Courant-sharp eigenvalues of a two-dimensional torus

In this note, we determine, in the case of the Laplacian on the flat two-dimensional torus (R/Z)(2), all the eigenvalues having an eigenfunction that satisfies Courant's theorem with equality (Courant-sharp situation). Following the strategy of angstrom. Pleijel (1956) [18], the proof is a combination of a lower bound (a la Weyl) of the counting function, with an explicit remainder term, and of a Faber-Krahn inequality for domains on the torus (deduced as in the work of P. Berard and D. Meyer from an isoperimetric inequality), with an explicit upper bound on the area. (C) 2015 Published by Elsevier Masson SAS on behalf of Academie des sciences.