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- W2002960883 abstract "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 Å. Pleijel (1956) [18], the proof is a combination of a lower bound (à 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. Bérard and D. Meyer from an isoperimetric inequality), with an explicit upper bound on the area. Cette note vise à déterminer quelles sont les valeurs propres du laplacien sur le tore plat (R/Z)2 qui ont une fonction propre réalisant le cas d'égalité dans le théorème de Courant (situation Courant-stricte). Nous suivons la stratégie de Å. Pleijel (1956) [18], qui associe une borne inférieure de type loi de Weyl pour la fonction de comptage et une inégalité de type Faber–Krahn. Comme dans les travaux de P. Bérard et D. Meyer, cette dernière est déduite d'une inégalité isopérimétrique, avec une condition de petitesse, ici explicitée, sur l'aire du domaine." @default.
- W2002960883 created "2016-06-24" @default.
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- W2002960883 date "2015-06-01" @default.
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- W2002960883 title "Courant-sharp eigenvalues of a two-dimensional torus" @default.
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- W2002960883 doi "https://doi.org/10.1016/j.crma.2015.03.014" @default.
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