Courant-sharp eigenvalues of the Laplacian on Euclidean domains

Let $\Omega \subset \mathbb{R}^n$, $n \geq 2$, be a bounded, connected, open set with Lipschitz boundary.
Let $u$ be an eigenfunction of the Laplacian on $\Omega$ with either a Dirichlet, Neumann or Robin boundary condition.
If an eigenfunction $u$ associated with the $k$--th eigenvalue has exactly $k$ nodal domains, then we call it a Courant-sharp eigenfunction. In this case, we call the corresponding eigenvalue a Courant-sharp eigenvalue.

We first discuss some known results for the Courant-sharp Dirichlet and Neumann eigenvalues of the Laplacian on Euclidean domains.

We then discuss whether the Robin eigenvalues of the Laplacian on the square are Courant-sharp.

This is based on joint work with B. Helffer (Université de Nantes).