We provide the existence of classical solutions to stationary mean-field game systems in the whole space 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 nonconvex 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.
Concentration of ground states in stationary mean-field games systems / Cesaroni, Annalisa; Cirant, Marco. - In: ANALYSIS & PDE. - ISSN 2157-5045. - 12:3(2019), pp. 737-787. [10.2140/apde.2019.12.737]
Concentration of ground states in stationary mean-field games systems
Cirant, Marco
2019-01-01
Abstract
We provide the existence of classical solutions to stationary mean-field game systems in the whole space 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 nonconvex 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.