The mean field games system: Carleman estimates, Lipschitz stability and uniqueness

Abstract An overdetermination is introduced in an initial condition for the second order mean field games system (MFGS). This makes the resulting problem close to the classical ill-posed Cauchy problems for PDEs. Indeed, in such a problem an overdetermination in boundary conditions usually takes place. A Lipschitz stability estimate is obtained. This estimate implies uniqueness. A new Carleman estimate is derived. This latter estimate is called “quasi-Carleman estimate”, since it contains two test functions rather than a single one in conventional Carleman estimates. These two estimates play the key role. Carleman estimates were not applied to the MFGS prior to the recent work of Klibanov and Averboukh in [M. V. Klibanov and Y. Averboukh, Lipschitz stability estimate and uniqueness in the retrospective analysis for the mean field games system via two Carleman estimates, preprint 2023, https://arxiv.org/abs/2302.10709].