Dirichlet eigenfunctions on the cube, sharpening the Courant nodal inequality

2015/06/18 by Bernard Helffer, Helffer, Bernard, Rola Kiwan +1
Mathematics · #35P15 #FOS: Mathematics #Spectral Theory (math.SP) #math.SP #msc:35P15

paper · pdf · doi:10.48550/arxiv.1506.05733

arxiv created 2015/06/18 · arxiv updated 2015/06/19

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, Bérard-Helffer, Helffer--Persson-Sundqvist, Léna, Leydold. All these results were devoted to (2D)-cases in open sets in \mathbb R² or in surfaces like \mathbb S² or \mathbb T². The aim of the current paper is to look for analogous results for domains in ℝ³ and, as Å.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.

Related

• On nodal domains in Euclidean balls
• Nodal domains in the square---the Neumann case
• Courant-sharp eigenvalues of the three-dimensional square torus
• Upper bounds for Courant-sharp Neumann and Robin eigenvalues
• Courant-sharp eigenvalues of a two-dimensional torus