First-Order Mean Field Games with Time-Measurable and $C ^{1,1}$ Hamiltonians: Analysis and Approximation

Elisabetta Carlini, Valentina Coscetti, Francisco J. Silva · 2026 · arXiv

DOI: 10.48550/arXiv.2610.28335 · open-access PDF

Abstract

We introduce and analyze a numerical framework for a class of first-order, non-local Mean Field Game (MFG) systems with two sources of low regularity: the time-dependent data are merely measurable, and the Hamiltonian is only of class $C^{1,1}$ with respect to the momentum variable. The proposed method combines a semi-Lagrangian approximation of the Hamilton--Jacobi--Bellman equation with a Lagrange--Galerkin discretization of the continuity equation. Temporal measurability is handled by averaging the data over the time-grid intervals and using the stability theory of $L^1$-viscosity solutions

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