Lipschitz stability for determination of states and inverse source problem for the mean field game equations

In a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$ over time interval $(0,T)$, we consider mean field game equations whose principal coefficients depend on the time and state variables with a general Hamiltonian. We attach the non-zero Robin boundary condition. We first prove the Lipschitz stability in $\mathrm{\Omega }\times (\epsilon ,T-\epsilon )$ with given $\epsilon >0$ for the determination of the solutions by Dirichlet data on arbitrarily chosen subboundary of $\mathrm{\partial }\mathrm{\Omega }$. Next we prove the Lipschitz stability for an inverse problem of determining spatially varying factors of source terms and a coefficient by extra boundary data and spatial data at an intermediate time.