On the Courant-sharp Dirichlet eigenvalues of Euclidean sets

Consider an arbitrary open set Omega in Euclidean space with finite Lebesgue measure.
Let lambda_k(Omega) denote the eigenvalues of the Dirichlet Laplace operator acting in L^2(Omega).
The Dirichlet eigenvalue lambda_k(Omega) is Courant-sharp if the corresponding eigenfunction has exactly k nodal domains.

In the first part of the talk we give an introduction to this topic, including a proof of Pleijel's theorem, and review
some recent results for the Courant-sharp Dirichlet eigenvalues of Euclidean sets.

In the second part of the talk we obtain an upper bound for the largest Courant-sharp Dirichlet eigenvalue of $\Omega$, and deduce an upper bound for the number of Courant-sharp Dirichlet eigenvalues of $\Omega$.

This is joint work with M.  van den Berg, and extends recent results due to P. Bérard and B. Helffer.