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.

W e are interested in two o verdetermined problems in spectral theory, k n o wn as Schi er's conjectures and related to the Pompeiu problem. We show the connection between these problems and the critical points of the eigenvalue functional with a volume constraint. In two dimensions, we use this fact to establish an integral identity satis ed by a conformal map.

We discuss a Schiffer's conjecture which is a symmetry problem for an overdetermined spectral p.d.e .. We show the connection between this problem and the critical points of the eigenvalue with a volume constraint as well as the Faber-Krahn inequality. We give two original proofs of these symmetry results in the case of the first eigenvalue. Keywords Domain derivative, first eigenvalue, Faber-Krahn inequality, continuous Steiner symmetrization This conjecture (SC)' has been more intensively studied than the previous one, because K. Malanowski et al. (eds.