Optimal convergence rate for periodic homogenization of convex Hamilton-Jacobi equations

In this paper, we show that the rate of convergence in periodic homogenization of convex Hamilton-Jacobi equations is always $O(\epsilon )$, which is optimal. This is a natural extension of a result concerning stable norms in metric geometry [4] that is essentially equivalent to the homogenization of convex static Hamilton-Jacobi equations. Another extremely interesting question in this direction is whether the $O(\epsilon )$ rate holds in the nonconvex setting. We present a special nonconvex example with $O(\epsilon )$ convergence rate, which relies on identifying the shape of the effective Hamiltonian and game theory interpretation formulas.