Differential and Variational Approach to First Order Mean Field Games in a Generalized Form
We investigate time dependent, first order Mean Field Games on the torus comparing, in a broad and general framework, the classical differential formulation , given by a Hamilton Jacobi equation coupled with a continuity equation, with a variational approach based on fixed points of a multivalued map acting on probability measures over trajectories. We prove existence of fixed points for very general Hamiltonians. When the Hamiltonian is differentiable with respect to the momentum, we show that the evaluation curve of any such fixed point solves a continuity equation driven by a vector field associated with the final condition in the Hamilton Jacobi equation. This field is defined without requiring additional regularity conditions on the value function solving the Hamilton--Jacobi equation. The field coincides with the classical vector field of Mean Field Systems at space--differentiability points of the value function lying in the space--time regions where optimal trajectories concentrate. Our analysis therefore provides a unified framework that bridges the differential and variational viewpoints in Mean Field Games, showing how aggregate optimality conditions naturally lead to continuity-equation descriptions under the sole assumption of differentiability of the Hamiltonian in the momentum variable.