  This paper studies a mean field game inspired by crowd motion in which agents
evolve in a compact domain and want to reach its boundary minimizing the sum of
their travel time and a given boundary cost. Interactions between agents occur
through their dynamic, which depends on the distribution of all agents.
  We start by considering the associated optimal control problem, showing that
semi-concavity in space of the corresponding value function can be obtained by
requiring as time regularity only a lower Lipschitz bound on the dynamics. We
also prove differentiability of the value function along optimal trajectories
under extra regularity assumptions.
  We then provide a Lagrangian formulation for our mean field game and use
classical techniques to prove existence of equilibria, which are shown to
satisfy a MFG system. Our main result, which relies on the semi-concavity of
the value function, states that an absolutely continuous initial distribution
of agents with an $L^p$ density gives rise to an absolutely continuous
distribution of agents at all positive times with a uniform bound on its $L^p$
norm. This is also used to prove existence of equilibria under fewer regularity
assumptions on the dynamics thanks to a limit argument.
