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Fully nonlinear second-order mean field games with nondifferentiable Hamiltonians

Thomas Sales, Iain Smears

TL;DR

This work establishes the well-posedness of fully nonlinear second-order mean field games with nondifferentiable Hamiltonians on bounded convex domains under Cordes-coefficient diffusion. It introduces a nonstandard variational-inequality (VI) reformulation that is equivalent to the original partial differential inclusion (PDI) and enables robust limit passages; existence is proved via Kakutani fixed point with a priori bounds, and uniqueness follows from a Lasry–Lions monotonicity condition on the coupling $F$. The authors also show that the MFG PDI/VI system arises as the limit of MFG PDEs with differentiable Hamiltonians and prove continuous dependence on data, including a regularization framework (Moreau–Yosida) and perturbations of $G$. Key technical contributions include a Cordes-based comparison principle for nondivergence form operators with discontinuous coefficients and a careful analysis of measurable selections from subdifferentials, expanding the scope of MFG theory to nondifferentiable Hamiltonians and more general domains.

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.

Fully nonlinear second-order mean field games with nondifferentiable Hamiltonians

TL;DR

This work establishes the well-posedness of fully nonlinear second-order mean field games with nondifferentiable Hamiltonians on bounded convex domains under Cordes-coefficient diffusion. It introduces a nonstandard variational-inequality (VI) reformulation that is equivalent to the original partial differential inclusion (PDI) and enables robust limit passages; existence is proved via Kakutani fixed point with a priori bounds, and uniqueness follows from a Lasry–Lions monotonicity condition on the coupling . The authors also show that the MFG PDI/VI system arises as the limit of MFG PDEs with differentiable Hamiltonians and prove continuous dependence on data, including a regularization framework (Moreau–Yosida) and perturbations of . Key technical contributions include a Cordes-based comparison principle for nondivergence form operators with discontinuous coefficients and a careful analysis of measurable selections from subdifferentials, expanding the scope of MFG theory to nondifferentiable Hamiltonians and more general domains.

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.

Paper Structure

This paper contains 26 sections, 18 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Fully nonlinear mean field games
  3. MFG with nondifferentiable Hamiltonians and main results
  4. Approach to the analysis and further contributions
  5. Notation and formulation of the problem
  6. The Hamiltonian and its subdifferential.
  7. Formulations of the problem
  8. Formulation of the PDI problem.
  9. Formulation of the VI problem.
  10. Cordes coefficients
  11. Main results
  12. Equivalence between the PDI and VI problems
  13. Existence of a solution
  14. Uniqueness of the solution
  15. Equivalence of the PDI and VI problems
  16. ...and 11 more sections

Key Result

Proposition 2.1

Let $\mathcal{A}$ be a compact metric space, and let $a\in C(\overline{\Omega}\times\mathcal{A};\mathbb{R}^{d\times d}_{\mathrm{sym},+})$ and $f\in C(\overline{\Omega}\times\mathcal{A})$. Let $H$ be defined by eqn:H_def. For each $M\in \mathbb{R}^{d\times d}_{\mathrm{sym}}$ and $x\in\overline{\Omega The set $\Lambda(x,M)$ is nonempty and compact in $\mathcal{A}$. For each $(x,M)\in \overline{\Omeg

Theorems & Definitions (52)

  • Example 2.1
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.1
  • Remark 2.2: Relation to other variational inequalities
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • ...and 42 more