Calculus of Variations and Geometric Measure Theory

A. Cesaroni - M. Cirant

Concentration of ground states in stationary Mean-Field Games systems

created by cesaroni on 30 May 2017
modified on 01 Jul 2021


Published Paper

Inserted: 30 may 2017
Last Updated: 1 jul 2021

Journal: Anal. PDE
Volume: 12
Number: 3
Pages: 737-787
Year: 2017


In this paper we provide the existence of classical solutions to stationary mean field game systems in the whole space $\mathbb{R}^N$, with coercive potential and aggregating local coupling, under general conditions on the Hamiltonian. The only structural assumption we make is on the growth at infinity of the coupling term in terms of the growth of the Hamiltonian. This result is obtained using a variational approach based on the analysis of the non-convex energy associated to the system. Finally, we show that in the vanishing viscosity limit mass concentrates around the flattest minima of the potential. We also describe the asymptotic shape of the rescaled solutions in the vanishing viscosity limit, in particular proving the existence of ground states, i.e. classical solutions to mean field game systems in the whole space without potential, and with aggregating coupling.