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This paper studies a mean field game inspired by crowd motion in which agentsevolve in a compact domain and want to reach its boundary minimizing the sum oftheir travel time and a given boundary cost. Interactions between agents occurthrough their dynamic, which depends on the distribution of all agents.  We start by considering the associated optimal control problem, showing thatsemi-concavity in space of the corresponding value function can be obtained byrequiring as time regularity only a lower Lipschitz bound on the dynamics. Wealso prove differentiability of the value function along optimal trajectoriesunder extra regularity assumptions.  We then provide a Lagrangian formulation for our mean field game and useclassical techniques to prove existence of equilibria, which are shown tosatisfy a MFG system. Our main result, which relies on the semi-concavity ofthe value function, states that an absolutely continuous initial distributionof agents with an $L^p$ density gives rise to an absolutely continuousdistribution 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 regularityassumptions on the dynamics thanks to a limit argument.

Sharp semi-concavity in a non-autonomous control problem and $L^p$ estimates in an optimal-exit MFG