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

Abstract

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.