  The time evolution of a collisionless plasma is modeled by the relativistic
Vlasov-Maxwell system which couples the Vlasov equation (the transport
equation) with the Maxwell equations of electrodynamics. We consider the case
that the plasma consists of $N$ particle species, the particles are located in
a bounded container $\Omega\subset\mathbb R^3$, and are subject to boundary
conditions on $\partial\Omega$. Furthermore, there are external currents,
typically in the exterior of the container, that may serve as a control of the
plasma if adjusted suitably. We do not impose perfect conductor boundary
conditions for the electromagnetic fields, but consider the fields as functions
on whole space $\mathbb R^3$ and model objects, that are placed in space, via
given matrix-valued functions $\varepsilon$ (the permittivity) and $\mu$ (the
permeability). A weak solution concept is introduced and existence of global in
time solutions is proved, as well as the redundancy of the divergence part of
the Maxwell equations in this weak solution concept.
