Further PDE methods for weak KAM theory

Evans, Lawrence C.
August 2009
Calculus of Variations & Partial Differential Equations;Aug2009, Vol. 35 Issue 4, p435
Academic Journal
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 s. 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.


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