By Gentile G., Mastropietro V., Procesi M.

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**Extra resources for Periodic solutions for completely resonant nonlinear wave equations**

**Example text**

P) Therefore for τ ≥ max{1, τ0 − 1} the Lebesgue measure of the set I1 is bounded by ε01+δ1 , with δ1 > 0. (p) Now we discuss how to bound the measure of the set I2 . 38) m m+ℓ for all p ≥ 1. By the parity properties of ω ˜ m without loss of generality we can confine ourselves to the case (p) (p) ˜ m )| < 1. Then the discussion proceeds as follows. 39) which implies for τ ≥ τ0 C0 2Cε0 |n| ≥ N1 ≡ 1/τ0 . 41) ≤ m |n|τ0 which gives m0 = m0 (n) = 2C|n|τ0 ε0 C0 . 43) so that one has to exclude no further value from E (p−1) , provided one takes τ ≥ τ0 .

Phys. 127 (1990), no. 3, 479–528. [33] A. Weinstein, Normal modes for nonlinear Hamiltonian systems, Invent. Math. 20 (1973), 47–57. [34] H. Whitney, Analytic extensions of differential functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), no. 1, 63–89.

56) (n)| ≤ C0 |n|−τ for p ≥ p0 by choosing p0 = p0 (n, j) ≤ const. log |n|. 18) are satisfied (p) with C0 replaced with 2C0 . By the definition of p0 all the intervals Ij (n) fall inside the union of (p0 ) (0) the intervals Jj (n), . . , Jj (p) meas ∪∞ p=0 I1 (n) as soon as p > p0 . 37) (p) 0 ≤ meas ∪pp=0 J1 ≤ p0 (n,1) ≤ const. |n|≥N0 p=0 (p) meas(I1 (n)) |n|≥N0 2C0 ≤ const. 55). Analogously one obtains the bounds meas(I2 ) ≤ Bε1+δ and meas(I3 ) ≤ Bε1+δ for the the Lebesgue 0 0 measures of the sets I2 and I3 (possibly redefining the constant B).

### Periodic solutions for completely resonant nonlinear wave equations by Gentile G., Mastropietro V., Procesi M.

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