  In this paper we study Mean Field Game systems under density constraints as
optimality conditions of two optimization problems in duality. A weak solution
of the system contains an extra term, an additional price imposed on the
saturated zones. We show that this price corresponds to the pressure field from
the models of incompressible Euler's equations {\`a} la Brenier. By this
observation we manage to obtain a minimal regularity, which allows to write
optimality conditions at the level of single agent trajectories and to define a
weak notion of Nash equilibrium for our model.
