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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On nodal domains in Euclidean balls

Authors: Bernard Helffer and Mikael Persson Sundqvist
Journal: Proc. Amer. Math. Soc. 144 (2016), 4777-4791
MSC (2010): Primary 35B05, 35P20, 58J50
Published electronically: April 20, 2016
MathSciNet review: 3544529
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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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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

Mikael Persson Sundqvist
Affiliation: Department of Mathematical Sciences, Lund University, Box 118, 221 00 Lund, Sweden

Keywords: Nodal domains, Courant theorem, ball, Dirichlet, Neumann
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
Article copyright: © Copyright 2016 American Mathematical Society