Interior nodal sets of Steklov eigenfunctions on surfaces

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 $H^0({\mathcal{S}}_{\lambda })\le C{\lambda }^2$. Furthermore, we obtain an upper bound for the measure of interior nodal sets $H^1({\mathcal{N}}_{\lambda })\le C{\lambda }^{\frac{3}{2}}$. Here those positive constants $C$ depend only on the surfaces.

Correct some typos

References in corpus (4)

• Measure of nodal sets of analytic Steklov eigenfunctions
• A lower bound for the nodal sets of Steklov eigenfunctions
• Lower bounds for interior nodal sets of Steklov eigenfunctions
• Spectral geometry of the Steklov problem