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Existence theory for non-separable mean field games in Sobolev spaces

The mean field games system is a coupled pair of nonlinear partial differential equations arising in differential game theory, as a limit as the number of agents tends to infinity. We prove existence and uniqueness of classical solutions for time-dependent mean field games with Sobolev data. Many works in the literature assume additive separability of the Hamiltonian, as well as further structure such as convexity and monotonicity of the resulting components. Problems arising in practice, however, may not have this separable structure; we therefore consider the non-separable problem. For our existence and uniqueness results, we introduce new smallness constraints which simultaneously consider the size of the time horizon, the size of the data, and the strength of the coupling in the system.