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We study the regularity and well-posedness of the local, first-orderforward-backward mean field games system, assuming a polynomially growing costfunction and a Hamiltonian of quadratic growth. We consider systems andterminal data that are strictly monotone in the density and study two differentregimes depending on whether there exists a lower bound for the running costfunction. The work relies on a transformation due to P.-L. Lions, which givesrise to an elliptic partial differential equation with oblique boundaryconditions, that is strictly elliptic when the coupling is unbounded frombelow. In this case, we prove that the solution is smooth. When the problem isdegenerate elliptic, we obtain existence and uniqueness of weak solutionsanalogous to those obtained by P. Cardaliaguet and P.J. Graber for the case ofa terminal condition that is independent of the density. The weak solutions areshown to arise as viscous limits of classical solutions to strictly ellipticproblems.

Classical and weak solutions to local first-order mean field games through elliptic regularity