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 ε ℝM →u(y) ε ℝN,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.
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Chierchia, L., Falcolini, C. A note on quasi-periodic solutions of some elliptic systems. Z. angew. Math. Phys. 47, 210–220 (1996). https://doi.org/10.1007/BF00916825
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DOI: https://doi.org/10.1007/BF00916825