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

Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-01962755
Contributor : Guilherme Mazanti <>
Submitted on : Sunday, November 3, 2019 - 5:28:14 PM
Last modification on : Saturday, October 3, 2020 - 4:15:31 AM

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• HAL Id : hal-01962755, version 2

### Citation

Samer Dweik, Guilherme Mazanti. Sharp semi-concavity in a non-autonomous control problem and $L^p$ estimates in an optimal-exit MFG. Nonlinear Differential Equations and Applications, Springer Verlag, 2020, 27 (2), pp.11. ⟨hal-01962755v2⟩