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, Berard–Helffer, for the problem with a Dirichlet boundary condition (which corresponds to $$h=+\infty $$). 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 (which corresponds to $$h=0$$) to the case where $$ h>0$$ is small.