Strong Stability of KAM Tori: from the Point of View of Viscosity Solutions of H-J Equations

It is well-known that a KAM torus can be considered as a graph of smooth viscosity solution. Salamon and Zehnder (Comment Math Helv 64:84–132, 1989) have proved that there exist invariant tori having prescribed Diophantine frequencies for nearly integrable and positively definite Lagrangian systems with associated Hamiltonian H, whose Diophantine index is τ. If the invariant torus is represented as in the cotangent bundle , then we can show that for any viscosity solution u (x, P), which satisfies the H-J Eq. (1.1),when is small enough. For the more exact form, please see Theorem 2 for details.