Sharp semi-concavity in a non-autonomous control problem and Lp estimates in an optimal-exit MFG

• Published in: Nonlinear differential equations and applications Vol. 27; no. 2
• Main Authors: Dweik, Samer, Mazanti, Guilherme
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 2020; Springer Nature B.V
• Subjects: Analysis; Concavity; Game theory; Mathematics; Mathematics and Statistics; Optimal control; Regularity; Travel time; 91A13; MFG system; 49N70; Semi-concavity of the value function; 93C15; estimate; 35B65; Mean field games; Non-autonomous optimal control; 49J15
• ISSN: 1021-9722; 1420-9004
• DOI: 10.1007/s00030-019-0612-4

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.