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We establish the stability under the formations of infimum and of convexcombinations of subsolutions of convex Hamilton-Jacobi equations, somecomparison and existence results for convex and coercive Hamilton-Jacobiequations with the Neumann type boundary condition as well as existence resultsfor the Skorokhod problem. We define the Aubry-Mather set associated with theNeumann type boundary problem and establish some properties of the Aubry-Matherset including the existence results for the ``calibrated'' extremals for thecorresponding action functional (or variational problem).

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