Some results about Schiffer's conjectures

We study two overdetermined problems in spectral theory, about the Laplace operator. These problems are known as Schiffer's conjectures and are related to the Pompeiu problem. We show the connection between these problems and the critical points of the functional eigenvalue with a volume constraint. We use this fact, together with the continuous Steiner symmetrization, to give another proof of Serrin's result for the first Dirichlet eigenvalue. In two dimensions and for a general simple eigenvalue, we obtain different integral identities and a new overdetermined boundary value problem.