Inserted: 31 mar 2017
Last Updated: 2 jun 2017
If you are the referee, or an interested reader: we mention in our paper the lecture of November 27, 2011, by P-L Lions (and we compare our results to what he does): this lecture does not exist, sorry... me meant 2009, not 2011.
We consider minimization problems for curves of measure, with kinetic and potential energy and a congestion penalization, as in the functionals that appear in Mean Field Games with a variational structure. We prove $L^\infty$ regularity results for the optimal density, which can be applied to the rigorous derivations of equilibrium conditions at the level of each agent's trajectory, via time-discretization arguments, displacement convexity, and suitable Moser iterations. Similar $L^\infty$ results have already been found by P.-L. Lions in his course on Mean Field Games, using a proof based on the use of a (very degenerate) elliptic equation on the dual potential (the value function) $\varphi$, in the case where the initial and final density were prescribed (planning problem). Here the strategy is highly different, and allows for instance to prove local-in-time estimates without assumptions on the initial and final data, and to insert a potential in the dynamics.