Courant-sharp eigenvalues of a two-dimensional torus
| Autor: | Corentin Léna |
|---|---|
| Přispěvatelé: | Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), ANR-12-BS01-0007,OPTIFORM,Optimisation de Formes(2012), European Project: 339958,EC:FP7:ERC,ERC-2013-ADG,COMPAT(2014), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11) |
| Rok vydání: | 2015 |
| Předmět: |
Pure mathematics
Clifford torus Nodal Domains 010103 numerical & computational mathematics 01 natural sciences Upper and lower bounds Mathematics - Spectral Theory Mathematics - Analysis of PDEs FOS: Mathematics [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] 35P05 35P20 58J50 Courant Theorem 0101 mathematics Remainder Spectral Theory (math.SP) Mathematics::Symplectic Geometry Eigenvalues and eigenvectors Mathematics 010102 general mathematics Mathematical analysis Partial Differential Equations Eigenvalues Torus General Medicine Mathematics::Spectral Theory Eigenfunction Analysis Spectral Theory Pleijel Theorem Isoperimetric inequality Laplace operator Analysis of PDEs (math.AP) [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP] |
| Zdroj: | Comptes Rendus. Mathématique Comptes Rendus. Mathématique, 2015, 353 (6), pp.535-539. ⟨10.1016/j.crma.2015.03.014⟩ Comptes Rendus. Mathématique, Académie des sciences (Paris), 2015, 353 (6), pp.535-539. ⟨10.1016/j.crma.2015.03.014⟩ |
| ISSN: | 1631-073X 1778-3569 |
| DOI: | 10.1016/j.crma.2015.03.014 |
| Popis: | Version abrégée pour publication - Shortened published version; International audience; 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. |
| Databáze: | OpenAIRE |
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