Courant-sharp eigenvalues of a two-dimensional torus
Résumé
In this paper, we determine, in the case of the Laplacian on the flat two-dimensional torus (R/Z) 2 , all the eigenvalues having an eigenfunction which satisfies Courant's theorem with equality (Courant-sharp situation). Following the strategy o A. Pleijel (1956), 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 Bérard-Meyer from an isoperimetric inequality), with an explicit upper bound on the area.
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