry and the Inner Radius of Nodal Domains
• Spectral geometry of the Steklov problem