The weak Pleijel theorem with geometric control
| Autor: | Pierre Bérard, Bernard Helffer |
|---|---|
| Přispěvatelé: | Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques Jean Leray (LMJL), Université de Nantes - Faculté des Sciences et des Techniques, Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS), Institut Fourier (IF ), Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN) |
| Rok vydání: | 2016 |
| Předmět: |
Mathematics - Differential Geometry
Open set FOS: Physical sciences Dirichlet Laplacian 01 natural sciences Omega Upper and lower bounds Mathematics - Spectral Theory Combinatorics Nodal domains [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] Simple (abstract algebra) 0103 physical sciences FOS: Mathematics Pleijel theorem 0101 mathematics Spectral Theory (math.SP) Mathematical Physics Eigenvalues and eigenvectors Mathematics Courant theorem 010102 general mathematics Order (ring theory) Statistical and Nonlinear Physics Mathematical Physics (math-ph) Mathematics::Spectral Theory Eigenfunction MSC 2010: 35P15 49R50 Differential Geometry (math.DG) [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] Bounded function 010307 mathematical physics Geometry and Topology [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP] |
| Zdroj: | Journal of Spectral Theory Journal of Spectral Theory, European Mathematical Society, 2016, 6 (4), pp.717--733 |
| ISSN: | 1664-039X 1664-0403 |
| DOI: | 10.4171/jst/138 |
| Popis: | Let $\Omega\subset \mathbb R^d\,, d\geq 2$, be a bounded open set, and denote by $\lambda\_j(\Omega), j\geq 1$, the eigenvalues of the Dirichlet Laplacian arranged in nondecreasing order, with multiplicities. The weak form of Pleijel's theorem states that the number of eigenvalues $\lambda\_j(\Omega)$, for which there exists an associated eigenfunction with precisely $j$ nodal domains (Courant-sharp eigenvalues), is finite. The purpose of this note is to determine an upper bound for Courant-sharp eigenvalues, expressed in terms of simple geometric invariants of $\Omega$. We will see that this is connected with one of the favorite problems considered by Y. Safarov. Comment: Revised Oct. 12, 2016. To appear in Journal of Spectral Theory 6 (2016) |
| Databáze: | OpenAIRE |
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