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We consider minimization problems for curves of measure, with kinetic andpotential energy and a congestion penalization, as in the functionals thatappear in Mean Field Games with a variational structure. We prove L infinityregularity results for the optimal density, which can be applied to therigorous derivations of equilibrium conditions at the level of each agent'strajectory, via time-discretization arguments, displacement convexity, andsuitable Moser iterations. Similar L infinity results have already been foundby P.-L. Lions in his course on Mean Field Games, using a proof based on theuse of a (very degenerate) elliptic equation on the dual potential (the valuefunction) phi, in the case where the initial and final density were prescribed(planning problem). Here the strategy is highly different, and allows forinstance to prove local-in-time estimates without assumptions on the initialand final data, and to insert a potential in the dynamics.

Optimal density evolution with congestion: L infinity bounds via flow interchange techniques and applications to variational Mean Field Games