  This paper considers a mean field game model inspired by crowd motion where
agents want to leave a given bounded domain through a part of its boundary in
minimal time. Each agent is free to move in any direction, but their maximal
speed is bounded in terms of the average density of agents around their
position in order to take into account congestion phenomena.
  After a preliminary study of the corresponding minimal-time optimal control
problem, we formulate the mean field game in a Lagrangian setting and prove
existence of Lagrangian equilibria using a fixed point strategy. We provide a
further study of equilibria under the assumption that agents may leave the
domain through the whole boundary, in which case equilibria are described
through a system of a continuity equation on the distribution of agents coupled
with a Hamilton--Jacobi equation on the value function of the optimal control
problem solved by each agent. This is possible thanks to the semiconcavity of
the value function, which follows from some further regularity properties of
optimal trajectories obtained through Pontryagin Maximum Principle. Simulations
illustrate the behavior of equilibria in some particular situations.
