*Accepted Paper*

**Inserted:** 22 nov 2018

**Last Updated:** 13 may 2019

**Journal:** Journal of the London Mathematical Society

**Year:** 2018

**Abstract:**

We consider variational Mean Field Games 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 is augmented by a pressure effect, i.e. 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 $H^1$ and $L^\infty$ 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.

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