Calculus of Variations and Partial Differential Equations

, Volume 35, Issue 4, pp 435–462

Further PDE methods for weak KAM theory


DOI: 10.1007/s00526-008-0214-1

Cite this article as:
Evans, L.C. Calc. Var. (2009) 35: 435. doi:10.1007/s00526-008-0214-1


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 σ. 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.

Mathematics Subject Classification (2000)


Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA