The Monge problem for supercritical Mane potentials on compact manifolds

We prove the existence of an optimal map for the Monge problem when the cost is a supercritical Mane potential on a compact manifold. Supercritical Mane potentials form a class of costs which generalize the Riemannian distances. We describe new links between this transportation problem and viscosity subsolutions of the Hamilton-Jacobi equation.