Dirichlet eigenfunctions on the cube, sharpening the Courant nodal inequality

TL;DR

This paper refines Courant's theorem for the Dirichlet Laplacian on the cube, proving that only the first two eigenvalues are Courant sharp, extending previous 2D results to three-dimensional domains.

Contribution

It establishes that in three-dimensional cubes, only the first two Dirichlet eigenvalues are Courant sharp, filling a gap in the understanding of nodal domain counts in 3D.

Findings

Only the first two eigenvalues are Courant sharp in the cube.

Extends Courant nodal domain results from 2D to 3D domains.

Provides a detailed analysis of eigenfunctions in the cube.

Abstract

This paper is devoted to the refine analysis of Courant's theorem for the Dirichlet Laplacian. Many papers (and some of them quite recent) have investigated in which cases this inequality in Courant's theorem is an equality: Pleijel, Helffer--Hoffmann-Ostenhof--Terracini, Helffer--Hoffmann-Ostenhof, B\'erard-Helffer, Helffer--Persson-Sundqvist, L\'ena, Leydold. All these results were devoted to $(2D)$-cases in open sets in $\mathbb R^2$ or in surfaces like $\mathbb S^2$ or $\mathbb T^2$. The aim of the current paper is to look for analogous results for domains in $\mathbb{R}^3$ and, as $\AA.$Pleijel was suggesting in his 1956 founding paper, for the simplest case of the cube. More precisely, we will prove that the only eigenvalues of the Dirichlet Laplacian which are Courant sharp are the two first eigenvalues.