Sharp semi-concavity in a non-autonomous control problem and $L^p$ estimates in an optimal-exit MFG - Archive ouverte HAL Access content directly
Journal Articles Nonlinear Differential Equations and Applications Year : 2020

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

Samer Dweik
• Function : Author
Guilherme Mazanti

#### 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.

### Dates and versions

hal-01962755 , version 1 (20-12-2018)
hal-01962755 , version 2 (03-11-2019)

### Identifiers

• HAL Id : hal-01962755 , version 2
• ARXIV :
• DOI :

### Cite

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, 2020, 27 (2), pp.11. ⟨10.1007/s00030-019-0612-4⟩. ⟨hal-01962755v2⟩

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