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On nodal domains in Euclidean balls

Helffer, Bernard and Sundqvist, Mikael Persson LU (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.

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author
organization
publishing date
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
  • scopus:84987842178
  • wos:000384000300022
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
2017-09-18 11:35:04
@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},
  keyword      = {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},
  volume       = {144},
  year         = {2016},
}