New estimates on the regularity of the pressure in density-constrained mean field games

We consider variational mean field games (MFGs) endowed with a constraint on the maximal density of the distribution of players. Minimizers of the variational formulation are equilibria for a game where both the running cost and the final cost of each player are augmented by a pressure effect, that is, a positive cost concentrated on the set where the density saturates the constraint. Yet, this pressure is a priori only a measure and regularity is needed to give a precise meaning to its integral on trajectories. We improve, in the limited case where the Hamiltonian is quadratic, which allows to use optimal transport techniques after time-discretization, the results obtained in a paper of the second author with Cardaliaguet and Mészáros. We prove (Formula presented.) and (Formula presented.) regularity under very mild assumptions on the data, and explain the consequences for the MFG, in terms of the value function and of the Lagrangian equilibrium formulation.