Courant-sharp Robin eigenvalues for the square -the case with small Robin parameter

K Gittins (1) , Bernard Helffer (2)

1 UNINE - Université de Neuchâtel = University of Neuchatel
2 LMJL - Laboratoire de Mathématiques Jean Leray

This article is the continuation of our first work on the determination of the cases where there is equality in Courant's Nodal Domain theorem in the case of a Robin boundary condition (with Robin parameter h). For the square, our first paper focused on the case where h is large and extended results that were obtained by Pleijel, Bérard-Helffer, for the problem with a Dirichlet boundary condition. There, we also obtained some general results about the behaviour of the nodal structure (for planar domains) under a small deformation of h, where h is positive and not close to 0. In this second paper, we extend results that were obtained by Helffer-Persson-Sundqvist for the Neumann problem to the case where h > 0 is small. MSC classification (2010): 35P99, 58J50, 58J37.

Mots clés: Courant-sharp; Robin eigenvalues; square *

Domaines: Mathématiques [math]; Théorie spectrale [math.SP]

K Gittins, Bernard Helffer. Courant-sharp Robin eigenvalues for the square -the case with small Robin parameter. Annales mathématiques du Québec, In press. ⟨hal-02369646⟩