Global-in-time regularity via duality for congestion-penalized Mean Field Games

Adam Prosinski (1) , Filippo Santambrogio (2)

1 University of Oxford
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay

After a brief introduction to one of the most typical problems in Mean Field Games, the congestion case (where agents pay a cost depending on the density of the regions they visit), and to its variational structure, we consider the question of the regularity of the optimal solutions. A duality argument, used for the first time in a paper by Y. Brenier on incompressible fluid mechanics, and recently applied to MFG with density constraints, allows to easily get some Sobolev regularity, locally in space and time. In the paper we prove that a careful analysis of the behaviour close to the final time allows to extend the same result including t = T .

Mots clés: mean field games; convex duality; Wasserstein space; Benamou-Brenier

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Adam Prosinski, Filippo Santambrogio. Global-in-time regularity via duality for congestion-penalized Mean Field Games. 2016. ⟨hal-01295289⟩