A Hessian-Dependent Functional With Free Boundaries and Applications to Mean-Field Games

We study a Hessian-dependent functional driven by a fully nonlinear operator. The associated Euler-Lagrange equation is a fully nonlinear mean-field game with free boundaries. Our findings include the existence of solutions to the mean-field game, together with Hölder continuity of the value function and improved integrability of the density. In addition, we prove the reduced free boundary is a set of finite perimeter. To conclude our analysis, we prove a \(\Gamma \) $\mathrm{\Gamma }$ -convergence result for the functional.