Non-coercive first order Mean Field Games

We study first order evolutive Mean Field Games where the Hamiltonian are non-coercive. This situation occurs, for instance, when some directions are “forbidden” to the generic player at some points. We establish the existence of a weak solution of the system via a vanishing viscosity method and, mainly, we prove that the evolution of the population's density is the push-forward of the initial density through the flow characterized almost everywhere by the optimal trajectories of the control problem underlying the Hamilton-Jacobi equation. As preliminary steps, we need to prove that the optimal trajectories for the control problem are unique (at least for a.e. starting points) and that the corresponding unique optimal control has a feedback expression in terms of the intrinsic gradient of the value function.