Homogenization of generalized characteristics associated to solutions of Hamilton-Jacobi equations

It was proved in Arch. Ration. Mech. Anal. 162 (2002), 1-23 that singularities of solutions for Hamilton-Jacobi equations will propagate along generalized characteristics. In this paper, we consider the homogenization of generalized characteristics. It is natural to think that (*) generalized characteristics associated to viscosity solutions of oscillatory Hamilton-Jacobi equations will converge to generalized characteristics associated to solutions of the effective equation. We show that this is indeed correct if the spatial dimension is 1. For high dimensions, we need to add some extra assumptions of singularities near generalized characteristics. We provide a counterexample that (*) fails without those assumptions. Some issues related to the weak KAM theory are also discussed.