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 $\mathcal{H}^{n-1}(\{u_\lambda = 0\}) \leq C \sqrt{\lambda}$.