Courant-sharp Robin eigenvalues for the square: The case of negative Robin parameter

Abstract

We consider the cases where there is equality in Courant’s nodal domain theorem for the Laplacian with a Robin boundary condition on the square. In our previous two papers, we treated the cases where the Robin parameter h > 0 is large, small respectively. In this paper we investigate the case where h < 0.

On Pleijel's nodal domain theorem for the Robin problem

“…A Robin eigenvalue is called Courant‐sharp if it has a corresponding eigenfunction with exactly $$k$$ nodal domains; note that an immediate consequence of the Pleijel theorem for $$C^{1,1}$$ domains is that the number of Courant‐sharp Robin eigenvlaues is finite. Gittins and Helffer [11, 12] studied upper bounds on the number of Courant‐sharp eigenvalues of the Robin problem on a square when the Robin parameter $$h$$ is constant. In particular, they show that the Robin Laplacian on a square with constant parameter $$h$$ has finitely many Courant‐sharp eigenvalues.…”