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 \(-\operatorname {div}(m D_pH(x, Du, m)) = 0\) in a domain \(\Omega \subset {\mathbb {R}}^d\) . 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 \(\Omega \) . 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.
