Regularity for Weak Solutions to First-Order Local Mean Field Games

Abstract

We establish interior regularity results for first-order, stationary, local mean-field game (MFG) systems. Specifically, we study solutions of the coupled system consisting of a Hamilton-Jacobi-Bellman equation $H(x,Du,m)=0$ and a transport equation $-\mathrm{div}(m{D}_{p}H(x,Du,m))=0$ in a domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^d$. Under suitable structural assumptions on the Hamiltonian $H$, without requiring monotonicity of the system, convexity of the Hamiltonian, separability in variables, or smoothness beyond basic continuity in $(p,m)$, we introduce a notion of weak solutions that allows the application of techniques from elliptic regularity theory. Our main contribution is to prove that the value function $u$ is locally Hölder continuous in $\hat{I}\text{c◯}$. The proof leverages the connection between first-order MFG systems and quasilinear equations in divergence form, adapting classical techniques to handle the specific structure of MFG systems.

25 pages