Weak-strong uniqueness for solutions to mean-field games

This paper addresses the crucial question of solution uniqueness in stationary first-order Mean-Field Games (MFGs). Despite well-established existence results, establishing uniqueness, particularly for weaker solutions in the sense of monotone operators, remains an open challenge. Building upon the framework of monotonicity methods, we introduce a linearization method that enables us to prove a weak-strong uniqueness result for stationary MFG systems on the d-dimensional torus. In particular, we give explicit conditions under which this uniqueness holds.