Concentration of ground states in stationary Mean-Field Games systems

Abstract

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.

44 pages. This new version has been obtained by merging the previous one with the paper arXiv:1802.02753

References in corpus (1)

• On Solutions of Mean Field Games with Ergodic Cost

Cited by in corpus (3)

• Maximal $L^q$-regularity for parabolic Hamilton-Jacobi equations and applications to Mean Field Games
• $1$-dimensional multi-agent optimal control with aggregation and distance constraints: qualitative properties and mean-field limit
• Ergodic Mean-Field Games with aggregation of Choquard-type