Calculus of Variations and Geometric Measure Theory

F. Santambrogio

A Modest Proposal for MFG with Density Constraints

created by santambro on 31 Oct 2011
modified on 01 Nov 2011


Submitted Paper

Inserted: 31 oct 2011
Last Updated: 1 nov 2011

Journal: proceedings of the workshop "Mean Field Games and related topics", Rome, May 2011
Year: 2011


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.