# Cantor families of periodic solutions for wave equations via by Berti M., Bolle P.

By Berti M., Bolle P.

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Topics in mathematical physics, general relativity, and cosmology in honor of Jerzy Plebański: proceedings of 2002 international conference, Cinvestav, Mexico City, 17-20 September 2002

Certainly one of smooth science's most famed and arguable figures, Jerzy Pleba ski was once a good theoretical physicist and an writer of many fascinating discoveries generally relativity and quantum conception. identified for his unprecedented analytic abilities, explosive personality, inexhaustible power, and bohemian nights with brandy, espresso, and massive quantities of cigarettes, he was once devoted to either technological know-how and artwork, generating innumerable handwritten articles - similar to monk's calligraphy - in addition to a set of oil work.

Symmetries, Integrable Systems and Representations

This quantity is the results of overseas workshops; countless research eleven – Frontier of Integrability – held at college of Tokyo, Japan in July twenty fifth to twenty ninth, 2011, and Symmetries, Integrable platforms and Representations held at Université Claude Bernard Lyon 1, France in December thirteenth to sixteenth, 2011.

Additional info for Cantor families of periodic solutions for wave equations via a variational principle

Example text

2. 17). 16). 21) with s∗ = 1 possesses a nontrivial “Mountain pass” critical set K∞ ⊂ SR∞ := { v H 1 = R∞ }. Next we fix the dimension N ∈ N of the finite dimensional subspace V1 in the orthogonal decomposition V = V1 ⊕ V2 , N depending only on ap (x). 7. 3 and its proof). 1, there are q ≤ q1 ≤ . . 1 holds. 4). 10) with ω(δ) = 1 + 2δ p−1 . 21) holds, with q = qM − p . e. 5). 2. If there is v ∈ V such that G(v) < 0, we may choose s∗ = −1. 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 .

8) σ/4,s for ε small enough because M ≥ ln2 1 → +∞ as 6L0 ε ε→0 and LM := L0 2M >> χM . 8), Lω rM ε exp (−χi ) ≤ Kε exp (−χM ) . 9) Lω w − εΠW Γ(δ, λ, v1 , w) ≤ K ε exp (−χM ) σ/4,s and since χM ≥ χ− ln2 (6L0 ε) = χ−(ln2 χ) lnχ˜ (6L0 ε) = (6L0 ε)− ln2 χ and, setting α := ln2 χ ∈ (0, 1), Lω w − εΠW Γ(λ, δ, v1 , w) ≤ Cε exp − σ/4,s for 0 < δ ≤ δ0 (γ, τ ) small enough. 41 1 (6L0 ε)α ≤ C ε exp (− C ) δα References [1] A. Ambrosetti, P. Rabinowitz, Dual Variational Methods in Critical Point Theory and Applications, Journ.

Let η have minimal period 2π. 3) T then h = 0. Proof. Step 1: For any y ∈ T and α < β h(s) ds = 0 . 3), for any real polynomial P (X) := k≥q ak X k divisible by X q , P η(y + s) − η(s) h(s) ds = 0 . 5) holds for any real polynomial ak X k . 6). By the Stone-Weierstrass theorem, the set A is dense in C([−M, M ], R) because it is a subalgebra with unity and A separates the points of [−M, M ] (take any X q with q odd). As a consequence for any continuous function g ∈ C(R) g η(y + s) − η(s) h(s) ds = 0 .