  We consider a typical problem in Mean Field Games: the congestion case, where
in the cost that agents optimize there is a penalization for passing through
zones with high density of agents, in a deterministic framework. This
equilibrium problem is known to be equivalent to the optimization of a global
functional including an $L^p$ norm of the density. The question arises as to
produce a similar model replacing the $L^p$ penalization with an $L^\infty$
constraint, but the simplest approaches do not give meaningful definitions.
Taking into account recent works about crowd motion, where the density
constraint $\rho\leq 1$ was treated in terms of projections of the velocity
field onto the set of admissible velocity (with a constraint on the divergence)
and a pressure field was introduced, we propose a definition and write a system
of PDEs including the usual Hamilton-Jacobi equation coupled with the
continuity equation. For this system, we analyze an example and propose some
open problems.
