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

We consider solutions satisfying the zero Neumann boundary condition and a linearized mean field game equation in $\mathrm{\Omega }\times (0,T)$ whose principal coefficients depend on the time and spatial variables with general Hamiltonian, where $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^d$ and $(0,T)$ is the time interval. 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 intermediate time.