Weak KAM aspects of convex Hamilton-Jacobi equations with Neumann type boundary conditions

We study convex Hamilton-Jacobi equations H(x, Du) = 0 and u(t) + H(x, Du) = 0 in a bounded domain Omega of R-n with the Neumann type boundary condition D(gamma)u = g in the viewpoint of weak KAM theory, where gamma is a vector field on the boundary partial derivative Omega pointing a direction oblique to partial derivative Omega. We establish the stability under the formations of infimum and of convex combinations of subsolutions of convex Hamilton-Jacobi equations, some comparison and existence results for convex and coercive Hamilton-Jacobi equations with the Neumann type boundary condition as well as existence results for the Skorokhod problem. We define the Aubry set associated with the Neumann type boundary problem and establish some properties of the Aubry set including the existence results for the "calibrated" extremals for the corresponding action functional (or variational problem). (C) 2010 Elsevier Masson SAS. All rights reserved.