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

TL;DR

This paper establishes a new sharp upper bound on the measure of the zero set of Dirichlet Laplace eigenfunctions in bounded domains, showing it does not exceed a constant times the square root of the eigenvalue.

Contribution

The paper proves a universal upper bound on the Hausdorff measure of nodal sets for Dirichlet eigenfunctions, valid for domains with minimal boundary regularity.

Findings

Hausdorff measure of nodal sets is bounded by C(Ω)√λ

Result holds for domains with C^1 boundary, even with smooth boundaries

Provides a sharp upper bound for the size of nodal sets

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.