2023/03/02 by Michael V. Klibanov, Klibanov, Michael V. · 2 citations
Economics, Econometrics and Finance · Engineering · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Stability and Controllability of Differential Equations #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2303.03928
openalex publication_date 2023/03/02 · openalex created_date 2023/03/10 · openalex updated_date 2026/07/28
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.