Perturbation theory for Dirichlet eigenvalues in perforated domains

In this talk I will present some recent results on asymptotics of eigenvalues of the Dirichlet Laplacian when a small compact set is removed from the initial domain. If the small set is concentrating at a point in some sense, the eigenvalue variation is proved to be strictly related to the vanishing order of one of the relative eigenfunctions at that point. A good understanding of this asymptotics leads to new issues, for instance optimal location or optimal shape of the hole (open problem) as well as possible ramification of multiple eigenvalues.