L. Chierchia et C. Falcolini, A NOTE ON QUASI-PERIODIC SOLUTIONS OF SOME ELLIPTIC-SYSTEMS, Zeitschrift fur angewandte Mathematik und Physik, 47(2), 1996, pp. 210-220
We extend a recent method of proof of a theorem by Kolmogorov on the c
onservation of quasi-periodic motion in Hamiltonian systems so as to p
rove existence of (uncountably many) real-analytic quasi-periodic solu
tions for elliptic systems Delta u = epsilon f(x)(u, y), where u: y is
an element of R(M) --> u(y) is an element of R(N), f = f(x, y) is a r
eal-analytic periodic Function and epsilon is a small parameter. Kolmo
gorov's theorem is obtained (in a special case) when M = 1 while the c
ase N = 1 is (a special case of) a theorem by J. Moser on minimal foli
ations of codimension 1 on a torus T-M + 1 In the autonomous case, f =
f(x), the above result holds for any epsilon.