Weak Kam Aspects of Convex Hamilton-jacobi Equations with Neumann Type Boundary Conditions

We study convex Hamilton-Jacobi equations H(x,Du) = a and ut+H(x,Du) = a in a bounded domain Ω of R with the Neumann type boundary condition Dγu = g in the viewpoint of weak KAM theory, where γ is a vector field on the boundary ∂Ω pointing a direction oblique to ∂Ω. We establish the stability under the formations of infimum and of convex combinations of subsolutions of convex HJ equations, some comparison and existence results for convex and coercive HJ equations with the Neumann type boundary condition as well as existence results for the Skorokhod problem. We define the Aubry-Mather set associated with the Neumann type boundary problem and establish some properties of the Aubry-Mather set including the existence results for the “calibrated” extremals for the corresponding action functional (or variational problem).