Courant-sharp eigenvalues of the three-dimensional square torus

Abstract

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