Calculus of Variations and Geometric Measure Theory

F. Santambrogio

Lecture notes on Variational Mean Field Games

created by santambro on 20 Apr 2020
modified on 15 Sep 2020


Accepted Paper

Inserted: 20 apr 2020
Last Updated: 15 sep 2020

Year: 2020

to appear as a chapter in Mean Field Games – Cetraro, Italy, 2019, Cardaliaguet and Porretta (Eds), Springer, C.I.M.E. Foundation Subseries.


These lecture notes aim at giving the details presented in the short course (6h) given in Cetraro, in the CIME School about MFG of June 2019. The topics which are covered concern first-order MFG with local couplings, and the main goal is to prove that minimizers of a suitably expressed global energy are equilibria in the sense that a.e. trajectory solves a control problem with a running cost depending on the density of all the agents. Both the case of a cost penalizing high densities and of an $L^\infty$ constraint on the same densities are considered. The details of a construction to prove that minimizers actually define equilibria are presented under a boundedness assumption of the running cost, which is proven in the relevant cases.