On nodal domains in Euclidean balls
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- by Bernard Helffer and Mikael Persson Sundqvist
- Proc. Amer. Math. Soc. 144 (2016), 4777-4791
- DOI: https://doi.org/10.1090/proc/13098
- Published electronically: April 20, 2016
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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 $\geq 2$. 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 $\mathbb R^d$, $d\geq 2$. It is the first result of this type holding in any dimension. The corresponding result for the Dirichlet Laplacian in the disc in $\mathbb R^2$ was obtained by B. Helffer, T. Hoffmann-Ostenhof and S. Terracini.
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Bibliographic Information
- Bernard Helffer
- Affiliation: Laboratoire de Mathématiques UMR CNRS 8628, Université Paris-Sud - Bât 425, F-91405 Orsay Cedex, France — and — Laboratoire de Mathématiques Jean Leray, Université de Nantes, France
- MR Author ID: 83860
- Email: bernard.helffer@math.u-psud.fr
- Mikael Persson Sundqvist
- Affiliation: Department of Mathematical Sciences, Lund University, Box 118, 221 00 Lund, Sweden
- MR Author ID: 1100569
- Email: mickep@maths.lth.se
- Received by editor(s): August 13, 2015
- Received by editor(s) in revised form: January 6, 2016
- Published electronically: April 20, 2016
- Communicated by: Michael Hitrik
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 4777-4791
- MSC (2010): Primary 35B05, 35P20, 58J50
- DOI: https://doi.org/10.1090/proc/13098
- MathSciNet review: 3544529