By Berti M., Bolle P.
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Univ. Chicago Press, 1999.  H. Brezis, J. M. Coron, L. Nirenberg, Free vibrations for a non-linear wave equation and a Theorem of P. Rabinowitz, Comm. Pure and Appl. Math. 33, 5, 667-684, 1980.  L. Chierchia, J. You, KAM tori for 1D nonlinear wave equations with periodic boundary conditions, Comm. Math. Phys. 211, 2000, no. 2, 497–525.  W. Craig, Probl`emes de petits diviseurs dans les ´equations aux d´eriv´ees partielles, Panoramas et Synth`eses, 9, Soci´et´e Math´ematique de France, Paris, 2000.
1) with frequencies ω < 1. 1 Clearly m∞ < +∞ because |ap (x)v p+1 | ≤ C|v|p+1 ∞ ≤C v |G(v)| ≤ Ω p+1 H1 , ∀v ∈ V . Let vn ∈ S be a maximizing sequence for G, namely G(vn ) → m∞ . 1) Since vn = 1, ∀n, we can assume that (up to subsequence) vn H1 v¯ ∈ V and, by the compact L∞ ∞ 1 H1 embedding H (T) → L (T), that vn → v¯. As a consequence ap (x)vnp → G(vn ) := Ω ap (x)¯ v p =: G(¯ v) . 2) we get G(¯ v ) = m∞ . Actually the maximum point v¯ ∈ S. Indeed, by the lower semicontinuity of the H 1 -norm for the weak topology, v¯ H 1 ≤ lim inf n vn H 1 = 1 .
1. Consider the orthogonal splitting W = W (n) ⊕ W (n)⊥ where W (n) = w ∈ W exp (ilt) wl (x) , W (n)⊥ = w ∈ W w= |l|≤Ln w= exp (ilt) wl (x) |l|>Ln with Ln := L0 2n for some large integer L0 , and denote by Pn : W → W (n) and Pn⊥ : W → W (n)⊥ the orthogonal projectors onto W (n) and W (n)⊥ . 12) imply • (P1) (Regularity) Γ(·, ·, ·, ·) ∈ C ∞ [0, δ0 ] × B(1) × B(2R∞ ; V1 ) × B(R∞ ; W ∩ Xσ,s ), Xσ,s . Moreover Dk Γ, ∀ k ≥ 0, are bounded on [0, δ0 ] × B(1) × B(2R∞ ; V1 ) × B(R∞ ; W ∩ Xσ,s ). • (P2) (Smoothing) ∀ w ∈ W (n)⊥ ∩ Xσ,s and ∀ 0 ≤ σ ≤ σ, w σ ,s ≤ exp (−Ln (σ − σ )) w The core of the Nash-Moser scheme is the invertibility of the linearized operators on W (n) Ln (δ, λ, v1 , w)[h] := Lω h − εPn ΠW Dw Γ(δ, λ, v1 , w)[h] .
Cantor families of periodic solutions for wave equations via a variational principle by Berti M., Bolle P.