Periodic solutions of the Vlasov-Poisson system with boundary conditions

We study the Vlasov-Poisson system with time periodic boundary conditions. For small data we prove existence of weak periodic solutions in any space d imension. In the one-dimensional case the result is stronger: we obtain exi stence of mild solution and uniqueness of this solution when the data are s mooth. It is necessary to impose a nonvanishing condition for the incoming velocities in order to control the lifetime of particles in the domain.