Perturbation Theory for Viscosity Solutions of Hamilton-Jacobi Equations and Stability of Aubry-Mather Sets

In this paper we study the stability of integrable Hamiltonian systems under small perturbations, proving a weak form of the KAM/Nekhoroshev theory for viscosity solutions of Hamilton-Jacobi equations. The main advantage of our approach is that only a finite number of terms in an asymptotic expansion are needed in order to obtain uniform control. Therefore there are no convergence issues involved. An application of these results is to show that Diophantine invariant tori and Aubry-Mather sets are stable under small perturbations.