Further PDE methods for weak KAM theory

Article

Abstract

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)

35J20 35F20 37J50 

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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