The sharp upper bound for the area of the nodal sets of Dirichlet Laplace eigenfunctions

Abstract

Let $\Omega$ be a bounded domain in ${\mathbb{R}}^n$ with $C^1$ boundary and let $u_{\lambda }$ be a Dirichlet Laplace eigenfunction in $\Omega$ with eigenvalue $\lambda$. We show that the $(n-1)$-dimensional Hausdorff measure of the zero set of $u_{\lambda }$ does not exceed $C(\Omega )\sqrt{\lambda }$. This result is new even for the case of domains with $C^{\infty }$-smooth boundary.