Pleijel’s nodal domain theorem for Neumann and Robin eigenfunctions
Annales de l'Institut Fourier, to appear, 19 p.

We show that equality in Courant’s nodal domain theorem can only be reached for a finite number of eigenvalues of the Neumann Laplacian, in an open, bounded, and connected subset of ℝ n with a C 1,1 boundary, when n≥2. This result is analogous to the theorem proved by Pleijel in 1956 for the Dirichlet Laplacian. We also show that the argument and the result extend to a class of Robin boundary conditions.

Nous montrons que le cas d’égalité dans le théorème de Courant n’est réalisé que pour un nombre fini de valeurs propres du laplacien de Neumann, dans un ouvert borné connexe de ℝ n à bord C 1,1 , lorsque n≥2. Ce résultat est analogue au théorème démontré par Pleijel en 1956 pour le laplacien de Dirichlet. Nous montrons de plus que la méthode de démonstration et le résultat peuvent être étendus à une classe de conditions au bord de Robin.

Received : 2016-12-14
Revised : 2017-11-09
Accepted : 2018-02-02
Published online : 2019-03-08
Classification:  35P05,  35P15,  35P20,  58J50
Keywords: Neumann eigenvalues, Robin eigenvalues, nodal domains, Courant’s theorem, Pleijel’s theorem 
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     author = {L\'ena, Corentin},
     title = {Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
}

Léna, Corentin. Pleijel’s nodal domain theorem for Neumann and Robin eigenfunctions. Annales de l'Institut Fourier, to appear, 19 p.

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