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 relate d 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. I n two dimensions and for a general simple eigenvalue, we obtain different i ntegral identities and a new overdetermined boundary value problem.