On the existence of classical solutions for stationary extended mean field games

Abstract

In this paper we consider extended stationary mean field games, that is mean-field games which depend on the velocity field of the players. We prove various a-priori estimates which generalize the results for quasi-variational mean field games in [GPSM12]. In addition we use adjoint method techniques to obtain higher regularity bounds. Then we establish existence of smooth solutions under fairly general conditions by applying the continuity method. When applied to standard stationary mean-field games as in [LL06a], [GSM11] or [GPSM12] this paper yields various new estimates and regularity properties not available previously. We discuss additionally several examples where existence of classical solutions can be proved.

Multi-population Mean Field Games systems with Neumann boundary conditions

“…In [9] existence is proved in the case M = 1 and with quadratic Hamiltonian by exploiting the Hopf-Cole change of variables, without any growth from above assumption on the cost; for more general Hamiltonians, a priori estimates on solutions are presented in [14]. A full proof of existence for general MFG systems in the one population case and periodic space is provided in [13], where more general estimates are supported by a continuation argument. However, some growth conditions of V with respect to m are required.…”

Existence of a solution to an equation arising from the theory of Mean Field Games

“…In the past five years alone, a large number of manuscripts have been devoted to it, revealing its importance, impact, and possible applications (see e.g. [7,8,15,[23][24][25][26][27][28]30,31,[33][34][35]39,[41][42][43][49][50][51][52]). In light of the publications [44][45][46][47], we restrict our study to the simplest framework of games: those with identical players.…”

Explicit solutions of one-dimensional, first-order, stationary mean-field games with congestion

“…For second-order problems, existence of stationary solutions was proven in [6]. In the stationary setting, existence and regularity of MFGs without congestion were considered in [10], [12], [14], [22], [23] (strong solutions) and [5], [19] (weak solutions). Many other stationary problems are examined in the literature, including obstacle problems [9], weaklycoupled systems [8], multi-populations [4], and logistic problems [13].…”