On nodal domains in Euclidean balls
(2016) In Proceedings of the American Mathematical Society 144(11). p.4777-4791- Abstract
Å. Pleijel (1956) has proved that in the case of the Laplacian with Dirichlet condition, the equality in the Courant nodal theorem (Courant sharp situation) can only be true for a finite number of eigenvalues when the dimension is ≥ 2. Recently Polterovich extended the result to the Neumann problem in two dimensions in the case when the boundary is piecewise analytic. A question coming from the theory of spectral minimal partitions has motivated the analysis of the cases when one has equality in Courant’s theorem. We identify the Courant sharp eigenvalues for the Dirichlet and the Neumann Laplacians in balls in ℝd, d ≥ 2. It is the first result of this type holding in any dimension. The corresponding result for the Dirichlet... (More)
Å. Pleijel (1956) has proved that in the case of the Laplacian with Dirichlet condition, the equality in the Courant nodal theorem (Courant sharp situation) can only be true for a finite number of eigenvalues when the dimension is ≥ 2. Recently Polterovich extended the result to the Neumann problem in two dimensions in the case when the boundary is piecewise analytic. A question coming from the theory of spectral minimal partitions has motivated the analysis of the cases when one has equality in Courant’s theorem. We identify the Courant sharp eigenvalues for the Dirichlet and the Neumann Laplacians in balls in ℝd, d ≥ 2. It is the first result of this type holding in any dimension. The corresponding result for the Dirichlet Laplacian in the disc in ℝ2 was obtained by B. Helffer, T. Hoffmann-Ostenhof and S. Terracini.
(Less)
- author
- Helffer, Bernard and Sundqvist, Mikael Persson LU
- organization
- publishing date
- 2016
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Ball, Courant theorem, Dirichlet, Neumann, Nodal domains
- in
- Proceedings of the American Mathematical Society
- volume
- 144
- issue
- 11
- pages
- 15 pages
- publisher
- American Mathematical Society (AMS)
- external identifiers
-
- wos:000384000300022
- scopus:84987842178
- ISSN
- 0002-9939
- DOI
- 10.1090/proc/13098
- language
- English
- LU publication?
- yes
- id
- a3860d10-cb38-4854-8de0-474edcfaccc7
- date added to LUP
- 2017-02-23 16:39:20
- date last changed
- 2024-04-28 07:34:38
@article{a3860d10-cb38-4854-8de0-474edcfaccc7,
abstract = {{<p>Å. Pleijel (1956) has proved that in the case of the Laplacian with Dirichlet condition, the equality in the Courant nodal theorem (Courant sharp situation) can only be true for a finite number of eigenvalues when the dimension is ≥ 2. Recently Polterovich extended the result to the Neumann problem in two dimensions in the case when the boundary is piecewise analytic. A question coming from the theory of spectral minimal partitions has motivated the analysis of the cases when one has equality in Courant’s theorem. We identify the Courant sharp eigenvalues for the Dirichlet and the Neumann Laplacians in balls in ℝ<sup>d</sup>, d ≥ 2. It is the first result of this type holding in any dimension. The corresponding result for the Dirichlet Laplacian in the disc in ℝ<sup>2</sup> was obtained by B. Helffer, T. Hoffmann-Ostenhof and S. Terracini.</p>}},
author = {{Helffer, Bernard and Sundqvist, Mikael Persson}},
issn = {{0002-9939}},
keywords = {{Ball; Courant theorem; Dirichlet; Neumann; Nodal domains}},
language = {{eng}},
number = {{11}},
pages = {{4777--4791}},
publisher = {{American Mathematical Society (AMS)}},
series = {{Proceedings of the American Mathematical Society}},
title = {{On nodal domains in Euclidean balls}},
url = {{http://dx.doi.org/10.1090/proc/13098}},
doi = {{10.1090/proc/13098}},
volume = {{144}},
year = {{2016}},
}