On nodal domains in Euclidean balls
(2016) In Proceedings of the American Mathematical Society 144(11). p.47774791 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 ℝ^{d}, 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 ℝ^{d}, 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^{2} was obtained by B. Helffer, T. HoffmannOstenhof 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

 scopus:84987842178
 wos:000384000300022
 ISSN
 00029939
 DOI
 10.1090/proc/13098
 language
 English
 LU publication?
 yes
 id
 a3860d10cb3848548de0474edcfaccc7
 date added to LUP
 20170223 16:39:20
 date last changed
 20180107 11:52:30
@article{a3860d10cb3848548de0474edcfaccc7, 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. HoffmannOstenhof and S. Terracini.</p>}, author = {Helffer, Bernard and Sundqvist, Mikael Persson}, issn = {00029939}, keyword = {Ball,Courant theorem,Dirichlet,Neumann,Nodal domains}, language = {eng}, number = {11}, pages = {47774791}, 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}, }