  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-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).
