Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions

Corentin Léna (1)

1 UNITO - Università degli studi di Torino = University of Turin

In this paper, 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 the case of an open, bounded and connected set in R n with a C 1,1 boundary. This result is analogous to Pleijel's nodal domain theorem for the Dirichlet Laplacian (1956). It confirms, in all dimensions, a conjecture formulated by Pleijel, which had already been solved by I. Polterovich for a two-dimensional domain with a piecewise-analytic boundary (2009). We also show that the argument and the result extend to a class of Robin boundary conditions.

Mots clés: Neumann eigenvalues; Robin eigenvalues; Nodal domains; Courant's theorem; Pleijel's theorem

Domaines: Equations aux dérivées partielles [math.AP]; Théorie spectrale [math.SP]

Corentin Léna. Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions. 2016. ⟨hal-01361943v2⟩