Small Oscillations in Some Nonlinear PDE's

In this paper we study existence of families of periodic orbits close to equilibria of Lagrangian PDE's. We begin by a short survey on the problem, then we illustrate a method recently introduced by the authors to construct small amplitude periodic solutions of some semilinear partial differential equations. As a difference with respect to standard ones it is based on contraction mapping principle, and avoids the use of KAM techinques. Applications to some concrete PDE's are also given. Among others we find small oscillations for a nonlinear plate equation in arbitrary space dimension.