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 and overdetermination in boundary conditions usually takes place. A Lipschitz stability estimate is obtained. This estimate implies uniqueness. A new Carleman estimate is derived. The second estimate the author calls "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 Aveboukh in arXiv: 2302.10709, 2023. Applications are discussed.