A Hessian-dependent functional with free boundaries and applications to mean-field games

TL;DR

This paper investigates a Hessian-dependent functional linked to fully nonlinear mean-field games with free boundaries, establishing existence, regularity, and convergence results for solutions and free boundary sets.

Contribution

It introduces a new Hessian-dependent functional, proves existence and regularity of solutions, and demonstrates $ ext{Gamma}$-convergence, advancing understanding of free boundary problems in mean-field games.

Findings

Existence of solutions to the mean-field game.

H"older continuity of the value function.

Reduced free boundary has finite perimeter.

Abstract

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 H\"older 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 $\Gamma$-convergence result for the functional.