Upper bounds for Courant-sharp Neumann eigenvalues

Let $\Omega$ be an open, bounded, connected set in $\R^n$, $n \geq 2$, with Lipschitz boundary.
We consider the eigenvalues of the Neumann Laplacian acting in $L^2(\Omega)$.
In particular, those that have a corresponding eigenfunction which achieves
equality in Courant's Nodal Domain theorem.
These eigenvalues are called Courant-sharp.

It was shown recently by C. Léna that an open, bounded, connected set in $\R^n$, $n \geq 2$,
with $C^{1,1}$ boundary has finitely many Courant-sharp Neumann eigenvalues.

We discuss upper bounds for the Courant-sharp Neumann eigenvalues of the Laplacian
of an open, bounded, connected set in $\R^n$, $n \geq 2$, with $C^2$ boundary.
In the case where the set is also convex, we present explicit upper bounds for
the Courant-sharp Neumann eigenvalues and the number of Courant-sharp Neumann
eigenvalues in terms of some of the geometric quantities of the set.

This is based on joint work with Corentin Léna (Stockholm University).