Lecture notes on variational mean field games

Abstract

These lecture notes aim at giving the details presented in the short course (6 h) 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∞ 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.