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Sharp semi-concavity in a non-autonomous control problem and \(\pmb {L^p}\) estimates in an optimal-exit MFG

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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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Acknowledgements

The authors would like to thank Filippo Santambrogio for the fruitful discussions that lead to this paper.

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  1. Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405, Orsay, France

    Samer Dweik & Guilherme Mazanti

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  1. Samer Dweik
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Correspondence to Samer Dweik.

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This work was partially supported by a public grant as part of the “Investissement d’avenir” project, reference ANR-11-LABX-0056-LMH, LabEx LMH, PGMO project VarPDEMFG. The first author was also partially supported by the by the French ANR project “GEOMETRYA”, reference ANR-12-BS01-0014, and by the Région Ile-de-France. The second author was also partially supported by the French ANR project “MFG”, reference ANR-16-CE40-0015-01, and by the Hadamard Mathematics LabEx (LMH) through the Grant No. ANR-11-LABX-0056-LMH in the “Investissement d’avenir” project.

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Dweik, S., Mazanti, G. Sharp semi-concavity in a non-autonomous control problem and \(\pmb {L^p}\) estimates in an optimal-exit MFG. Nonlinear Differ. Equ. Appl. 27, 11 (2020). https://doi.org/10.1007/s00030-019-0612-4

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