Boundary value problem for the three dimensional time periodic Vlasov-Maxwell system

In this work we study the existence of time periodic weak solution for the three dimensional Vlasov-Maxwell system with boundary conditions. The main idea consists of using the mass, momentum and energy conservation laws which allow us to obtain a priori estimates in the case of a star-shaped bounded spatial domain. We start by constructing time periodic smooth solutions for a regularized system. The existence for the Vlasov-Maxwell system follows by weak stability underuniform estimates. These results apply for both classical and relativistic cases and for systems with several species of particles.