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.