Locating the first nodal set in higher dimensions

Abstract

This paper estimates the location and the width of the nodal set of the first Neumann eigenfunctions on a smooth convex domain $\Omega \subset \mathbb R^n$, whose length is normalized to be 1 and whose cross-section is contained in a ball of radius $\epsilon$. In \cite{CJK2009}, an $O(\epsilon)$ bound was obtained by constructing a coordinate system. In this paper, we present a simpler method that does not require such a coordinate system. Moreover, in the special case $n = 2$, we obtain an $O(\epsilon^2)$ bound on the width of the nodal set, in analogy to the corresponding result in the Dirichlet case obtained in \cite{GJ1995}.