A priori estimates and large population limits for some nonsymmetric Nash systems with semimonotonicity

We address the problem of regularity of solutions to a family of semilinear parabolic systems of equations, which describe closed‐loop equilibria of some ‐player differential games with Lagrangian having quadratic behaviour in the velocity variable, running costs and final costs . By global (semi)monotonicity assumptions on the data and , and assuming that derivatives of in directions are of order for , we prove that derivatives of enjoy the same property. The estimates are uniform in the number of players . Such a behaviour of the derivatives of arise in the theory of Mean Field Games, though here we do not make any symmetry assumption on the data. Then, by the estimates obtained we address the convergence problem in a ‘heterogeneous’ Mean Field framework, where players all observe the empirical measure of the whole population, but may react differently from one another. We also discuss some results on the joint and vanishing viscosity limit.