Search

Your search keyword '"Courant theorem"' showing total 2 results
Start Over Descriptor "Courant theorem" Publisher european mathematical society - ems - publishing house gmbh
2 results on '"Courant theorem"'

1. Courant-sharp property for Dirichlet eigenfunctions on the Möbius strip.

Author

Bérard, Pierre, Helffer, Bernard, and Kiwan, Rola

Subjects
*EIGENFUNCTIONS, *EIGENVALUES, *SPECTRAL theory, *TRIANGLES, *TORUS
Abstract

The question of determining for which eigenvalues there exists an eigenfunction which has the same number of nodal domains as the label of the associated eigenvalue (Courant-sharp property) was motivated by the analysis of minimal spectral partitions. In previous works, many examples have been analyzed corresponding to squares, rectangles, disks, triangles, tori, . . . . A natural toy model for further investigations is the Mo¨ bius strip, a non-orientable surface with Euler characteristic 0, and particularly the "square" Mo¨ bius strip whose eigenvalues have higher multiplicities. In this case, we prove that the only Courant-sharp Dirichlet eigenvalues are the first and the second, and we exhibit peculiar nodal patterns. [ABSTRACT FROM AUTHOR]

Published
2021
Full Text
View/download PDF

2. The weak Pleijel theorem with geometric control

Author

Pierre Bérard, Bernard Helffer, 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), and Université de Nantes (UN)-Université de Nantes (UN)

Subjects
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]
Abstract

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)

Published
2016
Full Text
View/download PDF

Catalog

Books, media, physical & digital resources
See catalog results