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We investigate mean field game systems under invariance conditions for thestate space, otherwise called {\it viability conditions} for the controlleddynamics. First we analyze separately the Hamilton-Jacobi and the Fokker-Planckequations, showing how the invariance condition on the underlying dynamicsyields the existence and uniqueness, respectively in $L^\infty$ and in $L^1$.Then we apply this analysis to mean field games. We investigate further theregularity of solutions proving, under some extra conditions, that the valuefunction is (globally) Lipschitz and semiconcave. This latter regularityeventually leads the distribution density to be bounded, under suitableconditions. The results are not restricted to smooth domains.

Mean field games under invariance conditions for the state space