Regularity and Variationality of Solutions to Hamilton-Jacobi Equations. Part II: Variationality, Existence, Uniqueness

We formulate an Hamilton-Jacobi partial differential equation H(x, Du(x)) = 0 on a n dimensional manifold M, with assumptions of convexity of the sets {p : H(x, p) <= 0} subset of T*(x)M, for all x. We reduce the above problem to a simpler problem; this shows that u may be built using an asymmetric distance (this is a generalization of the "distance function" in Finsler geometry); this brings forth a 'completeness' condition, and a Hopf-Rinow theorem adapted to Hamilton-Jacobi problems. The 'completeness' condition implies that u is the unique viscosity solution to the above problem.