Viscous Stability of Quasi-periodic Lagrangian Tori

We consider a smooth Tonelli Lagrangian L : TTn → R and its viscosity solutions u(x, P ) characterized by the cell equation H(x, P + Dxu(x, P )) = H(P ), where H : T ∗Tn → R is the Hamiltonian associated with L. We will show that if P0 corresponds to a quasi-periodic Lagrangian invariant torus, then Dxu(x, P ) is Hölder continuous in P at P0 with Hölder exponent arbitrarily close to 1, and if both H and the torus are real analytic and the frequency vector of the torus is Diophantine, then Dxu(x, P ) is Lipschiz continuous in P at P0, i.e., there is a constant C > 0 such that ‖Du(x, P ) − Du(x, P0)‖∞ ≤ C‖P − P0‖ as ‖P − P0‖ 1. Similar P -regularity of the Peierls barrier for L will be obtained, and applications to viscosity solutions near KAM tori in configuration space in a nearly integrable Hamiltonian system will also be considered.