Pleijel’s nodal domain theorem for Neumann and Robin eigenfunctions
Annales de l'Institut Fourier, Volume 69 (2019) no. 1, pp. 283-301.

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 n with a C 1,1 boundary, when n2. 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 n à bord C 1,1 , lorsque n2. 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.

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Accepted:
Published online:
DOI: 10.5802/aif.3243
Classification: 35P05, 35P15, 35P20, 58J50
Keywords: Neumann eigenvalues, Robin eigenvalues, nodal domains, Courant’s theorem, Pleijel’s theorem
Mot clés : valeurs propres de Neumann, valeurs propres de Robin, domaines nodaux, théorème de Courant, théorème de Pleijel
Léna, Corentin 1

1 Grupo de Física Matemática Universidade de Lisboa Departamento de Matemática Faculdade de Cienciâs Campo Grande, Edifício C6 1749-016 Lisboa (Portugal)
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Léna, Corentin. Pleijel’s nodal domain theorem for Neumann and Robin eigenfunctions. Annales de l'Institut Fourier, Volume 69 (2019) no. 1, pp. 283-301. doi : 10.5802/aif.3243. http://www.numdam.org/articles/10.5802/aif.3243/

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