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

Let P be a bounded n-dimensional Lipschitz polytope, and let φλ be a Dirichlet Laplace eigenfunction in P corresponding to the eigenvalue λ. We show that the $(n-1)$-dimensional Hausdorff measure of the nodal set of φλ does not exceed C(P)√λ. Our result extends the previous ones in quaisconvex domains (including C¹ and convex domains) to general polytopes that are not necessarily quasiconvex.