The sharp estimate of nodal sets for Dirichlet Laplace eigenfunctions in polytopes

Abstract

Let $P$ be a bounded $n$-dimensional Lipschitz polytope, and let $\varphi_{\lambda}$ be a Dirichlet Laplace eigenfunction in $P$ corresponding to the eigenvalue $\lambda$. We show that the $(n-1)$-dimensional Hausdorff measure of the nodal set of $\varphi_{\lambda}$ does not exceed $C(P)\sqrt{\lambda}$. Our result extends the previous ones in quaisconvex domains (including $C^1$ and convex domains) to general polytopes that are not necessarily quasiconvex.