Dirichlet eigenfunctions in the cube, sharpening the Courant nodal inequality

Abstract

This paper is devoted to the refined analysis of Courant's theorem for the Dirichlet Laplacian in a bounded open set. Starting from the work by A. Pleijel in 1956, many papers have investigated in which cases the inequality in Courant's theorem is an equality. All these results were established for open sets in ${\mathbb{R}}^2$ or for surfaces like ${\mathbb{S}}^2$ or ${\mathbb{T}}^2$. The aim of the current paper is to look for the case of the cube in ${\mathbb{R}}^3$. We will prove that the only eigenvalues of the Dirichlet Laplacian which are Courant sharp are the two first eigenvalues.