Upper bounds for Courant-sharp Neumann and Robin eigenvalues

Abstract

We consider the eigenvalues of the Laplacian on an open, bounded, connected set in $\mathbb{R}^n$ with $C^2$ boundary, with a Neumann boundary condition or a Robin boundary condition. We obtain upper bounds for those eigenvalues that have a corresponding eigenfunction which achieves equality in Courant's Nodal Domain theorem. In the case where the set is also assumed to be convex, we obtain explicit upper bounds in terms of some of the geometric quantities of the set. Corrigendum. A previous version of this work was accepted and published by the "Bulletin de la Soci\'et\'e Math\'ematique de France" (see [2] in the bibliography of Appendix B). It contained a gap: the classical (Euclidean) Faber-Krahn inequality was applied in a setting where it might not hold. This version reproduces the previous one with the addition of a corrigendum in Appendix B that addresses the issue. All the results in Sections 2--8 and most of those in Section 9 are thus preserved.