Regularity of viscosity solutions near KAM torus (Q931530)

In the first part of this paper, the classical Lindstedt series expansion for perturbation of non-resonant integrable Hamiltonians used. In the last part the Theorems 3 and 4 obtained by \textit{D. A. Gomes} [SIAM J. Math. Anal. 35, No.~1, 135--147 (2003; Zbl 1034.37035)] are generalized. Based on them, weak regularity theorems on \(u^\varepsilon(x,P)\) and \(Du^\varepsilon(x,P)\) in a small neighborhood of \(P_\varepsilon\) are given. Here, \(u^\varepsilon(x,P)\) are the periodic viscosity solutions of the cell problem \[ H_\varepsilon(P+ D_x u^\varepsilon, x)=\overline H_\varepsilon(P). \] Some lemmas are presented in the Appendix.