Calculus of Variations and Geometric Measure Theory

P. Cardaliaguet - A. R. Mészáros - F. Santambrogio

First order Mean Field Games with density constraints: Pressure equals Price

created by mészáros on 07 Jul 2015
modified by santambro on 20 Jul 2016


Accepted Paper

Inserted: 7 jul 2015
Last Updated: 20 jul 2016

Journal: SIAM J. Contr. Opt.
Year: 2016


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 à 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.