Constrained Mean Field Games with Grushin type dynamics

This paper is devoted to a class of finite horizon deterministic mean field games with Grushin type dynamics, state constraints and nonlocal coupling. First, we consider the optimal control problem that each agent aims to solve when the evolution of the population is given and we establish some properties as: the existence of an optimal trajectory for any starting point $(x,t)$, the closed graph property for the multivalued map which associates to each point $(x,t)$ the set of optimal trajectories starting from that point, endowed with a suitable notion of convergence, the continuity of the value function. The main issue to overcome is due to the local interplay at boundary points between the set of state constraints and the degenerate dynamics. To this end, we shall point out two different sets of assumptions which are both sufficient for these properties. Afterwards, we tackle the mean field games; taking advantage of the aforementioned properties, we prove the existence of a relaxed equilibrium (which describes the evolution of the game in terms of a probability on the set of admissible trajectories) and derive the existence of a mild solution (which is a couple formed by the value function for the generic player and a family of time dependent measures on the state).