The talk is about two overdetermined problems in spectral theory, concerning the Laplacian operator. These problems are known as Schiffer's conjectures and they 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 the 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.