The existence of homogenization of Hamilton-Jacobi equations in the periodic setting has been established by Lions, Papanicolaou and Varadhan in 80's. However, it remains a major open problem to understand deep properties of the effective Hamiltonian that governs the homogenization limit. In this talk, I will focus on the mechanical Hamiltonian $H(p,x)={1\over 2}|p|^2+V(x)$ and present some recent progress together with close connections with the Aubry-Mather theory.
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