Further PDE methods for weak KAM theory

We introduce and make estimates for several new approximations that in appropriate asymptotic limits yield the key PDE for weak KAM theory, namely a Hamilton-Jacobi type equation for a potential u and a coupled transport equation for a measure sigma. We revisit as well a singular variational approximation introduced in Evans (Calc Vari Partial Differ Equ 17:159-177, 2003) and demonstrate approximate integrability of certain phase space dynamics related to the Hamiltonian flow. Other examples include a pair of strongly coupled PDE suggested by the Lions-Lasry theory (Lasry and Lions in Japan J Math 2:229-260, 2007) of mean field games and a new and extremely singular elliptic equation suggested by sup-norm variational theory.