Courant-sharp eigenvalues of a two-dimensional torus

Corentin Léna (1)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay

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.

Mots clés: Courant theorem; torus; Pleijel theorem; Nodal domains

Domaines: Equations aux dérivées partielles [math.AP]; Théorie spectrale [math.SP]

Corentin Léna. Courant-sharp eigenvalues of a two-dimensional torus. Comptes Rendus. Mathématique, 2015, 353 (6), pp.535-539. ⟨10.1016/j.crma.2015.03.014⟩. ⟨hal-01101926v3⟩