A note on quasi-periodic solutions of some elliptic systems

Abstract

We extend a recent method of proof of a theorem by Kolmogorov on the conservation of quasi-periodic motion in Hamiltonian systems so as to prove existence of (uncountably many) real-analytic quasi-periodic solutions for elliptic systems Δu=ɛf x (u, y), whereu ∶y e ℝ M →u(y) e ℝ N ,f=f(x, y) is a real-analytic periodic function and ɛ is a small parameter. Kolmogorov's theorem is obtained (in a special case) whenM=1 while the caseN=1 is (a special case of) a theorem by J. Moser on minimal foliations of codimension 1 on a torusT M +1. In the autonomous case,f=f(x), the above result holds for any ɛ.