Fully nonlinear second-order mean field games with nondifferentiable Hamiltonians

TL;DR

This paper establishes the existence and uniqueness of solutions for fully nonlinear second-order mean field games with nondifferentiable Hamiltonians, using a variational inequality reformulation and limit analysis from differentiable Hamiltonian systems.

Contribution

It introduces a novel reformulation of the PDI as a variational inequality, enabling existence proofs and limit analysis for nondifferentiable Hamiltonians in MFGs.

Findings

Existence of solutions under general conditions on coupling and sources.

Uniqueness of solutions for strictly monotone couplings.

Solutions can be obtained as limits of systems with differentiable Hamiltonians.

Abstract

We analyse fully nonlinear second-order mean field games (MFG) with nondifferentiable Hamiltonians, which take the form of a coupled system of a fully nonlinear Hamilton-Jacobi-Bellman equation and a Kolmogorov-Fokker-Planck partial differential inclusion (PDI) featuring the set-valued subdifferential of the Hamiltonian. We show the existence of solutions of some stationary MFG systems with quite general coupling operators and nonnegative distributional source terms, on general bounded convex domains, under the primary assumptions of uniform ellipticity and the Cordes condition on the diffusion coefficient. The existence proof is founded on an original, and equivalent, reformulation of the PDI as a nonstandard variational inequality (VI), that offers significant flexibility in passages to limits. Furthermore, the uniqueness of the solution of the PDI/VI system is proved in the case of strictly monotone couplings. We then show how the MFG PDI/VI system in the fully nonlinear setting can be obtained as the limit of a sequence of PDE systems with differentiable Hamiltonians, and we give further results on the continuous dependence of the solution.