  This paper is devoted to finite horizon deterministic mean field games in
which the state space is a network. The agents control their velocity, and when
they occupy a vertex, they can enter into any incident edge. The running and
terminal costs are assumed to be continuous in each edge but not necessarily
globally continuous on the network. A Lagrangian formulation is proposed and
studied. It leads to relaxed equilibria consisting of probability measures on
admissible trajectories. The existence of such relaxed equilibria is obtained.
The proof requires the existence of optimal trajectories and a closed graph
property for the map which associates to each point the set of optimal
trajectories starting from that point. To any relaxed equilibrium corresponds a
mild solution of the mean field game, i.e. a pair $(u,m)$ made of the value
function $u$ of a related optimal control problem, and a family $m= (m(t))_t$
of probability measures on the network. Given $m$, the value function $u$ is
characterized by a Hamilton-Jacobi problem on the network. Regularity
properties of $u$ and a weak form of a Fokker-Planck equation satisfied by $m$
are investigated.
