Courant-sharp eigenvalues of the three-dimensional square torus

TL;DR

This paper classifies all eigenvalues of the Laplacian on a 3D flat torus that meet Courant's nodal domain theorem with equality, using bounds, inequalities, and geometric analysis.

Contribution

It provides a complete characterization of Courant-sharp eigenvalues for the 3D flat torus, extending previous methods to a new geometric setting.

Findings

Identifies all Courant-sharp eigenvalues on the 3D torus.

Develops bounds and inequalities specific to the torus geometry.

Connects spectral properties with isoperimetric inequalities.

Abstract

In this paper, we determine, in the case of the Laplacian on the flat three-dimensional torus $(\mathbb{R}/\mathbb{Z})^3$, all the eigenvalues having an eigenfunction which satisfies the Courant nodal domains theorem with equality (Courant-sharp situation). Following the strategy of {\AA}. Pleijel (1956), the proof is a combination of an explicit lower bound of the counting function and a Faber-Krahn-type inequality for domains on the torus, deduced as, in the work of P. B\'erard and D. Meyer (1982), from an isoperimetric inequality. This inequality relies on the work of L. Hauswirth, J. Perez, P. Romon, and A. Ros (2004) on the periodic isoperimetric problem.