Viscosity solutions of Hamilton-Jacobi equations, and asymptotics for Hamiltonian systems

In this paper we apply the theory of viscosity solutions of Hamilton-Jacobi equations to understand the structure of certain Hamiltonian flows. In particular, we describe the asymptotic behavior of minimizing orbits, and prove analogs of the classical Hamilton-Jacobi integrability theory that hold under very general conditions. Then, combining partial differential equations techniques with dynamical systems ideas (Mather measures, ergodicity) we study solutions of time-independent Hamilton-Jacobi equation, namely, uniform continuity, difference quotients and non-uniqueness. Received: 16 October 2000 / Accepted: 23 February 2001 / Published online: 12 October 2001