On The Mean Field Games System With the Lateral Cauchy Data via Carleman Estimates

TL;DR

This paper develops new Carleman estimates to analyze the stability and uniqueness of solutions to the second order Mean Field Games system with lateral Cauchy data, providing insights into the system's robustness against data noise.

Contribution

The paper introduces two novel Carleman estimates that establish stability and uniqueness results for the MFG system with boundary data.

Findings

Derived two Hölder stability estimates for the MFG system

Proved the stability of solutions with respect to boundary data noise

Established uniqueness of solutions using new Carleman estimates

Abstract

The second order Mean Field Games system (MFGS) in a bounded domain with the lateral Cauchy data is considered. This means that both Dirichlet and Neumann boundary data for the solution the MFGS are given. Two H\"older stability estimates for two slightly diffeent cases are derived. These estimates indicate how stable the solution of the MFGS is with respect to the possible noise in the lateral Cauchy data. Our stability estimates imply uniqueness. The key mathematical apparatus is the apparatus of two new Carleman estimates.