Convergence rates in the periodic homogenization of vanishing-viscous Hamilton-Jacobi equations

We study convergence rates when periodic homogenization and vanishing viscosity occur at the same scale in Hamilton–Jacobi equations whose momentum Hessians are positive definite at every point. For bounded Lipschitz initial data and a class of smooth Hamiltonians, we establish an optimal rate $O(\epsilon \mathrm{\mid }\log \epsilon \mathrm{\mid })$ on fixed time intervals. The second-order term plays a key role: the elliptic cell problem provides higher derivative bounds for the effective Hamiltonian $\bar{H}$ and its Legendre dual $\bar{L}$. The proof is purely PDE, based on corrector expansions and viscosity comparison, while its guiding idea comes from the control interpretation of the Hopf–Lax formula. For a fixed target $(x,t)$, a minimizing origin selects a characteristic velocity and its dual momentum, these guide the construction of smooth comparison profiles, whose gradients supply the momentum argument of the first-order corrector. Thus we never need to differentiate the effective solution, even at points where it is nonsmooth.