Action potential and weak KAM solutions

For convex superlinear lagrangians on a compact manifold M we characterize the Peierls barrier and the weak KAM solutions of the Hamilton-Jacobi equation, as defined by A. Fathi [9], in terms of their values at each static class and the action potential defined by R. Mane [14]. When the manifold ill is non-compact, we construct weak KAM solutions similarly to Busemann functions in riemannian geometry. We construct a compactification Of M/(dc) by extending the Aubry set using these Busemann weak KAM solutions and characterize the set of weak KAM solutions using this extended Aubry set.