QUASI-PERIODIC OSCILLATIONS FOR WAVE EQUATIONS WITH PERIODIC FORCING. MASSIMILIANO BERTI, MICHELA PROCESI Abstract. Exis...

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QUASI-PERIODIC OSCILLATIONS FOR WAVE EQUATIONS WITH PERIODIC FORCING. MASSIMILIANO BERTI, MICHELA PROCESI Abstract. Existence of quasi-periodic solutions with two frequencies of completely resonant, periodically forced, nonlinear wave equations with periodic spatial boundary conditions is established. We consider both the cases the forcing frequency is (Case A) a rational number and (Case B) an irrational number.

Keywords: Nonlinear Wave Equation, Quasi-Periodic Solutions, Variational Methods, Lyapunov-Schmidt reduction, Infinite Dimensional Hamiltonian Systems.1 2000AMS subject classification: 35L05, 37K50, 58E05.

1. Introduction We present in this Note the results of [BPr] concerning existence of small amplitude quasi-periodic solutions for completely resonant forced nonlinear wave equations like vtt − vxx + f (ω1 t, v) = 0 (1.1) v(t, x) = v(t, x + 2π) where the nonlinear forcing term f (ω1 t, v) = a(ω1 t)v 2d−1 + O(v 2d ),

d > 1, d ∈ N+

is 2π/ω1 -periodic in time and the forcing frequency is: • A) ω1 ∈ Q • B) ω1 ∈ R \ Q. Existence of periodic solutions for completely resonant forced wave equations was first proved in the pioneering paper [R] (with Dirichlet boundary conditions) if the forcing frequency is a rational number (ω1 = 1 in [R]). This requires to solve an infinite dimensional bifurcation equation which lacks compactness property; see [BBi] and references therein for other results. If the forcing frequency is an irrational number existence of periodic solutions has been proved in [PY]: here the bifurcation equation is trivial but a “small divisors problem” appears. To prove existence of small amplitude quasi-periodic solutions for completely resonant PDE’s like (1.1) one has to deal both with a small divisor problem and with an infinite dimensional bifurcation equation. A major point is to understand from which periodic solutions of the linearized equation vtt − vxx = 0 , Massimiliano Berti, SISSA, via Beirut 2-4, Trieste, Italy, [email protected] . Michela Procesi, Universit` a di Roma 3, Largo S. Leonardo Murialdo, Roma, Italy, [email protected] . 1 Supported by M.I.U.R. “Variational Methods and Nonlinear Differential Equations”. 1

2

MASSIMILIANO BERTI, MICHELA PROCESI

the quasi-periodic solutions branch-off: indeed such linear equation possesses only 2πperiodic solutions of the form q+ (t + x)+ q− (t − x). For completely resonant autonomous PDE’s, existence of quasi-periodic solutions with 2-frequencies have been recently obtained in [P1] for the specific nonlinearities f = u3 + O(u5 ). Here the bifurcation equation is solved by ODE methods. In [BPr] we prove existence of quasi-periodic solutions with two frequencies ω1 , ω2 for the completely resonant forced equation (1.1) in both the two cases A) and B). The more interesting case is ω1 ∈ Q (case A) when the forcing frequency ω1 enters in resonance with the linear frequency 1. To find out from which solutions of the linearized equation quasi-periodic solutions of (1.1) branch-off, requires to solve an infinite dimensional bifurcation equation which can not be solved in general by ODE techniques (it is a system of integro-differential equations). However, exploiting the variational nature of equation (1.1) like in [BB1], the bifurcation problem can be reduced to finding critical points of a suitable action functional which, in this case, possesses the infinite dimensional linking geometry [BR]. On the other hand, we avoid the inherent small divisor problem by restricting the parameters to uncountable zero-measure sets as, e.g., in [P1], [BB1]. 1.1. Main results. We look for quasi-periodic solutions v(t, x) of (1.1) of the form v(t, x) = u(ω1 t, ω2 t + x) u(ϕ1 + 2k1 π, ϕ2 + 2k2 π) = u(ϕ1 , ϕ2 ),

∀k1 , k2 ∈ Z

with frequencies ω = (ω1 , ω2 ) = (ω1 , 1 + ε) imposing the frequency ω2 = 1 + ε to be close to the linear frequency 1. Therefore we get h i (1.2) ω12 ∂ϕ2 1 + (ω22 − 1)∂ϕ2 2 + 2ω1 ω2 ∂ϕ1 ∂ϕ2 u(ϕ) + f (ϕ1 , u) = 0 . We assume that the forcing term f : T × R → R f (ϕ1 , u) = a2d−1 (ϕ1 )u2d−1 + O(u2d ),

d ∈ N+ , d > 1

is analytic in u but has only finite regularity in ϕ1 . More precisely P k + • (H) f (ϕ1 , u) := ∞ 1 )u , d ∈ N , d > 1 and the coefficients ak (ϕ1 ) ∈ k=2d−1 ak (ϕP ∞ H 1 (T) verify, for some r > 0, k=2d−1 |ak |H 1 rk < ∞. The function f (ϕ1 , u) is not identically constant in ϕ1 . We look for solutions u of (1.2) in the multiplicative Banach algebra o n X X Hσ,s := u(ϕ) = uˆl eil·ϕ : uˆ∗l = uˆ−l and |u|σ,s := |ˆ ul |e|l2 |σ [l1 ]s < +∞ l∈Z2

where [l1 ] := max{|l1 |, 1} and σ > 0, s ≥ 0. In [BPr] we prove the following Theorems.

l∈Z2

QUASI PERIODIC OSCILLATIONS

3

Theorem A. Let ω1 = n/m ∈ Q. Assume that f satisfies assumption (H) and a2d−1 (ϕ1 ) 6= 0, ∀ϕ1 ∈ T. Let Bγ be the uncountable zero-measure Cantor set o n γ , ∀l1 , l2 ∈ Z \ {0} Bγ := ε ∈ (−ε0 , ε0 ) : |l1 + εl2 | > |l2 | where 0 < γ < 1/6. There exist constants σ > 0, s > 2, ε > 0, C > 0, such that ∀ε ∈ Bγ , |ε|γ −1 ≤ ε¯/m2 , there exists a classical solution u(ε, ϕ) ∈ Hσ,s of (1.2) with (ω1 , ω2 ) = (n/m, 1 + ε) satisfying 1 1 m2 |ε| 2(d−1) 2(d−1) (1.3) q¯ε (ϕ) ≤ C |ε| u(ε, ϕ) − |ε| γ ω13 σ,s for an appropriate function q¯ε ∈ Hσ,s \ {0} of the form q¯ε (ϕ) = q¯+ (ϕ2 )+ q¯− (2mϕ1 − nϕ2 ). As a consequence, equation (1.1) admits the quasi-periodic solution v(ε, t, x) := u(ε, ω1 t, x+ω2 t) with two frequencies (ω1 , ω2 ) = (n/m, 1+ε) and the map t → v(ε, t, ·) ∈ H σ (T) has the form2 m2 h i 2d−1 1 2(d−1) 2(d−1) |ε| q¯+ (x + (1 + ε)t) + q¯− ((1 − ε)nt − nx) =O . v(ε, t, x) − |ε| γ ω13 H σ (T) Remark that the bifurcation of quasi-periodic solutions u occurs both for ω2 > 1 and ω2 < 1; actually both Bγ ∩ (0, ε0 ) and Bγ ∩ (−ε0 , 0) are uncountable ∀ε0 > 0. At the first order the quasi-periodic solution v(ε, t, x) of equation (1.1) is the superposition of two waves traveling in opposite directions (in general, both components q+ , q− are non trivial). The bifurcation of quasi-periodic solutions looks quite different if ω1 is irrational. R 2π Theorem B. Let ω1 ∈ R\Q. Assume that f satisfies assumption (H), 0 a2d−1 (ϕ1 )dϕ1 6= 0 and f (ϕ1 , u) ∈ H s (T), s ≥ 1, for all u. Let Cγ ⊂ D ≡ (−ε0 , ε0 ) × (1, 2) be the uncountable zero-measure Cantor set ω1 γ ∈ / Q , |ω l + εl | > , (ε, ω ) ∈ D : ω ∈ / Q , 1 1 2 1 1 ω2 |l1 | + |l2 | . (1.4) Cγ := γ |ω1 l1 + (2 + ε)l2 | > , ∀ l1 ∈ Z \ {0} |l1 | + |l2 | Fix any 0 < s < s − 1/2. There exist R 2π positive constants ε, C, σ > 0, such that, −1 ∀(ε, ω1 ) ∈ Cγ with |ε|γ < ε and ε 0 a2d−1 (ϕ1 ) dϕ1 > 0, there exists a nontrivial solution u(ε, ϕ) ∈ Hσ,s of equation (1.2) with (ω1 , ω2 ) = (ω1 , 1 + ε) satisfying 1 1 |ε| 2(d−1) q ≤ C |ε| 2(d−1) (1.5) u(ε, ϕ) − |ε| ¯ (ϕ ) ε 2 γ σ,s for some function q¯ε (ϕ2 ) ∈ H σ (T) \ {0}. As a consequence, equation (1.1) admits the non-trivial quasi-periodic solution v(ε, t, x) := u(ε, ω1 t, x + ω2 t) with two frequencies (ω1 , ω2 ) = (ω1 , 1 + ε) and the map t → v(ε, t, ·) ∈ 2We

denote H σ (T) := {u(ϕ) =

P

l∈Z

u ˆl eilϕ : |u|H σ (T) :=

P

l∈Z

|ˆ ul |eσ|l| < +∞}.

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MASSIMILIANO BERTI, MICHELA PROCESI

H σ (T) has the form 1 2(d−1) q¯ε (x + (1 + ε)t) v(ε, t, x) − |ε|

H σ (T)

=O γ

−1

|ε|

2d−1 2(d−1)

.

Remark 1. Imposing in the definition of Cγ the condition ω1 /ω2 = ω1 /(1 + ε) ∈ Q we obtain, by Theorem B, the existence of periodic solutions of equation (1.1). They are reminiscent, in this completely resonant context, of the Birkhoff-Lewis periodic orbits with large minimal period accumulating at the origin, see [BaB]. In Theorem B, existence of quasi-periodic solutions could follow by other hypotheses on f . However the hypothesis that the leading term in the nonlinearity f is an odd power of u is not of technical nature. Actually, the following non-existence result holds: R 2π Theorem C. (Non-existence) Let f (ϕ1 , u) = a(ϕ1 )uD with D even and 0 a(ϕ1 ) dϕ1 6= 0. ∀R > 0 there exists ε0 > 0 such that ∀σ ≥ 0, s > s − 12 , ∀(ε, ω1 ) ∈ Cγ with |ε| < ε0 , equation (1.2) does not possess solutions u ∈ Hσ,s in the ball |u|σ,s ≤ R|ε|1/(D−1) . Theorem C also highlights that the existence of periodic solutions of nonlinear wave equations in the case of even powers, see [BB1], is due to the boundary effects imposed by the Dirichlet conditions. 2. Sketch of the proof of Theorem A We sketch now the proof of (the more difficult) Theorem A. To simplify notations we assume ω1 = 1, ε > 0 and a2d−1 (ϕ1 ) > 0. Instead of looking for solutions of equation (1.2) in a shrinking neighborhood of 0, it is a convenient devise to perform the rescaling u → δu

with

δ := |ε|1/2(d−1)

enhancing the relation between the amplitude δ and the frequency ω2 = 1 + ε. We obtain the equation (2.1)

Lε u + εf (ϕ1 , u, δ) = 0

where i h i h Lε := ∂ϕ2 1 + 2∂ϕ1 ∂ϕ2 + ε (2 + ε)∂ϕ2 2 + 2∂ϕ1 ∂ϕ2 ≡ L0 + εL1 and f (ϕ1 , δu) 2d−1 2d f (ϕ1 , u, δ) := 2(d−1) = a2d−1 (ϕ1 )u + δa2d (ϕ1 )u + . . . . δ Equation (2.1) is the Euler-Lagrange equation of the Lagrangian action functional Ψε ∈ C 1 (Hσ,s , R) defined by Z ε(2 + ε) 1 (∂ϕ1 u)2 + (∂ϕ1 u)(∂ϕ2 u) + (∂ϕ2 u)2 + ε(∂ϕ1 u)(∂ϕ2 u) − εF (ϕ1 , u, δ) Ψε (u) := 2 2 2 T ≡ Ψ0 (u) + εΓ(u, δ)

QUASI PERIODIC OSCILLATIONS

where F (ϕ1 , u, δ) :=

5

Ru

f (ϕ1 , ξ, δ)dξ and Z 1 Ψ0 (u) := (∂ϕ1 u)2 + (∂ϕ1 u)(∂ϕ2 u) 2 2 ZT (2 + ε) (∂ϕ2 u)2 + (∂ϕ1 u)(∂ϕ2 u) − F (ϕ1 , u, δ) . Γ(u, δ) := 2 T2 0

To find critical points of Ψε we perform a variational Lyapunov-Schmidt reduction inspired to [BB1], see also [AB]. The unperturbed functional Ψ0 : Hσ,s → R possesses an infinite dimensional linear space Q of critical points which are the solutions q of the equation L0 q = ∂ϕ1 ∂ϕ1 + 2∂ϕ2 q = 0 . The space Q can be written as n o X Q= q= qˆl eil·ϕ ∈ Hσ,s | qˆl = 0 for l1 (l1 + 2l2 ) 6= 0 l∈Z2

and, in view of the variational linking argument used for the bifurcation equation, we split Q as Q = Q+ ⊕ Q0 ⊕ Q− where3 Q+ Q0 Q−

n o n o σ := q ∈ Q : qˆl = 0 for l1 = 0 = q+ := q+ (ϕ2 ) ∈ H0 (T) n o := q0 ∈ R n o n o σ,s := q ∈ Q : qˆl = 0 for l1 + 2l2 = 0 = q− := q− (2ϕ1 − ϕ2 ), q− (·) ∈ H0 (T) .

We shall also use in Q the H 1 -norm | · |H 1 which is the natural norm for applying variational methods to the bifurcation equation. In order to prove analiticity (in ϕ2 ) of the solutions and to highlight the compactness of the problem we perform a finite dimensional Lyapunov-Schmidt reduction [BB2], decomposing Hσ,s = Q1 ⊕ Q2 ⊕ P where Q1 := Q1 (N ) :=

n

q=

X |l|≤N

il·ϕ

qˆl e

o

n o X il·ϕ ∈ Q , Q2 := Q2 (N ) := q = qˆl e ∈ Q |l|>N

and n o X il·ϕ P := p = pˆl e ∈ Hσ,s | pˆl = 0 for l1 (2l2 + l1 ) = 0 . l∈Z2 3H σ (T)

P denotes the functions of H σ (T) with zero average. H σ,s (T) := {u(ϕ) = l∈Z u ˆl eilϕ : P σ,s ∗ σ|l| s ul = u−l , |u|H σ,s (T) := l∈Z |ˆ ul |e [l] < +∞} and H0 (T) its functions with zero average. 0

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MASSIMILIANO BERTI, MICHELA PROCESI

Projecting equation (2.1) onto the closed subspaces Qi (i = 1, 2) and P , setting u = q1 + q2 + p with qi ∈ Qi and p ∈ P , we obtain h i f (ϕ , q + q + p, δ) =0 (Q1 ) L [q ] + Π 1 1 2 1 1 Q1 h i (Q2 ) L1 [q2 ] + ΠQ2 f (ϕ1 , q1 + q2 + p, δ) = 0 h i Lε [p] + εΠP f (ϕ1 , q1 + q2 + p, δ) = 0 (P ) where ΠQi : Hσ,s → Qi are the projectors onto Qi and ΠP : Hσ,s → P is the projector onto P . Step 1. (Solution of the (Q2 )-(P )-equations.) For ε ∈ Bγ , Lε restricted to P −1 has a bounded inverse satisfying |L−1 . Morevoer, also the operator ε [h]|σ,s ≤ |h|σ,s γ −2 L1 : Q2 → Q2 is invertible and |L−1 [h]| ≤ |h| N . σ,s σ,s 1 Fixed points of the nonlinear operator G : Q2 ⊕ P → Q2 ⊕ P defined by −1 f (ϕ , q + q + p, δ), −εL Π f (ϕ , q + q + p, δ) G(q2 , p; q1 ) := − L−1 Π P 1 1 2 Q2 1 1 2 ε 1 are solutions of the (Q2 )-(P )-equations. Using the Contraction Mapping Theorem, we can prove that ∀R > 0 there exist an integer N0 (R) ∈ N+ and positive constants ε0 (R) > 0, C0 (R) > 0 such that: ∀|q1 |H 1 ≤ 2R , ∀ε ∈ Bγ , |ε|γ −1 ≤ ε0 (R) , ∀N ≥ N0 (R) : 0 ≤ σN ≤ 1 , there exists a unique solution (q2 (q1 ), p(q1 )) := (q2 (ε, N, q1 ), p(ε, N, q1 )) ∈ Q2 ⊕ P of the (Q2 )-(P )-equations which satisfies |q2 (ε, N, q1 )|σ,s ≤ C0 (R)N −2 and |p(ε, N, q1 )|σ,s ≤ C0 (R)|ε|γ −1 . Step 2. (Solution of the (Q1 )-equation.) There remains to solve the finite dimensional (Q1 )-equation (2.2)

L1 [q1 ] + ΠQ1 f (ϕ1 , q1 + q2 (q1 ) + p(q1 ), δ) = 0 .

Actually (see [BB1]-[AB]) the bifurcation equation (2.2) is the Euler-Lagrange equation of the reduced Lagrangian action functional Φε,N : B2R ⊂ Q1 → R,

Φε,N (q1 ) := Ψε (q1 + q2 (q1 ) + p(q1 ))

which can be written as

Φε,N (q1 ) = const + ε Γ(q1 ) + Rε,N (q1 ) where

q 2d (2 + ε) (∂ϕ2 q1 )2 + (∂ϕ1 q1 )(∂ϕ2 q1 ) − a2d−1 (ϕ1 ) 1 2 2d T2 is the “Poincar´e-Melnikov” function [AB] and the remainder Z Rε,N (q1 ) := F (ϕ1 , q1 , δ = 0) − F (ϕ1 , q1 + q2 (q1 ) + p(q1 ), δ) Z

Γ(q1 ) :=

T2

1 + f (ϕ1 , q1 + q2 (q1 ) + p(q1 ), δ)(q2 (q1 ) + p(q1 )) 2

QUASI PERIODIC OSCILLATIONS

7

satisfies |Rε,N (q1 )| ≤ C1 (R)(δ + |ε|γ −1 + N −2 ) for some C1 (R) ≥ C0 (R). The problem of finding non-trivial solutions of the (Q1 )-equation reduces to find non-trivial critical points in B2R of the rescaled functional (still denoted Φε,N ) Z q 2d Φε,N (q1 ) = A(q1 ) − a2d−1 (ϕ1 ) 1 + Rε,N (q1 ) 2d T2 R 1 where the quadratic form A(q) := T2 2 (2 + ε)(∂ϕ2 q)2 + (∂ϕ1 q)(∂ϕ2 q) is positive definite on Q+ , negative definite on Q− and zero-definite on Q0 . For q1 = q+ + q0 + q− ∈ Q1 , α+ α− A(q1 ) = |q+ |2H 1 − |q− |2H 1 2 2 for suitable positive constants α+ , α− , bounded away from 0 by constants independent of ε. The geometry of Φε,N suggests to look for critical points of “linking type”. However we can not directly apply the linking theorem because Φε,N is defined only in B2R . To overcome this difficulty we extend Φε,N to the whole space in such a way that the e ε,N coincides with Γ(q1 ) outside B2R and so verifies the global hypothesis extension Φ of the linking theorem. R The final step is to prove (exploiting the homogeneity of the term a2d−1 q12d ) that ∃ R∗ > 0 independent on R, ε, N, γ and functions 0 < ε1 (R) ≤ ε0 (R), N1 (R) ≥ N0 (R) such that ∀|ε|γ −1 ≤ ε1 (R), N ≥ N1 (R) the functional e ε,N possesses a non trivial critical point q 1 ∈ Q1 with |q 1 |H1 ≤ R∗ . Φ ¯ := R∗ + 1, take |ε|γ −1 ≤ ε := ε(R) ¯ and 0 < σ < 1/N2 (R). ¯ The function Fix R h i 1/2(d−1) ¯ ¯ u(ε, ϕ) := |ε| q 1 + q2 (ε, N2 (R), q 1 ) + p(ε, N2 (R), q 1 ) is the solution of equation (1.2) in Theorem A. Remark 2. In the proof of Theorem B the bifurcation equation could be solved through variational methods as for Theorem A. However there is a simpler technique available. The bifurcation equation reduces, in the limit ε → 0, to a superquadratic Hamiltonian system with one degree of freedom. We prove existence of a non-degenerate, analytic solution by direct phase-space analysis. Therefore it can be continued by the Implicit function Theorem to an analytic solution of the complete bifurcation equation for ε small. References [AB] A. Ambrosetti, M. Badiale, Homoclinics: Poincar´e-Melnikov type results via a variational approach, Annales I. H. P. - Analyse nonlin., vol. 15, n.2, 1998, p. 233-252. [BaB] D. Bambusi, M. Berti, A Birkhoof-Lewis type theorem for some Hamiltonian PDE’s, to appear on SIAM Journal on Mathematical Analysis. [BR] V. Benci, P. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math. 52 (1979), no. 3, 241–273. [BBi] M. Berti, L. Biasco, Periodic solutions of nonlinear wave equations with non-monotone forcing terms, to appear on Rend. Mat. Acc. Naz. Lincei.

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[BB1] M. Berti, P. Bolle, Periodic solutions of nonlinear wave equations with general nonlinearities, Comm. Math. Phys. 243 (2003), no. 2, 315–328. [BB2] M. Berti, P. Bolle, Cantor families of periodic solutions for completely resonant non linear wave equations, preprint SISSA (2004). [BPr] M. Berti, M. Procesi, Quasi-periodic solutions of completely resonant forced wave equations, preprint Sissa. [PY] P.I. Plotinikov, L.N. Yungermann, Periodic solutions of a weakly nonlinear wave equation with an irrational relation of period to interval length, transl. in Diff. Eq. 24 (1988) n. 9 pp. 1059-1065. [P1] M. Procesi, Quasi-periodic solutions for completely resonant wave equations in 1D and 2D, to appear in Discr. Cont. Dyn. Syst. [R] P. Rabinowitz, Periodic solutions of nonlinear hyperbolic partial differential equations, Comm. Pure Appl. Math., 20 1967 145–205.

Keywords: Nonlinear Wave Equation, Quasi-Periodic Solutions, Variational Methods, Lyapunov-Schmidt reduction, Infinite Dimensional Hamiltonian Systems.1 2000AMS subject classification: 35L05, 37K50, 58E05.

1. Introduction We present in this Note the results of [BPr] concerning existence of small amplitude quasi-periodic solutions for completely resonant forced nonlinear wave equations like vtt − vxx + f (ω1 t, v) = 0 (1.1) v(t, x) = v(t, x + 2π) where the nonlinear forcing term f (ω1 t, v) = a(ω1 t)v 2d−1 + O(v 2d ),

d > 1, d ∈ N+

is 2π/ω1 -periodic in time and the forcing frequency is: • A) ω1 ∈ Q • B) ω1 ∈ R \ Q. Existence of periodic solutions for completely resonant forced wave equations was first proved in the pioneering paper [R] (with Dirichlet boundary conditions) if the forcing frequency is a rational number (ω1 = 1 in [R]). This requires to solve an infinite dimensional bifurcation equation which lacks compactness property; see [BBi] and references therein for other results. If the forcing frequency is an irrational number existence of periodic solutions has been proved in [PY]: here the bifurcation equation is trivial but a “small divisors problem” appears. To prove existence of small amplitude quasi-periodic solutions for completely resonant PDE’s like (1.1) one has to deal both with a small divisor problem and with an infinite dimensional bifurcation equation. A major point is to understand from which periodic solutions of the linearized equation vtt − vxx = 0 , Massimiliano Berti, SISSA, via Beirut 2-4, Trieste, Italy, [email protected] . Michela Procesi, Universit` a di Roma 3, Largo S. Leonardo Murialdo, Roma, Italy, [email protected] . 1 Supported by M.I.U.R. “Variational Methods and Nonlinear Differential Equations”. 1

2

MASSIMILIANO BERTI, MICHELA PROCESI

the quasi-periodic solutions branch-off: indeed such linear equation possesses only 2πperiodic solutions of the form q+ (t + x)+ q− (t − x). For completely resonant autonomous PDE’s, existence of quasi-periodic solutions with 2-frequencies have been recently obtained in [P1] for the specific nonlinearities f = u3 + O(u5 ). Here the bifurcation equation is solved by ODE methods. In [BPr] we prove existence of quasi-periodic solutions with two frequencies ω1 , ω2 for the completely resonant forced equation (1.1) in both the two cases A) and B). The more interesting case is ω1 ∈ Q (case A) when the forcing frequency ω1 enters in resonance with the linear frequency 1. To find out from which solutions of the linearized equation quasi-periodic solutions of (1.1) branch-off, requires to solve an infinite dimensional bifurcation equation which can not be solved in general by ODE techniques (it is a system of integro-differential equations). However, exploiting the variational nature of equation (1.1) like in [BB1], the bifurcation problem can be reduced to finding critical points of a suitable action functional which, in this case, possesses the infinite dimensional linking geometry [BR]. On the other hand, we avoid the inherent small divisor problem by restricting the parameters to uncountable zero-measure sets as, e.g., in [P1], [BB1]. 1.1. Main results. We look for quasi-periodic solutions v(t, x) of (1.1) of the form v(t, x) = u(ω1 t, ω2 t + x) u(ϕ1 + 2k1 π, ϕ2 + 2k2 π) = u(ϕ1 , ϕ2 ),

∀k1 , k2 ∈ Z

with frequencies ω = (ω1 , ω2 ) = (ω1 , 1 + ε) imposing the frequency ω2 = 1 + ε to be close to the linear frequency 1. Therefore we get h i (1.2) ω12 ∂ϕ2 1 + (ω22 − 1)∂ϕ2 2 + 2ω1 ω2 ∂ϕ1 ∂ϕ2 u(ϕ) + f (ϕ1 , u) = 0 . We assume that the forcing term f : T × R → R f (ϕ1 , u) = a2d−1 (ϕ1 )u2d−1 + O(u2d ),

d ∈ N+ , d > 1

is analytic in u but has only finite regularity in ϕ1 . More precisely P k + • (H) f (ϕ1 , u) := ∞ 1 )u , d ∈ N , d > 1 and the coefficients ak (ϕ1 ) ∈ k=2d−1 ak (ϕP ∞ H 1 (T) verify, for some r > 0, k=2d−1 |ak |H 1 rk < ∞. The function f (ϕ1 , u) is not identically constant in ϕ1 . We look for solutions u of (1.2) in the multiplicative Banach algebra o n X X Hσ,s := u(ϕ) = uˆl eil·ϕ : uˆ∗l = uˆ−l and |u|σ,s := |ˆ ul |e|l2 |σ [l1 ]s < +∞ l∈Z2

where [l1 ] := max{|l1 |, 1} and σ > 0, s ≥ 0. In [BPr] we prove the following Theorems.

l∈Z2

QUASI PERIODIC OSCILLATIONS

3

Theorem A. Let ω1 = n/m ∈ Q. Assume that f satisfies assumption (H) and a2d−1 (ϕ1 ) 6= 0, ∀ϕ1 ∈ T. Let Bγ be the uncountable zero-measure Cantor set o n γ , ∀l1 , l2 ∈ Z \ {0} Bγ := ε ∈ (−ε0 , ε0 ) : |l1 + εl2 | > |l2 | where 0 < γ < 1/6. There exist constants σ > 0, s > 2, ε > 0, C > 0, such that ∀ε ∈ Bγ , |ε|γ −1 ≤ ε¯/m2 , there exists a classical solution u(ε, ϕ) ∈ Hσ,s of (1.2) with (ω1 , ω2 ) = (n/m, 1 + ε) satisfying 1 1 m2 |ε| 2(d−1) 2(d−1) (1.3) q¯ε (ϕ) ≤ C |ε| u(ε, ϕ) − |ε| γ ω13 σ,s for an appropriate function q¯ε ∈ Hσ,s \ {0} of the form q¯ε (ϕ) = q¯+ (ϕ2 )+ q¯− (2mϕ1 − nϕ2 ). As a consequence, equation (1.1) admits the quasi-periodic solution v(ε, t, x) := u(ε, ω1 t, x+ω2 t) with two frequencies (ω1 , ω2 ) = (n/m, 1+ε) and the map t → v(ε, t, ·) ∈ H σ (T) has the form2 m2 h i 2d−1 1 2(d−1) 2(d−1) |ε| q¯+ (x + (1 + ε)t) + q¯− ((1 − ε)nt − nx) =O . v(ε, t, x) − |ε| γ ω13 H σ (T) Remark that the bifurcation of quasi-periodic solutions u occurs both for ω2 > 1 and ω2 < 1; actually both Bγ ∩ (0, ε0 ) and Bγ ∩ (−ε0 , 0) are uncountable ∀ε0 > 0. At the first order the quasi-periodic solution v(ε, t, x) of equation (1.1) is the superposition of two waves traveling in opposite directions (in general, both components q+ , q− are non trivial). The bifurcation of quasi-periodic solutions looks quite different if ω1 is irrational. R 2π Theorem B. Let ω1 ∈ R\Q. Assume that f satisfies assumption (H), 0 a2d−1 (ϕ1 )dϕ1 6= 0 and f (ϕ1 , u) ∈ H s (T), s ≥ 1, for all u. Let Cγ ⊂ D ≡ (−ε0 , ε0 ) × (1, 2) be the uncountable zero-measure Cantor set ω1 γ ∈ / Q , |ω l + εl | > , (ε, ω ) ∈ D : ω ∈ / Q , 1 1 2 1 1 ω2 |l1 | + |l2 | . (1.4) Cγ := γ |ω1 l1 + (2 + ε)l2 | > , ∀ l1 ∈ Z \ {0} |l1 | + |l2 | Fix any 0 < s < s − 1/2. There exist R 2π positive constants ε, C, σ > 0, such that, −1 ∀(ε, ω1 ) ∈ Cγ with |ε|γ < ε and ε 0 a2d−1 (ϕ1 ) dϕ1 > 0, there exists a nontrivial solution u(ε, ϕ) ∈ Hσ,s of equation (1.2) with (ω1 , ω2 ) = (ω1 , 1 + ε) satisfying 1 1 |ε| 2(d−1) q ≤ C |ε| 2(d−1) (1.5) u(ε, ϕ) − |ε| ¯ (ϕ ) ε 2 γ σ,s for some function q¯ε (ϕ2 ) ∈ H σ (T) \ {0}. As a consequence, equation (1.1) admits the non-trivial quasi-periodic solution v(ε, t, x) := u(ε, ω1 t, x + ω2 t) with two frequencies (ω1 , ω2 ) = (ω1 , 1 + ε) and the map t → v(ε, t, ·) ∈ 2We

denote H σ (T) := {u(ϕ) =

P

l∈Z

u ˆl eilϕ : |u|H σ (T) :=

P

l∈Z

|ˆ ul |eσ|l| < +∞}.

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H σ (T) has the form 1 2(d−1) q¯ε (x + (1 + ε)t) v(ε, t, x) − |ε|

H σ (T)

=O γ

−1

|ε|

2d−1 2(d−1)

.

Remark 1. Imposing in the definition of Cγ the condition ω1 /ω2 = ω1 /(1 + ε) ∈ Q we obtain, by Theorem B, the existence of periodic solutions of equation (1.1). They are reminiscent, in this completely resonant context, of the Birkhoff-Lewis periodic orbits with large minimal period accumulating at the origin, see [BaB]. In Theorem B, existence of quasi-periodic solutions could follow by other hypotheses on f . However the hypothesis that the leading term in the nonlinearity f is an odd power of u is not of technical nature. Actually, the following non-existence result holds: R 2π Theorem C. (Non-existence) Let f (ϕ1 , u) = a(ϕ1 )uD with D even and 0 a(ϕ1 ) dϕ1 6= 0. ∀R > 0 there exists ε0 > 0 such that ∀σ ≥ 0, s > s − 12 , ∀(ε, ω1 ) ∈ Cγ with |ε| < ε0 , equation (1.2) does not possess solutions u ∈ Hσ,s in the ball |u|σ,s ≤ R|ε|1/(D−1) . Theorem C also highlights that the existence of periodic solutions of nonlinear wave equations in the case of even powers, see [BB1], is due to the boundary effects imposed by the Dirichlet conditions. 2. Sketch of the proof of Theorem A We sketch now the proof of (the more difficult) Theorem A. To simplify notations we assume ω1 = 1, ε > 0 and a2d−1 (ϕ1 ) > 0. Instead of looking for solutions of equation (1.2) in a shrinking neighborhood of 0, it is a convenient devise to perform the rescaling u → δu

with

δ := |ε|1/2(d−1)

enhancing the relation between the amplitude δ and the frequency ω2 = 1 + ε. We obtain the equation (2.1)

Lε u + εf (ϕ1 , u, δ) = 0

where i h i h Lε := ∂ϕ2 1 + 2∂ϕ1 ∂ϕ2 + ε (2 + ε)∂ϕ2 2 + 2∂ϕ1 ∂ϕ2 ≡ L0 + εL1 and f (ϕ1 , δu) 2d−1 2d f (ϕ1 , u, δ) := 2(d−1) = a2d−1 (ϕ1 )u + δa2d (ϕ1 )u + . . . . δ Equation (2.1) is the Euler-Lagrange equation of the Lagrangian action functional Ψε ∈ C 1 (Hσ,s , R) defined by Z ε(2 + ε) 1 (∂ϕ1 u)2 + (∂ϕ1 u)(∂ϕ2 u) + (∂ϕ2 u)2 + ε(∂ϕ1 u)(∂ϕ2 u) − εF (ϕ1 , u, δ) Ψε (u) := 2 2 2 T ≡ Ψ0 (u) + εΓ(u, δ)

QUASI PERIODIC OSCILLATIONS

where F (ϕ1 , u, δ) :=

5

Ru

f (ϕ1 , ξ, δ)dξ and Z 1 Ψ0 (u) := (∂ϕ1 u)2 + (∂ϕ1 u)(∂ϕ2 u) 2 2 ZT (2 + ε) (∂ϕ2 u)2 + (∂ϕ1 u)(∂ϕ2 u) − F (ϕ1 , u, δ) . Γ(u, δ) := 2 T2 0

To find critical points of Ψε we perform a variational Lyapunov-Schmidt reduction inspired to [BB1], see also [AB]. The unperturbed functional Ψ0 : Hσ,s → R possesses an infinite dimensional linear space Q of critical points which are the solutions q of the equation L0 q = ∂ϕ1 ∂ϕ1 + 2∂ϕ2 q = 0 . The space Q can be written as n o X Q= q= qˆl eil·ϕ ∈ Hσ,s | qˆl = 0 for l1 (l1 + 2l2 ) 6= 0 l∈Z2

and, in view of the variational linking argument used for the bifurcation equation, we split Q as Q = Q+ ⊕ Q0 ⊕ Q− where3 Q+ Q0 Q−

n o n o σ := q ∈ Q : qˆl = 0 for l1 = 0 = q+ := q+ (ϕ2 ) ∈ H0 (T) n o := q0 ∈ R n o n o σ,s := q ∈ Q : qˆl = 0 for l1 + 2l2 = 0 = q− := q− (2ϕ1 − ϕ2 ), q− (·) ∈ H0 (T) .

We shall also use in Q the H 1 -norm | · |H 1 which is the natural norm for applying variational methods to the bifurcation equation. In order to prove analiticity (in ϕ2 ) of the solutions and to highlight the compactness of the problem we perform a finite dimensional Lyapunov-Schmidt reduction [BB2], decomposing Hσ,s = Q1 ⊕ Q2 ⊕ P where Q1 := Q1 (N ) :=

n

q=

X |l|≤N

il·ϕ

qˆl e

o

n o X il·ϕ ∈ Q , Q2 := Q2 (N ) := q = qˆl e ∈ Q |l|>N

and n o X il·ϕ P := p = pˆl e ∈ Hσ,s | pˆl = 0 for l1 (2l2 + l1 ) = 0 . l∈Z2 3H σ (T)

P denotes the functions of H σ (T) with zero average. H σ,s (T) := {u(ϕ) = l∈Z u ˆl eilϕ : P σ,s ∗ σ|l| s ul = u−l , |u|H σ,s (T) := l∈Z |ˆ ul |e [l] < +∞} and H0 (T) its functions with zero average. 0

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Projecting equation (2.1) onto the closed subspaces Qi (i = 1, 2) and P , setting u = q1 + q2 + p with qi ∈ Qi and p ∈ P , we obtain h i f (ϕ , q + q + p, δ) =0 (Q1 ) L [q ] + Π 1 1 2 1 1 Q1 h i (Q2 ) L1 [q2 ] + ΠQ2 f (ϕ1 , q1 + q2 + p, δ) = 0 h i Lε [p] + εΠP f (ϕ1 , q1 + q2 + p, δ) = 0 (P ) where ΠQi : Hσ,s → Qi are the projectors onto Qi and ΠP : Hσ,s → P is the projector onto P . Step 1. (Solution of the (Q2 )-(P )-equations.) For ε ∈ Bγ , Lε restricted to P −1 has a bounded inverse satisfying |L−1 . Morevoer, also the operator ε [h]|σ,s ≤ |h|σ,s γ −2 L1 : Q2 → Q2 is invertible and |L−1 [h]| ≤ |h| N . σ,s σ,s 1 Fixed points of the nonlinear operator G : Q2 ⊕ P → Q2 ⊕ P defined by −1 f (ϕ , q + q + p, δ), −εL Π f (ϕ , q + q + p, δ) G(q2 , p; q1 ) := − L−1 Π P 1 1 2 Q2 1 1 2 ε 1 are solutions of the (Q2 )-(P )-equations. Using the Contraction Mapping Theorem, we can prove that ∀R > 0 there exist an integer N0 (R) ∈ N+ and positive constants ε0 (R) > 0, C0 (R) > 0 such that: ∀|q1 |H 1 ≤ 2R , ∀ε ∈ Bγ , |ε|γ −1 ≤ ε0 (R) , ∀N ≥ N0 (R) : 0 ≤ σN ≤ 1 , there exists a unique solution (q2 (q1 ), p(q1 )) := (q2 (ε, N, q1 ), p(ε, N, q1 )) ∈ Q2 ⊕ P of the (Q2 )-(P )-equations which satisfies |q2 (ε, N, q1 )|σ,s ≤ C0 (R)N −2 and |p(ε, N, q1 )|σ,s ≤ C0 (R)|ε|γ −1 . Step 2. (Solution of the (Q1 )-equation.) There remains to solve the finite dimensional (Q1 )-equation (2.2)

L1 [q1 ] + ΠQ1 f (ϕ1 , q1 + q2 (q1 ) + p(q1 ), δ) = 0 .

Actually (see [BB1]-[AB]) the bifurcation equation (2.2) is the Euler-Lagrange equation of the reduced Lagrangian action functional Φε,N : B2R ⊂ Q1 → R,

Φε,N (q1 ) := Ψε (q1 + q2 (q1 ) + p(q1 ))

which can be written as

Φε,N (q1 ) = const + ε Γ(q1 ) + Rε,N (q1 ) where

q 2d (2 + ε) (∂ϕ2 q1 )2 + (∂ϕ1 q1 )(∂ϕ2 q1 ) − a2d−1 (ϕ1 ) 1 2 2d T2 is the “Poincar´e-Melnikov” function [AB] and the remainder Z Rε,N (q1 ) := F (ϕ1 , q1 , δ = 0) − F (ϕ1 , q1 + q2 (q1 ) + p(q1 ), δ) Z

Γ(q1 ) :=

T2

1 + f (ϕ1 , q1 + q2 (q1 ) + p(q1 ), δ)(q2 (q1 ) + p(q1 )) 2

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7

satisfies |Rε,N (q1 )| ≤ C1 (R)(δ + |ε|γ −1 + N −2 ) for some C1 (R) ≥ C0 (R). The problem of finding non-trivial solutions of the (Q1 )-equation reduces to find non-trivial critical points in B2R of the rescaled functional (still denoted Φε,N ) Z q 2d Φε,N (q1 ) = A(q1 ) − a2d−1 (ϕ1 ) 1 + Rε,N (q1 ) 2d T2 R 1 where the quadratic form A(q) := T2 2 (2 + ε)(∂ϕ2 q)2 + (∂ϕ1 q)(∂ϕ2 q) is positive definite on Q+ , negative definite on Q− and zero-definite on Q0 . For q1 = q+ + q0 + q− ∈ Q1 , α+ α− A(q1 ) = |q+ |2H 1 − |q− |2H 1 2 2 for suitable positive constants α+ , α− , bounded away from 0 by constants independent of ε. The geometry of Φε,N suggests to look for critical points of “linking type”. However we can not directly apply the linking theorem because Φε,N is defined only in B2R . To overcome this difficulty we extend Φε,N to the whole space in such a way that the e ε,N coincides with Γ(q1 ) outside B2R and so verifies the global hypothesis extension Φ of the linking theorem. R The final step is to prove (exploiting the homogeneity of the term a2d−1 q12d ) that ∃ R∗ > 0 independent on R, ε, N, γ and functions 0 < ε1 (R) ≤ ε0 (R), N1 (R) ≥ N0 (R) such that ∀|ε|γ −1 ≤ ε1 (R), N ≥ N1 (R) the functional e ε,N possesses a non trivial critical point q 1 ∈ Q1 with |q 1 |H1 ≤ R∗ . Φ ¯ := R∗ + 1, take |ε|γ −1 ≤ ε := ε(R) ¯ and 0 < σ < 1/N2 (R). ¯ The function Fix R h i 1/2(d−1) ¯ ¯ u(ε, ϕ) := |ε| q 1 + q2 (ε, N2 (R), q 1 ) + p(ε, N2 (R), q 1 ) is the solution of equation (1.2) in Theorem A. Remark 2. In the proof of Theorem B the bifurcation equation could be solved through variational methods as for Theorem A. However there is a simpler technique available. The bifurcation equation reduces, in the limit ε → 0, to a superquadratic Hamiltonian system with one degree of freedom. We prove existence of a non-degenerate, analytic solution by direct phase-space analysis. Therefore it can be continued by the Implicit function Theorem to an analytic solution of the complete bifurcation equation for ε small. References [AB] A. Ambrosetti, M. Badiale, Homoclinics: Poincar´e-Melnikov type results via a variational approach, Annales I. H. P. - Analyse nonlin., vol. 15, n.2, 1998, p. 233-252. [BaB] D. Bambusi, M. Berti, A Birkhoof-Lewis type theorem for some Hamiltonian PDE’s, to appear on SIAM Journal on Mathematical Analysis. [BR] V. Benci, P. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math. 52 (1979), no. 3, 241–273. [BBi] M. Berti, L. Biasco, Periodic solutions of nonlinear wave equations with non-monotone forcing terms, to appear on Rend. Mat. Acc. Naz. Lincei.

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[BB1] M. Berti, P. Bolle, Periodic solutions of nonlinear wave equations with general nonlinearities, Comm. Math. Phys. 243 (2003), no. 2, 315–328. [BB2] M. Berti, P. Bolle, Cantor families of periodic solutions for completely resonant non linear wave equations, preprint SISSA (2004). [BPr] M. Berti, M. Procesi, Quasi-periodic solutions of completely resonant forced wave equations, preprint Sissa. [PY] P.I. Plotinikov, L.N. Yungermann, Periodic solutions of a weakly nonlinear wave equation with an irrational relation of period to interval length, transl. in Diff. Eq. 24 (1988) n. 9 pp. 1059-1065. [P1] M. Procesi, Quasi-periodic solutions for completely resonant wave equations in 1D and 2D, to appear in Discr. Cont. Dyn. Syst. [R] P. Rabinowitz, Periodic solutions of nonlinear hyperbolic partial differential equations, Comm. Pure Appl. Math., 20 1967 145–205.

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