The Sharp Measure Upper Bound of the Nodal Sets of Neumann Laplace Eigenfunctions on C1,1 Domains

TL;DR

This paper establishes an upper bound proportional to the square root of the eigenvalue for the measure of the zero set of Neumann Laplace eigenfunctions in C^{1,1} domains, advancing understanding of eigenfunction nodal sets.

Contribution

It provides a sharp upper bound on the measure of nodal sets for Neumann eigenfunctions in C^{1,1} domains, extending previous results to this class of domains.

Findings

Hausdorff measure of zero set ≤ C√λ

Bound applies to C^{1,1} domains

Improves understanding of eigenfunction nodal sets

Abstract

Let {\Omega} be a bounded domain in R^n with C^{1,1} boundary and let u_{\lambda} be a Neumann 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\sqrt{\lambda}.