Hausdorff measure bound for the nodal sets of Neumann Laplace eigenfunctions

Abstract

We study the nodal sets of Neumann Laplace eigenfunctions in a bounded domain with ${\mathcal{C}}^{1,1}$ boundary. We show that for $u_{\lambda }$ such that $\Delta u_{\lambda }+\lambda u_{\lambda }=0$ with the Neumann boundary condition ${\partial }_{\nu }{u}_{\lambda }=0$, we have ${\mathcal{H}}^{n-1}(\{u_{\lambda }=0\})\le C\sqrt{\lambda }$.