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LECTURE NOTES on

NONLINEAR OSCILLATIONS OFHAMILTONIAN PDEs

Massimiliano Berti

26 November 2006

Introduction

Many Partial Differential Equations (PDEs) arising in physics can be seenas Infinite Dimensional Hamiltonian Systems

∂tu = J(∇uH)(u, t) , u ∈ H (1)

where the Hamiltonian function

H : H×R → R

is defined on an infinite dimensional Hilbert spaceH and J is a non-degenerateantisymmetric operator.

The main examples of Hamiltonian PDEs are:

a) The nonlinear wave equation

utt −∆u+ f(x, u) = 0 .

b) The nonlinear Schrodinger equation

i ∂tu−∆u+ f(x, |u|2)u = 0 .

c) The beam equationutt + uxxxx + f(x, u) = 0

and the higher dimensional membrane equation

utt + ∆2u+ f(x, u) = 0 .

d) The Euler equations of Hydrodynamics describing the evolution of aperfect, incompressible, irrotational fluid under the action of gravityand surface tension, as well as its approximate models like the Kortewegde Vries (KdV) equation, the Boussinesq equation, the Kadomtsev-Petviashvili equation.

2

3

For example the celebrated KdV equation

ut − uux + uxxx = 0

was introduced to describe the motion of the free surface of a fluid inthe small amplitude, long waves regime.

We start describing the Hamiltonian structure of the 1-dimensional Non-Linear Wave equation with Dirichlet boundary conditions

(NLW )

utt − uxx + f(t, x, u) = 0u(t, 0) = u(t, π) = 0

or periodic boundary conditions u(t, x + 2π) = u(t, x). In these notes weshall consider both the autonomous NLW (f independent of time) and thenon-autonomous NLW (forced case).

NLW can be seen as a second order Lagrangian PDE

utt = −(∇L2W )(u, t) (2)

defined on the infinite dimensional “configuration space” H10 (0, π) where

W (·, t) : H10 (0, π) → R

is the “potential energy”

W (u, t) :=

∫ π

0

(u2x

2+ F (t, x, u)

)dx

and (∂uF )(t, x, u) := f(t, x, u). Indeed, differentiating W and integrating byparts

duW (u, t)[h] =

∫ π

0

(uxhx + f(t, x, u)h

)dx =

∫ π

0

(− uxx + f(t, x, u)

)h dx ,

by the Dirichlet boundary conditions, so that the L2-gradient of W withrespect to u is

(∇L2W )(u, t) = −uxx + f(t, x, u)

and NLW writes as (2).The Lagrangian function of NLW is, as usual, the difference between the

kinetic and the potential energy

L(u, ut, t) :=

∫ π

0

u2t

2dx−W (u, t) =

∫ π

0

(u2t

2− u2

x

2− F (t, x, u)

)dx .

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The Nonlinear Wave equation (2) can be written also in Hamiltonian formut = ppt = −(∇L2W )(u, t) = uxx − f(t, x, u)

and so in the form (1) with Hamiltonian function H : H10 × L2 ×R → R

H(u, p, t) :=

∫ π

0

p2

2dx+W (u, t) =

∫ π

0

(p2

2+u2

x

2+ F (t, x, u)

)dx

defined on the infinite dimensional “phase-space” H10 × L2 × R, and the

operator J =

(0 I−I 0

)represents a symplectic structure: the variables

(u, p) are “Darboux coordinates”.

Similarly we can deal with the NLW equation under periodic boundaryconditions, choosing as configuration space H1(T) where T := R/2πZ.

The Hamiltonian structure of the nonlinear Schrodinger equation (b), thebeam and membrane equation (c), the KdV and the Euler equation (d) arepresented in the Appendix.

In these Notes we shall adopt the point of view of regarding the NonlinearWave equation (2) as an Infinite Dimensional Hamiltonian System, focusingin the search of invariant sets for the flow.

The analysis of the principle structures of an infinite dimensional phasespace such as equilibria, periodic orbits, embedded invariant tori, centermanifolds as well as stable and unstable manifolds, is an essential changeof paradigm in the study of hyperbolic equations with respect to the tra-ditional pursuit of the initial value problem (mainly studied for dispersiveequations).

Such “Dynamical Systems Philosophy” has allowed to find many newresults, known for finite dimensional Hamiltonian systems, for HamiltonianPDEs.

In the study of a complex Dynamical System the easiest invariant mani-folds to look for are the equilibria and, next, the periodic orbits.

The relevance of periodic solutions for understanding the dynamics of afinite dimensional Hamiltonian System, was first highlighted by Poincare.

In 1899 dans “Les methodes nouvelles de la Mecanique Celeste” Poincarewrote (apropos periodic solutions in the three-body problem)

D’ailleurs, ce qui nous rend ces solutions periodiques si precieuses, c’estqu’elles sont, pour ainsi dire, la seule breche par ou nous puissonsessayer de penetrer dans une place jusqu’ici reputee inabordable.

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At a first glance, their knowledge does not seem really interesting becausethe periodic orbits form a zero measure set in the phase space:

En effet, il y a une probabilite nulle pour que les conditions initiales dumouvement soient precisement celles qui correspondent a une solutionsperiodique.

But, as conjectured by Poincare, their knowledge is not irrelevant

“...voici un fait que je n’ai pu demontrer rigoureusement, mais qui me paraitpourtant tres vraisemblable. Etant donnees des equations de la formedefinie dans le n. 13 [the Hamilton’s equations] et une solution parti-culiere quelconque de ces equations, on peut toujours trouver une solu-tion periodique (dont la periode peut, il est vrai, etre tres longue), telleque la difference entre les deux solutions soit aussi petite qu’on le veut,pendant un temps aussi long qu’on le veut.” ([108], Tome 1, ch. III, a.36).

This conjecture was the main motivation for the systematic study of peri-odic orbits started with Poincare and then continued by Lyapunov, Birkhoff,Moser, Weinstein and many others.

The first existence results, based on the Poincare continuation method,regarded bifurcation of periodic orbits from elliptic (linearly stable) equi-libria under non-resonance conditions on the eigenfrequencies of the smalloscillations (Lyapunov Center theorem [75]). These periodic solutions arethe continuations of the linear normal modes having periods close to theperiods of the linearized equation.

Subsequently, more in the direction inspired by the conjecture of Poincare,Birkhoff and Lewis [30]-[32] proved that, for “sufficiently nonlinear” Hamil-tonian systems, there exist infinitely many periodic solutions with large min-imal period in any neighborhood of an elliptic periodic orbit.

Great progress in understanding the very complex orbit structure of anHamiltonian system was done by Kolmogorov [83], Arnold [8] and Moser [91](KAM theory) in the sixties: according to this theory, at least for sufficientlysmooth nearly integrable systems, a positive measure set of the phase spaceconsists of quasi-periodic solutions (KAM tori). The main difficulty for theirexistence proof is the presence of arbitrarily “small denominators” in theirapproximate expansion series.

From a dynamical point of view, these “small denominators” had alreadybeen highlighted by Poincare as the main source of chaotic dynamics due to

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very complex resonance type phenomena. Nowadays, this is the object ofstudy of “Arnold diffusion” [9].

Going back to periodic orbits, in the seventies, Weinstein [126], Moser[95] and Fadell-Rabinowitz [58], succeeded, with the aid of variational andtopological methods in bifurcation theory, to extend the Lyapunov Centertheorem removing the non-resonance condition on the linear eigenfrequencies.

About the conjecture of Poincare, some progress was achieved by Conleyand Zehnder [44] who proved that the KAM tori lie in the closure of theset of the periodic solutions1 (having, it is true, very long minimal period).As a consequence, the closure of the periodic orbits has positive Lebesguemeasure.

The conjecture of Poincare was positively answered only in a generic senseby Pugh and Robinson [110] in the early eighties: for a generic Hamiltonian(in the C2 topology) the periodic orbits are dense on a compact and regularenergy surface. On the other hand, for specific systems, the possibility ofapproximating any motion with periodic orbits still remains an open problem.

Finally, the search of periodic solutions in Hamiltonian systems gave riseto many new ideas and techniques in critical point theory, especially afterthe breakthrough of Rabinowitz [114] and Weinstein [127] on the existence ofa periodic solution on each compact and strictly convex energy hypersurface,see Ekeland [54], Hofer-Zehnder [69] and references therein.

Building on the experience gained from the qualitative study of finite di-mensional dynamical systems, the search of periodic solutions was regardedas a first step toward better understanding also the complicated flow evolu-tion of Hamiltonian PDEs.

In this direction Rabinowitz [115] and Brezis-Coron-Nirenberg [40] provedthe existence of periodic solutions for nonlinear wave equations via minimaxvariational methods (see Theorem 6.3.1 in the Appendix). Interestingly,these proofs work only to find periodic orbits with a rational frequency, thereason being that other periods give rise to a “small denominators” problemtype (remark 6.3.2).

Independently of these global results, in the early nineties the local bifur-cation theory of periodic and quasi-periodic solutions started to be extendedto “non-resonant” Hamiltonian PDEs like

utt − uxx + a1(x)u = a2(x)u2 + a3(x)u

3 + . . . (3)

by Kuksin, Wayne, Craig, Bourgain, Poeschel... followed by many others.

1For recent Birkhoff-Lewis type results concerning lower dimensional KAM tori -andapplications to the three body problem- we refer to [22].

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There exist periodic and quasi-periodic solutions close to the elliptic equi-librium u = 0?

It turns out that in infinite dimension also the periodic problem exhibitsa “small divisors difficulty”, whereas in finite dimension it arises only forfinding quasi-periodic solutions.

The first existence proofs of small amplitude periodic and quasi-periodicsolutions were obtained by Kuksin [84] and Wayne [125]. These results werebased on KAM theory and were initially limited to Dirichlet boundary con-ditions, because of the near coincidence of the linear eigenfrequencies underperiodic spatial boundary conditions.

In an effort to avoid these restrictions Craig-Wayne [49] introduced theLyapunov-Schmidt reduction method for Hamiltonian PDEs, extending theLyapunov Center theorem in presence of periodic boundary conditions. Lateron this method was generalized by Bourgain [33], [34] to construct also quasi-periodic solutions. For subsequent developments of the KAM approach, see[105], [43], [87], [85] and references therein.

All the previous results concerned “non-resonant” PDEs, like the nonlin-ear wave equation (3) with a1(x) 6≡ 0. The term a1(x) allows to verify, asin the Lyapunov Center theorem, suitable non-resonance conditions on thelinear eigenfrequencies of the small oscillations, and the bifurcation equationis finite dimensional2.

The aim of this course is to present recent bifurcation results of “Nonlin-ear Oscillations of Hamiltonian PDEs”, especially for “completely resonant”nonlinear wave equations (3) with

a1(x) ≡ 0

where infinite dimensional bifurcation phenomena appear jointly with smalldivisors difficulties. We shall deal with both autonomous and forced equa-tions. In the autonomous case our results can be seen as infinite dimensionalanalogues of the Weinstein-Moser and Fadell-Rabinowitz theorems. In [46]this case was pointed out as an important open question.

The plan of these lectures -which have a mainly didactical purpose- goesas follows:

Chapter 1) We shall start with an overview in finite dimension of the Lya-punov Center theorem, the Weinstein-Moser and the Fadell-Rabinowitz

2For non-resonant PDEs, also an extension of the Birkhoff-Lewis periodic orbit theoryhas been obtained, see [14], [29].

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theorems, concerning bifurcation of periodic solutions close to equilib-ria (nonlinear normal modes), respectively in the non-resonant and inthe resonant case.

Chapter 2) Next we shall pass to Infinite Dimensional Hamiltonian PDEsdescribing the “small divisors problem” and the presence of an infinitedimensional bifurcation phenomenon dealing with “completely reso-nant” autonomous NLW. The infinite dimensional bifurcation equationis solved via critical point theory, through a variant of the Ambrosetti-Rabinowitz Mountain Pass Theorem. The exposition is self-contained(in the Appendix we recall the main ideas in critical point theory). Thisinfinite resonance variational analysis is the main new contribution of[23]-[24]-[26] (see also [28]).

Chapter 3) contains a tutorial in Nash-Moser theory, which has beeninvented to overcome situations in which the standard Implicit Func-tion Theorem does not apply because the linearized operator is onlyinvertible with a “loss of derivatives” (due, for example, to the “smalldivisors”). We shall present first a simplified form of this theorem (ex-tracted by the final chapter of Zehnder in [100]) to highlight the mainfeatures of the method in an analytic setting. For completeness, wedeal also with a simple differentiable Nash-Moser theorem.

Chapter 4) Later we shall prove a Nash-Moser type theorem for com-pletely resonant nonlinear wave equations, implying the existence ofperiodic solutions for sets of frequencies of asymptotically full measure.We shall emphasize the inversion of the linearized equations which isthe most delicate step of any Nash-Moser Implicit Function Theorem.The required spectral analysis is new (see [25]) and different from theone of Craig-Wayne and Bourgain and allows to deal with nonlinearitieswith low regularity and without symmetry assumptions.

Chapter 5) Finally we shall present new existence results of [21] for peri-odic solutions in the forced case (with rational period). The bifurcationequation (which now, unlike the autonomous case, is completely de-generate) is solved via purely variational methods (not of Nash-Mosertype). This approach is based on constrained minimization and it isinspired as in Rabinowitz [111] by elliptic regularity theory.

Appendix) We present the Hamiltonian formulation of some PDEs includ-ing the Euler’s equation of hydrodynamics. We also discuss the basicnotions in critical point theory which are used throughout these Lecture

9

Notes. Finally we prove the Rabinowitz [115] result about existence ofone 2π-periodic solution of a superlinear NLW via minimax methods,following the proof of Brezis-Coron-Nirenberg [40].

This monograph is based on the material of a series of PhD courses theauthor delivered at the International School for Advanced Studies (SISSA)in the last academic years. Part of the results discussed here have appearedin joint works with Luca Biasco and Philippe Bolle. I wish to warmly thankthese friends for many hours of enjoyable work together.

This text is far from being an exhaustive treatise on the theory of Hamil-tonian Partial Differential equations, but, rather, an introduction to the re-search in this fascinating and rapidly growing field.

General references:

For finite dimensional Dynamical Systems:

Moser J., Zehnder E., Notes on Dynamical Systems, Courant Lecture Notes2005 (and ICTP Lecture Notes 1991).

For infinite dimensional Hamiltonian systems:

Kuksin S., Nearly integrable infinite-dimensional Hamiltonian systems, Lec-ture Notes in Mathematics, 1556, Springer-Verlag, Berlin, 1993.

Craig W., Problemes des petits diviseurs dans les equations aux derivees par-tielles, Panoramas et Syntheses, 9, Societe Mathematique de France, Paris,2000.

For bifurcation theory:

Ambrosetti A., Prodi G., A Primer of Nonlinear Analysis, Cambridge Univ.Press.

Nirenberg L., Topics in Nonlinear Functional Analysis, Courant Institute ofMathematical Sciences, New York University.

For the Nash-Moser Implicit Function Theorem:

the last chapter by Zehnder in the previous book of Nirenberg.

For critical point theory:

Rabinowitz P., Minimax methods in critical point theory with applications todifferential equations, CBMS Regional Conference Series in Mathematics, 65.American Mathematical Society, 1986.

Struwe M., Variational methods. Applications to nonlinear partial differen-tial equations and Hamiltonian systems, 34. Springer-Verlag, Berlin, 1996.

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Acknowledgments: I wish to express my deep graditude to Prof. A. Am-brosetti for his constant encouragement and for very useful comments whichimproved the presentation of these Lecture Notes.

I wish to thank him also for his invitation to teach this course at the SISSASpring School, “Variational Problems in Nonlinear Analysis”, Trieste-2005,at the SISSA PhD spring semester 2006, and at the “School on NonlinearDifferential Equations”, ICTP, Trieste, October 2006.

I warmly thank D. Bambusi, L. Biasco, P. Bolle, W. Craig, M. Procesi forvery stimulating and fruitful discussions and for sharing much work together.I’m also deeply indebted to A. Abbondandolo, V. Coti Zelati, M. Groves, forcarefully reading the manuscript.

I’m very grateful also to all the students for the high interest they demon-strated, and the numerous questions they posed, in particular P. Baldi, S.Paleari, N. Visciglia. I thank L. Di Gregorio for the pictures.

Napoli, 20 November 2006.

Address: Massimiliano Berti, Dipartimento di Matematica e Applicazioni“R. Caccioppoli”, Universita di Napoli “Federico II”, via Cintia, 80126Napoli, Italy, [email protected] .

Contents

Index 11

1 Finite dimension 131.1 The Lyapunov Center Theorem . . . . . . . . . . . . . . . . . 171.2 The Theorems of Weinstein-Moser and Fadell-Rabinowitz . . . 20

1.2.1 The variational Lyapunov-Schmidt reduction . . . . . . 261.2.2 Proof of the Weinstein-Moser Theorem 1.2.3 . . . . . . 281.2.3 Proof of the Fadell-Rabinowitz Theorem 1.2.4 . . . . . 31

2 Infinite dimension 392.0.4 The Lyapunov Center Theorem for PDEs . . . . . . . . 41

2.1 Completely resonant wave equations . . . . . . . . . . . . . . 432.2 The case p odd . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.2.1 The range equation . . . . . . . . . . . . . . . . . . . . 472.2.2 The bifurcation equation . . . . . . . . . . . . . . . . . 522.2.3 The Mountain Pass argument . . . . . . . . . . . . . . 54

2.3 The case p even . . . . . . . . . . . . . . . . . . . . . . . . . . 582.4 Multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.5 The small divisor problem . . . . . . . . . . . . . . . . . . . . 64

3 A tutorial in Nash-Moser theory 683.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.2 An analytic Nash-Moser Theorem . . . . . . . . . . . . . . . . 693.3 A differentiable Nash-Moser Theorem . . . . . . . . . . . . . . 76

4 Application to the NLW 824.0.1 The 0th order bifurcation equation . . . . . . . . . . . 84

4.1 The finite dimensional reduction . . . . . . . . . . . . . . . . . 854.1.1 Solution of the (Q2)-equation . . . . . . . . . . . . . . 86

4.2 Solution of the range equation . . . . . . . . . . . . . . . . . . 894.2.1 The Nash-Moser scheme . . . . . . . . . . . . . . . . . 92

4.3 Solution of the (Q1)-equation . . . . . . . . . . . . . . . . . . 104

11

CONTENTS 12

4.4 The linearized operator . . . . . . . . . . . . . . . . . . . . . . 1084.4.1 Decomposition of Ln(ε, v1, w) . . . . . . . . . . . . . . 1084.4.2 Step 1: Inversion of D . . . . . . . . . . . . . . . . . . 1114.4.3 Step 2: Inversion of Ln . . . . . . . . . . . . . . . . . . 114

5 Forced vibrations 1215.1 The forcing frequency ω ∈ Q . . . . . . . . . . . . . . . . . . 1215.2 The variational Lyapunov-Schmidt reduction . . . . . . . . . . 124

5.2.1 The range equation . . . . . . . . . . . . . . . . . . . . 1265.2.2 The bifurcation equation . . . . . . . . . . . . . . . . . 128

5.3 Monotone f . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1305.3.1 Step 1: the L∞-estimate . . . . . . . . . . . . . . . . . 1305.3.2 Step 2: the H1-estimate . . . . . . . . . . . . . . . . . 133

5.4 Non-Monotone f . . . . . . . . . . . . . . . . . . . . . . . . . 1355.4.1 Step 1: the L2k-estimate . . . . . . . . . . . . . . . . . 1405.4.2 Step 2: the L∞-estimate. . . . . . . . . . . . . . . . . . 1415.4.3 Step 3: The H1-estimate. . . . . . . . . . . . . . . . . 1435.4.4 The “maximum principle” . . . . . . . . . . . . . . . . 144

6 Appendix 1496.1 Hamiltonian PDEs . . . . . . . . . . . . . . . . . . . . . . . . 149

6.1.1 The Euler equations of hydrodynamics . . . . . . . . . 1506.2 Critical point theory . . . . . . . . . . . . . . . . . . . . . . . 154

6.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 1546.2.2 Minima . . . . . . . . . . . . . . . . . . . . . . . . . . 1556.2.3 The minimax idea . . . . . . . . . . . . . . . . . . . . . 1576.2.4 The Mountain Pass Theorem . . . . . . . . . . . . . . 159

6.3 Free vibrations of NLW: a global result . . . . . . . . . . . . . 1636.4 Approximation of irrationals by rationals . . . . . . . . . . . . 167

6.4.1 Continued fractions . . . . . . . . . . . . . . . . . . . . 1706.5 The Banach algebra property of Xσ,s . . . . . . . . . . . . . . 172

Chapter 1

Finite dimension

Let consider a finite dimensional dynamical system

x = f(x) , x ∈ Rn (1.1)

where the vector fieldf ∈ C2(Rn,Rn)

possesses an equilibrium at x = 0, i.e f(0) = 0.

• Question: there exist periodic solutions of (1.1) close to x = 0?

The first thing is to study the linearized system

x = Ax, A := (Dxf)(0) . (1.2)

A necessary condition to find periodic solutions of the nonlinear system(1.1) close to the equilibrium x = 0 is the presence of purely imaginaryeigenvalues of A. Indeed, if all the eigenvalues λ = α± iβ of A have non zeroreal part (i.e. the matrix A is hyperbolic), then the Hartmann-Grobmanntheorem (see e.g.[65]) ensures a topological conjugacy between the linear andthe non-linear flows close to the equilibrium, see figure 1.1. In this case, sincethe linear system (1.2) does not possess periodic solutions, neither does thenonlinear system (1.1).

Therefore, if A has the eigenvalues

λ1, . . . , λn

not necessarily distinct from each other, we shall assume that

λ1 = iω , λ2 = −iω

13

CHAPTER 1. FINITE DIMENSION 14

x’=Ax x’=f(x)

E

EE

Es s

u u

W

W

s

u

Figure 1.1: The Hartmann Grobmann theorem.

is a pair of purely imaginary eigenvalues (A is a real matrix) with complexeigenvectors

ξ = e1 + ie2 , ξ = e1 − ie2 , e1, e2 ∈ Rn .

Then Ae1 = −ωe2, Ae2 = ωe1 and the real plane

E := 〈e1, e2〉 ⊂ Rn

is invariant for the linear dynamics of (1.2) and it is filled up1 by the (2π/ω)-periodic solutions

x(t) = Re(c(e1 + ie2)e

iωt), c ∈ C,

see figure 1.2.

Definition 1.0.1 The matrix A is called elliptic iff all its eigenvalues arepurely imaginary. In this case the equilibrium x = 0 is called elliptic.

We could expect periodic solutions of (1.1) close to E. However, in gen-eral, periodic orbits need not exist in the nonlinear system as the following

1If ω = 0, i.e. if A is singular, the plane E is filled up by equilibria. in the sequelwe shall give sufficient conditions to find periodic solutions of (1.1) in the case Dxf(0) isnon-singular.


e

e1

2

E

ω

Figure 1.2: The invariant plane E of the linear system.

simple example shows x1 = −x2 + (x2

1 + x22)x1

x2 = x1 + (x21 + x2

2)x2 .(1.3)

Here (x1, x2) = (0, 0) is an equilibrium of (1.3) and the eigenvalues of thelinearized system x1 = −x2, x2 = x1, are ±i. But, for any solution x(t) =(x1(t), x2(t)) of (1.3),

d

dt(x2

1(t) + x22(t)) = 2(x2

1(t) + x22(t))

2

so that, if a periodic solution x(t) exists, whatever its period T is,

0 =

∫ T

0

d

dt(x2

1(t) + x22(t)) dt =

∫ T

0

2(x21(t) + x2

2(t))2 dt

and then x(t) ≡ 0.

Exercise: Make the phase portrait of (1.3) (hint: use polar coordinates).

The previous example shows that -in order to find periodic solutions of(1.1)- the class of the vector fields f(x) must be restricted.

Two typical conditions are assumed:

(i) Systems with a prime integral (for example Hamiltonian Systems);

(ii) Reversible systems.

In these lectures we shall consider only the class (i).

Definition 1.0.2 A smooth function G : Rn → R is a prime integral of(1.1) iff G(x(t)) = constant along any solution x(t) of (1.1). Equivalently

∇G(x) · f(x) = 0 , ∀x ∈ Rn . (1.4)


x

x

x(0)

x(t)

G(x)=G(x(0))

Figure 1.3: Any orbit evolves within the level set G(x) = G(x(0)).

In other words any orbit evolves within the level set G(x) = G(x(0)),see figure 1.3.

The main example of systems possessing a first integral are the au-tonomous Hamiltonian systems

(HS) x = J∇H(x) , x ∈ R2n

where

J =

(0 I−I 0

)∈ Mat(2n× 2n)

is the standard symplectic matrix and the function

H : R2N → R

is called the Hamiltonian of (HS). The HamiltonianH itself is a prime integralof (HS) because

d

dtH(x(t)) = (∇H(x(t)), J∇H(x(t))) = 0

by the antisymmetry of J .

We first analyze the form of such prime integral.

Lemma 1.0.1 If G is a C2 prime integral of (1.1), f(0) = 0 and A :=Dxf(0) is non-singular (i.e. det A 6= 0), then (∇G)(0) = 0 and (D2G)|E(0) =ρ I for some ρ ∈ R. As a consequence, if (D2G)|E(0) is non singular, then(D2G)|E(0) is either positive or negative definite.


Proof. Differentiating (1.4) we find

(D2G)(x)ξ · f(x) + (∇G)(x) · (Dxf)(x)ξ = 0 , ∀ξ ∈ Rn . (1.5)

Therefore, as f(0) = 0, (∇G)(0) · Aξ = 0,∀ξ ∈ Rn, and (∇G)(0) = 0 sinceA is nonsingular.

Differentiating (1.5) at x = 0 we find

2(D2G)(0)ξ · Aξ = 0 , ∀ξ ∈ Rn . (1.6)

Restricting (1.6) to E we find, ∀ξ = ξ1e1 + ξ2e2, ξi ∈ R

0 = (D2G)(0)ξ · Aξ

= ω[− ξ2

1(D2G)(0)e1 · e2 + ξ1ξ2[(D

2G)(0)e1 · e1 − (D2G)(0)e2 · e2]

+ ξ22(D

2G)(0)e1 · e2],

i.e. (D2G)(0)e1 · e2 = 0, (D2G)(0)e1 · e1 = (D2G)(0)e2 · e2 = ρ. Therefore

(D2G)(0)ξ · ξ = ρ(ξ21 + ξ2

2) , ∀ξ ∈ E

and, if (D2G)|E(0) is non singular, then ρ 6= 0 and (D2G)|E(0) is eitherpositive or negative definite.

1.1 The Lyapunov Center Theorem

We can now state the first continuation theorem of periodic orbits close toan equilibrium.

Theorem 1.1.1 (Lyapunov 1907 [75]) Let λ1 = iω, λ2 = −iω 6= 0 be apair of purely imaginary eigenvalues of the non-singular matrix A := Dxf(0)and E be the real eigenspace corresponding to λ1, λ2.

If the other eigenvalues of A, λk, k = 3, . . . , n satisfy the “non-resonancecondition”

λk

λ1

6∈ Z (1.7)

and if G is a C2 prime integral of (1.1) with (D2G)|E(0) 6≡ 0, then, for everysmall ε > 0, there exists a unique T (ε)-periodic solution p(ε, t) of (1.1) onthe level set2 G(x) = G(0) + ε2 with period T (ε) near 2π/ω.

As ε→ 0 the solution p(ε, t)L∞→ 0 and T (ε) → 2π/ω.


0

e

e

2

1 E

ε,x(t)=p( t)

Figure 1.4: The Lyapunov periodic solutions p(t, ε) fill out a 2-dimensionalembedded disk which is tangent to the plane E at x = 0.

The Lyapunov periodic orbits p(t, ε) of Theorem 1.1.1 are parametrized bythe value G(p(t, ε)) = G(0) + ε2 of the first integral G; other parametriza-tions are possible, see [6]. Note that, in general, the Lyapunov periodicorbits are not parametrized by their frequency: it may happen that all theirfrequencies are the same, like e.g. in linear systems.

Remark 1.1.1 (Regularity) If f,G ∈ Cr+1 the Lyapunov periodic solu-tions fit together into a two dimensional embedded invariant manifold whichis Cr and tangent to the plane E at x = 0. If f and G are analytic, then theembedding is analytic, see [97].

Remark 1.1.2 (Multiplicity) By theorem 1.1.1, if there are s pairs ofpurely imaginary eigenvalues

λj = ±iωj , j = 1, 2, . . . , s , (1.8)

we find s distinct families of periodic solutions having periods close to 2π/ωj

under the non-resonance conditions

λk

λj

6∈ Z ∀k 6= j , j = 1, 2, . . . , s . (1.9)

2By Lemma 1.0.1 (D2G)|E(0) is either positive or negative definite; to fix notations weassume (D2G)|E(0) > 0. In the case (D2G)|E(0) < 0 the periodic solution p(ε, t) lies inthe level set G(x) = G(0)− ε2.


Remark 1.1.3 (Existence) If there are s pairs of purely imaginary eigen-values λj = ±iωj with 0 < ω1 ≤ . . . ≤ ωs−1 < ωs (namely λs is simple),then the Lyapunov Center theorem ensures always the persistence of the pe-riodic orbits with the shortest period (highest frequency ωs). Indeed thenon-resonance condition (1.9) is satisfied because

λk

λs

=iωk

iωs

< 1 ∀k = 1, . . . , s− 1

and λj/(iωs) 6∈ Z for the other eigenvalues λj /∈ iR .The other families need not exist, as the following example [98] shows:

z1 = i(jz1 + zj2)

z2 = i(−z2 + jz1zj−12 ) , j ∈ 2, 3, . . .

(1.10)

where in complex notation zk = xk + iyk ∈ C. The function

G = j|z1|2 − |z2|2

is a prime integral of (1.10) with non singular D2G(0). Moreover

d

dtIm(z1z2) = (|z2|2 + j2|z1|2)|z2|2j−2 ≥ 0

so that periodic orbits have z2 = 0 and therefore

z1 = ceijt , z2 = 0

gives the shortest period solutions.

Exercise: Find explicitly the Lyapunov periodic solutions of the Hamil-tonian system in (R6,

∑3i=1 dqi ∧ dpi) with Hamiltonian

H =1

2ω(q2

1 + p21) +

1

2(q2

2 + p22)−

1

2(q2

3 + p23) + (p2

1 + q21)(q2q3 + p2p3)

when the non-resonance condition 1 6= lω, ∀l ∈ Z, holds.

We shall not prove the Lyapunov Center Theorem which, thanks to thenon-resonance condition (1.7) and the existence of the prime integral G, ulti-mately reduces to the implicit function theorem, see e.g. [98] (as a particularcase of the Poincare continuation theorem) and [6] (as a particular case ofthe Hopf bifurcation theorem).


1.2 The Theorems of Weinstein-Moser and

Fadell-Rabinowitz

If the non-resonance condition (1.7) among the eigenvalues of A is violated(resonant case), no periodic solutions except the equilibrium point need exist.

An example due to Moser [96] is provided by the Hamiltonian system in(R4,

∑2i=1 dxi ∧ dyi) generated by the Hamiltonian

H =x2

1 + y21

2− x2

2 + y22

2+

(x2

1 + y21 + x2

2 + y22

)(y1y2 − x1x2) . (1.11)

(x, y) = 0 is an elliptic equilibrium and the eigenvalues of the linearizedsystem are ±i,±i (not simple). But

d

dt(x1y2 + y1x2) = −4(y1y2 − x1x2)

2 −(x2

1 + y21 + x2

2 + y22

)2

so that the unique periodic solution is x = y = 0.

Remark 1.2.1 It may be useful to understand the choice of system (1.11) tointroduce action-angle coordinates xi =

√2Ii cosϕi, yi =

√2Ii sinϕi, i = 1, 2.

The Hamiltonian in (1.11) takes the form

H = I1 − I2 − 4(I1 + I2)√I1I2 cos(ϕ1 + ϕ2)

which can not be transformed in Birkhoff normal form because the frequencyvector ω = (1,−1) of the harmonic oscillators “enters in resonance” withcos(k · ϕ), k = (1, 1) (resonance of order 2), see [10].

We also remark that the quadratic part of the Hamiltonian (1.11),

H2 :=x2

1 + y21

2− x2

2 + y22

2,

is neither positive or negative definite, and that its signature signH2 = 0 (thesignature is the difference between the dimension of the maximal subspacewhere H2 is positive definite and the dimension of the maximal subspacewhere H2 is negative definite).

In contrast, two remarkable theorems by Weinstein [126]-Moser [95] andFadell-Rabinowitz [58] prove the existence of periodic solutions under theassumptions respectively

(WM) (D2H)(0) > 0 (or < 0 ) , (FR) sign(D2H)(0) 6= 0 .


Note that (WM) is a stronger condition than (FR) and that, as the example(1.11) shows, the assumption (FR) is necessary.

Under (WM) the Weinstein-Moser theorem finds periodic solutions ofthe Hamiltonian system (HS) with fixed energy, while, assuming (FR), theFadell-Rabinowitz theorem proves the existence of solutions with fixed pe-riod3.

Let us give the precise statements of these theorems.

Theorem 1.2.1 (Weinstein 1973-Moser 1976) Let H ∈ C2(R2n,R),(∇H)(0) = 0 and D2H(0) > 0. For all ε small enough there exist, on eachenergy surface H(x) = H(0) + ε2 at least n geometrically distinct periodicsolutions of the Hamiltonian system (HS).

O

H(x)

H=H(0)

H=H(0)+ε

D H(0)>02

2

Figure 1.5: The Weinstein-Moser Theorem.

By the assumptions of the Weinstein-Moser theorem 1.2.1 the point x = 0is an elliptic equilibrium. Indeed the Hamiltonian of the linearized systemx = J(D2H)(0)x is H2 := (1/2)D2H(0)x · x. Since the level sets of H2 areellipsoids, all the solutions of the linearized system are bounded and thereforethe eigenvalues of the matrix J(D2H)(0) must be all purely imaginary.

Theorem 1.2.1 was first proved by Weinstein [126]. Moser [95] proved theextension described in remark 1.2.4 below.

Remark 1.2.2 Moser [95] proved also an analogue of Theorem 1.2.1 fornon-Hamiltonian systems: if (1.1) has a positive definite first integral G, i.e.(D2G)(0) > 0, then there exists at least one periodic solution on each levelG(x) = G(0) + ε2. The difference w.r.t the Lyapunov Center Theorem1.1.1 is that the non-resonance condition (1.7) is not assumed.

3Assuming just (FR), existence of periodic solutions with fixed energy was more re-cently proved by Bartsch [16].


Theorem 1.2.2 (Fadell-Rabinowitz 1978) Let H ∈ C2(R2n,R) with(∇H)(0) = 0. Assume that any non-zero solution of the linearized systemx = J(D2H)(0)x is T -periodic (and not constant). If

signD2H(0) = 2ν 6= 0

then, either

(i) x = 0 is a non-isolated T -periodic solution of the Hamiltonian system(HS),

or

(ii) ∃k,m ∈ N with k +m ≥ |ν| and a left neighborhood, Ul, resp. a rightneighborhood, Ur, of T in R such that ∀λ ∈ Ul \ T, resp. Ur \ T,there exist at least k, resp. m, distinct, non-trivial λ-periodic solutionsof the Hamiltonian system (HS). The L∞- norm of the solutions tendsto 0 as λ→ T .

Remark 1.2.3 Unlike the Lyapunov Center Theorem (see remark 1.1.1) theperiodic solutions of Theorems 1.2.1 and 1.2.2 do not in general vary con-tinuously with respect to the parameters ε (energy) and λ (period). This willcause a serious difficulty for infinite dimensional PDEs, see Chapter 4 andespecially remark 4.3.3.

For the extension to Hamiltonian PDEs, it will be more convenient tocount the number of periodic solutions with a given frequency, as in theFadell-Rabinowitz theorem, and not with a given energy.

Remark 1.2.4 In the Weinstein-Moser theorem, resp. Fadell-Rabinowitz,one could assume a weaker condition. Suppose R2n = E ⊕ F where E andF are invariant subspaces of x = J(D2H)(0)x, and all its solutions in Eare T -periodic, while none of its solutions in F \ 0 have this period. Toconclude the existence of periodic solutions it is sufficient that (D2H)(0)|E >0 (Weinstein-Moser), respectively sign(D2H)(0)|E 6= 0 (Fadell-Rabinowitz).

The number of solutions found in Theorems 1.2.1, 1.2.2 is in generaloptimal.

A first trivial example showing the optimality of the estimate in theWeinstein-Moser Theorem stated like in 1.2.1 is provided by the quadraticHamiltonian

H2 =n∑

j=1

ωj

x2j + y2

j

2


X

T

T

λ

Figure 1.6: The bifurcation diagram of the Fadell-Rabinowitz Theorem. Theupper diagram represents case (i): there exists a sequence of non-trivialperiodic solutions of (HS) tending to 0 with period equal to T (this caseholds for example if H ≡ H(0) + (1/2)(D2H)(0)x · x is quadratic). In case(ii) there exist non trivial periodic solutions with period λ near T . Suchsolutions of (HS) could not depend continuosly w.r.t the period λ.

where the frequency vector ω := (ω1, . . . , ωn) of the decoupled harmonicoscillators satisfy the non-resonance condition

ω · k 6= 0 ∀k ∈ Zn \ 0 . (1.12)

In this case the only periodic solutions of (HS) have the form(xj, yj) = Aj(cos(ωjt+ ϕj), sin(ωjt+ ϕj)) , xi = yi = 0 ,∀i 6= j

for j = 1, . . . , n, and Aj is uniquely fixed by the energy level.

However, the previous example is not really interesting: under the non-resonance condition (1.12) the continuation of the periodic solutions can beproved just by the Lyapunov Center Theorem (see remark 1.1.2).


There is also the following “completely resonant” example (for n = 2)

H =(x2

1 + y21) + (x2

2 + y22)

1 + (x21 + y2

1)− (x22 + y2

2)

where (D2H)(0) > 0 and the eigenvalues of the linearized system are allequals to ±i. Consider an energy level H = h. It is described also as

(x21 + y2

1)(1− h) + (x22 + y2

2)(1 + h) = h . (1.13)

In other words the energy surface is also the energy surface of the quadraticHamiltonian in the left hand side of (1.13). But if two Hamiltonians sharethe same energy level, they possess (on this level) the same trajectories up toa reparametrization of time4. Then all the orbits of the Hamiltonian systemgenerated by H on the surface H = h are harmonic oscillations and, if

1− h

1 + h/∈ Q ,

then the system has exactly only n = 2 periodic orbits.This example can also show the optimality of the estimate on the number

of periodic solutions at fixed period given by the Fadell-Rabinowitz theorem,at least when D2H(0) is positive definite.

Exercise: Prove the last statement.

We shall prove a simplified version of the Theorems of Weinstein-Moserand Fadell-Rabinowitz without obtaining the optimal multiplicity results.We shall prove:

Theorem 1.2.3 (Weinstein-Moser) Let H ∈ C2(R2n,R), (∇H)(0) = 0and D2H(0) > 0. For all ε small enough there exist, on each energy surfaceH(x) = H(0)+ ε2 at least 2 geometrically distinct periodic solutions of theHamiltonian system (HS).

We shall prove also a simpler Fadell-Rabinowitz theorem assuming thatD2H(0) > 0. In this case the eigenvalues of the linearized Hamiltonian vectorfield J(D2H)(0),

±iω1, . . . ,±iωn ,

4Assume M = H(x) = h = F (x) = f with ∇H(x) 6= 0, ∇F (x) 6= 0 on M . Then∇F (x) = λ(x)∇H(x) with λ(x) 6= 0 on M . If φt denotes the flow of the Hamiltonianvector field J∇H(x) and ψs the flow of J∇F (x), then we have, on M , ψs(x) = φt(x),where t(s, x) solves dt/ds = λ(φt(x)), t(0, x) = 0.


are purely imaginary. For definiteness we suppose5

ω1 = ω2 = . . . = ωn = 1 (1.14)

so that the linearized system

x = J(D2H)(0)x (1.15)

possesses a 2n-dimensional linear space of periodic solutions with the sameminimal period 2π.

Theorem 1.2.4 (Fadell-Rabinowitz) Under the assumptions above, ei-ther

(i) x = 0 is a non-isolated 2π-periodic solution of (HS);

or

(ii) there is a one sided neighborhood U of 1 such that ∀λ ∈ U \ 1 theHamiltonian system (HS) possesses at least two distinct non-trivial2πλ-periodic solutions;

or

(iii) there is a neighborhood U of 1 such that ∀λ ∈ U \ 1 the Hamiltoniansystem (HS) possesses at least one non-trivial 2πλ-periodic solution.

Theorem 1.2.4 is a particular case of a general bifurcation result for po-tential operators due to Rabinowitz [113], see also [116].

Both the Weinstein-Moser theorem and the Fadell-Rabinowitz theoremfollow by arguments of variational bifurcation theory.

The Weinstein-Moser theorem, thanks to the energy constraint, ulti-mately reduces to find critical points of a smooth functional defined on a(2n− 1)-dimensional sphere, invariant under the S1-action induced by timetranslations. Existence of a maximum and a minimum is obvious. Multiplic-ity of critical points is a consequence of the S1-symmetry invariance, see e.g.[116].

The Fadell-Rabinowitz theorem is more subtle because there is no en-ergy constraint, and one has to look for critical points of a reduced actionfunctional defined in a open neighborhood of the origin.

5In the general Fadell-Rabinowitz Theorem 1.2.2 the periodic solutions of the linearizedequation x = JD2H(0)x need not to have the same minimal period T . For example itcould be ωj = j ∈ N, j = 1, . . . , n, as for the completely resonant NLW, see (2.6).


1.2.1 The variational Lyapunov-Schmidt reduction

Suppose, for simplicity,

H(0) = 0 and D2H(0) = I , (1.16)

i.e. (1.14) (we shall prove also the Weinstein-Moser theorem in this resonantcase which contains the essence of the problem).

Normalizing the period we look for 2π-periodic solutions of

Jx+ λ∇H(x) = 0 (1.17)

(hence x(t/λ) is a 2πλ-periodic solution of (HS)).

Equation (1.17) is the Euler-Lagrange equation of the C1-action func-tional

Ψ(λ, ·) : H1(T) → R , T := (R/2πZ)

defined by

Ψ(λ, x) :=

∫T

(1

2Jx(t) · x(t) + λH(x(t))

)dt

because, ∀h ∈ H1(T),

(DxΨ)(λ, x)[h] =

∫T

(1

2Jx · h+

1

2Jh · x+ λ∇H(x) · h

)dt

=

∫T

(Jx+ λ∇H(x)

)· h dt (1.18)

integrating by parts and using the antisymmetry of J .

To find critical points of the highly indefinite functional Ψ we perform aLyapunov-Schmidt reduction decomposing

H1(T) := V ⊕ V ⊥

whereV :=

v ∈ H1(T) | v = J(D2H)(0)v

is a 2n-dimensional6 linear space (by (1.14)) and

V ⊥ :=w ∈ H1(T) |

∫T

w · v dt = 0 , ∀v ∈ V.

6The map V 3 v → v(0) ∈ R2n is an isomorphism. On the finite dimensional space Vthe H1-norm ‖v‖H1 is equivalent to the euclidean norm |v(0)|.


Projecting (1.17), for x = v + w, v ∈ V , w ∈ V ⊥, yieldsΠV (J(v + w) + λ∇H(v + w)) = 0 bifurcation equationΠV ⊥(J(v + w) + λ∇H(v + w)) = 0 range equation

where ΠV , ΠV ⊥ denote the projectors on V , resp. V ⊥.

We solve first the range equation with the standard implicit functiontheorem, finding a solution w(λ, v) ∈ V ⊥ for v ∈ Br(0) (≡ ball in V of radiusr centered at zero), r > 0 small enough, and λ sufficiently close to 1.

Indeed the nonlinear operator

F : R× V × V ⊥ → V ⊥

with range in V ⊥ := w ∈ L2(T) |∫Tw · v dt = 0 , ∀v ∈ V , defined by

F(λ, v, w) := ΠV ⊥(J(v + w) + λ∇H(v + w))

vanishes at F(1, 0, 0) = 0 and its partial derivative

(DwF)(1, 0, 0)[W ] = ΠV ⊥(JW +D2H(0)W ) , ∀W ∈ V ⊥

is an isomorphism between V ⊥ and V ⊥. The solution w(λ, v) ∈ V ⊥ of therange equation is a C1 function w.r.t. (λ, v).

We havew(λ, 0) = 0 (1.19)

and(Dvw)(1, 0) = −(DwF)−1(1, 0, 0)(DvF)(1, 0, 0) = 0 (1.20)

because

(DvF)(1, 0, 0)[h] = ΠV ⊥(Jh+D2H(0)h) = 0 , ∀h ∈ V .

By (1.19)-(1.20) it follows

w(λ, v) = o(‖v‖H1) as v → 0 (1.21)

uniformly for λ near 1.

It remains to solve the finite dimensional bifurcation equation

ΠV (J(v + w(λ, v)) + λ∇H(v + w(λ, v))) = 0 . (1.22)

Equation (1.22) is still variational, being the Euler-Lagrange equation of the“reduced action functional” Φ : Br(0) ⊂ V → R defined by

Φ(λ, v) := Ψ(λ, v + w(λ, v)) . (1.23)


Indeed ∀h ∈ V ,

(DvΦ)(λ, v)[h] = (DxΨ)(λ, v + w(λ, v))[h+ (Dvw)(λ, v)[h]

]= (DxΨ)(λ, v + w(λ, v))[h] (1.24)

(1.18)=

∫T

ΠV (J(v + w) + λ∇H(v + w)) · h (1.25)

and in (1.24) we have used that

(DxΨ)(λ, v + w(λ, v))[W ] = 0 , ∀W ∈ V ⊥ (1.26)

since w(λ, v) solves the range equation and (Dvw)(λ, v)[h] ∈ V ⊥.

1.2.2 Proof of the Weinstein-Moser Theorem 1.2.3

We need to find critical points of the action functional

Ψ(λ, x) :=

∫T

1

2Jx(t) · x(t) + λ

∫T

(H(x(t))− ε2) dt (1.27)

which differs just for the “energy” constant ε2 by the action functional givenabove; with a slight abuse of notations we shall use the same symbol fordenoting both (clearly their corresponding Euler-Lagrange equations are thesame).

We do not fix the period 2πλ but we impose that

(DvΦ)(λ, v)[v] = 0 (1.28)

(the radial derivative of the reduced action functional Φ vanishes).

Lemma 1.2.1 Equation (1.28) can be solved for λ = λ(v), ∀v ∈ Br(0), forr > 0 small enough. The period λ(v) → 1 as v → 0.

Proof. By (1.25) and a Taylor expansion

(DvΦ)(λ, v)[v](1.25)=

∫T

J(v + w(λ, v)) · v + λ∇H(v + w(λ, v)) · v

=

∫T

(Jv + λD2H(0)v) · v +R(λ, v)

= (λ− 1)

∫T

D2H(0)v · v +R(λ, v) (1.29)


(because v ∈ V ) where, by (1.21),

R(λ, v) := λ

∫T

∇H(v + w(λ, v)) · v −D2H(0)v · v +

∫T

Jw(λ, v) · v

= o(‖v‖2) (1.30)

(with its derivatives) uniformly in λ close to 1.By (1.29), equation (1.28) amounts to solve

λ = 1− R(λ, v)

Q(v)(1.31)

where, ∀v 6= 0, using that D2H(0)v · v is constant along the solutions of(1.15),

Q(v) =

∫T

D2H(0)v · v = 2πD2H(0)v(0) · v(0) = 2π|v(0)|2 > 0

supposing (1.16).By these estimates, the term R(λ, v)/Q(v) → 0 (with its derivatives) as

v → 0.By the implicit function theorem we can solve (1.31) finding λ(v) for v

small. Clearly λ(v) → 1 as v → 0.

Next we consider, for ε > 0 small enough, the sphere-like manifold7

Sε :=v ∈ Br(0) |

∫T

H(v + w(λ(v), v)) dt = 2πε2

and the functionalI(v) := Φ(λ(v), v) .

Lemma 1.2.2 A critical point v ∈ Sε of I : Sε → R is a critical point of

Φ(λ(v), ·) : Br(0) ⊂ V → R

(with fixed period λ(v)). As a consequence x := v + w(λ(v), v) is a 2πλ(v)-periodic solution of (HS) with energy H(x) = ε2.

Proof. Differentiating in the definition of I yields, ∀h ∈ TvSε,

DI(v)[h] = (∂λΦ)(λ(v), v)dλ(v)[h] + (DvΦ)(λ(v), v)[h] . (1.32)

7Sε is radially diffeomorphic to a sphere because D2H(0) > 0 and (1.21).


υ−

Tυ−Sε

O.

Figure 1.7: The sphere-like manifold Sε .

Now

(∂λΦ)(λ, v)(1.23)= (∂λΨ)(λ, v + w(λ, v)) + (DxΨ)(λ, v + w(λ, v))[∂λw]

(1.27)=

∫T

(H(v + w(λ, v))− ε2) dt (1.33)

because (DxΨ)(λ, v + w(λ, v))[∂λw] = 0 by (1.26) and ∂λw ∈ V ⊥.Furthermore

(∂λΦ)(λ(v), v) = 0

because the term (1.33) vanishes for λ := λ(v) being v ∈ Sε. We concludeby (1.32) that

DI(v)[h] = (DvΦ)(λ(v), v)[h] , ∀h ∈ TvSε . (1.34)

By (1.28) we have also (DvΦ)(λ(v), v)[v] = 0 and, since

TvSε ⊕ 〈v〉 = V ,

we deduce that v is a critical point of Φ(λ(v), ·) : Br(0) ⊂ V → R.Finally, since x = v + w(λ(v), v) is a solution of (HS), its energy H(v +

w(λ(v), v))(t) is constant in time and, since v ∈ Sε, it is equal to ε2.

To conclude the proof of the Weinstein-Moser Theorem 1.2.3, since themanifold Sε is compact, I|Sε possesses at least one maximum and one min-imum, implying the existence of at least 2 geometrically distinct periodicsolutions of the Hamiltonian system (HS) with energy ε2 and with periodsclose to 1.


1.2.3 Proof of the Fadell-Rabinowitz Theorem 1.2.4

To prove Theorem 1.2.4 we have to find non-trivial critical points of thereduced action functional Φ(λ, ·) defined in (1.23) near v = 0 for λ near 1(fixed period).

By (1.21) the functional Φ(λ, ·) possesses a strict local minimum or maxi-mum at v = 0, according λ > 1 or λ < 1, because (recall H(0) = ∇H(0) = 0)

Φ(λ, v) =

∫T

1

2Jv · v +

λ

2(D2H)(0)v · v + o(‖v‖2)

=(λ− 1)

2

∫ 2π

0

(D2H)(0)v(t) · v(t) dt+ o(‖v‖2)

=(λ− 1)

22π(D2H)(0)v(θ) · v(θ) + o(‖v‖2) , ∀θ ∈ T (1.35)

being D2H(0)v · v constant along the solutions of (1.15).

If v = 0 is not an isolated critical point of Φ(1, ·) then alternative (i) ofTheorem 1.2.4 holds. Thus, for what follows, we assume v = 0 is an isolatedcritical point of Φ(1, ·). Consequently v = 0 is either

(a) a strict local maximum or minimum for Φ(1, ·);

(b) Φ(1, ·) takes on both positive and negative values near v = 0.

Case (a) leads to alternative (ii) and Case (b) leads to alternative (iii) ofTheorem 1.2.4. The figure 1.8 gives the idea of the existence proof.

Case (a): Suppose v = 0 is a strict local maximum of Φ(1, ·) (to handlethe case of a strict local minimum just replace Φ with −Φ).

Since Φ(1, 0) = 0, for r > 0 small enough,

∃β > 0 such that Φ|∂Br(1, ·) ≤ −2β

By continuity, for λ near 1,

Φ|∂Br(λ, ·) ≤ −β .

Take λ > 1. By (1.35), Φ(λ, ·) has a local minimum at v = 0. More preciselythere exists ρ ∈ (0, r) such that

Φ(λ, v) ≥ α(λ) > 0 , ∀v ∈ ∂Bρ , (1.36)

see figure 1.9.


Case(a)

Case(b)

υ υ

υ

Φ(1,υ) Φ(λ,υ)

λ>1

Figure 1.8: For λ 6= 1 the functional Φ(λ, ·) possesses non-trivial criticalpoints close to v = 0.

The maximum

c(λ) := maxv∈Br

Φ(λ, v) ≥ α(λ) > 0

is attained in the interior of Br because Φ(λ, ·) is negative on ∂Br. Further-more, the maximum is not attained in v = 0 because Φ(λ, 0) = 0.

To find another non trivial critical point of Φ(λ, ·) we define the MountainPass critical level

c(λ) := infγ∈Γ

maxt∈[0,1]

Φ(λ, γ(t))

where the minimax class Γ is

Γ :=γ ∈ C([0, 1], Br) | γ(0) = 0 and γ(1) ∈ ∂Br

. (1.37)

The Mountain Pass Theorem of Ambrosetti-Rabinowitz [7] can not be di-rectly applied because Φ(λ, ·) is defined only in a neighborhood of 0. How-ever, since

c(λ) ≥ α(λ) > 0 > −β ≥ Φ(λ, ·)|∂Br ,


−

−

−

− −−

−

−

−

−

−

−−

−

−

−

ρ..

+

+

++

+

++

r

+

O

Figure 1.9: Case (a): the level sets of the functional Φ(λ, ·).

it is easy to adapt its proof showing that c(λ) is a critical value.Applying Theorem 6.2.5 in the Appendix to Φ(λ, ·) we conclude that there

exists a Palais-Smale sequence vn ∈ Br at the level c(λ), i.e. such that

Φ(λ, vn) → c(λ) , ∇Φ(λ, vn) → 0 . (1.38)

By compactness, up to subsequence, vn → v ∈ Br. Actually v ∈ Br(0) \ 0because Φ(λ, v) = c(λ) > 0 and Φ(λ, ·)|∂Br < 0, Φ(λ, 0) = 0. The point v isa non trivial critical point of Φ(λ, ·) at the level c(λ).

If c(λ) < c(λ) then Φ(λ, ·) has two distinct critical points. If c(λ) = c(λ)then c(λ) equals the maximum of Φ(λ, ·) over every curve in Γ. Thereforethere are infinitely many maxima, for n ≥ 2, and at least 2 for n = 1. In anycase alternative (ii) holds.

Case (b). Again the idea is to reduce the proof to an argument of “Moun-tain Pass” type. In this case, since Φ(1, ·) takes on both positive and neg-ative values near v = 0, the functional Φ(λ, ·) possesses the Mountain-Passgeometry both for λ > 1 and λ < 1. The construction is more subtle thanthe previous case because a level set of Φ(1, ·) no longer bounds a compactneighborhood of v = 0.

Let consider the negative gradient flowddtη(t, v) = −∇Φ(1, η(t, v))

η(0, v) = v(1.39)

which has a unique solution local in time for all v ∈ V near zero (by (1.24)the vector field ∇Φ(1, v) ∈ C1 because w(λ, v) ∈ C1).


In the sequel we take r > 0 small enough so that Φ(1, ·) has no criticalpoints in Br except v = 0, i.e. the only rest point of ∇Φ(1, ·) in Br is v = 0.

In order to apply a deformation argument the main issue is the construc-tion of an “isolating Conley set”:

Proposition 1.2.1 ([113]-[116]) There exists an open neighborhood Q ⊂Br of v = 0 and constants c+ > 0 > c− such that ∀v ∈ ∂Q either

• (i) Φ(1, v) = c+, or

• (ii) Φ(1, v) = c−, or

• (iii) η(t, v) ∈ ∂Q, ∀t near 0.

O

Q

c

c

c

c

+

+

− −

S

S

+

−

T−

Figure 1.10: The isolating “Conley set”.

Proof. The open neighborhood Q will be constructed taking a sufficientlysmall ball Bε ⊂ Br and letting it evolve in the future and in the past by thenegative gradient flow generated by (1.39), see precisely the definition (1.40).

We follow the exposition in [116]. Let us first define

S+ :=v ∈ Br : η(t, v) ∈ Br ∀t > 0

and

S− :=v ∈ Br : η(t, v) ∈ Br ∀t < 0

.

Lemma 1.2.3 S+, S− 6= ∅.


Proof. Let us verify the S+ case. Let vm ∈ Br be a sequence such thatvm → 0 and Φ(1, vm) > 0. Consider the solution η(t, vm) of (1.39).

We claim that there exists a largest tm < 0 such that η(tm, vm) =: ym ∈∂Br. If not, the drawback orbit passing through vm at t = 0 is contained inBr, so it must have a cluster point which is a critical point of Φ(1, ·) in Br

with positive critical value. But, by assumption, there are no other criticalpoints of Φ(1, ·) in Br except v = 0.

Furthermore tm → −∞ as m→ +∞, because v = 0 is an equilibrium ofthe C1 vector field −∇Φ(1, v).

Let y be a limit point of ym. Then η(t, y) ∈ Br for all t > 0 and soy ∈ S+.

Remark 1.2.5 By the negative gradient structure of (1.39) there results

S+ ≡v ∈ Br : η(t, v) ∈ Br ∀t > 0 and η(t, v)

t→+∞−→ 0.

Indeed, if η(t, v) remains in Br for all t > 0, η(t, v) has to converge to someconnected component of the critical set of Φ(1, ·) in Br, i.e. to v = 0, by theassumption on r > 0. Similarly

S− ≡v ∈ Br : η(t, v) ∈ Br ∀t < 0 and η(t, v)

t→−∞−→ 0.

Let define Ac := v ∈ V : Φ(1, v) ≤ c.

Lemma 1.2.4 There exists 0 < ε < r such that if c := maxv∈BεΦ(1, v) and

v ∈ Bε\S− then, as t→ −∞, the orbit η(t, v) leaves Ac ∩Br via Ac\∂Br.

Proof. By contradiction. There are sequences εm → 0, vm ∈ Bεm\S− andtm < 0 such that zm := η(tm, vm) ∈ ∂Br (we take tm to be the largest timesuch that η(tm, vm) ∈ ∂Br) and

minv∈Bεm

Φ(1, v) ≤ Φ(1, zm) ≤ maxv∈Bεm

Φ(1, v) .

Therefore, up to subsequence, zm → z ∈ ∂Br and Φ(1, z) = 0. Arguing asin the proof of Lemma 1.2.3 we deduce that z ∈ S+. By remark 1.2.5 weget η(t, z) → 0 as t → +∞. Hence Φ(1, η(t, z)) → Φ(1, 0) ≡ 0 as t → +∞.However Φ(1, z) = 0 and the function t→ Φ(1, η(t, z)) is strictly decreasing.This contradiction proves the lemma.

Let ε be as in Lemma 1.2.4 and define

c+ := maxv∈Bε

Φ(1, v) > 0 , c− := minv∈Bε

Φ(1, v) < 0 .

By the previous lemma we get


Lemma 1.2.5 ∀v ∈ Bε\S− there is a corresponding t−(v) < 0 such thatΦ(1, η(t−(v), v)) = c+ and η(t−(v), v) /∈ ∂Br.

Similarly ∀v ∈ Bε\S+ there is a corresponding t+(v) > 0 such thatΦ(1, η(t+(v), v)) = c− and η(t+(v), v) /∈ ∂Br.

Finally define

Q :=η(t, v) : v ∈ Bε and t−(v) < t < t+(v)

(1.40)

where, if v ∈ S− then t−(v) := −∞, and, if v ∈ S+ then t+(v) := +∞.

Lemma 1.2.6 The function t−(·) : Bε → R ∪ −∞ is upper semicontinu-ous; t+(·) : Bε → R ∪ +∞ is lower semicontinuous.

Proof. Let us verify the t+ case.a) We claim that, if v ∈ Bε\S+ then ∀ vn → v ⇒ lim inf t+(vn) ≥ t+(v).

If not, there exists vn → v, vn ∈ Bε\S+, such that τ := lim inf t+(vn) <t+(v). Since the function t→ Φ(1, η(t, v)) is strictly decreasing we get

Φ(1, η(τ, v)) > Φ(1, η(t+(v), v)) = c− . (1.41)

However c− = Φ(1, η(t+(vn), vn)), ∀n. Up to subsequence t+(vn) → τ , and,therefore, by the continuity of the flow, we get

c− = lim Φ(1, η(t+(vn), vn)) = Φ(1, η(τ, v)) > c−

by (1.41). Contradiction.Actually it is true that t+ is continuous on Bε\S+.

b) We claim that, if v ∈ Bε ∩ S+ then ∀ vn → v ⇒ lim t+(vn) = +∞.If not, there exists vn → v and T < +∞ such that t+(vn) < T . Clearly

vn ∈ Bε\S+ (because t+(v) = +∞ on S+).We have Φ(1, η(t+(vn), vn)) = c− < 0, ∀n. Up to subsequence t+(vn) →

τ ≤ T , and, therefore, Φ(1, η(τ, v)) = c− < 0. This contradicts the fact thatΦ(1, η(t, v)) ≥ 0, ∀t > 0 (since v ∈ S+, then η(t, v) → 0 as t → +∞ andt→ Φ(1, η(t, v)) is strictly decreasing).

Proof of Proposition 1.2.1 completed. We first verify that Q is anopen neighborhood of v = 0. Clearly 0 ∈ Bε ⊂ Q. Next, if z ∈ Q then z =η(t, v) for some v ∈ Bε and some t ∈ (t−(v), t+(v)). Take δ small such thatBδ(v) ⊂ Bε and such that, for all w ∈ Bδ(v), t

−(w) < t < t+(w) (which existsbecause t+(·) is lower semicontinuous and t−(·) is upper semicontinuous).

The set η(t, Bδ(v)) ⊂ Q is a neighborhood of z ∈ Q. Therefore Q is open.


Let us show that if v ∈ ∂Q then either (i) or (ii) hold, either the possibility(iii) occurs.

Let v ∈ ∂Q. Hence there exist vn := η(tn, xn) ∈ Q, xn ∈ Bε, tn ∈(t−(xn), t+(xn)) such that vn → v. Calling Ov := η(t, v) | t ∈ R the orbitof η(·, v), it is easy to see that, passing to the limit, Ov ∩ Bε 6= ∅, namelythere esist x ∈ Bε and τ ∈ R such that v = η(τ, x).

Then, either Φ(1, v) = c± and case (i) or (ii) holds; either Φ(1, v) ∈(c−, c+). In this latter case τ ∈ (t−(x), t+(x)). Hence x ∈ ∂Bε for otherwisev ∈ Q. Furthermore Φ(1, η(s, x)) ∈ (c−, c+) for s near τ . It is clear thatη(s, x) ∈ ∂Q for s near τ . Indeed η(s, x) = limn η(s, xn), xn ∈ Bε, xn → x,and therefore s ∈ (t−(x), t+(x)).

The proof of proposition 1.2.1 is complete.

Conclusion of Case (b). Define the Mountain Pass level

c(λ) := infγ∈Γ

maxt∈[0,1]

Φ(λ, γ(t))

whereΓ :=

γ ∈ C([0, 1], Q) | γ(0) = 0 and γ(1) ∈ T−

(1.42)

andT− :=

v ∈ S− : Φ(1, v) = c−

6= ∅ .

For λ > 1, as in (1.36) we conclude c(λ) ≥ α(λ) > 0. Following the proofof the Mountain Pass Theorem we can now obtain the existence of a Palais-Smale sequence vn ∈ Q at the level c(λ), i.e. such that

Φ(λ, vn) → c(λ) , ∇Φ(λ, vn) → 0 . (1.43)

The argument is based on a deformation argument using that −∇Φ(1, v) isa pseudo-gradient vector field for Φ(λ, v) for all v ∈ ∂Q and constructing,thanks to (iii), a pseudo-gradient vector field which leaves Q invariant underits flow, see for the proof remark 6.2.4.

By compactness vm → v ∈ Q. Actually v ∈ intQ because ∇Φ(λ, v) 6= 0∀v ∈ ∂Q for λ sufficiently close to 1. Again v 6= 0 because Φ(λ, v) = c(λ) > 0and Φ(λ, 0) = 0.

The proof of the Fadell-Rabinowitz Theorem 1.2.4 is complete.

Remark 1.2.6 So far, periodic solutions close to the equilibrium were ob-tained from assumptions on the linearized vector field only. The periodic so-lutions found are the continuations of the linear normal modes having periodsclose to the periods of the linearized equation. If, in contrast, the Hamiltonian


vector field f(x) is “sufficiently smooth and nonlinear”, then an abundance ofperiodic solutions with large period are expected (Birkhoff-Lewis orbits [32]),see also [22]. An extension of these results to Hamiltonian PDEs has beengiven in [14], [29].

Before concluding this Chapter we remark that global variational methodsfor finding periodic solutions of Hamiltonian systems as critical points of theaction functional has been successfully used since the pioneering papers byRabinowitz [114] and Weinstein [127] where, independently, the existenceof at least one periodic solution has been established on each compact andstrictly convex energy hypersurface. We refer to the books by Ekeland [54],Mawhin-Willem [88], and Hofer-Zehnder [69].

We also quote the long standing conjecture stating the existence of atleast n geometrically distinct periodic orbits (closed characteristics) on anycompact and convex energy hypersurface (this is a non perturbative analogueof the Weinstein-Moser theorem).

Ekeland and Hofer [55] have proved the existence of 2 closed characteris-tics (theorem 1.2.3 can be seen as a particular case of this result).

The best results available at the moment are by Long and Zhu [74] wherethe existence of at least [n/2] + 1 closed characteristics has been proved.

We shall not however pursue on this path but we shall describe the ex-tensions of the Lyapunov, Weinstein-Moser and Fadell-Rabinowitz theoremsregarding bifurcation of small amplitude periodic solutions of HamiltonianPDEs.

Chapter 2

Infinite dimension

We want to extend the local bifurcation theory of periodic solutions describedin the previous chapter to infinite dimensional Hamiltonian PDEs.

Let consider the autonomous nonlinear wave equationutt − uxx + a1(x)u = a2(x)u

2 + a3(x)u3 + . . .

u(t, 0) = u(t, π) = 0(2.1)

which possesses the equilibrium solution u ≡ 0.As in the previous chapter we pose the following

• Question: there exist periodic solutions of (2.1) close to u = 0?

The first step is again to study the linearized equationutt − uxx + a1(x)u = 0u(t, 0) = u(t, π) = 0 .

(2.2)

The Sturm-Liouville operator −∂xx+a1(x) possesses a basis ϕjj≥1 of eigen-vectors with real eigenvalues λj

(−∂xx + a1(x))ϕj = λjϕj , λj → +∞ . (2.3)

The ϕj are orthonormal with respect to the L2 scalar product.In this basis equation (2.2) reduces to infinitely many decoupled linear

oscillators: u(t, x) =∑

j uj(t)ϕj(x) is a solution of (2.2) iff

uj + λjuj = 0 j = 1, 2, . . . . (2.4)

If −∂xx + a1(x) is positive definite, all its eigenvalues λj > 0 are positive1

and u = 0 looks like an “infinite dimensional elliptic equilibrium” for (2.2)

1If λj < 0 (there are at most finitely many negative eigenvalues) then the correspondinglinear equation (2.4) describes an harmonic repulsor (hyperbolic directions).

39

CHAPTER 2. INFINITE DIMENSION 40

X X X

uu

u

ω ω

u

1

1 j

j

j

jϕ (x)ϕ

1(x)

1

Figure 2.1: The basis of eigenvectors

with linear frequencies of oscillations

ωj :=√λj ,

see figure 2.1. The quadratic Hamiltonian which generates (2.2),

H2(u, p) =

∫ π

0

p2

2+u2

x

2+ a1(x)

u2

2dx ,

where p := ut, is positive definite and, in coordinates, writes

H2 =∑j≥1

p2j + λju

2j

2

where pj := uj ∈ l2 (Plancharel Theorem).The general solution of (2.2) is therefore given by the linear superposition

of infinitely many oscillations of amplitude aj, frequency ωj and phase θj onthe normal modes ϕj:

u(t, x) =∑j≥1

aj cos(ωjt+ θj)ϕj(x) .

Hence all solutions of (2.2) are either periodic in time, either quasi-periodic,either almost-periodic.

A solution u is periodic when each of the frequencies ωj for which theamplitude aj is nonzero (active frequencies) is an integer multiple of a basicfrequency ω0:

ωj = ljω0 , lj ∈ Z .


In this case u is 2π/ω0 periodic in time.The solution u is quasi-periodic with a m-dimensional frequency base if

there is a m-dimensional frequency vector ω0 ∈ Rm with rationally inde-pendent components (i.e. ω0 · k 6= 0, ∀k ∈ Zm \ 0) such that the activefrequencies satisfy

ωj := lj · ω0 , lj ∈ Zm .

A solution is called almost periodic otherwise, namely if there is not a finitenumber of base frequencies.

It is a natural question to ask whether some of these periodic, quasi-periodic, or almost periodic solutions of the linear equation (2.2) persists inthe non-linear equation (2.1).

2.0.4 The Lyapunov Center Theorem for PDEs

Bifurcation of small amplitude periodic and quasi-periodic solutions of (2.1)was indeed obtained by Kuksin [84] and Wayne [125] extending KAM theory.Next Craig-Wayne [49] introduced the Lyapunov-Schmidt reduction methodfor Hamiltonian PDEs to handle the nonlinear wave equation (2.1) with peri-odic spatial boundary conditions where some resonance phenomena appearsdue to the near coincidence of pairs of linear frequencies.

We start describing the Craig-Wayne result [49] which is an extension ofthe Lyapunov Center Theorem to the nonlinear wave equation (2.1).

The main difficulty to overcome is the appearance of a

(i) “small divisors” problem

(which in finite dimension arises only for the search of quasi-periodic solu-tions).

To explain how it arises, recall the key non-resonance hypothesys (1.7) inthe Lyapunov Center Theorem2

ωj − lω1 6= 0 , ∀l ∈ Z , ∀j = 2, . . . , n .

Hence, in finite dimension, for any ω sufficiently close to ω1, the same con-dition ωj − lω 6= 0, ∀l ∈ Z, ∀j = 2, . . . , n, holds and the standard implicitfunction theorem can be applied.

In contrast, the eigenvalues of the Sturm-Liouville problem (2.3) growpolynomially3 like λj ≈ j2 + O(1) for j → +∞ (as it is seen by lower and

2The eigenvalues of the finite dimensional matrix A of the previous chapter correspondshere to ±iωj , j = 1, . . ..

3For example the eigenvalues of −∂xx +m are λj = j2 +m with eigenvectors sin(jx).


upper comparison with the operator with constant coefficients, see e.g. [107]),and therefore ωj = j + o(1). As a consequence, in infinite dimensions, whenω1 /∈ Q, the set

ωj − lω1 , ∀l ∈ Z , j = 2, 3, . . .

accumulates to zero and the non-resonance condition

ωj − lω1 6= 0 , ∀l ∈ Z , j = 2, 3, . . . (2.5)

is not sufficient to apply the standard implicit function theorem.

This is the “small divisors” problem (this name is due by the fact thatsuch quantities appears as denominators) which shall be described in moredetail at the end of this Chapter and in Chapter 4.

Nevertheless, replacing (2.5) with some stronger condition, persistence ofa large Cantor like set of small amplitude periodic solutions of (2.1) can beensured.

A typical result is the following

Theorem 2.0.5 (Craig-Wayne [49]) Let

f(x, u) := a1(x)u− a2(x)u2 − a3(x)u

3 + . . .

be a function analytic in the region (x, u) | |Im x| < σ , |u| < 1 and oddf(−x,−u) = −f(x, u). Among this class of nonlinearities there is an opendense set F (in C0-topology) such that, ∀f ∈ F , there exist a Cantor like setC ⊂ [0, r∗) of positive measure and a C∞ function Ω(r) with Ω(0) = ω1 suchthat ∀r ∈ C, there exists a periodic solution u(t, x; r) of (2.1) with frequencyΩ(r). These solutions are analytic in (x, t) and satisfy

|u(t, x; r)− r cos(Ω(r)t)ϕ1(x)| ≤ Cr2 , |Ω(r)− ω1| < Cr2 .

The Lyapunov solutions u(t, x; r) are parametrized with the amplituder, but also the corresponding set of frequencies Ω(r), r ∈ C, has positivemeasure.

The conditions on the terms a1(x), a2(x), a3(x), etc. are, roughly, thefollowings: first a condition on a1(x) to avoid primary resonances on thelinear frequencies ωj (which depend on a1), see the non-resonance condition(2.5); next a condition of genuine nonlinearity placed upon a2(x), a3(x) isrequired to solve the 2-dimensional bifurcation equation. Some “partiallyresonant” PDE where the bifurcation equation is 2m dimensional has beenstudied in [50].


Remark 2.0.7 To prove existence of quasi-periodic solutions with m-frequencies

u(t, x) = U(ωt, x) , ω ∈ Rm ,

where U(·, x) : Tm → R, the main difficulty w.r.t. the periodic case reliesin a more complicated geometry of the numbers ω · l − ωj, l ∈ Zm, j ∈ N.Existence of quasi-periodic solutions with the Lyapunov-Schmidt approachhas been proved by Bourgain [33], [34]. For existence results via the KAMapproach see e.g. [105], [43], [87], [85] and references therein.

The “completely resonant” case

a1(x) ≡ 0

whereωj = j , ∀j ∈ N (2.6)

(infinitely many resonance relations among the linear frequencies) was leftan open problem. In this case all the solutions of (2.2) are 2π-periodic. Thisis the analogous situation considered in finite dimension by the Weinstein-Moser and Fadell-Rabinowitz theorems. For infinite dimensional Hamilto-nian PDEs, aside the small divisor problem (i), this leads to the furthercomplication of an infinite dimensional bifurcation phenomenon.

In the sequel we shall discuss the extension to Hamiltonian PDEs ofthe results of Weinstein-Moser and Fadell-Rabinowitz. The required infiniteresonance variational analysis is the main contribution of [23]-[24] and [26].

2.1 Completely resonant wave equations

From now on we shall consider the completely resonant PDEutt − uxx = f(x, u)u(t, 0) = u(t, π) = 0

(2.7)

wheref(x, u) = ap(x)u

p +O(up+1) , p ≥ 2 .

The linearized equation at u = 0utt − uxx = 0u(t, 0) = u(t, π) = 0

(2.8)


possesses an infinite dimensional linear space of periodic solutions with thesame period 2π. Indeed any solution of (2.8) can be written as

v(t, x) =∑j≥1

aj cos(jt+ θj) sin(jx)

(Fourier method) and it is 2π-periodic in time.It shall be also convenient to express the solutions of the wave equation

(2.8) using the D’Alembert method, namely representing

v(t, x) = η(t+ x)− η(t− x) (2.9)

where η is any 2π-periodic function, as the linear superposition of two waveswith the opposite profile η traveling in opposite directions.

Lemma 2.1.1 Any solution of (2.8) can be written in the form (2.9).

Proof. The general solution of utt − uxx = 0 can be written as u(t, x) =p(t + x) + q(t − x) for any p, q : R → R (in the “characteristic” variabless+ := t+x, s− := t−x, the equation (∂tt−∂xx)u = (∂t +∂x) (∂t−∂x)u = 0reads ∂s+ ∂s−u = 0).

Imposing the Dirichlet boundary conditionsu(t, 0) = p(t) + q(t) = 0 , ∀tu(t, π) = p(t+ π) + q(t− π) = 0 , ∀t

we get q(t) = −p(t) and p(t+π) = −q(t−π) = p(t−π), ∀t. This last relationshows that p(·) is a 2π-periodic function and we have written u(t, x) = p(t+x)− p(t− x) in the form (2.9).

For proving existence of small amplitude periodic solutions of (2.7), thenew difficulty to overcome is:

(ii) the presence of an infinite dimensional space of periodic solutions of thelinear equation (2.8) with the same period: which solutions of (2.8) canbe continued to solutions of the nonlinear equation (2.7)?

The first existence results for completely resonant wave equations wereobtained by Lidskij-Shulman [77] and Bambusi-Paleari [15] for f(u) = u3.The small divisor problem (i) is bypassed imposing on the frequency ω astrongly non-resonance condition (see (2.16) below) which is satisfied on azero measure set Wγ. The choice of the nonlinearity f(u) = u3 is due to themethod used to solve the bifurcation equation.


In [23], solving the infinite dimensional bifurcation equation via min-maxvariational methods, we proved existence of periodic solutions of (2.7), forthe same zero measure set of frequencies Wγ, but for a general nonlinearity.For simplicity we now present these results for the nonlinearities

f(u) := aup , a 6= 0 , p ≥ 2

where p is either an odd or an even integer.

We start with the easier case when p is odd.

2.2 The case p odd

Since the nonlinear wave equation (2.7) is autonomous the period of thesolutions is a priori-unknown and we introduce it as a free parameter.

Normalizing the period, we look for 2π-periodic solutions ofω2utt − uxx = aup

u(t, 0) = u(t, π) = 0(2.10)

in the Banach algebra4

X :=u ∈ H1(Ω,R)∩L∞(Ω,R)

∣∣∣ u(t, 0) = u(t, π) = 0, u(−t, x) = u(t, x)

where Ω := (R/2πZ)× [0, π], endowed with norm5

‖u‖ := |u|∞ + ‖u‖H1 .

We denote |u|L2 := (∫

Ω|u|2)1/2 the L2-norm and (u,w)L2 :=

∫Ωuw the L2-

scalar product.We can look for solutions even in time because equation (2.7) is reversible.

We shall make assumptions on the frequency ω in (2.16).

Remark 2.2.1 Any u ∈ X can be developed in Fourier series as

u(t, x) =∑

l≥0,j≥1

ulj cos lt sin jx

and its H1-norm and scalar product are

‖u‖2H1 :=

∫Ω

u2t + u2

x =π2

2

∑l≥0,j≥1

u2lj(j

2 + l2) (2.11)

4With respect to multiplication of functions, i.e. ‖u1u2‖ ≤ C‖u1‖ ‖u2‖ ∀u1, u2 ∈ X.5Note that H1(Ω) does not embed into L∞(Ω) since Ω is a bidimensional domain.


(u,w)H1 := (u,w) :=

∫Ω

utwt + uxwx =π2

2

∑l≥0,j≥1

uljwlj(l2 + j2) ∀u,w ∈ X .

Instead of looking for solutions of (2.10) taking value in a shrinking neigh-borhood of 0 it is a convenient devise to perform the rescaling

u→ δu , δ > 0

obtaining ω2utt − uxx = εaup

u(t, 0) = u(t, π) = 0(2.12)

whereε := δp−1 .

Equation (2.12) is the Euler-Lagrange equation of the Lagrangian actionfunctional Ψ ∈ C1(X,R) defined by

Ψ(u) := Ψω,ε(u) :=

∫ 2π

0

dt

∫ π

0

[ω2

2u2

t −1

2u2

x + εF (u)]dx (2.13)

where

F (u) :=

∫ u

0

f(s) ds = aup+1

p+ 1.

To find critical points of Ψ, we perform a Lyapunov-Schmidt reduction de-composing

X = V ⊕W

where

V :=v =

∑j≥1

aj cos(jt) sin jx∣∣∣ aj ∈ R,

∑j≥1

j2a2j < +∞

=

v = η(t+ x)− η(t− x)

∣∣∣ η(·) ∈ H1(T), η odd

(2.14)

are the solutions in X of the linear equation (2.8) and

W :=w ∈ X

∣∣∣ (w, v)L2 = 0, ∀ v ∈ V

= ∑

l≥0,j≥1

wlj cos(lt) sin jx ∈ X∣∣∣ wjj = 0 ∀j ≥ 1

.

W is also the H1-orthogonal of V in X.


Remark 2.2.2 On V the norm ‖ ‖ is equivalent to the H1-norm ‖ ‖H1 since

|v|∞ ≤ C‖v‖H1 .

Moreover, recalling (2.14), the embedding

(V, ‖ · ‖H1) → (V, | · |∞) (2.15)

is compact because H1(T) → L∞(T) is compact.

Projecting (2.12), for u := v + w with v ∈ V , w ∈ W , yieldsω2vtt − vxx = εΠV f(v + w) bifurcation equationω2wtt − wxx = εΠWf(v + w) range equation

where ΠV : X → V and ΠW : X → W are the (continuous) projectorsrespectively on V and W .

The existence of a solution w := w(ε, ω, v) of the range equation can notbe derived for any ω ≈ 1, ε ≈ 0 (as in finite dimension) by the standardimplicit function theorem at ω = 1, ε = 0. Indeed, even tough (∂tt− ∂xx)

−1 :W → W is compact, it gains just 1 derivative (see Lemma 5.2.1) while theterm (ω2 − 1)∂tt loses 2 derivatives.

2.2.1 The range equation

We first solve the range equation assuming that the frequency ω satisfies thefollowing non-resonance condition

ω ∈ Wγ :=ω ∈ R

∣∣∣ |ωl − j| ≥ γ

l, ∀(l, j) ∈ N×N , j 6= l

(2.16)

for some γ > 0, introduced in [15].For example, by Liouville Theorem 6.4.3, a quadratic irrational ω (namely

an irrational root of a second order polynomial with integer coefficients)belongs to Wγ for some γ := γ(ω) > 0.

Remark that, by Dirichlet Theorem 6.4.1, the condition in (2.16) is thestrongest possible requirement of this kind: if τ < 1, there are no real num-bers ω such that |ωl − j| ≥ (γ/lτ ), ∀l 6= j. This follows also from (6.36).Finally, still by Dirichlet Theorem (or (6.36)), the set Wγ is empty6 if γ ≥ 1.

6Actually the setWγ is empty if γ ≥ 1/√

5, because, by Hurwitz Theorem (see Theorem2F in [117]), if x is irrational, there exist infinitely many distinct rational numbers p/qsuch that |x− p/q| < 1/

√5 q2.


Lemma 2.2.1 For 0 < γ ≤ 1/4 the set Wγ is uncountable, has zero measureand accumulates to ω = 1 both from the left and from the right.

Proof. We claim that, if γ ∈ (0, 1/4], the set Wγ contains the uncountablymany irrational numbers ω such that its continued fraction expansion is

ω = [1, a1, a2, . . .] := 1 +1

a1 +1

a2 + . . .

for every a1 ∈ N andai ∈ 1, 2 , ∀i ≥ 2 , (2.17)

(we refer to Appendix 6.4 for the basic notions about continued fractions).All these numbers ω belong to the open neighborhood (1, 1 + a−1

1 ) and, fora1 → +∞, they accumulate to ω = 1.

Let us prove the previous claim. Take any pairs of integers (l, j) ∈ N×N,l 6= j. If

|ωl − j| ≥ 1

2l

then the condition in the definition ofWγ, |ωl−j| ≥ γ/l, is obviously satisfiedbecause γ ≤ 1/4. On the other hand, if

|ωl − j| < 1

2l,

then, by a Theorem of Legendre (see Theorem 6.4.4), j/l coincides with oneof the convergents of ω

1 , [1, a1] , [1, a1, a2] , . . . , [1, a1, a2, . . . , an] , . . .

see the Definition 6.4.2.The quotient j/l is different from the first convergent of ω, i.e. j/l 6= 1,

because l 6= j by the definition of Wγ. Therefore

j

l= [1, a1, a2, . . . , an] =

pn

qnfor some n ≥ 1

and, since qn and pn are relatively prime by (6.35), qn divides l and l ≥ qn.As a consequence, in order to get |ωl−j| ≥ γ/l with γ ∈ (0, 1/4], it is enoughto prove that

|ωqn − pn| ≥1

4qn, ∀n ≥ 1 . (2.18)


Every convergent pn/qn of ω satisfies, see (6.37),∣∣∣ω − pn

qn

∣∣∣ ≥ 1

q2n(an+1 + 2)

(2.19)

whence, since, by (2.17), an+1 ∈ 1, 2, ∀n ≥ 1,∣∣∣ω − pn

qn

∣∣∣ ≥ 1

q2n(2 + 2)

=1

4q2n

proving (2.18).In conclusion, we have proved that

∀(l, j) ∈ N×N , l 6= j , |ωl − j| ≥ γ

lwith γ ∈ (0, 1/4] ,

namely that ω ∈ Wγ.

Remark 2.2.3 If in (2.17) we take ai ∈ 1, 2, . . . , N, ∀i ≥ 2, we deduce by(2.19) that ω ∈ Wγ for all γ ∈ (0, 1/(2 +N)].

Similarly, the uncountably many irrational numbers

ω := [0, 1, a2, a3, a4, . . . ]

for every a2 ∈ N and with a3, a4, . . . ∈ 1, 2 belong to the set

Wγ ∩(1− 1

a2 + 1, 1

)with γ ∈ (0, 1/4] .

For a2 → +∞ these numbers accumulate to ω = 1 from the left.

Exercise If ω ∈ Wγ, ω > 1, then also 2− ω < 1 belongs to Wγ. This givesan alternative proof that Wγ accumulates to ω = 1 also from the left.

Lemma 2.2.2 Let ω ∈ Wγ ∩ (1/2, 3/2) for some γ > 0. Then the operatorLω := ω2∂tt − ∂xx : D(Lω) ⊂ W → W has a bounded inverse defined by

L−1ω w :=

∑j≥1,l≥0,l 6=j

wlj

−ω2l2 + j2cos(lt) sin(jx) , ∀w ∈ W (2.20)

satisfying

‖L−1ω w‖ ≤ C

γ‖w‖ (2.21)

for a positive constant C independent of γ and ω.


Proof. The eigenvalues −ω2l2 + j2 |l ≥ 0, j ≥ 1, l 6= j of (Lω)|W satisfy,for ω ∈ Wγ

|−ω2l2 +j2| = |ωl−j|(ωl+j) ≥ γ

l(ωl+j) ≥ γω ≥ γ

2∀l 6= j , l ≥ 1 (2.22)

for ω > 1/2 (for l = 0 the inequality (2.22) is trivial). Therefore, by (2.20)and (2.11)

‖L−1ω w‖H1 ≤ C

γ‖w‖H1 . (2.23)

We claim also that

|L−1ω w|∞ ≤ C

γ‖w‖H1 . (2.24)

The inequalities (2.23)-(2.24) imply (2.21).The key observation to prove (2.24) is that the lower bound (2.22) can

be significantly improved if j is not the closest integer e(l) to ωl, defined by|e(l)− ωl| = minj∈N |j − ωl|. By (2.20)

|L−1ω w|∞ ≤

∑l≥0,j≥1,j 6=l

|wl,j||ωl − j|(ωl + j)

= S1 + S2

where

S1 :=∑

l≥0,j≥1,j 6=l,j 6=e(l)

|wl,j||ωl − j|(ωl + j)

and

S2 :=∑

l≥0,e(l) 6=l

|wl,e(l)||ωl − e(l)|(ωl + e(l))

.

Estimate of S1. For j 6= e(l) we have

|j − ωl| ≥ |j − e(l)| − |e(l)− ωl| ≥ |j − e(l)| − 1

2≥ |j − e(l)|

2.

Moreover, since |e(l)− ωl| < 1/2, and ω ≥ 1/2, it results 4ωl ≥ e(l) + l andhence

4(j + ωl) ≥ j + e(l) + l ≥ |j − e(l)|+ l .

Defining wlj by wlj = 0 if j ≤ 0 or j = l, we get

S1 ≤∑

l≥0,j∈Z,j 6=e(l)

8|wl,j||j − e(l)|(|j − e(l)|+ l)

≤ 8R1|w|L2


by the Cauchy-Schwarz inequality where

R21 :=

∑l≥0,j∈Z,j 6=e(l)

1

(j − e(l))2(|j − e(l)|+ l)2=

∑l≥0,j∈Z,j 6=0

1

j2(|j|+ l)2

≤∑

l≥0,j∈Z,j 6=0

1

j2(1 + l)2<∞ .

Estimate of S2. Using (2.22)

S2 ≤2

γ

∑e(l) 6=l

|wl,e(l)| ≤C

γ‖w‖H1

by the Cauchy-Schwarz inequality.

Fixed points of the nonlinear operator G : W → W defined by

G(ε, ω;w) := εL−1ω ΠWf(v + w)

are solutions of the range equation.

Lemma 2.2.3 (Solution of the range equation) Assume ω ∈ Wγ. ∀R >0, ∃ ε0(R) > 0, C0(R) > 0 such that ∀v ∈ B2R := v ∈ V | ‖v‖H1 ≤ 2R,∀0 ≤ ε ≤ ε0(R) there exists a unique solution w(ε, v) ∈ W of the rangeequation satisfying

‖w(ε, v)‖ ≤ C0(R)ε

γ. (2.25)

Moreover the map v → w(ε, v) is in C1(B2R,W ).

Proof. By the Banach algebra property of X and (2.21)

‖G(ε, ω;w)‖ ≤ εC

γ‖v + w‖p ≤ ε

C ′

γ((2R)p + ‖w‖p) , ∀v ∈ B2R .

Hence, for ρ := 2(C ′/γ)ε(2R)p and ∀0 ≤ ε ≤ ε0(R) so small that ρ ≤ 2R,G(ε, ω;Bρ) ⊂ Bρ where Bρ := w ∈ W | ‖w‖ ≤ ρ. Similarly we can prove

‖DG(ε, ω;w)‖ ≤ 1

2, ∀v ∈ B2R

∀0 ≤ ε ≤ ε0(R) small enough. We conclude by the Contraction mappingtheorem. The regularity of w(ε, v) w.r.t to v follows by the Implicit FunctionTheorem.


2.2.2 The bifurcation equation

It remains to solve the infinite dimensional bifurcation equation

ω2vtt − vxx = εΠV f(v + w(ε, v)) . (2.26)

Equation (2.26) is the Euler-Lagrange equation of the reduced Lagrangianaction functional Φε : B2R → R defined by

Φε(v) := Ψ(v + w(ε, v))

where Ψ is the Lagrangian action functional introduced in (2.13).Indeed, w solves the range equation iff it is a critical point of the restricted

functional w → Ψ(v + w) ∈ R, i.e. if and only if

DΨ(v + w)[h] =

∫Ω

ω2wtht − wxhx + εf(v + w)h = 0, ∀h ∈ W (2.27)

(by orthogonality∫

Ωvtht =

∫Ωvxhx = 0). Therefore, ∀h ∈ V ,

DΦε(v)[h] = DΨ(v + w(ε, v))[h+Dw(ε, v)[h]

]= DΨ(v + w(ε, v))[h]

=

∫Ω

ω2vtht − vxhx + εΠV f(v + w(ε, v))h (2.28)

because Dw(ε, v)[h] ∈ W , (2.27) and∫

Ωwtht =

∫Ωwxhx = 0, ∀h ∈ V , see

figure 2.2.The variational nature of the bifurcation equation dealing with variational

problems is a general fact, see e.g [4], [116].

Remark 2.2.4 By (2.28), since w(ε, ·) ∈ C1(B2R,W ), DΦε ∈ C1(B2R, V )and Φε ∈ C2(B2R,R).

To find critical points of Φε we expand

Φε(v) =

∫Ω

ω2

2(vt + (w(ε, v))t)

2 − 1

2(vx + (w(ε, v))x)

2 + εF (v + w(ε, v))

=

∫Ω

ω2

2v2

t −v2

x

2+ εF (v + w(ε, v))− ε

1

2f(v + w(ε, v))w(ε, v)

because by orthogonality ∫Ω

vtwt =

∫Ω

vxwx = 0


W

O

w

Vυ

εw (υ) ∆ψ(υ+w)

υ +

Figure 2.2: The variational reduction: the gradient ∇Ψ(v + w(ε, v)) lies inthe direction of V .

and, inserting h = w(ε, v) in (2.27),∫Ω

ω2w2t − w2

x + εf(v + w(ε, v))w(ε, v) = 0 .

Hence, using that |vt|2L2 = |vx|2L2 = ‖v‖2H1/2,

Φε(v) =ω2 − 1

4‖v‖2

H1+ε

∫Ω

[F (v+w(ε, v))−1

2f(v+w(ε, v))w(ε, v)

]. (2.29)

To find non-trivial critical points of (2.29) we have to impose a suitablerelation between the frequency ω and the amplitude ε (ω must tend to 1 asε→ 0). Imposing the “frequency-amplitude” relation

ω2 − 1

2= −ε sign(a)

(recall f(u) := aup) we get by (2.29)

Φε(v) = −ε sign(a)[1

2‖v‖2

H1 − |a|∫

Ω

vp+1

p+ 1+Rε(v)

]where the remainder Rε ∈ C2(B2R,R) (recall remark (2.2.4)),

Rε(v) := −|a|∫

Ω

1

p+ 1

[(v + w(ε, v))p+1 − vp+1

]− 1

2(v + w(ε, v))pw(ε, v)


satisfies, by Lemma 2.2.3,

|Rε(v)| , |(∇Rε(v), v)| ≤ C1(R)ε

γ. (2.30)

The functional (that we still denote by Φε)

Φε(v) :=1

2‖v‖2

H1 − |a|∫

Ω

vp+1

p+ 1+Rε(v) (2.31)

possesses a local minimum at the origin and one can think to prove existenceof non-trivial critical points via the Mountain Pass Theorem of Ambrosettiand Rabinowitz [7] (Theorem 6.2.4 in the Appendix), see figure 2.3. Notethat since p is odd, the functional

∫Ωvp+1 > 0, ∀v 6= 0.

O

V

R1B (H )

εΦ (υ)

Figure 2.3: The mountain Pass geometry of Φε

However, we can not apply directly the Mountain Pass Theorem 6.2.4since Φε is defined only in a neighborhood of the origin.

2.2.3 The Mountain Pass argument

Step 1: Extension of Φε. We define the extended action functional Φε ∈C2(V,R) as

Φε(v) :=‖v‖2

H1

2− |a|

∫Ω

vp+1

p+ 1+ Rε(v)

where Rε : V → R is

Rε(v) := λ(‖v‖2

H1

R2

)Rε(v)


and λ : [0,+∞) → [0, 1] is a smooth, non-increasing, cut-off function suchthat

λ(x) =

1 for |x| ≤ 10 for |x| ≥ 4

and |λ′(x)| < 1. By definition

Φε(v) =

Φε(v) for ‖v‖H1 ≤ R‖v‖2

H1

2− |a|

∫Ω

vp+1

p+ 1for ‖v‖H1 ≥ 2R .

Furthermore by (2.30) and the definition of λ

|Rε(v)| , |(∇Rε(v), v)| ≤ C1(R)ε

γ. (2.32)

Step 2: Φε verifies the geometrical hypotheses of the Mountain Pass Theo-rem 6.2.4.

Let 0 < ρ < R. For all ‖v‖H1 = ρ

Φε(v) =‖v‖2

H1

2− |a|

∫Ω

vp+1

p+ 1+Rε(v) ≥

1

2ρ2 − κ1ρ

p+1 − εγ−1C1(R) .

Fix ρ > 0 such that (ρ2/2)− κ1ρp+1 ≥ ρ2/4. For 0 < εγ−1C1(R) ≤ ρ2/8 we

get

Φε(v) ≥1

8ρ2 > 0 if ‖v‖H1 = ρ . (2.33)

For any ‖v0‖H1 = 1 there exists t large enough such that

Φε(tv0) =t2

2− |a| t

p+1

p+ 1

∫Ω

v0p+1 < 0

because p ≥ 3 is an odd integer (in contrast, if p is even, then∫

Ωvp+1 = 0

∀v ∈ V , see (2.40)).

We define the minimax class

Γ :=γ ∈ C([0, 1], V ) | γ(0) = 0 , γ(1) = v

where v := tv0 (so Φε(v) < 0), and the Mountain Pass level

cε := infγ∈Γ

maxs∈[0,1]

Φε(γ(s)) . (2.34)

Since any path γ ∈ Γ intersects the sphere v ∈ V | ‖v‖H1 = ρ, by (2.33)

cε ≥ min‖v‖H1=ρ

Φε(v) ≥1

8ρ2 > 0 . (2.35)


By the Mountain Pass Theorem 6.2.4 we deduce the existence of a Palais-Smale sequence for Φε at the level cε > 0, namely: there exists a sequencevn ∈ V such that

Φε(vn) → cε , ∇Φε(vn) → 0 . (2.36)

Our aim is to prove that the Palais-Smale sequence vn converges, up tosubsequence, to some non-trivial critical point v∗ 6= 0 in some open ball of Vwhere Φε and Φε coincide.

We need some bounds for the H1-norm of vn independent of ε.

Step 3: Confinement of the Palais-Smale sequence. By the definition (2.34)and (2.32)

cε ≤ maxs∈[0,1]

Φε(sv) ≤ maxs∈[0,1]

[s2

2‖v‖2

H1 − |a|sp+1

p+ 1

∫Ω

vp+1]

+ 1 =: κ (2.37)

for 0 < C1(R)γ−1ε < 1.

Now we can prove the confinement property of the Palais Smale sequencevn at the level cε.

Φε(vn)− (∇Φε(vn), vn)

p+ 1=

(1

2− 1

p+ 1

)‖vn‖2

H1 + Rε(vn)− (∇Rε(vn), vn)

p+ 1(2.32)

≥(1

2− 1

p+ 1

)‖vn‖2

H1 − 2C1(R)γ−1ε

≥(1

2− 1

p+ 1

)‖vn‖2

H1 − 1 (2.38)

for 0 < 2C1(R)γ−1ε ≤ 1. By (2.36) and (2.37), for n large

Φε(vn)− (∇Φε(vn), vn)

p+ 1≤ (cε + 1) + ‖vn‖H1 ≤ κ+ 1 + ‖vn‖H1

and, by (2.38), we derive

κ+ 1 + ‖vn‖H1 ≥(1

2− 1

p+ 1

)‖vn‖2

H1 − 1

whence‖vn‖H1 ≤ R∗

for a suitable constant R∗ independent of ε.


Step 4: Existence of a nontrivial critical point. Let us fix

R := R∗ + 1 and take 0 < ε ≤ ε1(R) .

By the previous Step, definitively, vn ∈ BR and so Φε(vn) = Φε(vn). Hence

∇Φε(vn) = ∇Φε(vn) = vn −∇G(vn) +∇Rε(vn) → 0 (2.39)

where we have set G(v) := |a|∫

Ωvp+1/(p+ 1).

Lemma 2.2.4 (Compactness) ∇G : V → V and ∇Rε : BR → V arecompact operators.

Proof. Let ‖vk‖H1 be bounded. Then, up to subsequence, vk v ∈ Vweakly in H1 and vk → v in |·|∞ by the compact embedding (2.15). Moreoversince also wk := w(ε, vk) is bounded we can assume that wk w weakly inH1 and wk → w in | · |Lq norm for all q <∞. Since

(∇G(v), h) =

∫Ω

|a|vph ,

it follows easily that ∇G(vk) → ∇G(v) in V . We claim that ∇Rε(vk) → Rwhere

(R, h) =

∫Ω

|a|(v + w)ph− |a|vph = (∇Rε(v), h) .

Indeed, since wk → w in | · |Lq , it converges (up to a subsequence) alsoa.e.. We can deduce, by the Lebesgue dominated convergence theorem, that(vk +wk)

p → (v+w)p in L2 since (vk +wk)p → (v+w)p a.e. and (vk +wk)

p

is bounded in L∞. Hence, since

(∇Rε(vk), h) =

∫Ω

|a|(vk + wk)ph− |a|vp

kh ,

∇Rε(vk) → R.

Since the Palais-Smale sequence vn is bounded in H1 and ∇G and ∇Rε

are compact, by Lemma 6.2.1, vn is precompact and therefore converges toa nontrivial critical point vε of Φε.

Moreover lim infε→0 ‖vε‖H1 ≥ ρ > 0 because7

Φε(vε) = limn→∞

Φε(vn) = cε ≥ρ2

8> 0

7Actually vε shall converge for ε→ 0 to the Mountain Pass critical set of 2−1‖v‖2H1 −(p+ 1)−1

∫Ωvp+1, see [24], [26].


by (2.36)-(2.35).

In conclusion we have found a non-trivial solution

u = δ(vε + w(ε, vε)

)∈ X

of (2.10).

We have finally proved the following theorem

Theorem 2.2.1 ([23]) (Existence, p odd) Let f(u) = aup (a 6= 0) for anodd integer p ≥ 3. There exists C2(f) > 0 such that, ∀ω ∈ Wγ satisfying|ω − 1| ≤ C2(f) and ω < 1 if a > 0 (resp. ω > 1 if a < 0), equation (2.7)possesses at least one, non trivial, small amplitude 2π/ω-periodic solutionuω 6= 0. The solution uω converges to zero like ‖uω‖ ≤ C|ω − 1|1/(p−1) asω → 1.

a>0 a<0

ω ω

ω=1ω=1

Figure 2.4: The bifurcation diagram of Theorem 2.2.1. Since equation (2.7)is autonomous there is a whole circle of periodic solutions uω(·+θ, x),∀θ ∈ R.In general uω does not vary smoothly with the frequency ω.

Remark 2.2.5 (Regularity) By a bootstrap argument the periodic solutionuω of Theorem 2.2.1 is in C2, see [23]. Actually we could also prove that uω

is analytic, see next chapters.

2.3 The case p even

The case f(u) = aup with p even integer is more difficult because∫Ω

vp+1 ≡ 0 , ∀v ∈ V (2.40)


and so the development (2.31) does not imply any more the Mountain Passgeometry of Φε. (2.40) follows by a more general lemma:

Lemma 2.3.1 Let ϕ1, . . . , ϕ2k+1 ∈ V . Then∫Ω

ϕ1 · . . . · ϕ2k+1 = 0 . (2.41)

In particular, if v ∈ V then v2k ∈ W .

Proof. By the change of variables (t, x) 7→ (t, π − x) and periodicity,∫Ω

ϕ1 · . . . · ϕ2k+1 =

∫ π

0

∫T

2k+1∏j=1

(ηj(t+ x)− ηj(t− x)

)dtdx

=

∫ π

0

∫T

2k+1∏j=1

(ηj(t+ π − x)− ηj(t− π + x)

)dtdx

=

∫ π

0

∫T

2k+1∏j=1

(ηj(t− x)− ηj(t+ x)

)dtdx

= (−1)2k+1

∫Ω

ϕ1 · . . . · ϕ2k+1 ,

which implies (2.41).

To find critical points of Φε we have to develop at higher orders in ε thenon-quadratic term in (2.29). We find∫

Ω

F (v + w(ε, v))− 1

2f(v + w(ε, v))w(ε, v) =

a

2

∫Ω

vpw(ε, v) +R1 (2.42)

where, setting wε := w(ε, v),

R1 :=a

p+ 1

∫Ω

[(v + wε)

p+1 − vp+1 − (p+ 1)vpwε

]− a

2

∫Ω

[(v + wε)p − vp]wε

= O(‖wε‖2) = O(ε2)

Substituting the expression

w(ε, v) = εaL−1ω ΠWv

p + εaL−1ω ΠW

[(v + w(ε, v))p − vp

]= εaL−1

ω vp +R2 ,


where, by (2.25),

R2 := εaL−1ω ΠW

[(v + w(ε, v))p − vp

]= O(ε‖wε‖) = O(ε2) ,

into (2.29), we get

Φε(v) :=ω2 − 1

4‖v‖2

H1 + ε2a2

2

∫Ω

vpL−1ω vp +R3

where

R3 := εa

2

∫Ω

vpR2 + εR1 = O(ε3).

Setting the frequency-amplitude relation

ω2 − 1

2:= −ε2

we reduce to find critical points of

Φε(v) =1

2‖v‖2

H1 −a2

2

∫Ω

vpL−1ω vp + ε−2R3(v)

=1

2‖v‖2

H1 −a2

2

∫Ω

vpL−1vp +Rε(v) (2.43)

where L−1 : W → W is the inverse of the D’Alambertian operator L :=L1 := ∂tt − ∂xx, and the remainder

Rε := R3 ε−2 +

∫Ω

vp(L−1ω − L−1)vp (2.44)

satisfies an estimate like in (2.30)

Rε(v) , (∇Rε(v), v) = O(ε) . (2.45)

To prove the estimate (2.45) for the second term in (2.44), develop in Fourierseries vp := r =

∑l 6=j rlj cos(lt) sin(jx) (recall that vp ∈ W ). Hence

L−1vp =∑j 6=l

rlj

−l2 + j2cos(lt) sin(jx) , L−1

ω vp =∑j 6=l

rlj

−ω2l2 + j2cos(lt) sin(jx),

yielding∫Ω

vp(L−1ω − L−1)vp = c

∑j 6=l

(ω2 − 1)r2ljl

2

(ω2l2 − j2)(l2 − j2)

≤ Cε

γ

∑j 6=l

r2lj l

2 ≤ C ′ ε

γ‖r‖2

H1 ≤ C ′ ε

γ‖vp‖2


using |ω2l2− j2| ≥ γ/2, |l2− j2| ≥ 1 and Cauchy-Schwartz. The estimate for(∇Rε(v), v) can be proved similarly.

Now, to show that Φε in (2.43) has the Mountain Pass geometry it remainsonly to check that

∫ΩvpL−1vp 6≡ 0. This is proved with the methods of

characteristics.

Lemma 2.3.2∫

ΩvpL−1vp > 0, ∀v 6= 0.

Proof. Write v = η(t + x) − η(t − x) ∈ V and vp = m(t + x, t − x) withm(s1, s2) = (η(s1)− η(s2))

p. z := L−1(vp) is a solution of ztt − zxx = vp

z(t, 0) = z(t, π) = 0z(t+ 2π, x) = z(t, x) .

Making the change of variables (s1, s2) = (t+x, t−x) such equation becomes(for z(t, x) = M(t+ x, t− x)) ∂s2s1M(s1, s2) = (1/4)m(s1, s2)

M(s1, s1) = M(s1, s1 − 2π) = 0M(s1 + 2π, s2 + 2π) = M(s1, s2) ∀s2 ≤ s1 ≤ s2 + 2π .

We claim that a solution of this problem is

M(s1, s2) :=1

8

∫Qs1,s2

m(ξ1, ξ2) dξ1 dξ2 (2.46)

where

Qs1,s2 :=

(ξ1, ξ2) ∈ R2 | s1 ≤ ξ1 ≤ s2 + 2π, s2 ≤ ξ2 ≤ s1

.

Indeed, differentiating w.r.t. s1

∂s1M(s1, s2) =1

8

∫ s2+2π

s1

m(ξ1, s1) dξ1 −1

8

∫ s1

s2

m(s1, ξ2) dξ2.

Differentiating w.r.t. s2, since m(s1, s2) = (η(s1)− η(s2))p with p even, and

η is 2π-periodic,

∂s2∂s1M =1

8m(s2 + 2π, s1) +

1

8m(s1, s2) =

1

4m(s1, s2).

From (2.46), M ≥ 0 and, by Lemma 2.3.3 below,∫Ω

vpL−1vp =1

2

∫T2

m(s1, s2)M(s1, s2) ≥ 0 .

If∫

ΩvpL−1vp = 0 then m(s1, s2) = 0 or M(s1, s2) = 0 for all s2 ≤ s1 ≤

s2 + 2π. If M(s1, s2) = 0 then m(ξ1, ξ2) = 0 for all (ξ1, ξ2) ∈ Qs1,s2 and som(s1, s2) = 0. In both cases m(s1, s2) = 0 and so v = 0.


Lemma 2.3.3 Let m : R2 → R be 2π-periodic w.r.t. both variables. Then∫ 2π

0

∫ π

0

m(t+ x, t− x) dt dx =1

2

∫ 2π

0

∫ 2π

0

m(s1, s2) ds1 ds2. (2.47)

Proof. Make the change of variables (s1, s2) = (t + x, t − x) and use theperiodicity of m.

With the same Mountain Pass type argument developed in section 2.2 wecan prove:

Theorem 2.3.1 [23] (Existence, p even) Let f(u) = aup (a 6= 0) foran even integer p. There exists C3(f) > 0 such that, ∀ω ∈ Wγ, ω < 1,with |ω − 1| ≤ C3(f) equation (2.7) possesses at least one non trivial, smallamplitude 2π/ω-periodic solution uω 6= 0. The solution uω converges to zerolike ‖uω‖ ≤ C|ω − 1|1/2(p−1) for ω → 1.

Wω

ω=1

γ

Figure 2.5: The Cantor families of solutions for p even. Also here uω is notin general smooth w.r.t ω.

Remark 2.3.1 This higher difficulty for finding periodic solutions when thenonlinearity f is non-monotone is a common feature for nonlinear wave equa-tions, see e.g. [111], [40], [41], [45]. In physical terms there is no “confine-ment effect” due to the potential. Actually, in [27]-[28] a non-existence result


is proved for even power nonlinearities in the case of spatial periodic bound-ary conditions. This highlights that the term

∫ΩvpL−1vp, which is responsible

for the existence result of Theorem 2.3.1, is a “boundary effect” due to theDirichlet conditions.

Remark 2.3.2 In [27]-[28] existence of quasi-periodic solutions with 2 fre-quencies for completely resonant nonlinear wave equations with spatial pe-riodic boundary condition has been obtained. After a similar variationalLyapunov-Schmidt reduction the reduced action functional turns out to havethe infinite dimensional linking geometry, [19].

2.4 Multiplicity

As in the Weinstein-Moser and Fadell-Rabinowitz theorems one could expectmultiplicity of periodic solutions. This could be proved exploiting the S1

symmetry of the action functional induced by time translations. However, inthis case we can provide a much more detailed picture of these solutions.

Critical points of the reduced action functional Φε restricted to each sub-space Vn ⊂ V formed by the functions

v = η(n(t+ x))− η(n(t− x)) ∈ V , n ∈ N

which are 2π/n-periodic, are critical points of Φε on the whole V .As in the previous section we can prove that, for each n ≥ 1, the reduced

action functional Φε|Vnpossesses a Mountain Pass critical point vn.

It turns out that the larger n is, the smaller |ω − 1| has to be (i.e. ε). Inthis way, for ω ∈ Wγ sufficiently close to 1 we can prove the existence of alarge number Nω of 2π/ω-periodic solutions

u1, . . . , un, . . . , uNω

where Nω → +∞ as ω → 1. Making refined estimates we can obtain8

Nω ≈√γτ/|ω − 1|

for some τ ∈ [1, 2].These periodic solutions un have increasing energies (in n), rotating with

faster and faster frequency: actually we can prove that the minimal period ofthe n-th solution un is 2π/nω. This, in particular, proves that the solutionsun are geometrically distinct.

8This estimate for Nω is in general the optimal one. Note indeed that we can alwaysperform a finite dimensional reduction to an N dimensional subspace of V if N2(ω2−1) ≈γ.


Theorem 2.4.1 ([24]) (Multiplicity, p odd) Let f(u) = aup (a 6= 0) foran odd integer p ≥ 3. Then there exists a positive constant C4 := C4(f) suchthat, ∀ω ∈ Wγ and ∀n ∈ N\0 satisfying

|ω − 1|n2

γ≤ C4 (2.48)

and ω < 1 if a > 0 (resp. ω > 1 if a < 0), equation (2.7) possesses at leastone periodic classical C2 solution with minimal period 2π/(nω).

As a consequence there exist u1, . . . , un, . . . , uNω , with9

Nω :=[ γC4

|ω − 1|

]− 1 ,

geometrically distinct periodic solutions of (2.7) with the same period 2π/ω.

Theorem 2.4.2 ([24]) (Multiplicity, p even) Let f(u) = aup (a 6= 0) foran even integer p. Then there exists a positive constant C5 depending onlyon f such that, ∀ω ∈ Wγ with ω < 1, ∀n ≥ 2 such that

(|ω − 1|n2)1/2

γ≤ C5 (2.49)

equation (2.7) possesses at least one periodic classical C2 solution with min-imal period 2π/(nω).

As a consequence there exist u2, . . . , un, . . . , uNω , with

Nω :=[ (γC5)

2

|ω − 1|

]− 1 ,

geometrically distinct periodic solutions of (2.7) with the same period 2π/ω.If p = 2 the existence result holds true for n = 1 as well.

2.5 The small divisor problem

We now want to discuss the “small divisors” problem (i) to prove existenceof periodic solutions for positive (actually asymptotically full) measure set offrequencies ω.

For this we have to relax the non-resonance condition in (2.16) requiring

ω ∈ Wγ,τ :=|ωl − j| ≥ γ

lτ, ∀j 6= l , l ≥ 1 , j ≥ 1

(2.50)

9[x] ∈ Z denotes the integer part of x ∈ R.


for some τ > 1, γ > 0 (note that τ = 1 in (2.16)).

By standard measure estimates the set Wγ,τ has positive measure, actu-ally has asymptotically full density as ω → 1. The Proof is instructive.

Proposition 2.5.1 For any γ > 0, τ > 1, it results

limη→0

|Wγ,τ ∩ (1, 1 + η)||η|

= 1 . (2.51)

Proof. To fix the ideas suppose η > 0. Note that, since l 6= j, if 1 ≤ l ≤ (2/3η), then ∀ω ∈ (1, 1 + η)

|ωl − j| = |(l − j) + (ω − 1)l| ≥ 1− |η|l ≥ 1

3

and the condition in (2.50) is trivially satisfied (γ < 1/3).We have to estimate the measure of the complementary set

Wcγ,τ ∩ (1, 1 + η) =

⋃(j,l)∈S

Rjl (2.52)

where

Rjl :=(jl− γ

lτ+1,j

l+

γ

lτ+1

)∩ (1, 1 + η)

and

S :=

(j, l) ∈ N×N , j 6= l , l ≥ 2

3η

.

Note that if (j, l) does not satisfy

(1− 4η)l < j < (1 + 4η)l

thenRjl = ∅ .

So we can restrict in (2.52) to the subset of indexes

S :=

(j, l) ∈ N×N , j 6= l , l ≥ 2

3η, (1− 4η)l < j < (1 + 4η)l

finding the estimate

|Wcγ,τ ∩ (1, 1 + η)| ≤

∑(j,l)∈eS

|Rjl|

≤∑

l≥2/3η,

∑(1−4η)l<j<(1+4η)l

2γ

lτ+1

≤∑

l≥2/3η

8ηl2γ

lτ+1= 16ηγ

∑l≥2/3η

1

lτ

= O(γητ )


because τ > 1. Hence

limη→0

|Wcγ,τ ∩ (1, 1 + η)|

|η|= 0

and the estimate (2.51) follows.

For ω satisfying only (2.50), the moduli of the eigenvalues of (Lω)|W canbe bounded from below just as

|ω2l2 − j2| = |ωl − j|(ωl + j) ≥ γ

lτ(ωl + j) ≥ γ

lτ−1. (2.53)

(actually for infinitely many (ln, jn) the estimate (2.53) is accurate).As a consequence the operator

(Lω)−1|Ww =

∑j≥1,l≥0,l 6=j

wlj

−ω2l2 + j2cos lt sin jx , ∀w ∈ W

does not send (W, ‖ · ‖) into itself. Actually, by (2.53), the operator (Lω)−1|W

“loses (τ − 1)-derivatives” and the usual Picard iteration method employedin the standard implicit function Theorem can not lead to an approximationof the solution, since after finitely many steps all derivatives are exhausted.

One could think to work in spaces of C∞ or analytic functions. Forexample, in the sequel we shall deal with the space of functions

Xσ,s :=u(t, x) =

∑l∈Z

exp (ilt) ul(x) | ul ∈ H10 ((0, π),R) , ul(x) = u−l(x)

and ‖u‖2σ,s :=

∑l∈Z

exp (2σ|l|)(l2s + 1)‖ul‖2H1 < +∞

(2.54)

which are analytic in time (σ > 0) and valued in H10 ((0, π),R). More pre-

cisely, for σ > 0, s ≥ 0, Xσ,s is the space of all even, 2π-periodic in timefunctions with values in H1

0 ((0, π),R), which have a bounded analytic ex-tension in the complex strip |Im t| < σ with trace function on |Im t| = σbelonging to Hs(T, H1

0 ((0, π),C)).

By (2.53), for w ∈ W ∩Xσ,s, L−1ω w ∈ Xσ′,s just for σ′ < σ and

‖L−1ω h‖2

σ′,s = c∑|l|6=j

exp (2σ′|l|)(l2s + 1)w2

ljj2

(−ω2l2 + j2)2

(2.53)

≤ c∑|l|6=j

exp (2σ′|l|)(l2s + 1)l2(τ−1)w2

ljj2

γ2


= c∑|l|6=j

exp (−2(σ − σ′)|l|)l2(τ−1) exp (2σ|l|)(l2s + 1)w2

ljj2

γ2

≤ C‖h‖2

σ,s

γ2(σ − σ′)2(τ−1)

because

exp (−2(σ − σ′)y) y2(τ−1) ≤ C(τ)

(σ − σ′)2(τ−1), ∀y ≥ 0 .

The resulting “Cauchy-type” estimate for the inverse of (Lω)|W

‖L−1ω h‖σ′,s ≤

C

γ(σ − σ′)(τ−1)‖h‖σ,s σ′ < σ . (2.55)

In this case, however, the estimates for the approximate solutions obtainedby the Picard iteration scheme shall diverge, see remark 3.2.1.

In the next chapter we shall present another iteration scheme (Nash-Moser) to overcome such difficulty due to the small divisors.

Chapter 3

A tutorial in Nash-Mosertheory

3.1 Introduction

The classical implicit function theorem is concerned with the solvability ofthe equation

F(x, y) = 0 (3.1)

where F : X × Y → Z is a smooth map, X, Y , Z are Banach spaces, andthere exists (x0, y0) ∈ X × Y such that

F(x0, y0) = 0 .

If x is close to x0 we want to solve (3.1) finding y = y(x).The main assumption of the classical implicit function theorem is that

the partial derivative (DyF)(x0, y0) : Y → Z possesses a bounded inverse

(DyF)−1(x0, y0) ∈ L(Z, Y ) .

Note that, if (DyF)(x0, y0) ∈ L(Y, Z) is injective and surjective, by theopen mapping theorem, the inverse operator (DyF)−1(x0, y0) : Z → Y isautomatically continuous.

As we saw in the previous Chapter there are situations where

(DyF)(x0, y0) has an unbounded inverse

(for example the image (DyF)(x0, y0)[Y ] is only dense in Z).

An approach to these class of problems has been proposed by Nash inthe pioneering paper [100], for proving that any Riemannian manifold can be

68

CHAPTER 3. A TUTORIAL IN NASH-MOSER THEORY 69

isometrically embedded in RN for N sufficiently large. Subsequently, Moser[90] has highlighted the main features of the technique in an abstract set-ting, being able to cover problems arising from Celestial Mechanics and par-tial differential equations [91]-[92]-[93]. Further extensions and applicationswere made by Gromov [64], by Zehnder [131] to small divisors problems, byHormander [70] to problems in gravitation, by Sergeraert [121] to catastro-phe theory, by Schaeffer [118] to free boundary problems in electromagnetics,by Beale [18] in water waves, by Hamilton [66] to foliations, by Klainermann[81]-[82] to Cauchy problems, to mention just a few, showing the power andversatility of the technique.

The main idea is to replace the usual Picard iteration method with amodified Newton iteration scheme. Roughly speaking, the advantage is that,since this latter scheme is quadratic (see remark 3.2.1 and 3.3.2), the iteratesshall converge to the expected solution at a super-exponential rate. Thisaccelerated speed of convergence is sufficiently strong to compensate thedivergences in the scheme due to the “loss of derivatives”.

There are many ways to present the Nash-Moser theorems, according tothe applications one has in mind. We shall prove first a very simple “an-alytic” Implicit Function Theorem (inspired to Theorem 6.1 by Zehnder in[100], see also [51]) to highlight the main features of the method in an ab-stract “analytic” setting (i.e. with estimates which can be typically obtainedin Banach scales of analytic functions). In our next application to the non-linear wave equation, indeed, we shall be able to prove, with a variant ofthis scheme, existence of analytic (in time) solutions of (NLW) for positivemeasure sets of frequencies.

Next, for completeness, we present also a Nash-Moser theorem in a dif-ferentiable setting (i.e. modeled for applications on spaces of functions withfinite differentiability like, for example, Banach scales of Sobolev spaces). Toavoid technicalities we present it in the form of an inversion type theorem asin Moser [90].

3.2 An analytic Nash-Moser Theorem

Consider three one parameter families of Banach spaces

Xσ , Yσ , Zσ , 0 ≤ σ ≤ 1

with norms | · |σ such that (Banach scales)

∀0 ≤ σ ≤ σ′ ≤ 1 |x|σ ≤ |x|σ′ ∀x ∈ Xσ′


(analogously for Yσ, Zσ) so that

∀0 ≤ σ ≤ σ′ ≤ 1 X1 ⊆ Xσ′ ⊆ Xσ ⊆ X0

(the same for Yσ, Zσ).

Example: The Banach spaces of analytic functions

Xσ :=f : Td → R , f(ϕ) :=

∑k

fkeik·ϕ | |f |σ :=

∑k

|fk|eσ|k| < +∞.

LetF : X0 × Y0 → Z0

be a mapping defined on the largest spaces of the scales.Suppose there exists (x0, y0) ∈ X1×Y1 (in the smallest spaces) such that

F(x0, y0) = 0 . (3.2)

Assume thatF(Bσ) ⊂ Zσ ∀0 ≤ σ ≤ 1 (3.3)

where Bσ is the neighborhood of (x0, y0)

Bσ := BσR(x0)×Bσ

R(y0) ⊂ Xσ × Yσ

andBσ

R(x0) :=x ∈ Xσ| |x− x0|σ < R

analogously for Bσ

R(y0) ⊂ Yσ.

We shall make the following hypotheses in which K and τ are fixed positiveconstants.

(H1) (Taylor Estimate) ∀0 < σ ≤ 1, ∀x ∈ BσR(x0) the map F(x, ·) :

BσR(y0) → Zσ is differentiable and, ∀(x, y), (x, y′) ∈ Bσ,∣∣∣F(x, y′)−F(x, y)− (DyF)(x, y)[y′ − y]

∣∣∣σ≤ K|y′ − y|2σ .

Condition (H1) is clearly satisfied if F(x, ·) ∈ C2(BσR(y0), Zσ) and D2

yyF(x, ·)is uniformly bounded for x ∈ Bσ

R(x0).

(H2) (Right Inverse of loss τ) ∀0 < σ ≤ 1, ∀(x, y) ∈ Bσ there is a linearoperator L(x, y) ∈ L(Zσ, Yσ′), ∀σ′ < σ, such that ∀z ∈ Zσ

(DyF)(x, y) L(x, y)z = z

in Zσ′ and ∣∣∣L(x, y)[z]∣∣∣σ′≤ K

(σ − σ′)τ|z|σ . (3.4)


The operator L(x, y) is the right inverse of (DyF)(x, y) in the sense that

(DyF)(x, y) L(x, y) is the continuous injection Zσi→ Zσ′

∀σ′ < σ. Estimate (3.4) is a typical “Cauchy-type” estimate (like (2.55)) foroperators acting somewhat as differential operators of order τ in scales ofBanach spaces of analytic functions.

Theorem 3.2.1 Let F satisfy (3.2),(3.3), (H1)-(H2). If x ∈ BσR(x0) for

some σ ∈ (0, 1] and |F(x, y0)|σ is sufficiently small1, then there exists asolution

y(x) ∈ Bσ/2R (y0) ⊂ Yσ/2

of the equationF(x, y(x)) = 0 .

Proof. We define the Newton iteration schemeyn+1 = yn − L(x, yn)F(x, yn)y0 := y0 ∈ Y1 ⊆ Yσ

(3.5)

for n ≥ 0. Throughout the induction proof we will verify at each step (see theClaim below) that yn belongs to the domain of F(x, ·), L(x, ·) and thereforeyn+1 is well defined.

Since the inverse operator L(x, yn) “loses analyticity” (hypothesis (H2))the iterates yn will belong to larger and larger spaces Yσn .

To quantify this phenomenon, let us define the sequence

σ0 := σ ∈ (0, 1] , σn+1 := σn − δn

where the “loss of analyticity” at each step of the iteration is

δn :=δ0

n2 + 1

and δ0 > 0 is small enough so that the “total loss of analyticity”∑n≥0

δn =∑n≥0

δ0n2 + 1

<σ

2(3.6)

(therefore σn > σ/2, ∀n ≥ 0).

We claim the following:

1quantified in (3.7); this latter condition defines a neighborhood of x0 in Xσ.


Claim: Take χ := 3/2 and define2

ρ := ρ(K,R, τ, σ) := min√eK,min

n≥0

(δτn+1

K2e(2−χ)χn

),

R/2∑∞k=0 e

−χk

> 0 .

If

|F(x, y0)|σ < minρe−1 ,

δτ0

Kρe−1,

δτ0

K

R

2

, (3.7)

then the following statements hold true for all n ≥ 0 :

(n; 1) (x, yn) ∈ Bσn and |F(x, yn)|σn ≤ ρe−χn,

(n; 2) |yn+1 − yn|σn+1 ≤ ρe−χn,

(n; 3) |yn+1 − y0|σn+1 < R/2.

Before proving the Claim, let us conclude the proof of Theorem 3.2.1.By (n; 2) the sequence yn ∈ Yσ/2 is a Cauchy sequence (in the largest

space Yσ/2). Indeed, for any n > m

|yn − ym|σ/2 ≤n−1∑k=m

|yk+1 − yk|σ/2 ≤n−1∑k=m

|yk+1 − yk|σk+1

(k;2)

≤n−1∑k=m

ρe−χk → 0 for n,m→ +∞ .

Hence yn converges in Yσ/2 to some y(x) ∈ Yσ/2. Actually y(x) ∈ Bσ/2R/2(y0) ⊂

Bσ/2R by (n; 3). Finally, by the continuity of F with respect to the second

variable and (n; 1)

F(x, y(x)) = limn→∞

F(x, yn) = 0

implying that y(x) is a solution of F(x, y) = 0.

Let’s now prove the Claim. Its proof proceeds by induction. First, let usverify it for n = 0. It reduces to the smallness condition (3.7) for |F(x, y0)|σ.

2We have ρ > 0 because the sequence of positive numbers

δτn+1e

(2−χ)χn

= δτ0

e12 (3/2)n

(1 + (n+ 1)2)τ→ +∞ as n→ +∞ .


(0; 1) By assumption x ∈ BσR(x0) so that (x, y0) ∈ Bσ0 := Bσ. By (3.3) we

have that F(x, y0) ∈ Zσ and |F(x, y0)|σ ≤ ρe−1 follows by (3.7).

(0; 2)-(0; 3) Since (x, y0) ∈ Bσ, by (3.5) and (H2),

|y1 − y0|σ1 = |L(x, y0)F(x, y0)|σ1 ≤K

(σ0 − σ1)τ|F(x, y0)|σ .

Under the smallness condition (3.7) we have verified both (0; 2)-(0; 3).

Now, suppose (n; 1)-(n; 2)-(n; 3) are true. By (n; 3),

yn+1 ∈ Bσn+1

R (y0)

and so(x, yn+1) ∈ Bσn+1 .

Hence F(x, yn+1) ∈ Zσn+1 (by (3.3)) and, by (H2),

yn+2 := yn+1 − L(x, yn+1)F(x, yn+1) ∈ Yσn+2

is well defined.Set for brevity

Q(y, y′) := F(x, y′)−F(x, y)− (DyF)(x, y)[y′ − y] . (3.8)

By a Taylor expansion

|F(x, yn+1)|σn+1 =∣∣∣F(x, yn) + (DyF)(x, yn)[yn+1 − yn] +Q(yn, yn+1)

∣∣∣σn+1

(3.5)= |Q(yn, yn+1)|σn+1

(H1)

≤ K|yn+1 − yn|2σn+1(3.9)

(n;2)

≤ Kρ2e−2χn

. (3.10)

By (3.10) the claim (n+ 1; 1) is verified whenever

Kρ2e−2χn

< ρe−χn+1

which holds true for any n ≥ 0 if

ρ < minn≥0

( 1

Ke(2−χ)χn

)=

√e

K. (3.11)


Now

|yn+2 − yn+1|σn+2

(3.5)= |L(x, yn+1)F(x, yn+1)|σn+2

(H2)

≤ K

(σn+1 − σn+2)τ|F(x, yn+1)|σn+1

(3.9)

≤ K2

(σn+1 − σn+2)τ|yn+1 − yn|2σn+1

(3.12)

(n;2)

≤ K2

(σn+1 − σn+2)τρ2e−2χn

and therefore the claim (n+ 1; 2) is verified whenever

K2

(σn+1 − σn+2)τρ2e−2χn

< ρe−χn+1

which holds true, for any n ≥ 0, if

ρ < minn≥0

(δτn+1

K2e(2−χ)χn

). (3.13)

Finally

|yn+2 − y0|σn+2 ≤n+1∑k=0

|yk+1 − yk|σn+2 ≤n+1∑k=0

|yk+1 − yk|σk+1

(k;2)

≤n+1∑k=0

ρe−χk

< ρ∞∑

k=0

e−χk

which implies (n+ 1; 3) assuming

ρ <R/2∑∞

k=0 e−χk . (3.14)

In conclusion, if ρ > 0 is small enough (depending on K, τ , R, σ) accordingto (3.11)-(3.13)-(3.14) the claim is proved.

This completes the proof.

Remark 3.2.1 The key point of the Nash-Moser scheme is the estimate

|yn+2 − yn+1|σn+2 ≤K2

δτn+1

|yn+1 − yn|2σn+1(3.15)

see (3.12). Even though δn → 0, this quadratic estimate ensures that thesequence of numbers |yn+1− yn|σn+1 tends to zero at a super-exponential rate


(see (n; 2)) if |y1 − y0|σ1 is sufficiently small. Note that the Picard iterationscheme would yield just |yn+2 − yn+1|σn+2 ≤ Cδ−τ

n+1|yn+1 − yn|σn+1, i.e. thedivergence of the estimates.

Clearly, the drawback to get (3.15) is to invert the linearized operators ina whole neighborhood of (x0, y0), see (H2). This is the most difficult step toapply the Nash-Moser method in concrete situations, see e.g. section 4.4.

Remark 3.2.2 The hypotheses in Theorem 3.2.1 could be considerably weak-ened, see [100]. For example in (H1) one could assume a loss of analyticity3

also in the quadratic part of the Taylor expansion∣∣∣F(x, y′)−F(x, y)− (DyF)(x, y)[y′ − y]∣∣∣σ′≤ K

(σ − σ′)α|y′ − y|2σ

∀σ′ < σ and some α > 0 (independent of σ).Furthermore one could assume in (H2) just4 the existence of an “approx-

imate right inverse”, namely ∀z ∈ Zσ∣∣∣((DyF)(x, y) L(x, y)− I)[z])∣∣∣

σ′≤ K

(σ − σ′)τ|F(x, y)|σ|z|σ (3.16)

Remark that L(x, y) is an exact inverse at the solutions F(x, y) = 0.Furthermore in the statement of Theorem 3.2.1 it is possible to get better

and quantitative estimates.

Since we have not assumed the existence of the left inverse of (DyF)(x, y)in the assumptions of Theorem 3.2.1, uniqueness of the solution y(x) can notbe expected (it could lack also in the linear problem).

Local uniqueness follows assuming the existence of a left inverse:

(H2)′ ∀0 < σ ≤ 1, ∀(x, y) ∈ Bσ there is a linear operator ξ(x, y) ∈L(Zσ, Yσ′), ∀σ′ < σ, such that, ∀h ∈ Yσ

ξ(x, y) (DyF)(x, y)[h] = h

in Yσ′ and ∀z ∈ Zσ ∣∣∣ξ(x, y)[z]∣∣∣σ′≤ K

(σ − σ′)τ|z|σ . (3.17)

3In the application considered in the next chapter the quadratic part Q satisfies (H1),i.e. it does not lose regularity.

4In the application to the (NLW) we can find an inverse operator satisfying (H2).


The operator ξ(x, y) is the left inverse of (DyF)(x, y) in the sense that ξ(x, y)(DyF)(x, y) is the continuous injection Yσ

i→ Yσ′ , ∀σ′ < σ.

Theorem 3.2.2 (Uniqueness) Let F satisfy (3.3), (H1)-(H2)′. Let (x, y),(x, y′) ∈ Bσ be solutions of F(x, y) = 0, F(x, y′) = 0. If |y − y′|σ is smallenough (depending on K, τ , σ) then y = y′ in Yσ/2.

Proof. Setting h := y − y′ ∈ Yσ we have

|h|σ′(H2)′

= |ξ(x, y) (DyF)(x, y)[h]|σ′(3.17)

≤ K

(σ − σ′)τ|(DyF)(x, y)[h]|σ

≤ K

(σ − σ′)τ|Q(y, y′)|σ (3.18)

since F(x, y) = 0, F(x, y′) = 0 and recalling the definition of Q(y, y′) in(3.8).

By (3.18) and (H1) we get

|h|σ′(H1)

≤ K2

(σ − σ′)τ|y′ − y|2σ =

K2

(σ − σ′)τ|h|2σ , ∀σ′ < σ

whence, for σ′ := σn+1, σ := σn, δn := σn − σn+1,

|h|σn+1 ≤ K2δ−τn |h|2σn

, ∀n ≥ 0 .

These last estimates imply that, if |h|σ = |y − y′|σ (σ = σ0) is sufficientlysmall (depending on K, τ , σ), then |h|σn ≤ ρe−χn

, ∀n ≥ 0, and therefore|h|σ/2 = 0, namely h = y − y′ = 0 in Yσ/2.

3.3 A differentiable Nash-Moser Theorem

The iterative scheme (3.5) can not work to prove a Nash-Moser implicit func-tion theorem in spaces, say, of class Ck, because, due to the loss of deriva-tives of the inverse linearized operators, after a fixed number of iterationsall derivatives will be exhausted. The scheme has to be modified applying asequence of “smoothing” operators which regularize yn+1 − yn at each step.

To avoid technicalities, we present the ideas of the Nash-Moser differen-tiable theory in the form of an inversion type theorem (as in Moser [90])rather than an Implicit function type theorem.


To make it precise, consider a Banach scale (Ys)s≥0 satisfying

Ys′ ⊂ Ys ⊂ Y0 , ∀s′ ≥ s ≥ 0

(with norms |y|s ≤ |y|s′ , ∀y ∈ Ys′), equipped with a family of “smoothing”linear operators

S(t) : Y0 → Y∞ :=⋂s≥0

Ys , t ≥ 0

such that|S(t)u|s+r ≤ Cs,r t

r|u|s , ∀u ∈ Ys (3.19)

|(I − S(t))u|s ≤ Cs,r t−r|u|s+r , ∀u ∈ Ys+r (3.20)

for some positive constants Cs,r. The meaning of (3.20) is that, functionsu which are already smooth (u ∈ Ys+r), are approximated by S(t)u, in theweaker norm | |s, with a higher degree of accuracy.

For the construction of these smoothing operators for concrete Banachscales see for example Schwartz [119] or Zehnder in [100] (via convolutionswith suitable “mollifiers”).

Remark 3.3.1 Estimates (3.19)-(3.20) are the usual ones in the Sobolevscale

Ys :=f(ϕ) :=

∑k

fkeik·ϕ | |f |2s :=

∑k

|fk|2(1 + |k|2s) < +∞

for the projector SN on the first N Fourier-modes

SN

( ∑k

fkeik·x

):=

∑|k|≤N

fkeik·x

(when t := N is an integer).

Exercise: On a scale (Xs)s≥0 equipped with smoothing operators (S(t))t≥0,the following convexity inequality holds: for all 0 ≤ λ1 ≤ λ2, α ∈ [0, 1] andu ∈ Xλ2:

|u|λ ≤ Kλ1,λ2|u|1−αλ1

|u|αλ2, λ = (1− α)λ1 + αλ2 . (3.21)

(This is the basis of the Gagliardo [61]- Nirenberg [101]- Moser [92] interpo-lation estimates in Sobolev spaces, see [124] for a modern account.)

We make the following assumptions where α, K, τ are fixed positive con-stants.


(H1) (Tame estimate) F : Ys+α → Ys, ∀s ≥ 0, satisfies5

|F(y)|s ≤ K(1 + |y|s+α) , ∀y ∈ Ys+α .

(H2) (Taylor estimate) F : Ys+α → Ys, ∀s ≥ 0, is differentiable and|(DF)(y)[h]|s ≤ K|h|s+α ,∣∣∣F(y′)−F(y)− (DF)(y)[y′ − y]

∣∣∣s≤ K|y′ − y|2s+α .

(H3) (Right Inverse of loss τ) ∀y ∈ Y∞ there is a linear operator L(y) ∈L(Ys+τ , Ys), ∀s ≥ 0, i.e.

|L(y)[h]|s ≤ K|h|s+τ , ∀h ∈ Ys+τ ,

such thatDF(y) L(y)[h] = h , ∀h ∈ Ys+τ ,

in Ys−α (the operator DF(y) L(y) is the inclusion i : Ys+τ → Ys−α).

Hypothesys (H1)-(H2)-(H3) state, roughly, that F , DF , respectivelyL, act somewhat as differential operators of order α, respectively τ .

Theorem 3.3.1 Let F satisfy (H1)-(H2)-(H3) and fix any s0 > α + τ . If|F(0)|s0+τ is sufficiently small (depending on α, τ , K, s0) then there existsa solution y ∈ Ys0 of the equation F(y) = 0.

Proof. Consider the iterative schemeyn+1 = yn − S(Nn)L(yn)F(yn)y0 := 0

(3.22)

whereNn := eλχn

, Nn+1 = Nχn , χ := 3/2

for some λ large enough, depending on α, τ , K, s0, to be chosen later.

By (3.22), the increment yn+1−yn ∈ Y∞, ∀n ≥ 0, and, therefore, yn ∈ Y∞,∀n ≥ 0 (because y0 := 0 ∈ Y∞). Furthermore

|yn+1 − yn|s0

(3.22)= |S(Nn)L(yn)F(yn)|s0

(3.19)

≤ C0Nα+τn |L(yn)F(yn)|s0−α−τ

(H3)

≤ C0Nα+τn K|F(yn)|s0−α (3.23)

5Differential operators F of order α satisfy the “tame” property (H1), i.e. |F(y)|sgrows at most linearly with the higher norm | · |s+α. This apparently surprising factfollows by the interpolation inequalities (3.21), see [100], [66].


Claim: Take β := 15(α+ τ) and suppose

|F(0)|s0+τ < e−λ4(α+τ)/KCs0,0 . (3.28)

There is λ := λ(τ, α,K, s0) ≥ 1, such that the following statements hold truefor all n ≥ 0:

• (n;1) Bn ≤ N νn = eλχnν , ν := 4(τ + α),

• (n;2) |yn+1 − yn|s0 ≤ N−νn = e−λχnν.

Statement (0; 1) is verified by

B0 := |L(0)F(0)|s0+β

(H3)

≤ K|F(0)|s0+β+τ ≤ eλν

which holds true for λ := λ(s0, α, τ,K) large enough.

Statement (0; 2) follows by

|y1 − y0|s0

(3.22)= |S(N0)L(0)F(0)|s0

(3.19)

≤ Cs0,0|L(0)F(0)|s0

(H3)

≤ Cs0,0K|F(0)|s0+τ

(3.28)< e−λν .

Now suppose (n; 1)-(n; 2) are true. To prove (n+ 1; 1) write

Bn+1

(3.27)

≤ C3

(1 +

n∑k=0

N τ+αk Bk

) (n;1)

≤ C3

(1 +

n∑k=0

e(τ+α+ν)λχk)

= C3

(1 + e(τ+α+ν)λχn

n∑k=0

e−(τ+α+ν)λ(χn−χk))

≤ C3

(1 + e(τ+α+ν)λχn

n∑k=0

e−5(τ+α)(χn−χk))

≤ C4e(τ+α+ν)λχn

< eνλχn+1

for some C4 := C4(α, τ,K, s0) > 0 and λ := λ(α, τ,K, s0) ≥ 1 sufficientlylarge (because ν(χ− 1) > τ + α).

Remark 3.3.2 The main novelty w.r.t to the analytic scheme -compare (3.25)with (3.15)- is to prove that the term Nα+τ

n N−βn−1Bn−1 in (3.25) is super-

exponentially small. This follows, for β large, by (n;1), implying that |yn+1−yn|s0 still converges to zero at a super-exponential rate if |y1 − y0|s0 is suffi-ciently small, statement (n; 2).


Let us prove (n+ 1; 2). Recalling that Nn := eλχnwe have

|yn+2 − yn+1|s0

(3.25)

≤ C1eλ(α+τ)χn+1

e−λβχn

Bn + C1eλ(α+τ)χn+1|yn+1 − yn|2s0

(n;1),(n;2)

≤ C1eλ(α+τ)χn+1

e−λβχn

eλνχn

+ C1eλ(α+τ)χn+1

e−2νλχn

≤ e−λχn+1ν

once we impose

C1eλχn(χ(α+τ)−β+ν) <

e−λχn+1ν

2, C1e

λχn(χ(α+τ)−2ν) <e−λχn+1ν

2.

These inequalities are satisfied, for λ large enough depending on α, τ , K, s0,because

β − ν(1 + χ2)− χ(α+ τ) > 0 and (2− χ)ν − χ(α+ τ) > 0

for β := 15(α+ τ), ν := 4(α+ τ), χ = 3/2.This concludes the proof of the Claim.

By (n; 2) the sequence yn is a Cauchy sequence in Ys0 and therefore yn →y ∈ Ys0 . By (3.24), (n; 1)-(n; 2), |F(yn)|s0−α → 0 and, therefore, by thecontinuity of F(·), we deduce F(y) = 0.

Remark 3.3.3 Clearly much weaker conditions could be assumed. First ofall conditions (H1)-(H2)-(H3) need to hold just on a neighborhood of y0 = 0.Next, we could allow the constant K := K(| |s0) to depend on the weakernorm | · |s0. The inverse could be substitute by an approximate right inverseas in (3.16).

Chapter 4

Application to the NLW

We want now to prove existence of periodic solutions of the completely res-onant1 nonlinear wave equation NLW

utt − uxx = a(x)up

u(t, 0) = u(t, π) = 0(4.1)

where p ≥ 2, p ∈ N, for sets of frequencies ω close to 1 with asymptoticallyfull measure.

Normalizing the period t → ωt and rescaling the amplitude u → δu,δ > 0, as in Chapter 2, we look for 2π-periodic solutions of

ω2utt − uxx = εa(x)up

u(t, 0) = u(t, π) = 0(4.2)

where ε := δp−1.In order to apply a Nash-Moser scheme we look for solutions of (4.2) in

the space Xσ,s introduced in (2.54), formed by functions analytic in time(σ > 0) and valued in H1

0 (0, π).

We shall fix s > 1/2 so that Xσ,s is a multiplicative Banach algebra (seeAppendix 6.5), namely

‖u1u2‖σ,s ≤ C‖u1‖σ,s‖u2‖σ,s , ∀u1, u2 ∈ Xσ,s ,

1Under spatial periodic boundary conditions Bourgain [35] proved, when f = u3 +O(u4), existence of periodic solutions, branching off exact traveling wave solutions ofutt−uxx +u3 = 0. Recently Yuan [129] has proved, still for periodic boundary conditions,existence of certain types of quasi-periodic solutions. For Dirichlet boundary conditionsperiodic solutions have been proved by Gentile-Mastropietro-Procesi [62] for analytic f =u3 +O(u5), using the Lindsted series techniques.

82

CHAPTER 4. APPLICATION TO THE NLW 83

and the Nemitsky operator induced on Xσ,s by f(x, u) = a(x)up satisfies,taking a(x) ∈ H1(0, π),

‖f(u)‖σ,s ≤ C‖u‖pσ,s (4.3)

and it is in C∞(Xσ,s, Xσ,s) (actually it is analytic, see [107]).

Projecting (4.2) according to the orthogonal decomposition

Xσ,s = (V ∩Xσ,s)⊕ (W ∩Xσ,s)

where V is defined in (2.14) and

W :=w =

∑l∈Z

exp(ilt) wl(x) ∈ X0,s | w−l = wl

and

∫ π

0

wl(x) sin(lx) dx = 0, ∀l ∈ Z

yields (using that vtt = vxx)(ω2 − 1)

2∆v = εΠV f(v + w) bifurcation equation

Lωw = εΠWf(v + w) range equation(4.4)

where∆v := vxx + vtt , Lω := ω2∂tt − ∂xx . (4.5)

As explained at the end of Chapter 2, in order to find periodic solutions forpositive measure sets of frequencies ω we have to solve the range equation

Lωw − εΠWf(v + w) = 0

finding w = w(ε, v) via a Nash-Moser type iteration scheme.Two are the main difficulties in this program:

(i) Prove the invertibily of the linearized operators

h 7→ L(ε, v, w)[h] := Lωh− εΠWf′(v + w)h

obtained at any step of the Nash-Moser iteration (with a loss of ana-lyticity like in hypotheses (H2) of Theorem 3.2.1).

This invertibility property does not hold for all the values of the param-eters (ε, v). The eigenvalues λlj(ε, v), l ≥ 0, j ≥ 1 of L(ε, v, w) are, ingeneral, dense on R (as the spectrum of the unperturbed operator Lω) andthey depend very sensitively on (ε, v). The linear operator L(ε, v, w) shall


be invertible only outside the “resonant web” of parameters (ε, v) where atleast one λlj(ε, v) is zero. As a consequence the Nash-Moser iteration schemeshall converge to a solution w(ε, v) just on a complicated Cantor-like set, seeTheorem 4.2.1.

The presence of these “Cantor gaps” will have painful implications forsolving next the bifurcation equation, see remark 4.3.3.

The invertibility of the linear operator L(ε, v, w) is obtained by the newmethod developed in [25], which is different from the approach of Craig-Wayne [49] and Bourgain [33]-[34]. The method in [25] requires less regularityand it does not demand oddness assumptions on the nonlinearity.

The second main difficulty which appears is related to the fact that weare dealing with a completely resonant PDE and the solutions v ∈ V of thelinear equation (2.8) form an infinite dimensional space:

(ii) If v ∈ V ∩ Xσ,s then the solution w(ε, v) of the range equation, ob-tained through the Nash-Moser iteration will have a lower regularity,e.g. w(ε, v) ∈ Xσ/2,s. Therefore in solving next the bifurcation equa-tion for v ∈ V , the best estimate we can obtain is v ∈ V ∩ Xσ/2,s+2,which makes the scheme incoherent.

This second problem does not arise for non-resonant or partially resonantHamiltonian PDEs like utt−uxx+a1(x)u = a2(x)u

2+. . . where the bifurcationequation is finite dimensional [49]-[50].

4.0.1 The 0th order bifurcation equation

To deal with this infinite dimensional bifurcation problem we shall considerhere the simplest situation which occurs when

ΠV (a(x)vp) 6≡ 0 (4.6)

or, equivalently2

∃v ∈ V such that

∫Ω

a(x)vp+1 6= 0 . (4.7)

For definiteness we shall assume that ∃v ∈ V such that∫

Ωa(x)vp+1 > 0. In

this case we set the “frequency-amplitude” relation3

ω2 − 1

2= −ε

2Condition (4.6) is verified if and only if a(π − x) 6≡ (−1)pa(x), see Lemma 7.1 of [25].If (4.6) is violated, as for f(u) = au2, then the right hand side of equation (4.9) vanishes.In this case the correct 0th-order non-trivial bifurcation equation is not (4.9) and another“frequency-amplitude” relation is required, as in section 2.3.

3In the case ∃v ∈ V such that∫Ωa(x)vp+1 < 0 we have to set ω2 − 1 = 2ε.


and system (4.4) becomes−∆v = ΠV f(v + w) bifurcation equationLωw = εΠWf(v + w) range equation .

(4.8)

When ε = 0 system (4.8) reduces to w = 0 and to the “0th-order bifurcationequation”

−∆v = ΠV (a(x)vp) (4.9)

which is the Euler-Lagrange equation of the functional Φ0 : V 7→ R

Φ0(v) :=‖v‖2

H1

2−

∫Ω

a(x)vp+1

p+ 1(4.10)

where ‖v‖2H1 :=

∫Ωv2

t + v2x.

By the Mountain Pass theorem [7] (applied as in Chapter 2) there existsat least one nontrivial critical point of Φ0, i.e. a solution of (4.9).

Due to the fact that the range equation can be solved by w(ε, v) only fora Cantor-like set of parameters (ε, v) the solution of the bifurcation equationby means of variational methods is a very difficult stuff, see remark 4.3.3 and[26]. Therefore we shall confine here to the simplest situation when at leastone solution v ∈ V of (4.9) is non degenerate in V , i.e.

ker Φ′′0|V (v) = 0 (4.11)

(note that vt /∈ ker Φ′′0|V (v) because the functions of V are even in time). This

non-degeneracy condition is somehow analogous to the “Arnold condition”in KAM theory.

4.1 The finite dimensional reduction

To overcome the second difficulty (ii) we perform a further Lyapunov-Schmidtreduction to a finite dimensional bifurcation equation on a subspace of V ofdimension N depending only on v and the nonlinearity a(x)up.

Introducing the decomposition

V = V1 ⊕ V2

where V1 :=v ∈ V | v =

∑Nl=1 cos(lt)ul sin(lx) , ul ∈ R

V2 :=

v ∈ V | v =

∑l≥N+1 cos(lt)ul sin(lx) , ul ∈ R


and settingv := v1 + v2 , v1 ∈ V1, v2 ∈ V2 ,

system (4.4) is equivalent to−∆v1 = ΠV1f(v1 + v2 + w) (Q1)−∆v2 = ΠV2f(v1 + v2 + w) (Q2)Lωw = εΠWf(v1 + v2 + w) range equation

(4.12)

where ΠVi: Xσ,s 7→ Vi denotes the projector on Vi (i = 1, 2).

Our strategy to find solutions of system (4.12) is the following:

Solution of the (Q2)-equation. We solve first the (Q2)-equation obtain-ing v2 = v2(v1, w) ∈ V2∩Xσ,s+2 when w ∈ W ∩Xσ,s, provided we have chosenN := N large enough and 0 ≤ σ ≤ σ small enough, depending only on v andthe nonlinearity a(x)up.

Solution of the range equation. Next we solve the range equation, ob-taining w = w(ε, v1) ∈ W ∩Xσ/2,s by means of a Nash-Moser type iterationscheme for (ε, v1) belonging to some Cantor-like set B∞.

Solution of the (Q1)-equation. Finally, we shall solve the finite dimen-sional (Q1)-equation in section 4.3.

Remark 4.1.1 Under the non-degeneracy condition (4.11) we could alsosolve first the infinite dimensional bifurcation equation in (4.4) and next therange equation. We prefer to proceed as below because this scheme allows todeal also with degenerate cases, see [26].

4.1.1 Solution of the (Q2)-equation

By the regularizing property of the Laplacian

(−∆)−1 : V ∩Hk(Ω) 7→ V ∩Hk+2(Ω) , ∀k ≥ 0 ,

and a direct bootstrap argument, the solution v ∈ V of equation (4.9), sat-isfies4

v ∈ V⋂k≥0

Hk(Ω) ≡ V ∩ C∞(Ω) .

In particular‖v‖0,s < R (4.13)

for some finite constant R > 0.

The next Lemma is the crucial step to overcome the difficulty mentionedin (ii).

4Even if a(x) ∈ H1((0, π),R) only, because the projection ΠV has a regularizing effectin the variable x.


Proposition 4.1.1 (Solution of the (Q2)-equation) There exists

N := N(R, a(x), p) ∈ N

such that

∀ 0 ≤ σ ≤ σ :=ln 2

N, ∀ ‖v1‖0,s ≤ 2R , ∀‖w‖σ,s ≤ 1 ,

there exists a unique

v2 = v2(v1, w) ∈ V2 ∩Xσ,s+2

with ‖v2(v1, w)‖σ,s ≤ 1 which solves the (Q2)-equation. The function v2(·, ·)is C∞ and Dkv2 are bounded on bounded sets.

Furthermore, the following “projection property” holds

v2(ΠV1 v, 0) = ΠV2 v . (4.14)

If, in addition, w ∈ Xσ,s′ for some s′ ≥ s, then v2(v1, w) ∈ Xσ,s′+2 and

‖v2(v1, w)‖σ,s′+2 ≤ K(s′, ‖w‖σ,s′) . (4.15)

Proof. Fixed points of the nonlinear operatorN (v1, w, ·) : V2 7→ V2 definedby

N (v1, w, v2) := (−∆)−1ΠV2

(a(x)(v1 + w + v2)

p)

are solutions of the (Q2)-equation. Let

B :=v2 ∈ V2 ∩Xσ,s | ‖v2‖σ,s ≤ 1

.

We claim that there exists N := N(R, a(x), p) ∈ N such that ∀0 ≤ σ ≤ σ :=ln 2/N , ‖v1‖0,s ≤ 2R, ‖w‖σ,s ≤ 1, the operator N (v1, w, ·) is a contraction inB:

• (i) ‖v2‖σ,s ≤ 1 ⇒ ‖N (v1, w, v2)‖σ,s ≤ 1;

• (ii) v2, v2 ∈ B ⇒ ‖N (v1, w, v2)−N (v1, w, v2)‖σ,s ≤ 12‖v2 − v2‖σ,s.

Let us prove (i). ∀u ∈ Xσ,s,∥∥∥(−∆)−1ΠV2u∥∥∥

σ,s≤ C

(N + 1)2‖u‖σ,s


and so, ∀‖w‖σ,s ≤ 1, ‖v1‖0,s ≤ 2R,∥∥∥N (v1, w, v2)∥∥∥

σ,s≤ C

(N + 1)2

∥∥∥a(x)(v1 + v2 + w)p∥∥∥

σ,s

(4.3)

≤ C ′

(N + 1)2

(‖v1‖p

σ,s + ‖v2‖pσ,s + ‖w‖p

σ,s

)≤ C ′

(N + 1)2

((4R)p + ‖v2‖p

σ,s + 1)

(4.16)

for some C ′ := C ′(a(x), p), because

‖v1‖σ,s ≤ exp (σN)‖v1‖0,s ≤ 4R

for all 0 ≤ σ ≤ (ln 2/N).Therefore, by (4.16), defining

N :=√C ′((4R)p + 2) ,

we have ∀‖v1‖0,s ≤ 2R, ∀‖w‖σ,s ≤ 1, ∀0 ≤ σ ≤ σ := ln 2/N ,

‖v2‖σ,s ≤ 1 ⇒∥∥∥N (v1, w, v2)

∥∥∥σ,s≤ C ′

(N + 1)2

((4R)p + 1 + 1

)≤ 1

and (i) follows. Property (ii) can be proved similarly and the existence of aunique solution v2(v1, w) ∈ B follows by the Contraction Mapping Theorem.

Next v2(v1, w) ∈ Xσ,s+2 by a bootstrap argument using the regularizationproperty of the Laplacian.

The C∞ smoothness of v2(·, ·) follows by the Implicit Function Theoremand the C∞ smoothness of N , see [25] for details.

Let us prove (4.14). We may assume that N has been chosen so largethat

‖ΠV2 v‖0,s ≤1

2.

Since v solves equation (4.9), ΠV2 v solves the (Q2)-equation associated with(v1, w) = (ΠV1 v, 0) and ‖ΠV1 v‖0,s ≤ R by (4.13). Hence

ΠV2 v = N (ΠV1 v, 0,ΠV2 v)

is a fixed point of N (ΠV1 v, 0, ·) in B (for σ = 0) and, by its uniqueness,ΠV2 v = v2(ΠV1 v, 0).

To prove (4.15) we exploit that v2(v1, w) ∈ Xσ,s solves

v2 = (−∆)−1ΠV2

(a(x)(v1 + w + v2)

p).


Suppose that w ∈ Xσ,s′ for some s′ ≥ s. Then, by a direct bootstrap argu-ment, using the regularizing properties of (−∆)−1, the fact that v1 ∈ Xσ,r,∀r ≥ s (because V1 is finite dimensional), the Banach algebra property ofXσ,r, we derive that v2(v1, w) ∈ Xσ,s′+2 and ‖v2(v1, w)‖σ,s′+2 ≤ K(s′, ‖w‖σ,s′).

Remark 4.1.2 Proposition 4.1.1 solves the difficulty mentioned in (ii) be-cause v2(v1, w) has always the same smoothness of w ∈ Xσ,s (actually it gainsalso two derivatives, w ∈ Xσ,s+2). Therefore during the Nash-Moser iterationthe “tail” v2(v1, wn) will “adjust” its smoothness as the iterated wn (whichdecreases its analyticity).

Remark 4.1.3 We perform the finite dimensional reduction in B(2R;V1) :=v1 ∈ V1 | ‖v1‖0,s ≤ 2R because we expect solutions of the (Q1)-equationclose to ΠV1 v and ‖ΠV1 v‖0,s ≤ R. This will be observed in section 4.3.

We stress that in the sequel we shall consider as fixed the constants Nand σ which depend only on v and the nonlinearity a(x)up.

4.2 Solution of the range equation

By the previous section we are reduced to solve the range equation withv2 = v2(v1, w), namely

Lωw = εΠW Γ(v1, w) (4.17)

whereΓ(v1, w) := f

(v1 + w + v2(v1, w)

). (4.18)

The solution w = w(ε, v1) of the range equation (4.17) will be obtained bymeans of a Nash-Moser iteration scheme for (ε, v1) belonging to a Cantor-likeset of parameters.

Theorem 4.2.1 (Solution of the range-equation) For ε0 > 0 smallenough, there exists

w(·, ·) ∈ C∞(A0,W ∩Xσ/2,s

)where A0 := [0, ε0]×B(2R;V1), satisfying∥∥∥w(ε, v1)

∥∥∥σ/2,s

≤ Cε

γ,

∥∥∥Dkw(ε, v1)∥∥∥

σ/2,s≤ C(k) (4.19)


and the “large” Cantor set B∞ ⊂ A0 defined below, such that

∀(ε, v1) ∈ B∞ , w(ε, v1) solves the range equation (4.17) .

The Cantor set B∞ writes explicitly

B∞ :=

(ε, v1) ∈ A0 :∣∣∣ω(ε)l − j − ε

M(v1, w(ε, v1))

2j

∣∣∣ ≥ 2γ

(l + j)τ,∣∣∣ω(ε)l − j

∣∣∣ ≥ 2γ

(l + j)τ,∀j ≥ 1 , ∀l ≥ 1

3ε, l 6= j

⊂ A0

where ω(ε) =√

1− 2ε and

M(v1, w) :=1

|Ω|

∫Ω

(∂uf)(x, v1 + w + v2(v1, w)

)dxdt. (4.20)

As explained in the introduction, to get the invertibility of the linearizedoperators we have to excise values of the parameters (ε, v1). For this reason,it is more convenient to apply a slight modification of the analytic Nash-Moser scheme described in section 3.2, introducing, as in section 3.3, also asequence of “smoothing” operators via finite dimensional Fourier truncations,as in remark 3.3.1.

The advantage is that we avoid to invert the infinite dimensional oper-ator L(ε, v, w), replacing it by a sequence of finite dimensional matrices ofincreasing dimensions, imposing at each step a finite number of conditionson (ε, v1), see property (P3) below.

We consider the orthogonal splitting

W = W (n) ⊕W (n)⊥

whereW (n) =

w ∈ W

∣∣∣ w =∑|l|≤Ln

exp (ilt) wl(x)

(4.21)

W (n)⊥ =w ∈ W

∣∣∣ w =∑|l|>Ln

exp (ilt) wl(x)

(4.22)

andLn := L02

n

with L0 ∈ N large enough. We denote by

Pn : W → W (n) and P⊥n : W → W (n)⊥

the orthogonal projectors onto W (n) and W (n)⊥.

The existence of a solution of the range equation is based on the followingproperties (P1)-(P2)-(P3).


• (P1) (Regularity) Γ(·, ·) ∈ C∞(B(2R;V1) × B(1;W ∩ Xσ,s), Xσ,s

).

Moreover DkΓ are bounded on B(2R;V1)×B(1;W ∩Xσ,s).

(P1) is a consequence of the C∞-regularity of the Nemitsky operatorinduced by f(u) on Xσ,s and of the C∞-regularity of the map v2(·, ·) provedin Proposition 4.1.1.

• (P2) (Smoothing estimate) ∀ w ∈ W (n)⊥ ∩Xσ,s and ∀ 0 ≤ σ′ ≤ σ,‖w‖σ′,s ≤ exp (−Ln(σ − σ′))‖w‖σ,s.

The standard property (P2) follows from

‖w‖2σ′,s =

∑|l|>Ln

exp (2σ′|l|)(l2s + 1)‖wl‖2H1

=∑|l|>Ln

exp (−2(σ − σ′)|l|) exp (2σ|l|)(l2s + 1)‖wl‖2H1

≤ exp (−2(σ − σ′)Ln)‖w‖2σ,s .

The next property (P3) is the invertibility property of the linearized op-erator

Ln(ε, v1, w)[h] := Lωh− εPnΠWDwΓ(v1, w)[h] , (4.23)

∀h ∈ W (n).

• (P3) (Invertibility of Ln) Let γ ∈ (0, 1), τ ∈ (1, 2). Suppose ∃C1 > 0 such that∥∥∥(∂uf)(v1 + w + v2(v1, w))

∥∥∥σ,s+

2τ(τ−1)2−τ

≤ C1 . (4.24)

For ε0 > 0 small enough, if (ε, v1) belongs to

An(w) :=

(ε, v1) ∈ [0, ε0]×B(2R, V1) | |ω(ε)l − j| ≥ γ

(l + j)τ,∣∣∣ω(ε)l − j − ε

M(v1, w)

2j

∣∣∣ ≥ γ

(l + j)τ, ∀(j, l) ∈ N×N

l 6= j,1

3ε< l, l ≤ Ln, j ≤ 2Ln

(4.25)

then Ln(ε, v1, w) is invertible on W (n) and∥∥∥L−1n (ε, v1, w)[h]

∥∥∥σ,s≤ K

γ(Ln)τ−1‖h‖σ,s (4.26)

for some K > 0 (independent of n).


Remark 4.2.1 An estimate like (4.26) which holds for any n ≥ 0 is tanta-mount to say that the inverse of the infinite dimensional operator L(ε, v1, w)[h]:= Lωh−εΠWDwΓ(v1, w)[h] satisfies ‖L−1(ε, v1, w)[h]‖σ,s ≤ K ′γ−1‖h‖σ,s+τ−1,i.e. it loses (τ−1) derivatives. This can be seen via a “dyadic decomposition”using that Ln = L02

n, Ln+1 = 2Ln.

Property (P3) is the real core of the convergence proof and where theanalysis of the small divisors enters into play. Property (P3) is proved insection 4.4.

4.2.1 The Nash-Moser scheme

Following the proof of the “analytic” Nash-Moser Theorem 3.2.1 we firstdefine the sequence

σ0 := σ , σn+1 = σn − γn , ∀ n ≥ 0 , (4.27)

whereγn :=

γ0

n2 + 1

denotes the “loss of analyticity” at each step of the iteration, and we chooseγ0 > 0 small such that the “total loss of analyticity”∑

n≥0

γn =∑n≥0

γ0

n2 + 1≤ σ

2. (4.28)

By (4.27) and (4.28)

σn >σ

2> 0 , ∀n .

Proposition 4.2.1 (Induction) ∃ L0 := L0(γ, τ) > 0, ε0 := ε0(γ, τ) > 0,such that ∀εγ−1 < ε0, ∀n ≥ 0 there exists a solution wn = wn(ε, v1) ∈ W (n)

of the equation

(Pn) Lωwn − εPnΠW Γ(v1, wn) = 0

defined inductively for

(ε, v1) ∈ An ⊆ An−1 ⊆ . . . ⊆ A1 ⊆ A0 := [0, ε0]×B(2R;V1)

whereAn := An−1 ∩ An(wn−1) 6= ∅ (4.29)

and the set An(wn−1) is defined by (4.25).


It results wn =∑n

i=0 hi with hi ∈ W (i) satisfying

‖hi‖σi,s ≤ Cε

γexp(−χi) , χ =

3

2(4.30)

wn(·, ·) ∈ C∞(An,W(n)) and∥∥∥wn(ε, v1)

∥∥∥σn,s

≤ ε

γK1 ,

∥∥∥Dkwn(ε, v1)∥∥∥

σn,s≤ K1(k) (4.31)

for any k ≥ 0. Furthermore there exists wn(·, ·) ∈ C∞ (A0, W(n)) such that

wn(ε, v1) = wn(ε, v1) , ∀(ε, v1) ∈n⋂

i=0

Ai (4.32)

where, for some ν := ν(γ, τ) > 0 small enough,

Ai :=

(ε, v1) ∈ Ai | dist((ε, v1), ∂Ai) ≥2ν

L3i

⊂ Ai

∀i = 0, . . . , n. The function wn satisfies∥∥∥wn(ε, v1)∥∥∥

σn,s≤ ε

γC ,

∥∥∥Dkwn(ε, v1)∥∥∥

σn,s≤ C(k) (4.33)

for any k ≥ 0.

Proof. The proof proceeds by induction.

First step: initialization. Let L0 be given. For any

|ω − 1|L0 ≤1

2

then Lω|W (0) is invertible and

‖L−1ω h‖σ0,s ≤ 2‖h‖σ0,s , ∀h ∈ W (0)

(this is the finite dimensional situation like in Chapter 1). Indeed the eigen-values of Lω|W (0) are −ω2l2 + j2, ∀ 0 ≤ l ≤ L0, j ≥ 1, j 6= l, and

| − ω2l2 + j2| = | − ωl + j|(ωl + j)

≥(|j − l| − |ω − 1|L0

)(ωl + j) ≥ 1− 1

2.

Therefore, using Property (P1), ∀εγ−1 < ε1(γ, L0) small enough, ∀v1 ∈B(2R;V1), equation (P0) has a unique solution w0(ε, v1) ∈ W (0) satisfying

‖w0(ε, v1)‖σ0,s ≤ K0ε .


Moreover w0(·, ·) ∈ C∞(A0,W(0)) and ‖Dkw0(ε, v1)‖σ0,s ≤ C(k).

Second step: iteration. Suppose we have already defined a solution wn ∈W (n) of equation (Pn) satisfying the properties stated in the Proposition. Wewant to define

wn+1 = wn + hn+1 , hn+1 ∈ W (n+1)

as an exact solution of the equation

(Pn+1) Lωwn+1 − εPn+1ΠW Γ(v1, wn+1) = 0 .

For h ∈ W (n+1) we develop

Lω(wn + h)− εPn+1ΠW Γ(v1, wn + h) = Lωwn − εPn+1ΠW Γ(v1, wn)

+ Lωh− εPn+1ΠWDwΓ(v1, wn)[h]

+ R(h)

= rn + Ln+1(ε, v1, wn)[h] +R(h)

where, since wn solves equation (Pn),rn := Lωwn − εPn+1ΠW Γ(v1, wn) = −εP⊥

n Pn+1ΠW Γ(v1, wn)

R(h) := −εPn+1ΠW

(Γ(v1, wn + h)− Γ(v1, wn)−DwΓ(v1, wn)[h]

).

Hence equation (Pn+1) amounts to solve

Ln+1(ε, v1, wn)[h] = −rn −R(h) . (4.34)

The rest rn is “super-exponentially” small because

‖rn‖σn+1,s

(P2)

≤ ε C exp (−Lnγn)∥∥∥Pn+1ΠW Γ(v1, wn)

∥∥∥σn,s

≤ ε C ′ exp (−Lnγn)∥∥∥Γ(v1, wn)

∥∥∥σn,s

(P1)

≤ ε C ′′ exp (−Lnγn) (4.35)

using also (4.31).The term R(h) is “quadratic” in h, since, by (P1) and (4.31),‖R(h)‖σn+1,s ≤ Cε ‖h‖2

σn+1,s

‖R(h)−R(h′)‖σn+1,s ≤ Cε (‖h‖σn+1,s + ‖h′‖σn+1,s) ‖h− h′‖σn+1,s(4.36)

for all ‖h‖σn+1,s, ‖h′‖σn+1,s small enough.


Lemma 4.2.1 There is C1 > 0 such that ∀‖v1‖0,s ≤ 2R, ∀n ≥ 0∥∥∥(∂uf)(v1 + wn + v2(v1, wn)

)∥∥∥σn+1,s+

2τ(τ−1)2−τ

≤ C1 . (4.37)

Proof. Since wn =∑n

i=0 hi

‖wn‖σn+1,s+2τ(τ−1)

2−τ

≤n∑

i=0

‖hi‖σi,s+2τ(τ−1)

2−τ

(∗)≤

n∑i=0

C(τ)‖hi‖σi,s

(σi − σn+1)2τ(τ−1)

2−τ

(4.30)

≤ Cε

γ

n∑i=0

exp (−χi)

γ2τ(τ−1)/(2−τ)i

≤ Cε

γ

+∞∑i=0

exp (−χi)(1 + i2

γ0

) 2τ(τ−1)2−τ

=ε

γK

since σi − σn+1 ≥ γi := γ0/(1 + i2), ∀i = 0, . . . , n and using in (∗) theelementary inequality

maxk≥1

kα exp−(σi − σn+1)k ≤C(α)

(σi − σn+1)α.

By (4.15) also‖v2(v1, wn)‖

σn+1,s+2τ(τ−1)

2−τ

≤ K ′

and (4.37) holds by the Banach algebra property of Xσ,s+2τ(τ−1)/(2−τ).

By the previous Lemma, for

(ε, v1) ∈ An+1 := An ∩ An+1(wn) (4.38)

property (P3) applies: the linear operator Ln+1(ε, v1, wn) is invertible and∥∥∥Ln+1(ε, v1, wn)−1∥∥∥

σn+1,s≤ C

γ(Ln+1)

τ−1, ∀(ε, v1) ∈ An+1 . (4.39)

Remark 4.2.2 Note that An ≡ . . . ≡ A1 ≡ A0 if L02n < 1/3ε.

Remark 4.2.3 To show that A∞ := ∩n≥0An is not empty but, actually, itis a “large” set, the key ingredient to exploit is again the “super-exponential”convergence (4.30) of the approximate solutions wn. The method we choosehere is the following: we define a map w(·, ·) on the whole set A0 (see (4.58))and a set B∞ ⊂ A∞ depending only on w, see Lemma 4.2.5. In Proposition4.3.1 we prove that B∞ -and so A∞- is a “large” set. An advantage ofthis approach is the very explicit expression of the Cantor set B∞. This isespecially exploited in the fine measure estimate of [26].


By (4.34), equation (Pn+1) is equivalent to find a fixed point

h = G(ε, v1, wn, h) , h ∈ W (n+1) , (4.40)

of the nonlinear operator

G(ε, v1, wn, h) := −Ln+1(ε, v1, wn)−1(rn +R(h)

).

Lemma 4.2.2 (Contraction) There exist L0(γ, τ) > 0, ε0(L0, γ, τ) > 0,such that, ∀εγ−1 < ε0, the operator G(ε, v1, wn, ·) is, for any n ≥ 0, a con-traction in the ball

B(ρn+1;W(n+1)) :=

h ∈ W (n+1) | ‖h‖σn+1,s ≤ ρn+1 :=

ε

γexp (−χn+1)

.

Proof. We first prove that G(ε, v1, wn, ·) maps the ball B(ρn+1;W(n+1))

into itself.∥∥∥G(ε, v1, wn, h)∥∥∥

σn+1,s=

∥∥∥Ln+1(ε, v1, wn)−1(rn +R(h)

)∥∥∥σn+1,s

(4.39)

≤ C

γ(Ln+1)

τ−1(‖rn‖σn+1,s + ‖R(h)‖σn+1,s

)≤ C ′

γ(Ln+1)

τ−1(ε exp (−Lnγn) + ε ‖h‖2

σn+1,s

)by (4.35) and (4.36). Therefore, if ‖h‖σn+1,s ≤ ρn+1, then∥∥∥G(ε, v1, wn, h)

∥∥∥σn+1,s

≤ C ′

γ(Ln+1)

τ−1ε(

exp (−Lnγn) + ρ2n+1

)≤ ρn+1

provided that

C ′ ε

γ(Ln+1)

τ−1 exp (−Lnγn) ≤ ρn+1

2(4.41)

and

C ′ ε

γ(Ln+1)

τ−1ρn+1 ≤1

2. (4.42)

(4.41) becomes, for ρn+1 := εγ−1 exp (−χn+1),

C ′(Ln+1)τ−1 exp (−Lnγn) ≤ 1

2exp (−χn+1)

which, for Ln := L02n, γn := γ0/(1+n2) and L0 := L0(γ, τ) > 0 large enough,

is satisfied ∀n ≥ 0. (4.42) becomes

C ′ ε2

γ2

(L0(γ, τ)2

n+1)τ−1

exp (−χn+1) ≤ 1

2


which is satisfied for εγ−1 ≤ ε0(L0, γ, τ) small enough, ∀n ≥ 0.With similar estimates, by (4.36), we get ∀h, h′ ∈ B(ρn+1;W

(n+1)),∥∥∥G(ε, v1, wn, h′)− G(ε, v1, wn, h)

∥∥∥σn+1,s

≤ 1

2‖h− h′‖σn+1,s

again for L0 large enough and εγ−1 ≤ ε0(L0, γ, τ) small enough, uniformlyin n.

By the Contraction Mapping Theorem we deduce the existence of a uniquehn+1 ∈ W (n+1) solving (4.40) and satisfying∥∥∥hn+1

∥∥∥σn+1,s

≤ ρn+1 =ε

γexp (−χn+1) . (4.43)

Furthermore, by an elementary bootstrap, since L−1n+1(ε, v1, wn) gains two

spatial derivatives, each hn(t, ·) ∈ H3(0, π) and wn is a classical solution ofequation (Pn). This completes the existence proof.

Finally wn+1 =∑n+1

i=0 hi, hi ∈ W (i), satisfies

‖wn+1,s‖σn+1,s ≤n+1∑i=0

‖hi‖σi,s ≤+∞∑i=0

Cε

γexp (−χi) = K1

ε

γ

and the left estimate of (4.31) holds.

Remark 4.2.4 A difference with respect to the “quadratic” Nash-Moser (plusthe truncation) scheme in [49], is that hn(ε, v1) is found as an exact solu-tion of equation (Pn), and not just as a solution of the linearized equationrn + Ln+1(ε, v1, wn)[h] = 0. This is more convenient to prove the regularityof hn(·, ·) in the next Lemma.

Lemma 4.2.3 (Estimates for the derivatives) hn(·, ·) ∈ C∞(An,W(n)),

∀n ≥ 0, and ∥∥∥Dkhn(ε, v1)∥∥∥

σn,s≤ [K1(k, χ)]n+1 exp(−χn) (4.44)

for some χ ∈ (1, χ).As a consequence wn(·, ·) ∈ C∞(An,W

(n)) and (4.31) holds.

Proof. By the first step in the proof of Proposition 4.2.1, h0 = w0 ∈C∞(A0,W

(0)) and ‖Dkw0(ε, v1)‖σ0,s ≤ C(k).Next, assume by induction that hn(·, ·) ∈ C∞(An,W

(n)). We shall provethat hn+1(·, ·) ∈ C∞(An+1,W

(n+1)).


hn+1(ε, v1) is a solution of

(Pn+1) Un+1

(ε, v1, hn+1(ε, v1)

)= 0

whereUn+1(ε, v1, h) := Lω(wn + h)− εPn+1ΠW Γ(v1, wn + h) .

We claim that DhUn+1(ε, v1, hn+1) = Ln+1(ε, v1, wn+1) (= Ln+1(wn+1) forshort) is invertible and∥∥∥(

DhUn+1(ε, v1, hn+1))−1∥∥∥

σn+1,s≤ C ′

γ(Ln+1)

τ−1 . (4.45)

By the Implicit Function Theorem, hn+1(·, ·) ∈ C∞(An+1,W(n+1)). To prove

(4.45) we first recall that Ln+1(wn) is invertible and (4.39) holds. We have∥∥∥Ln+1(wn+1)− Ln+1(wn)∥∥∥

σn+1,s=

∥∥∥εPn+1ΠW

(DwΓ(v1, wn+1)

−DwΓ(v1, wn))∥∥∥

σn+1,s

(P1)

≤ Cε ‖hn+1‖σn+1,s

(4.43)

≤ Cε2

γexp(−χn+1) . (4.46)

Therefore

Ln+1(wn+1) = Ln+1(wn)[Id+Ln+1(wn)−1

(Ln+1(wn+1)−Ln+1(wn)

)](4.47)

is invertible whenever (recall (4.39) and (4.46))∥∥∥Ln+1(wn)−1(Ln+1(wn+1)− Ln+1(wn)

)∥∥∥σn+1,s

≤ C

γ(Ln+1)

τ−1 ε2

γexp(−χn+1)

<1

2(4.48)

provided that εγ−1 is small enough (note that (Ln+1)τ−1 = (L02

n+1)τ−1 <<exp(χn+1) for n large). Furthermore, by (4.47), (4.39), (4.48), estimate (4.45)holds.

We now prove in detail estimate (4.44) for k = 1. Let us call λ := (ε, v1) ∈An+1. Differentiating equation (Pn+1) with respect to λ we obtain

(P ′n+1) Ln+1(ε, v1, wn+1)

[∂λhn+1(ε, v1)

]= −(∂λUn+1)

(ε, v1, hn+1(ε, v1)

)


and therefore∥∥∥∂λhn+1

∥∥∥σn+1,s

(4.45)

≤ C

γ(Ln+1)

τ−1∥∥∥(∂λUn+1)(ε, v1, hn+1)

∥∥∥σn+1,s

. (4.49)

Since wn solves equation (Pn), that is Lωwn = εPnΠW Γ(v1, wn),

Un+1(ε, v1, h) = Lωh+ ε(PnΠW Γ(v1, wn)− Pn+1ΠW Γ(v1, wn + h))

and we can write

(∂λUn+1)(ε, v1, h) = (∂λUn+1)(ε, v1, 0) + r(ε, v1, h) (4.50)

where

(∂λUn+1)(ε, v1, 0) = (Pn − Pn+1)ΠW∂λ[εΓ(v1, wn(ε, v1)]

= −εP⊥n Pn+1ΠW

[(∂λΓ)(v1, wn) + (∂wΓ)(v1, wn)[∂λwn]

]−∂λ(ε)P

⊥n Pn+1Γ(v1, wn) (4.51)

and

r(ε, v1, h) := εPn+1ΠW

[(∂λΓ)(v1, wn)− (∂λΓ)(v1, wn + h)

]+ εPn+1ΠW

[(∂wΓ)(v1, wn)− (∂wΓ)(v1, wn + h)

][∂λwn]

+ ∂λ(Lω(ε)h) + ∂λ(ε)Pn+1ΠW (Γ(v1, wn)− Γ(v1, wn + h))

with ∂λ(Lω(ε)h) = 0, ∂λ(ε) = 0 if λ 6= ε and ∂ε(Lω(ε)h) = −2htt (recall thatω2 = 1− 2ε). Therefore∥∥∥r(ε, v1, h)

∥∥∥σn+1,s

≤ Cε ‖h‖σn+1,s

(1 + ‖∂λwn‖σn+1,s

)+ CL2

n+1‖h‖σn+1,s.

(4.52)We now estimate (∂λUn+1)(ε, v1, 0). By (4.51)∥∥∥(∂λUn+1)(ε, v1, 0)

∥∥∥σn+1,s

(P2)

≤ exp(−Lnγn)[ε∥∥∥(∂λΓ)(v1, wn) +

(∂wΓ)(v1, wn)[∂λwn]∥∥∥

σn,s+

∥∥∥Γ(v1, wn)∥∥∥

σn,s

](P1)

≤ C exp(−Lnγn)(1 + ‖∂λwn‖σn,s

). (4.53)

Combining (4.49), (4.50), (4.52), (4.53) and (4.43)∥∥∥∂λhn+1

∥∥∥σn+1,s

≤ C

γ(Ln+1)

τ+1( εγ

exp(−χn+1) + exp(−Lnγn))

×(1 + ‖∂λwn‖σn,s

)≤ C(χ) exp(−χn+1)

(1 +

n∑i=0

‖∂λhi‖σi,s

)(4.54)


for any χ ∈ (1, χ). Reasoning by induction (4.54) implies (4.44) for k = 1.

We now define, by interpolation, say, a C∞-extension wn(ε, v1) of wn(ε, v1).

Lemma 4.2.4 (Whitney C∞ Extension) Let

Ai :=

(ε, v1) ∈ Ai | dist((ε, v1), ∂Ai) ≥2ν

L3i

⊂ Ai

where ν := ν(γ, τ) > 0 is some small constant specified later, see Lemma

4.2.5. There exists hi ∈ C∞(A0,W(i)) satisfying

‖hi‖σi,s ≤εK

γexp(−χi) (4.55)

for some χ ∈ (1, χ), such that

wn :=n∑

i=0

hi ∈ C∞(A0,W(n))

satisfies (4.32) and (4.33).

Proof. Let ϕ : R × V1 → R+ be a C∞ function supported in the openball B(0, 1) of center 0 and radius 1 with

∫R×V1

ϕ dµ = 1. Here µ is the pull

back of the Lebesgue measure in RN+1 onto R× V1.Let ϕi : R× V1 → R+ be the “mollifier”

ϕi(λ) :=(L3

i

ν

)N+1

ϕ(L3

i

νλ)

(here λ := (ε, v1)) which is a C∞ function satisfying

suppϕi ⊂ B(0,

ν

L3i

)and

∫R×V1

ϕi dµ = 1 . (4.56)

Next we define ψi : R× V1 → R+ as

ψi(λ) :=(ϕi ∗ χA∗i

)(λ) =

∫R×V1

ϕi(λ− η)χA∗i(η) dµ(η)

where χA∗iis the characteristic function of the set

A∗i :=

λ = (ε, v1) ∈ Ai | dist(λ, ∂Ai) ≥

ν

L3i

⊂ Ai ,

namely χA∗i(λ) := 1 if λ ∈ A∗

i , and χA∗i(λ) := 0 if λ /∈ A∗

i .


The function ψi is C∞ and, ∀k ∈ N, ∀λ ∈ R× V1,

|Dkψi(λ)| =∣∣∣ ∫

R×V1

Dkϕi(λ− η)χA∗i(η) dµ(η)

∣∣∣≤

∫R×V1

∣∣∣(L3i

ν

)k(L3i

ν

)N+1

(Dkϕ)(L3

i

ν(λ− η)

)∣∣∣ dµ(η)

=(L3

i

ν

)k∫

R×V1

|Dkϕ| dµ =(L3

i

ν

)k

C(k) (4.57)

where C(k) :=∫R×V1

|Dkϕ| dµ. Furthermore, by (4.56) and the definition of

A∗i and Ai,

0 ≤ ψi(λ) ≤ 1 , suppψi ⊂ intAi and ψi(λ) = 1 if λ ∈ Ai .

Finally we can define

w0(λ) := w0(λ) , wi+1(λ) := wi(λ) + hi+1(λ) ∈ W (i+1)

where

hi+1(λ) :=

ψi+1(λ)hi+1(λ) if λ ∈ Ai+1

0 if λ /∈ Ai+1

is in C∞(A0,W(i+1)) because supp ψi+1 ⊂ int Ai+1 and, by Lemma 4.2.3,

hi+1 ∈ C∞(Ai+1,W(i+1)).

Therefore we have wn(λ) :=∑n

i=0 hi(λ), wn ∈ C∞(A0,W(n)) and (4.32)

holds. By the bounds (4.57) and (4.44) we get (4.55) and∥∥∥Dkhi(λ)∥∥∥

σi,s≤ C(k, χ)i

(L3i+1

ν

)k

exp(−χi) ≤ K(k)

νkexp(−χi)

for some 1 < χ < χ and some positive constant K(k) large enough. Theestimates (4.33) follow.

The proof of Proposition 4.2.1 is complete.

Proof of Theorem 4.2.1. The sequence wn (with all its derivatives)converges uniformly in A0 for the norm ‖ ‖σ/2,s, to some function

w(ε, v1) ∈ C∞(A0,W ∩Xσ/2,s) (4.58)

which, by (4.33), satisfies (4.19) and∥∥∥w(ε, v1)− wn(ε, v1)∥∥∥

σ/2,s

(4.55)

≤ Cε

γexp(−χn) . (4.59)


Remark 4.2.5 If (ε, v1) /∈ A∞ := ∩n≥0An then w(ε, v1) =∑

n≥1 hn(ε, v1) isa finite sum, see the proof of Lemma 4.2.4.

To get the explicit Cantor set B∞, let consider

Bn :=

(ε, v1) ∈ A0 |∣∣∣ω(ε)l − j − ε

M(v1, w(ε, v1))

2j

∣∣∣ ≥ 2γ

(l + j)τ,

|ω(ε)l − j| ≥ 2γ

(l + j)τ, ∀(j, l) ∈ N×N such that

l 6= j ,1

3ε< l, l ≤ Ln, j ≤ 2Ln

.

Note that Bn does not depend on the approximate solution wn but only onthe fixed function w.

Lemma 4.2.5 If νγ−1 > 0 and εγ−1 are small enough, then

Bn ⊂ An, ∀n ≥ 0 .

Proof. We shall prove the Lemma by induction. First B0 ⊂ A0. Supposenext that Bn ⊂ An holds. In order to prove that Bn+1 ⊂ An+1, take any(ε, v1) ∈ Bn+1. We have to justify that the ball

B((ε, v1),

2ν

L3n+1

)⊂ An+1 . (4.60)

First, since Bn+1 ⊂ Bn ⊂ An, then (ε, v1) ∈ An. Hence, since Ln+1 > Ln,

B((ε, v1),

2ν

L3n+1

)⊂ An .

Let (ε′, v′1) ∈ B((ε, v1), 2ν/L3n+1). Since (ε, v1) ∈ An we have wn(ε, v1) =

wn(ε, v1). Moreover by (4.31) ‖Dwn‖σ/2,s ≤ C. By (4.59)∥∥∥wn(ε′, v′1)− w(ε, v1)∥∥∥

σ/2,s≤

∥∥∥wn(ε′, v′1)− wn(ε, v1)∥∥∥

σ/2,s

+∥∥∥wn(ε, v1)− w(ε, v1)

∥∥∥σ/2,s

≤ 2νC

L3n+1

+Cε

γexp(−χn) .


Hence, by the definition of Bn, setting ω′ :=√

1− 2ε′∣∣∣ω′l − j − ε′M(v′1, wn(ε′, v′1))

2j

∣∣∣ ≥∣∣∣ωl − j − ε

M(v1, w(ε, v1))

2j

∣∣∣− lCν

L3n+1

− Cεν

L3n+1

− Cε2

γexp(−χn)

≥ 2γ

(l + j)τ− Cν

L2n+1

− Cε2

γexp(−χn)

≥ γ

(l + j)τ

for all 1/3ε < l < Ln+1, l 6= j, j ≤ 2Ln+1, whenever

γ

(3Ln+1)τ≥ C

( ν

L2n+1

+ε2

γexp(−χn)

). (4.61)

(4.61) holds true, for εγ−1 and νγ−1 small, for all n ≥ 0, because τ < 2 andlimn→∞ Lτ

n+1 exp(−χn) = 0. Then (4.60) is proved.

Corollary 4.2.1 (Solution of the range equation) If

(ε, v1) ∈ B∞ :=⋂n≥0

Bn ⊂⋂n≥0

An ⊂⋂n≥0

An ,

then w(ε, v1) ∈ Xσ/2,s is a solution of the range equation (4.17).

Proof. If (ε, v1) ∈ B∞ then (ε, v1) ∈ ∩ni=0Ai, ∀n ≥ 0, and so wn(ε, v1) =

wn(ε, v1) solves equation (Pn)

Lωwn = εPnΠW Γ(v1, wn) = εΠW Γ(v1, wn)− εP⊥n ΠW Γ(v1, wn) . (4.62)

Since∥∥∥P⊥n ΠW Γ(v1, wn)

∥∥∥σ/2,s

(P2)

≤ C exp(− Ln(σn − (σ/2))

)∥∥∥Γ(v1, wn)∥∥∥

σn,s

(P1)

≤ C exp(− γ0

L02n

(n2 + 1)

)being σn− (σ/2) ≥ γn := γ0/(n

2 +1), the right hand side of (4.62) convergesin Xσ/2,s to εΠW Γ(v1, w(ε, v1)).

Next subtracting the equations (Pi) and (Pi−1) we derive

Lωhi = εPi−1ΠW

(Γ(v1, wi)− Γ(v1, wi−1)

)+ εP⊥

i−1PiΠW Γ(v1, wi)


and so∥∥∥Lωw − Lωwn

∥∥∥σ/2,s

≤∑i>n

‖Lωhi‖σ/2,s

≤ εC∑i>n

‖hi‖σ/2,s + exp(− Li(σi − (σ/2))

)≤ εC

∑i>n

exp(−χi) + exp(− Li(σi − (σ/2))

)which tends to 0 for n→ +∞. Hence the left hand side in (4.62) convergesin Xσ/2,s to Lωw and the corollary is proved.

What just remains to prove is that

B∞ :=⋂n≥0

Bn 6= ∅ ,

and that, actually, B∞ is a “large” set.

4.3 Solution of the (Q1)-equation

Once the (Q2)-equation and the range equation are solved (with “gaps” forthe latter) the last step is to find solutions of the finite dimensional (Q1)-equation

−∆v1 = ΠV1G(ε, v1) (4.63)

whereG(ε, v1) := f

(v1 + w(ε, v1) + v2(v1, w(ε, v1))

)such that (ε, v1) belong to the Cantor set B∞.

For ε = 0 the (Q1)-equation (4.63) reduces to

−∆v1 = ΠV1G(0, v1) = ΠV1

(a(x)(v1 + v2(v1, 0))p

). (4.64)

Lemma 4.3.1 v1 := ΠV1 v ∈ B(R;V1) is a non-degenerate solution of (4.64).

Proof. By (4.14) and since v solves (4.9), v1 solves (4.64). Assume thath1 ∈ V1 is a solution of the linearized equation at v1 of (4.64), i.e.

−∆h1 = ΠV1

(pa(x)(v1 + v2(v1, 0))p−1(h1 + h2)

)(4.65)


where h2 := Dv1v2(v1, 0)[h1] ∈ V2. By the definition of v2

−∆v2(v1, 0) = ΠV2

(a(x)(v1 + v2(v1, 0))p

), ∀v1 ∈ B(2R, V1) ,

whence, differentiating at v1,

−∆h2 = ΠV2

(pa(x)(v1 + v2(v1, 0))p−1(h1 + h2)

). (4.66)

Summing (4.65) and (4.66), h = h1 + h2 ∈ V is a solution of Φ′′(v)[h] = 0.By (4.11), h = 0 and hence h1 = 0.

By Lemma 4.3.1 and since

(ε, v1) 7→ −∆v1 − ΠV1G(ε, v1)

is in C∞([0, ε0) × B(2R, V1);V1), we can solve the equation (4.63) by theImplicit Function Theorem, finding a C∞ path

ε 7→ v1(ε) ∈ B(2R, V1)

of solutions of (4.63) with v1(0) = v1.By Theorem 4.2.1 (and Proposition 4.1.1), the function

u(ε) := ε1

p−1

[v1(ε) + v2

(v1(ε), w(ε, v1(ε))

)+ w(ε, v1(ε))

]∈ Xσ/2,s (4.67)

is a solution of equation (4.1) if ε belongs to the Cantor-like set

C :=ε ∈ [0, ε0) | (ε, v1(ε)) ∈ B∞

.

Remark 4.3.1 Actually u is a classical solution because

uxx(t, x) = ω2utt(t, x)− f(x, u(t, x)) ∈ H10 (0, π)

∀t ∈ T and so u(t, ·) ∈ H3(0, π) ⊂ C2(0, π).

The smoothness of v1(·) actually implies that the set C has full densityat the origin, namely

limη→0+

|C ∩ [0, η)|η

= 1 . (4.68)

Geometrically this estimate exploits the structure of the Cantor set B∞ andthat the curve of solutions ε 7→ v1(ε) cross transversally B∞ (it is a graph).

Proposition 4.3.1 The Cantor set C verifies the measure estimate (4.68).


ε

ε

v (ε)

0

1

Figure 4.1: The Cantor set where the range equation is solved.

Proof. Let 0 < η < ε0. We have to estimate the complementary set

Cc ∩ (0, η) =ε ∈ (0, η)

∣∣∣ ∣∣∣ω(ε)l − j − εm(ε)

2j

∣∣∣ < 2γ

(l + j)τ

or∣∣∣ω(ε)l − j

∣∣∣ < 2γ

(l + j)τfor some l >

1

3ε, l 6= j

where

m(ε) := M(v1(ε), w(ε, v1(ε)))

is a function in C1([0, ε0),R) and we recall that the function M(·, ·) is definedin (4.20).

We can writeCc ∩ (0, η) =

⋃(l,j)∈IR

(Rl,j ∪ Sl,j

)(4.69)

where

Sl,j :=ε ∈ (0, η) |

∣∣∣ω(ε)l − j − εm(ε)

2j

∣∣∣ < 2γ

(l + j)τ

Rl,j :=

ε ∈ (0, η) |

∣∣∣ω(ε)l − j∣∣∣ < 2γ

(l + j)τ

and

IR :=

(l, j) | l, j > 1

3η, l 6= j,

j

l∈ [1− 4η, 1 + 4η]


(note indeed that Rj,l = Sj,l = ∅ unless j/l ∈ [1− 4η, 1 + 4η]).By proposition 2.5.1 we already know that∣∣∣ ⋃

(l,j)∈IR

Rl,j

∣∣∣ = o(η)

for η → 0. Here we show that also∣∣∣ ⋃(l,j)∈IR

Sl,j

∣∣∣ = o(η) .

This is a consequence of the measure estimate

|Slj| = O( γ

lτ+1

)(4.70)

which follows by the C1-smoothness of m(ε), because, setting

flj(ε) := ω(ε)l − j − εm(ε)

2j,

we have

|∂εflj(ε)| =∣∣∣ −l√

1− 2ε− m(ε) + εm′(ε)

2j

∣∣∣ ≥ 1

2l

for any ε ∈ (0, η) and η small enough.Since, for a given l, the number of j for which (l, j) ∈ IR is O(ηl),

|Cc ∩ (0, η)| ≤∑

(l,j)∈IR

|Sj,l| ≤ C∑

l≥1/3η

ηl × γ

lτ+1≤ Cγητ

which concludes the proof.

The non-degeneracy condition (4.11) on the nonlinearity can be verifiedon several examples5, yielding the following theorem

Theorem 4.3.1 ([25]) Let

f(x, u) =

a2u2 a2 6= 0

a3(x)u3 〈a3〉 := (1/π)

∫ π

0a3(x) 6= 0

a4u4 a4 6= 0 .

Then, s > 1/2 being given, there exist ε0 > 0, σ > 0 and a C∞-curve[0, ε0) 3 ε 7→ u(ε) ∈ Xσ/2,s with the following properties:

5In the cases f = a2u2, a4u

4 the condition (4.6) is violated and, as in section 2.3,the correct 0-th order bifurcation equation is the Euler-Lagrange equation of 1

2‖v‖H21−

a2p

∫ΩvpL−1vp, p = 2, 4.


• (i)∥∥∥u(ε)− ε

1p−1 v

∥∥∥σ/2,s

= O(ε2

p−1 ) for some v ∈ V ∩Xσ,s, v 6= 0;

• (ii) There exists a Cantor set C ⊂ [0, ε0) of asymptotically full measure,i.e. satisfying (4.68), such that, ∀ ε ∈ C, u(ε)(ω(ε)t, x) is a 2π/ω(ε)-periodic classical solution of (4.1) with frequency respectively

ω =

√

1− 2ε2√1− 2εsign〈a3〉√1− 2ε2 .

Remark 4.3.2 (Multiplicity) To get multiplicity of periodic solutions wecan look for 2π/n time-periodic solutions of (4.2) as well. We can prove[25]-[11] that there exists n0 ∈ N and a Cantor-like set C of asymptoticallyfull measure, such that ∀ε ∈ C, equation (4.1) possesses 2π/(nω(ε))-periodic,geometrically distinct, solutions un for every

n0 ≤ n ≤ N(ε) where limε→0

N(ε) = +∞ .

Each un is in particular 2π/ω(ε)-periodic.

Remark 4.3.3 (Weaken the non-degeneracy condition) We could solvethe bifurcation equation with variational methods as in Chapter 2, proving theexistence of Mountain Pass solutions v1(ε) of the (Q1)-equation ∀ε > 0 small.However, since the section Eε := v1 | (ε, v1) ∈ B∞ has “gaps” (except forω in the zero measure set Wγ of Chapter 2), the big difficulty is to provethat (ε, v1(ε)) ∈ B∞ for a large set of ε’s. Although B∞ is in some sensea “large” set, this property is not obvious: the complementary of B∞ is ac-tually arcwise connected! And the critical point v1(ε) could depend (withouta non-degeneracy condition) in a highly irregular way on ε. Results in thisdirection have been obtained in [26] using variational methods for parameterdepending families of functionals possessing the Mountain Pass geometry.

4.4 The linearized operator

We prove in this section the key property (P3) on the inversion of the linearoperator Ln(ε, v1, w) defined in (4.23).

4.4.1 Decomposition of Ln(ε, v1, w)

We can write, recalling (4.18),


Ln(ε, v1, w)[h] := Lωh− εPnΠWDwΓ(v1, w)[h]

= Lωh− εPnΠW ((∂uf)(v1 + w + v2(v1, w))(h+ ∂wv2[h]))

= Lωh− εPnΠW (b(t, x)h)− εPnΠW

(b(t, x)∂wv2[h]

)(4.71)

where, for brevity,

b(t, x) := (∂uf)(x, v1 + w + v2(v1, w)) . (4.72)

To invert Ln it is convenient to perform a Fourier expansion and represent theoperator Ln as a matrix. The main difference with respect to the procedureof Craig-Wayne-Bourgain [49]-[34] is that we shall develop Ln only in time-Fourier basis and not in the time and spatial Fourier basis eilt sin(jx), l ∈ Z,j ≥ 1. This is more convenient to deal with nonlinearities f(x, u) with finiteregularity in x and without oddness assumptions.

In time-Fourier basis the operator6

Lω := −ω2∂tt + ∂xx

is represented by the diagonal matrix (of spatial operators)

Lω ≡

ω2L2

n + ∂xx 0 . . . 0 0. . . . . . . . .0 0 ω2k2 + ∂xx 0. . . . . .0 0 0 ω2L2

n + ∂xx

because

h =∑|k|≤Ln

exp (ikt)hk(x) ⇒ Lωh =∑|k|≤Ln

exp (ikt)(ω2k2 + ∂xx)hk(x) .

The operator h 7→ PnΠW (b(t, x) h) is the composition of the multiplicationoperator for the function

b(t, x) =∑l∈Z

exp(ilt)bl(x)

with the projectors ΠW and Pn.

6In order to have the positive signs below it is the opposite of Lω defined in (4.5). Asthere is no risk of confusion we denote it in the same way.


The projector ΠW : Xσ,s 7→ W is given, ∀ h =∑

k∈Z exp (ikt)hk ∈ Xσ,s, by

(ΠWh)(t, x) =∑k∈Z

exp (ikt)(πkhk)(x) (4.73)

whereπk : H1

0 ((0, π);R) 7→ 〈sin kx〉⊥

is the L2-orthogonal projector.

∀h =∑

|k|≤Lnexp(ikt)hk we have

PnΠW (b(t, x) h) =∑

|k|,|l|≤Ln

exp(ilt)πl(bl−k(x)hk(x))

so that it is represented by the matrix which has in row l and column k thespace operator πl(bl−k(x)[· ]), namely

PnΠW (b ·) ≡

π−Lnb0 π−Lnb−1 . . . π−Lnb−2Ln

. . .πlb0 . . . πlbl−k

. . . . . .

. . . . . . . . .πLnb2Ln . . . πLnb1 πLnb0

As usual, in Fourier basis, the multiplication operator is described by a“Toepliz matrix” (constant terms along the parallels to the diagonal). Notealso that the zero time Fourier coefficient

b0(x) =1

2π

∫ 2π

0

b(t, x) dt

is the time average of b(t, x).Distinguishing the “diagonal” matrix D

D ≡

ω2L2

n + ∂xx − επ−Lnb0 0 . . . 0 0. . . . . . . . .0 0 ω2k2 + ∂xx − επkb0 0. . . . . .0 0 0 ω2L2

n + ∂xx − επLnb0

and the “off-diagonal Toepliz” matrix M1

M1 ≡

0 π−Lnb−1 . . . π−Lnb−2Ln

. . .0 . . . πlbl−k

. . .πLnb2Ln . . . πLnb1 0


we decomposeLn(ε, v1, w) = D − εM1 − εM2

where Dh := Lωh− εPnΠW (b0(x) h)M1h := PnΠW (b(t, x) h)M2h := PnΠW (b(t, x) ∂wv2[h])

(4.74)

andb(t, x) := b(t, x)− b0(x)

has zero time-average. Note that we do not decompose the term M2 in adiagonal and “off-diagonal term”.

To invert Ln we first (Step 1) prove that the diagonal part D is invertible,see Corollary 4.4.2. Next (Step 2) we prove that the “off-diagonal Toepliz”operator εM1 (Lemma 4.4.6) and εM2 (Lemma 4.4.7) are small enough withrespect to D, yielding the invertibility of the whole Ln.

4.4.2 Step 1: Inversion of D

In time-Fourier basis the operator D is represented by the diagonal matrix

D =

D−Ln 0 . . . 0 0. . . . . . 00 0 Dk 0. . . . . .0 0 0 DLn

where each

Dk : D(Dk) ⊂ 〈sin kx〉⊥ 7→ 〈sin kx〉⊥

is the Sturm-Liouville type operator

Dku := ω2k2u+ ∂xxu− επk(b0(x) u) .

We can diagonalize in space each Dk with respect to the scalar product (forε|b0|∞ < 1)

〈u, v〉ε :=

∫ π

0

uxvx + εb0(x)uv dx .

Its associated norm is equivalent to the H1-norm

‖u‖2H1

(1− ε |b0|∞

)≤ ‖u‖2

ε ≤ ‖u‖2H1

(1 + ε |b0|∞

)(4.75)

because∫ π

0u2 ≤

∫ π

0u2

x, ∀u ∈ H10 (0, π).


Lemma 4.4.1 (Sturm-Liouville) The operator −∂xx + επk(b0(x) ·) actingon 〈sin kx〉⊥ possesses a 〈 , 〉ε-orthonormal basis of eigenvectors (ϕk,j)j≥1,j 6=|k|with eigenvalues

λk,j = λk,j(ε, v1, w) = j2 + εM(v1, w) +O(ε‖b0‖H1

j

), j 6= |k| (4.76)

λk,j = λ−k,j, ϕ−k,j = ϕk,j, where

M(v1, w) =1

π

∫ π

0

b0(x) dx =1

4π

∫Ω

b(t, x) dxdt

was defined in (4.20).

Proof. It is a standard fact of perturbation theory for the eigenvalues ofself-adjoint operators, see e.g. [80]. The derivative with respect to ε at ε = 0of an eigenvalue λkj(ε) is obtained restricting the derivative of the operatorsw.r.t to ε, here πk(b0(x) ·), to the unperturbed eigenspace, here sin(jx):

∂ελkj(ε)|ε=0 =1

π

∫ π

0

sin(jx)πk(b0(x) sin(jx)) =1

π

∫ π

0

sin2(jx)b0(x)

=1

π

∫ π

0

(1− cos(2jx))b0(x) =1

π

∫ π

0

b0(x) +O(‖b0‖H1

j

)and integrating by parts. For details see Lemma 4.1 in [25].

Corollary 4.4.1 (Diagonalization of D) The operator D is the diagonaloperator diagω2k2 − λk,j in the basis

cos(kt)ϕk,j ; 0 ≤ k ≤ Ln , j ≥ 1, j 6= k .

Imposing the “non-resonance conditions” of property (P3) we get a lowerbound for the modulus of the eigenvalues of Dk.

Lemma 4.4.2 (Lower bound for the eigenvalues of D) There is c > 0such that if (ε, v1) ∈ An(w) and ε0 is small enough then ∀ 1 ≤ |k| ≤ Ln

αk := minj≥1,j 6=|k|

|ω2k2 − λk,j| ≥c γ

|k|τ−1> 0 . (4.77)

Moreover α0 ≥ 1/2.


Proof. Since α−k = αk it is sufficient to consider k ≥ 0. The estimate forα0 is trivial. Next

First Case: 0 < k ≤ 1/3ε. Since |ω − 1| ≤ 2ε and k 6= j we have

|ωk − j| ≥ |k − j| − |ω − 1|k ≥ 1− 2ε1

3ε=

1

3.

Therefore |ω2k2 − j2| = |ωk − j|(ωk + j) ≥ 1/3, and, by (4.76), for ε small,

|ω2k2 − λkj| ≥1

3≥ c γ

kτ−1

(the singular sites appear only for k > 1/3ε).

Second Case: k > 1/3ε. By a Taylor expansion

|ω2k2 − λk,j|(4.76)=

∣∣∣ω2k2 − j2 − εM(v1, w) +O(ε‖b0‖H1

j

)∣∣∣≥

∣∣∣(ωk −√j2 + εM(v1, w)

)(ωk +

√j2 + εM(v1, w)

)∣∣∣− Cε

j

≥∣∣∣ωk − j − ε

M(v1, w)

2j+O

(ε2

j3

)∣∣∣ωk − Cε

j

≥∣∣∣ωk − j − ε

M(v1, w)

2j

∣∣∣ ωk − C ′(ε2k

j3+ε

j

)≥ γωk

(k + j)τ− C

(ε2k

j3+ε

j

), (4.78)

since (ε, v1) ∈ An(w).If αk := minj≥1,j 6=k |ω2k2− λk,j| is attained at j = j(k) then |ωk− j| ≤ 1

(provided ε is small enough). Therefore, using 1 < τ < 2 and |ω − 1| ≤ 2ε,we can derive (4.77) from (4.78).

Lemma 4.4.3 Suppose αk 6= 0. Then Dk is invertible and ∀u =∑

j 6=|k| ujϕk,j ∈〈sin kx〉⊥

|Dk|−1/2u :=∑j 6=|k|

uj ϕk,j√|ω2k2 − λk,j|

satisfies ∥∥∥|Dk|−1/2u∥∥∥

H1≤ 2√αk

‖u‖H1 . (4.79)


Proof. Using that ϕk,j is an orthonormal basis for the 〈 , 〉ε scalar product∥∥∥ |Dk|−1/2u∥∥∥2

ε=

∥∥∥ ∑j 6=|k|

uj ϕk,j√|ω2k2 − λk,j|

∥∥∥2

ε=

∑j 6=|k|

|uj|2

|ω2k2 − λk,j|

≤ 1

αk

∑j 6=|k|

|uj|2 =‖u‖2

ε

αk

.

Since the norms ‖ · ‖ε and ‖ · ‖H1 are equivalent, (4.79) follows.

Corollary 4.4.2 (Estimate of |D|−1/2) If (ε, v1) ∈ An(w) and ε0 is smallenough, then D is invertible and ∀s′ ≥ 0∥∥∥|D|−1/2h

∥∥∥σ,s′

≤ C√γ‖h‖σ,s′+ τ−1

2∀h ∈ W (n). (4.80)

Proof. Since

|D|−1/2h :=∑|k|≤Ln

exp(ikt)|Dk|−1/2hk

using (4.79) and (4.77)∥∥∥|D|−1/2h∥∥∥2

σ,s′=

∑|k|≤Ln

exp(2σ|k|)(1 + k2s′)∥∥∥|Dk|−1/2hk

∥∥∥2

H1

≤∑|k|≤Ln

exp(2σ|k|)(1 + k2s′)4

αk

‖hk‖2H1

≤ 8‖h0‖2H1 + C

∑0<|k|≤Ln

exp(2σ|k|)(1 + k2s′)|k|τ−1

γ‖hk‖2

H1

≤ C ′

γ‖h‖2

σ,s′+ τ−12

proving (4.80).

4.4.3 Step 2: Inversion of LnTo show the invertibility of Ln it is convenient to write

Ln = D − εM1 − εM2 = |D|1/2(U − εR1 − εR2

)|D|1/2

whereU := |D|−1/2D|D|−1/2 = |D|−1D

andRi := |D|−1/2Mi|D|−1/2, i = 1, 2 .


Lemma 4.4.4 (Estimate of ‖U−1‖) U is invertible and, ∀s′ ≥ 0,∥∥∥U−1h∥∥∥

σ,s′= ‖h‖σ,s′

(1 +O(ε‖b0‖H1)

)∀ h ∈ W (n). (4.81)

Proof. The eigenvalues of Uk := |Dk|−1Dk are sign(ω2l2 − λlj) = ±1 andtherefore Uk is invertible and

‖U−1k u‖ε = ‖u‖ε , ∀u ∈ 〈sin kx〉⊥ .

By (4.75),‖U−1

k u‖H1 = ‖u‖ε(1 +O(ε‖b0‖H1)) .

Therefore U is invertible and (4.81) holds.

The estimate of the “off-diagonal” operator R1 requires a careful analysisof the “small divisors”.

Lemma 4.4.5 (Analysis of the small divisors) Let (ε, v1) ∈ An(w) andε0 small. There exists C > 0 such that, ∀l 6= k,

1

αkαl

≤ C|k − l|2

τ−1β

γ2ετ−1where β :=

2− τ

τ∈ (0, 1) . (4.82)

Proof. To obtain (4.82) we distinguish different cases.

• First case, sites far away the diagonal: |k− l| ≥ (1/2)[max(|k|, |l|)]β.

Then (αkαl)−1 ≤ C|k − l|2

τ−1β /γ2.

Estimating both αk, αl with the worst possible lower bound (4.77),

αk ≥c

γ|k|τ−1, αl ≥

c

γ|l|τ−1,

we obtain

1

αkαl

≤ C|k|τ−1|l|τ−1

γ2≤ C

[max(|k|, |l|)]2(τ−1)

γ2≤ C ′ |k − l|2

τ−1β

γ2.

The other cases to consider (sites close to the diagonal)

0 < |k − l| < 1

2

[max(|k|, |l|)

]β

(4.83)

are the most dangerous.


We note that, in this situation, sign(l) = sign(k) and, to fix the ideas, weassume in the sequel that l, k ≥ 0 (the estimate for k, l < 0 is the same, sinceαkαl = α−kα−l). In this case, k, l ≥ 0 are of the same order, namely

l

2≤ k ≤ 2l and

k

2≤ l ≤ 2k .

Indeed, if k > l, then k − l < kβ/2 ≤ k/2, since β < 1, and therefore k ≤ 2l.If k < l then l − k < lβ/2 ≤ l/2 whence k ≥ l/2.

• Second case: 0 < |k − l| < (1/2)[max(|k|, |l|)]β and (|k| ≤ 1/3ε or|l| ≤ 1/3ε). Then (αkαl)

−1 ≤ C/γ.

Suppose, for example, that 0 ≤ k ≤ 1/3ε. We claim that, if ε is smallenough, then αk ≥ (k + 1)/8. Indeed, ∀j 6= k,

|ωk − j| = |ωk − k + k − j| ≥ |k − j| − |ω − 1| |k| ≥ 1− 2ε k ≥ 1

3.

Therefore ∀0 ≤ k ≤ 1/3ε, ∀j 6= k, j ≥ 1,

|ω2k2 − j2| = |ωk − j| |ωk + j| ≥ ωk + 1

3≥ k + 1

6

and so

αk := minj≥1,k 6=j

∣∣∣ω2k2 − λk,j

∣∣∣= min

j≥1,k 6=j

∣∣∣ω2k2 − j2 − εM(v1, w) +O(ε‖b0‖H1

j

)∣∣∣≥ k + 1

6− ε C ≥ k + 1

8. (4.84)

Next, we estimate αl. If also 0 ≤ l ≤ 1/3ε then, arguing as above, αl ≥(l + 1)/8 ≥ 1/8 and therefore (αkαl)

−1 ≤ 64.If l > 1/3ε, we estimate αl ≥ cγ/lτ−1 by the lower bound (4.77), and

(4.84) implies, since l ≤ 2k,

1

αkαl

≤ Clτ−1

(k + 1)γ≤ C ′

γ

kτ−1

(k + 1)≤ C ′

γ

because 1 < τ < 2.

In the remaining cases we consider

|k − l| < 1

2

[max(|k|, |l|)

]β

and both |k|, |l| > 1

3ε.

We have to distinguish two sub-cases. For this, ∀k ∈ Z, let j = j(k) ≥ 1 bean integer such that αk := minn6=|k| |ω2k2−λk,n| = |ω2k2−λk,j|. Analogouslylet i = i(k) ≥ 1 be an integer such that αl = |ω2l2 − λl,i|.


• Third case: 0 < |k − l| < (1/2)[max(|k|, |l|)]β, |k|, |l| > 1/3ε andk − l = j − i. Then (αkαl)

−1 ≤ C/γετ−1.

Indeed∣∣∣(ωk − j)− (ωl − i)∣∣∣ = |ω(k − l)− (j − i)| = |ω − 1||k − l| ≥ ε

2

and therefore |ωk − j| ≥ ε/4 or |ωl − i| ≥ ε/4. Assume for instance that|ωk − j| ≥ ε/4. Then

|ω2k2 − j2| = |ωk − j| |ωk + j| ≥ εωk

4≥ ε(1− 2ε)

k

4

and so, for ε small enough, |αk| ≥ εk/8. Hence, estimating αl ≥ cγ/lτ−1

with the worst possible lower bound (4.77), yields

1

αkαl

≤ Clτ−1

γεk≤ C ′

γk2−τε≤ C ′

γετ−1

because l ≤ 2k and k > 1/3ε.

• Fourth case: 0 < |k− l| < (1/2)[max(k, l)]β, k, l > 1/3ε and k− l 6=j − i. Then (αkαl)

−1 ≤ C/γ2.

Using that ω is γ-τ -Diophantine, we get∣∣∣(ωk − j)− (ωl − i)∣∣∣ =

∣∣∣ω(k − l)− (j − i)∣∣∣ ≥ γ

|k − l|τ≥ Cγ

[max(k, l)]βτ

≥ C

2

( γ

kβτ+

γ

lβτ

)so that

|ωk − j| ≥ Cγ

2kβτor |ωl − i| ≥ Cγ

2lβτ.

Assume, for instance, the first inequality. Therefore

|ω2k2 − j2| = |ωk − j| |ωk + j| ≥ C ′γk1−βτ = C ′γkτ−1

since β := (2 − τ)/τ . Hence, for ε small enough, αk ≥ C ′γkτ−1/2 and,estimating αl with the worst possible lower bound (4.77), we deduce

1

αkαl

≤ Clτ−1

γ2kτ−1≤ C ′

γ2

because l ≤ 2k.

Collecting the estimates of all the previous cases, (4.82) follows.


Remark 4.4.1 The analysis of the small divisors in the Cases II)-III)-IV) ofthe previous Lemma corresponds, in the language of [49]-[46], to the propertyof “separation of the singular sites”.

Lemma 4.4.6 (Bound of the off-diagonal operator R1) Assume (4.24).

There exists C > 0 such that∥∥∥R1h∥∥∥

σ,s+ τ−12

≤ CC1

ετ−12 γ

‖h‖σ,s+ τ−12

∀h ∈ W (n) . (4.85)

Proof. We recall that

R1h := |D|−1/2PnΠW

(b(t, x) |D|−1/2h

)so that, for h ∈ W (n),

(R1h)(t, x) =∑|k|≤Ln

(R1h)k(x) exp(ikt)

with

(R1h)k = |Dk|−1/2πk

(b |D|−1/2h

)k

= |Dk|−1/2πk

[ ∑|l|≤Ln

bk−l|Dl|−1/2hl

]. (4.86)

Set Bm := ‖bm(x)‖H1 . From (4.86), (4.79), and since the zero time-Fouriercoefficient of b is B0 = 0,∥∥∥(R1h)k

∥∥∥H1≤ C

∑|l|≤Ln,l 6=k

Bk−l√αk√αl

‖hl‖H1 . (4.87)

Hence, by (4.82)∥∥∥(R1h)k

∥∥∥H1≤ C

γετ−12

Sk where Sk :=∑|l|≤Ln

Bk−l|k − l|τ−1

β ‖hl‖H1 . (4.88)

By (4.88), setting

S(t) :=∑|k|≤Ln

Sk exp(ikt)


(with S−k = Sk) and s′ := s+ (τ − 1)/2,∥∥∥R1h∥∥∥2

σ,s′=

∑|k|≤Ln

exp (2σ|k|)(k2s′ + 1)∥∥∥(R1h)k

∥∥∥2

H1

≤ C2

γ2ετ−1

∑|k|≤Ln

exp (2σ|k|)(k2s′ + 1)s2k

=C2

γ2ετ−1‖S‖2

σ,s′ . (4.89)

It turns out thatS = Pn(bc)

where

b(t) :=∑l∈Z

|l|τ−1

β Bl exp(ilt) and c(t) :=∑|l|≤Ln

‖hl‖H1 exp(ilt) .

Therefore, by (4.89) and since s′ > 1/2,∥∥∥R1h∥∥∥

σ,s′≤ C

γετ−12

‖bc‖σ,s′ ≤C

γετ−12

‖b‖σ,s′‖c‖σ,s′

≤ C

γετ−12

‖b‖σ,s′+ τ−1β‖h‖σ,s′ (4.90)

since ‖b‖σ,s′ ≤ ‖b‖σ,s′+ τ−1β

and ‖c‖σ,s′ = ‖h‖σ,s′ .

Now, since 0 < β < 1

‖b‖σ,s+ τ−12

+ τ−1β≤ ‖b‖

σ,s+2(τ−1)

β

≤ ‖b‖σ,s+

2τ(τ−1)2−τ

≤ C1 (4.91)

by (4.24). (4.90) and (4.91) prove (4.85).

The estimate of R2 is derived by the regularizing property∥∥∥∂wv2[u]∥∥∥

σ,s+2≤ C‖u‖σ,s (4.92)

of Proposition 4.1.1: by (4.80) the “loss of τ − 1 derivatives” due to |D|−1/2

applied twice, is compensated by the gain of 2 derivatives due to ∂wv2.

Lemma 4.4.7 (Estimate of R2) Assume (4.24). There exists a constantC > 0 such that∥∥∥R2h

∥∥∥σ,s+ τ−1

2

≤ CC1

γ‖h‖σ,s+ τ−1

2∀h ∈ W (n).


Proof. Using that τ < 3 to get (4.93)∥∥∥R2h∥∥∥

σ,s+ τ−12

(4.80)

≤ C√γ

∥∥∥M2|D|−1/2h∥∥∥

σ,s+τ−1

=C√γ

∥∥∥PnΠW

(b ∂wv2

[|D|−1/2h

])∥∥∥σ,s+τ−1

≤ C√γ‖b‖σ,s+τ−1

∥∥∥∂wv2

[|D|−1/2h

]∥∥∥σ,s+τ−1

≤ C ′√γ‖b‖σ,s+τ−1

∥∥∥∂wv2

[|D|−1/2h

]∥∥∥σ,s+2

(4.93)

(4.92)

≤ C√γ‖b‖σ,s+τ−1

∥∥∥|D|−1/2h∥∥∥

σ,s

(4.80)

≤ CC1

γ‖h‖σ,s+ τ−1

2

since ‖b‖σ,s+τ−1 ≤ ‖b‖σ,s+

2τ(τ−1)2−τ

≤ C1 by (4.91).

Proof of property (P3) completed. By Lemma 4.4.4 the operatorU is invertible in Xσ,s+ τ−1

2and, by Lemmas 4.4.6 and 4.4.7, provided ε is

small enough,

ε∥∥∥U−1R1

∥∥∥L(Xσ,s+(τ−1)/2)

, ε∥∥∥U−1R2

∥∥∥L(Xσ,s+(τ−1)/2)

<1

4.

Therefore alsoU − εR1 − εR2 : Xσ,s+ τ−1

2→ Xσ,s+ τ−1

2

has a bounded inverse:∥∥∥(U − εR1 − εR2)−1h

∥∥∥σ,s+ τ−1

2

=∥∥∥(I − εU−1R1 − εU−1R2)

−1U−1h∥∥∥

σ,s+ τ−12

≤ 2‖U−1h‖σ,s+ τ−12≤ C‖h‖σ,s+ τ−1

2.

We finally deduce∥∥∥L−1n h

∥∥∥σ,s

=∥∥∥|D|−1/2(U − εR1 − εR2)

−1|D|−1/2h∥∥∥

σ,s(4.80)

≤ C√γ

∥∥∥(U − εR1 − εR2)−1|D|−1/2h

∥∥∥σ,s+ τ−1

2

(4.94)

≤ C ′√γ

∥∥∥|D|−1/2h∥∥∥

σ,s+ τ−12

(4.80)

≤ C ′′

γ‖h‖σ,s+τ−1

≤ C ′′

γ(Ln)τ−1‖h‖σ,s

because h ∈ W (n). This completes the proof of property (P3).

Chapter 5

Forced vibrations

We discuss in this Chapter the problem of forced vibrations: we look fornontrivial periodic solutions of the equation

utt − uxx = εf(t, x, u)u(t, 0) = u(t, π) = 0

(5.1)

where the nonlinearity is T -periodic in time

f(t+ T, x, u) = f(t, x, u) . (5.2)

We look for T -periodic solutions.

It is clear that the bifurcation problem will be drastically different ac-cording if the forcing frequency ω := 2π/T verifies either

• (i) ω ∈ Q

• (ii) ω ∈ R \Q.

This second case (ii) leads to a small divisor problem similar to the onediscussed in the previous Chapters. Some existence results in this directionhave been obtained in [104] (for ω given and ε small enough) and, morerecently, in [59] and [12] also for ε = 1 and ω large (rapid vibrations).

We shall now be concerned with the case the forcing frequency ω is arational number, i.e. with case (i).

5.1 The forcing frequency ω ∈ Q

For simplicity of exposition we shall assume

T = 2π , i.e. ω = 1 , (5.3)

121

CHAPTER 5. FORCED VIBRATIONS 122

and so we look for nontrivial 2π-periodic solutions of (5.1).

The spectrum of the D’Alembertian operator in the space of 2π-periodicin time functions with spatial Dirichlet boundary conditions is

σ(∂tt − ∂xx) =− l2 + j2 | l ∈ Z , j ∈ N

⊂ Z

and it is therefore formed by integer numbers.Zero is an eigenvalue of infinite multiplicity (when |l| = j) but the spec-

trum is not dense in R and, in particular, the other eigenvalues are wellseparated from 0 if |l| 6= j. This is the big difference with respect to case (ii)when ω is an irrational number.

For this reason the main difficulty for proving existence of periodic solu-tions of (5.1) in the case (i) does not rely in solving the range equation, butin solving the bifurcation equation which has an intrinsic lack of compactness(see remark 5.2.2).

The first breakthrough regarding problem (5.1)-(5.2)-(5.3) was achievedby Rabinowitz in [111] where, using methods inspired by the theory of ellipticregularity, existence and regularity of solutions was proved for nonlinearitiessatisfying the strongly monotonicity assumption

(∂uf)(t, x, u) ≥ β > 0 . (5.4)

Theorem 5.1.1 (Rabinowitz [111]) Let f ∈ Ck satisfy (5.2)-(5.3)-(5.4).Then ∀ε small enough, there exists a 2π-periodic solution u ∈ Hk of (5.1).

Other existence results of weak and classical solutions have been obtained,still for strongly monotone nonlinearities, in [76]-[53]-[41].

Subsequently, Rabinowitz [112] was able to prove existence of weak solu-tions (actually a continuum branch) for a class of weakly monotone nonlin-earities like

f(t, x, u) = u2k+1 +G(t, x, u)

whereG(t, x, u2) ≥ G(t, x, u1) if u2 ≥ u1 .

The weak solutions obtained in [112] are only continuous functions. In gen-eral, more regularity is not expected: Brezis and Nirenberg [41] proved -butonly for strictly monotone nonlinearities- that any 2π-periodic L∞-solution of(5.1) is smooth, even in the nonperturbative case ε = 1, whenever the nonlin-earity f is smooth (remark the difference with the autonomous case discussedin the previous Chapter where actually analytic solutions were found).

The monotonicity assumption (strong or weak) on the nonlinearity is the


key property in [111]-[112] for overcoming the lack of compactness in theinfinite dimensional bifurcation equation, see section 5.3.

On the other hand, little is known about existence and regularity of so-lutions without the monotonicity of f .

The first existence results of weak solutions for non-monotone forcingterms were obtained by Willem [128], Hofer [68] and Coron [45] for nonlin-earities like

f(t, x, u) = g(u) + h(t, x)

where g(u) satisfies suitable linear growth conditions and ε = 1 (non pertur-bative case).

Existence of weak solutions is proved, in [128]-[68], for a set of h densein L2, although explicit criteria that characterize such h are not provided.The infinite dimensional bifurcation problem is overcome by assuming non-resonance hypotheses between the asymptotic behavior of g(u) and the spec-trum of the D’Alembertian operator.

On the other side, Coron [45] finds weak solutions assuming the additionalsymmetry h(t, x) = h(t + π, π − x) and restricting to the space of functionssatisfying u(t, x) = u(t + π, π − x), where the Kernel of the d’Alembertianoperator reduces to 0.

Exercise: Prove the last statement: the unique solution of the linear homo-geneous wave equation (2.8) satisfying the symmetry condition

u(t, x) = u(t+ π, π − x) (5.5)

is u = 0.Hint: it is convenient to use the D’Alembert representation (2.9).

More recently, existence results of periodic solutions for concave and con-vex nonlinearities has been obtained in [17].

In [20]-[21] existence and regularity of solutions of (5.1) has been provedfor a large class of non-monotone forcing terms f(t, x, u), including, for ex-ample,

f(t, x, u) = ±u2k + h(t, x)

and h > 0, see section 5.4.

In this Chapter we shall first prove the Rabinowitz Theorem 5.1.1, seesection 5.2. Next we shall prove some existence results of forced vibrationsfor non-monotone nonlinearities extracted from [21].


5.2 The variational Lyapunov-Schmidt reduc-

tion

In view of the variational argument that we shall use to solve the bifurcationequation we look for solutions

u : Ω := T× (0, π) → R

of (5.1) in the Banach space

E := H1(Ω) ∩ C1/20 (Ω)

where H1(Ω) is the usual Sobolev space and C1/20 (Ω) is the space of all the

1/2-Holder continuous functions u : Ω −→ R satisfying

u(t, 0) = u(t, π) = 0

endowed with norm1

‖u‖E := ‖u‖H1(Ω) + ‖u‖C1/2(Ω)

where‖u‖2

H1(Ω) := ‖u‖2L2(Ω) + ‖ux‖2

L2(Ω) + ‖ut‖2L2(Ω)

and

‖u‖C1/2(Ω) := ‖u‖C0(Ω) + sup(t,x) 6=(t1,x1)

|u(t, x)− u(t1, x1)|(|t− t1|+ |x− x1|)1/2

.

Critical points of the Lagrangian action functional Ψ ∈ C1(E,R)

Ψ(u) := Ψ(u, ε) :=

∫Ω

[u2t

2− u2

x

2+ εF (t, x, u)

]dtdx

where

F (t, x, u) :=

∫ u

0

f(t, x, ξ)dξ

are weak solutions of (5.1).

For ε = 0, the critical points of Ψ in E reduce to the solutions of thelinear equation (2.8) and form the space

V := N ∩H1(Ω) ⊂ E (5.6)

1The choice of this norm is motivated by the regularizing property of the inverse of theD’Alembertian operator proved in Lemma 5.2.1.


where

N :=v(t, x) = v(t+ x)− v(t− x) =: v+(t, x)− v−(t, x)

such that v ∈ L2(T) and

∫ 2π

0

v(s) ds = 0

is the L2-closure of the classical solutions of the linear equation (2.8).It results V ⊂ E because any function v ∈ H1(T) is 1/2-Holder contin-

uous. The only difference between the space V introduced in (2.14) withrespect to V defined in (5.6) is that the functions v in (2.14) are not neces-sarily even in time. For notational convenience we denote these spaces in thesame way.

DecomposeE = V ⊕W

whereW := N⊥ ∩ E

and

N⊥ :=h ∈ L2(Ω) |

∫Ω

hv = 0, ∀v ∈ N

=h =

∑j≥1,j 6=|l|

hljeilt sin jx with ‖h‖2

L2 = π2∑

|hlj|2 <∞.

Projecting (5.1), for u = v + w with v ∈ V , w ∈ W , yields0 = ΠN f(v + w) bifurcation equationwtt − wxx = εΠN⊥ f(v + w) range equation

where ΠN and ΠN⊥ are the projectors from L2(Ω) onto N and N⊥, and f(u)denotes the Nemitski operator induced on E by the nonlinearity

f(u)(t, x) := f(t, x, u).

In the case the nonlinearity is strictly monotone the usual approach of[111]-[53]-[6] is to solve first the bifurcation equation finding its unique solu-tion v = v(w), and, next, to solve the range equation.

On the contrary, for a non-monotone nonlinearity the bifurcation equationcan not in general be solved uniquely (see remark 5.4.3). Therefore, we shallsolve first the range equation and thereafter the bifurcation equation.

In the case of monotone f this approach makes also some technical aspectof the proof easier than in [111].


5.2.1 The range equation

We first study the invertibility properties of the D’Alambertian operator := ∂tt − ∂xx restricted to N⊥.

Lemma 5.2.1 The inverse of the D’Alambertian operator −1 : N⊥ → Wdefined by

−1f :=∑

j≥1,j 6=|l|

flj

−l2 + j2eilt sin jx , ∀f ∈ N⊥

is a bounded operator, i.e. satisfies∥∥∥−1f∥∥∥

E≤ C ‖f‖L2 (5.7)

for a suitable positive constant C.

Proof. Let h := −1f . For any fixed x ∈ (0, π) we have∥∥∥∂th(·, x)∥∥∥2

L2(T)= 2π

∑l∈Z

l2|hl(x)|2 (5.8)

where

hl(x) :=∑j 6=|l|

flj

−l2 + j2sin jx

satisfies

|hl(x)|2 ≤( ∑

j 6=|l|

|flj||j2 − l2|

)2

≤∑j 6=|l|

|flj|2∑j 6=|l|

1

|j2 − l2|2(5.9)

by Cauchy-Schwartz. By (5.8)-(5.9),∥∥∥∂th(·, x)∥∥∥2

L2(T)≤ 2π

∑l∈Z

( ∑j 6=|l|

|flj|2)( ∑

j 6=|l|

l2

|j2 − l2|2). (5.10)

Nowl

j2 − l2=

1

2(j − l)− 1

2(j + l)

whence, for any l,∑j 6=|l|

l2

|l2 − j2|2≤

∑j 6=|l|

1

(j − l)2+

1

(j + l)2≤ 4

∑j≥1

1

j2=: M2


and, by (5.10),∥∥∥∂th(·, x)∥∥∥2

L2(T)≤ 2πM2

∑l∈Z

∑j 6=|l|

|flj|2 =2M2

π‖f‖2

L2(Ω) . (5.11)

As a consequence, for any fixed x, by Holder inequality

|h(t, x)− h(t′, x)| ≤( ∫ t′

t

|∂th(τ, x)|2 dτ)1/2√

|t− t′|

(5.11)

≤ M√

2π−1‖f‖L2(Ω)

√|t− t′| (5.12)

and h(·, x) is 1/2-Holder continuous, uniformly in x.Similarly, for any fixed t,∥∥∥∂xh(t, ·)

∥∥∥2

L2(0,π)≤ M2

2π‖f‖2

L2(Ω) (5.13)

and so

|h(t, x′)− h(t, x)| ≤ M√2π‖f‖L2(Ω)

√|x− x′| (5.14)

uniformly in t. By (5.12)-(5.14) we deduce

‖h‖C1/2(Ω) ≤ C‖f‖L2

and integrating (5.11) and (5.13) we also get

‖∂th‖2L2(Ω), ‖∂xh‖2

L2(Ω) ≤ C ′‖f‖2L2(Ω)

implying (5.7).

Remark 5.2.1 The previous lemma could also be proved via the integralrepresentation formula for −1 given in [76]:

−1f = −1

2

∫ π

x

∫ t−x+ξ

t+x−ξ

f(ξ, τ) dτdξ + c(π − x)

π(5.15)

where

c :=1

2

∫ π

0

∫ t+ξ

t−ξ

f(ξ, τ) dτdξ ≡ 1

2

∫ π

0

∫ ξ

−ξ

f(ξ, τ) dτdξ ≡ const

is a constant independent of t (because f ∈ N⊥ ).Using (5.15) it follows that −1 is a bounded operator also between the

spacesHk(Ω) −→ Hk+1(Ω) , Ck(Ω) −→ Ck+1(Ω) . (5.16)


Fixed points w ∈ W of

w = ε−1ΠN⊥f(v + w)

are solutions of the range equation.

Applying the Contraction Mapping Theorem as in Lemma 2.2.3 we have:

Lemma 5.2.2 (Solution of the range equation) ∀R > 0, ∃ ε0(R) > 0,C0(R) > 0, such that

∀ |ε| ≤ ε0(R) and ∀ v ∈ N with ‖v‖L∞ ≤ 2R

there exists a unique solution w(ε, v) ∈ W of the range equation satisfying

‖w(ε, v)‖E ≤ C0(R)|ε| . (5.17)

Moreover the map (ε, v) → w(ε, v) is C1(‖v‖L∞ ≤ 2R,W ).

Exercise: Prove the existence, for ε small enough, of a unique 2π-periodicsolution in E of

utt − uxx = ε(f(u) + cos(2t) sin(x))u(t, 0) = u(t, π) = 0 .

Hint: the subspace of functions in E satisfying (5.5) is invariant under theNemitsky operator induced by the nonlinearity .

5.2.2 The bifurcation equation

Once the range equation has been solved by w(ε, v) ∈ W there remains theinfinite dimensional bifurcation equation

ΠN f(v + w(ε, v)) = 0

which is equivalent, since V is dense in N with the L2-norm, to∫Ω

f(v + w(ε, v))φ = 0 ∀φ ∈ V . (5.18)

By the variational Lyapunov-Schmidt reduction (see the same arguments insection 2.2) (5.18) is the Euler-Lagrange equation of the reduced Lagrangianaction functional2

Φ :‖v‖H1 < 2R

→ R

2Since ‖v‖L∞(Ω) ≤ ‖v‖H1(Ω) the functional Φ(v) is well defined for any ‖v‖H1(Ω) < 2Rby Lemma 5.2.2.


defined byΦ(v) := Φ(ε, v) := Ψ(v + w(ε, v), ε)

which can be written (like in (2.29) with ω = 1) as

Φ(v) = ε

∫Ω

[F (v + w(ε, v))− 1

2f(v + w(ε, v))w(ε, v)

]dtdx . (5.19)

Lemma 5.2.3 The following continuity property holds:

‖vn‖H1 , ‖v‖H1 ≤ R , vnL∞−→ v =⇒ Φ(vn) −→ Φ(v) . (5.20)

Proof. Setting wn := w(ε, vn) and w := w(ε, v), we have∣∣∣∣∫Ω

F (vn + wn)− F (v + w)

∣∣∣∣ ≤ max |f(t, x, u)|∫

Ω

|vn − v + wn − w|

≤ C0(R)(‖vn − v‖L1 + ‖wn − w‖L1

)−→ 0

as n→∞, by the fact that ‖vn − v‖L∞ → 0 and therefore, by Lemma 5.2.2,‖w(ε, vn)−w(ε, v)‖E → 0. An analogous estimate holds for the second termin the integral (5.19).

Φ lacks compactness properties and to find its critical points we can notrely on critical point theory.

Remark 5.2.2 In the corresponding reduced Lagrangian action functionalin (2.29) for the autonomous case, it is the “elliptic” term ‖v‖2

H1 whichintroduces compactness in the problem allowing to find Mountain-pass criticalpoints. This term is also the responsible for the regularity of the solutions.

In the case f is strictly monotone, it is natural to try to minimize Φ,because F is strictly convex in u.

However, we do not apply the direct methods of the calculus of variations(see the Appendix 6.2.2) because, without assuming any growth conditionon the nonlinearity f , the functional Φ could neither be well defined on anyLp-space.

Following [111] we make a constrained minimization:

Lemma 5.2.4 ∀R > 0, the functional Φ attains minimum in

BR :=v ∈ V, ‖v‖H1 ≤ R

.


Proof. Let vn ∈ BR be a minimizing sequence Φ(vn) → infBRΦ. Since vn

is bounded in V , up to a subsequence vnH1

v and, by the compact embedding

(2.15), vnL∞−→ v. Therefore by (5.20) v is a minimum point of Φ restricted

to BR.

Since v could belong to the boundary ∂BR we only have the variationalinequality

DvΦ(v)[φ] =

∫Ω

f(v + w(ε, v))φ ≤ 0 (5.21)

for any admissible variation φ ∈ V , namely for any φ ∈ V such that

v + θφ ∈ BR , ∀θ < 0

sufficiently small.A sufficient condition for φ ∈ V to be an admissible variation is

〈v, φ〉H1 > 0 (5.22)

because

‖v + θφ‖2H1 = ‖v‖2

H1 + 2θ〈v, φ〉H1 + θ2‖φ‖2H1

≤ R2 + 2θ〈v, φ〉H1 + θ2‖φ‖2H1 < R2

for θ < 0 small enough.

The heart of the existence proof of Theorem 5.1.1 is to obtain, choosingsuitable admissible variations, the a-priori estimate ‖v‖H1 < R∗ for someR∗ > 0 (independent of ε), i.e. to show that v is an inner minimum point ofΦ in BR∗ .

It is here where the monotonicity plays its role.

5.3 Monotone f

In the sequel κi will denote positive constants independent of ε (possiblydepending on the nonlinearity).

5.3.1 Step 1: the L∞-estimate

Using the variational inequality (5.21) and (5.17)∫Ω

f(v)φ =

∫Ω

f(v + w(ε, v))φ+

∫Ω

[f(v)− f(v + w(ε, v))]φ(5.23)

(5.21)

≤∫

Ω

[f(v)− f(v + w(ε, v))]φ(5.17)

≤ |ε|C1(R)‖φ‖L1


O

R

ϕ

υ−

Figure 5.1: The variational inequality.

where C1(·) is a suitable increasing function depending on f .Then there exists a decreasing function 0 < ε1(·) ≤ ε0(·) such that∫

Ω

f(v)φ ≤ ‖φ‖L1 for |ε| ≤ ε1(R) . (5.24)

We now choose

φ = q(v+)− q(v−) = q(ˆv(t+ x))− q(ˆv(t− x)) ∈ V

where

q(λ) := qM(λ) :=

0 , if |λ| ≤Mλ−M , if λ ≥Mλ+M , if λ ≤M

and3

M :=1

2‖ˆv‖L∞(T) . (5.25)

We can assume M > 0, i.e. v is not identically zero.

Lemma 5.3.1 ([111]) φ ∈ V is an admissible variation and

v(t, x)φ(t, x) ≥ 0 , ∀(t, x) ∈ Ω . (5.26)

Proof. By Lemma 2.3.3, for p, q ∈ L1(T)∫Ω

p(t+ x)q(t− x) dt dx =1

2

∫ 2π

0

p(s)ds

∫ 2π

0

q(s)ds . (5.27)

3Note ‖v‖L∞(T) ≤ ‖v‖L∞(Ω) ≤ 2‖v‖L∞(T), ∀v ∈ N ∩ L∞.


Using repeatedly (5.27) and since ˆv has zero average

〈v, φ〉H1 = 〈v+ − v−, q(v+)− q(v−)〉H1 =

∫Ω

v+q(v+) + v−q(v−)

+ 2

∫Ω

q′(v+)[(v+)2

t + (v+)2x

]> 0

because q′ ≥ 0 and v±q(v±) > 0 in a positive measure set, being q a monotoneodd function and by our choice of M .

Finally, since q is a monotone function,

vφ = (v+ − v−)(q(v+)− q(v−)) ≥ 0

proving (5.26).

We now estimate from below the left hand side in (5.23). By the meanvalue Theorem, the strongly monotonicity assumption (5.4), and (5.26)∫

Ω

f(v)φ =

∫Ω

[f(t, x, 0) + fu(intermediate point)v

]φ

≥∫

Ω

f(t, x, 0)φ+ β

∫Ω

vφ

whence

β

∫Ω

vφ ≤∫

Ω

|f(t, x, 0)||φ|+∫

Ω

f(v)φ

(5.24)

≤ max |f(t, x, 0)|‖φ‖L1 + ‖φ‖L1 := κ1‖φ‖L1 . (5.28)

Now, since ˆv has zero average, using (5.27) and again Lemma 2.3.3∫Ω

vφ =

∫Ω

v+q(v+) + v−q(v−)(2.47)= π

∫ 2π

0

ˆv(s)q(ˆv(s))ds

≥ πM‖q(ˆv)‖L1(T) (5.29)

since λq(λ) ≥M |q(λ)|. Next∫Ω

|φ| ≤∫

Ω

|q(v+)|+ |q(v−)| (2.47)= 2π‖q(ˆv)‖L1(T) (5.30)

and (5.29)-(5.30) imply∫Ω

vφ ≥ 1

2M‖φ‖L1

(5.25)=

1

4‖v‖L∞‖φ‖L1 . (5.31)

By (5.28) and (5.31) we finally deduce

‖v‖L∞ ≤ 4κ1

βfor |ε| ≤ ε1(R) . (5.32)


5.3.2 Step 2: the H1-estimate

The H1-estimate is carried out inserting in the variational inequality (5.21)the variation

φ := −D−hDhv

where

(Dhf)(t, x) :=f(t+ h, x)− f(t, x)

his the difference quotience. Note that φ is admissible because

〈−D−hDhv, v〉H1 = 〈Dhv, Dhv〉H1 > 0

by the formula for integration by parts∫Ω

f(D−hg) = −∫

Ω

(Dhf)g , ∀f, g ∈ L2(Ω) . (5.33)

We shall use the following technical Lemma.

Lemma 5.3.2 Let f ∈ L2(Ω) have a weak derivative ft ∈ L2(Ω). Then

‖Dhf‖L2(Ω) ≤ ‖ft‖L2(Ω) (5.34)

and

DhfL2

−→ ft as h −→ 0 . (5.35)

Proof. To prove (5.34) assume temporarily f is smooth. From the funda-mental Theorem of calculus

(Dhf)(t, x) =f(t+ h, x)− f(t, x)

h=

∫ 1

0

ft(t+ hs, x)ds .

By Cauchy-Schwartz inequality, Fubini Theorem and periodicity∫Ω

|Dhf(t, x)|2dtdx =

∫ π

0

∫ 2π

0

∣∣∣ ∫ 1

0

ft(t+ hs, x)ds∣∣∣2dtdx

≤∫ π

0

∫ 2π

0

∫ 1

0

|ft(t+ hs, x)|2dsdtdx

=

∫ π

0

∫ 1

0

∫ 2π

0

|ft(t+ hs, x)|2dtdsdx

=

∫ π

0

∫ 1

0

∫ 2π

0

|ft(t, x)|2dtdsdx

=

∫ 1

0

∫ π

0

∫ 2π

0

|ft(t, x)|2dtdxds = ‖ft‖2L2(Ω) .


Inequality (5.34) is valid, for any f having a weak derivative ft ∈ L2(Ω), byapproximation.

Proof of (5.35). We first show the weak L2-convergence. Let φ ∈ C1(Ω).By (5.33) and the Lebesgue dominated convergence Theorem∫

Ω

(Dhf)φ = −∫

Ω

f(D−hφ)h→0−→ −

∫Ω

fφt =

∫Ω

ftφ (5.36)

since f has a weak derivative ft.Since, by (5.34), Dhf is bounded in L2, and (5.36) holds in the dense

subset C1(Ω) ⊂ L2(Ω), we conclude DhfL2

ft.By the weakly lower semicontinuity of the norm and (5.34)

‖ft‖L2(Ω) ≤ lim inf ‖Dhf‖L2(Ω)

(5.34)

≤ ‖ft‖L2(Ω)

implyinglim ‖Dhf‖L2(Ω) = ‖ft‖L2(Ω) . (5.37)

The weak convergence DhfL2

ft and (5.37) imply the strong convergence(5.35).

By the variational inequality (5.21), denoting w := w(ε, v),

0(5.21)

≥∫

Ω

f(v + w)φ(5.33)=

∫Ω

Dhf(v + w)Dhv

h→0−→∫

Ω

∂t

(f(v + w)

)vt

=

∫Ω

(ft(v + w) + fu(v + w)(vt + wt)

)vt

applying the previous Lemma (since v ∈ H1 and w ∈ E).Therefore, using the L∞-estimate (5.32) for v (which is independent on

R) and (5.17), we derive∫Ω

fu(v + w)v2t ≤

∣∣∣ ∫Ω

(ft(v + w) + fu(v + w)wt

)vt

∣∣∣ ≤ κ2‖vt‖L2 (5.38)

for |ε| ≤ ε2(R) ≤ ε1(R).By the strongly monotonicity assumption fu ≥ β > 0∫

Ω

fu(v + w)v2t ≥

∫Ω

βv2t


implying, from (5.38), ‖vt‖L2 < κ2/β and, in conclusion,

‖v‖H1 < κ3 , ∀ |ε| ≤ ε2(R) .

Proof of Theorem 5.1.1 completed. Take

R∗ := κ3 and ε∗ := ε2(R∗) .

Therefore, ∀ |ε| ≤ ε∗ ,‖v(ε)‖H1 < R∗

is an interior minimum point of Φ in BR∗ := ‖v‖H1 < R∗ and

u := v(ε) + w(ε, v(ε)) ∈ E

is a weak solution of (5.1).

Using similar techniques (inspired by regularity theory), inserting suitablevariations in the Euler Lagrange equation∫

Ω

f(v + w(ε, v))φ = 0 , ∀φ ∈ V

and exploiting the regularizing property (5.16) of −1, one could obtain L∞

and H1-estimates for the higher order derivatives of v and w(ε, v), provingthe higher regularity of the solution u, see [111]-[21].

We leave as an exercise the proof of the unicity of the solution u (in aball of fixed radius independent of ε).

5.4 Non-Monotone f

In this section we shall prove the following existence Theorem of forced vi-brations of (5.1) for non-monotone nonlinearities.

Theorem 5.4.1 ([21]) Let

f(t, x, u) = u2k + h(t, x) (5.39)

where h ∈ N⊥ satisfies h(t, x) > 0 (or h(t, x) < 0) a.e. in Ω.

(i) (Existence) For all ε small enough, there exists at least one weak solutionu ∈ E of (5.1) with ‖u‖E ≤ C|ε|.(ii) (Regularity) If h ∈ Hj(Ω)∩Cj−1(Ω), j ≥ 1, then u ∈ Hj+1(Ω)∩Cj

0(Ω).


Theorem 5.4.1 is a Corollary of a more general result which holds fornon-monotone nonlinearities like, for example,

f(t, x, u) =

±(sinx) u2k + h(t, x)± u2k + u2k+1 + h(t, x)

and also without any growth condition for f , see Theorems 1-2 in [20]-[21].

Remark 5.4.1 The hypothesis h > 0 enables to prove the existence of aminimum of the reduced action functional Φ.

Remark 5.4.2 The assumption h ∈ N⊥ is not of technical nature: if h /∈N⊥, periodic solutions of (5.1) do not exist in any fixed ball ‖u‖L∞ ≤ Rfor ε small. Indeed, let u := uε = vε + wε, vε ∈ V , wε ∈ W , be a weaksolution of (5.1) with ‖uε‖L∞ ≤ R. Since wε satisfies the range equation wε =ε−1ΠN⊥(u2k

ε +h), then ‖wε‖L∞ ≤ C(R)|ε|. Moreover ΠN((vε +wε)2k +h) =

0, and since ΠNv2kε = 0 by Lemma 2.3.1,∥∥∥ΠNh

∥∥∥L2

=∥∥∥ΠN((vε + wε)

2k − v2kε )

∥∥∥L2→ 0

as ‖wε‖L∞ → 0. Therefore ΠNh = 0 and h ∈ N⊥.

Remark 5.4.3 (Multiplicity) For non-monotone nonlinearities f unicityof the solutions is not expected. For example, let f be like in (5.39) andh(t, x) := −(v0(t, x))

2k for some v0 ∈ V \ 0. By Lemma 2.3.1, h ∈ N⊥.Equation (5.1) possesses, beyond the ε-small solution u of Theorem 5.4.1,also the other two (not small) solutions ±v0.

The difficulty for dealing non-monotone nonlinearities is well highlightedfor nonlinearities f like in (5.39). In this case the variational inequality (5.21)vanishes identically for ε = 0, because∫

Ω

(v2k + h(t, x)

)φ 6≡ 0, ∀φ ∈ V

by Lemma 2.3.1 and h ∈ N⊥.Therefore, for deriving, if ever possible, the required a-priori estimates,

we have to develop the variational inequality (5.21) at higher orders in ε.

Perform in (5.1) the change of variables

u→ ε(H + u) (5.40)


where H is a weak solutions ofHtt −Hxx = hH(t, 0) = H(t, π) = 0H(t+ 2π, x) = H(t, x)

(5.41)

and, since h > 0 in Ω, we can always choose

H(t, x) > 0 , ∀(t, x) ∈ Ω

by the “maximum principle” theorem proved in subsection 5.4.4 (Proposition4.11 of [21]).

Therefore (renaming ε2k → ε) we look for solutions ofutt − uxx = εf(t, x, u)u(t, 0) = u(t, π) = 0u(t+ 2π, x) = u(t, x)

where the nonlinear forcing term is now (still called f)

f(t, x, u) := (H + u)2k .

Implementing a variational Lyapunov-Schmidt reduction as in section 5.2, wefind the existence of a constrained minimum v ∈ BR, where the variationalinequality ∫

Ω

(H + v + w(ε, v))2kφ ≤ 0 (5.42)

is satisfied for any admissible variation φ ∈ V .

As in section 5.3 the required a-priori estimate for the H1-norm of vis be proved in several steps inserting into the variational inequality (5.42)suitable admissible variations. We shall derive, first, an L2k-estimate for v(it is needed at least when k ≥ 2) next, an L∞-estimate, and, finally, theH1-estimate.

We shall make the proof only for the more difficult case k ≥ 2.

The following key “coercivity” estimate shall be heavily exploited.

Lemma 5.4.1 (Coercivity estimate) Let B ∈ C(Ω) with B > 0 in Ω.∀ v ∈ N ∩ L2k(Ω) ∫

Ω

Bv2k ≥ ck(B)

∫Ω

v2k (5.43)

where

ck(B) :=1

4kminΩαk

B > 0 , αk :=1

4(1 + 2k)(5.44)

andΩα := T× (απ, π − απ) ⊆ Ω .


The inequality (5.43) is not trivial because B can vanish at the boundaryof ∂Ω (i.e. B(t, 0) = B(t, π) = 0). It holds true because v(t, x) = v(t+ x)−v(t−x) is the linear superposition of two traveling waves which “spend a lotof time” far from the boundary x = 0, x = π.

Lemma 5.4.1 is a consequence of several Lemmas.

Lemma 5.4.2 For p, q ∈ L1(T)∫Ωα


2

∫ 2π

0

p(s)ds

∫ 2π

0

q(s)ds

− 1

2

∫ 2απ

−2απ

∫ 2π

0

p(y)q(z + y)dydz (5.45)

and∫Ωα

p(t+ x) dt dx =

∫Ωα

p(t− x) dt dx = π(1− 2α)

∫ 2π

0

p(s)ds . (5.46)

For f, g : R → R continuous,∫Ωα

f(p(t+ x))g(p(t− x)) dt dx =

∫Ωα

f(p(t− x))g(p(t+ x)) dt dx . (5.47)

∀a, b ∈ R(a− b)2k ≥ a2k + b2k − 2k(a2k−1b+ ab2k−1) . (5.48)

Proof. By the periodicity in t∫Ωα

p(t+ x)q(t− x)dtdx =

∫eΩα

p(t+ x)q(t− x)dtdx

where Ωα := απ < x < π(1− α),−x < t < −x+ 2π. Under the change of

variables s+ := t+ x, s− := t− x the domain Ωα transforms into0 < s+ < 2π, s+ − 2π(1− α) < s− < s+ − 2πα

and therefore∫

Ωα


2

∫ 2π

0

ds+ p(s+)

∫ s+−2απ

−2π+s++2απ

q(s−)ds−

=1

2

∫ 2π

0

ds+ p(s+)( ∫ 2π

0

q(s)ds −∫ s++2απ

s+−2απ

q(s−)ds−

)=

1

2

∫ 2π

0

p(s)ds

∫ 2π

0

q(s)ds

−1

2

∫ 2π

0

ds+ p(s+ )

∫ 2απ

−2απ

q(s+ + z)dz


and we obtain (5.45) by Tonelli’s Theorem (calling s+ = y).Formula (5.46) follows by (5.45) setting q 6= 1 .Since the change of variables (t, x) 7→ (t, π − x) leaves the domain Ωα

unchanged and using the periodicity of p∫Ωα

f(p(t+ x))g(p(t− x))dtdx =

∫Ωα

f(p(t+ π − x))g(p(t− π + x)) dt dx

=

∫ π−απ

απ

∫ 2π

0

f(p(t+ π − x))g(p(t+ π + x))dtdx

=

∫ π−απ

απ

∫ 2π

0

f(p(t− x))g(p(t+ x))dtdx

proving (5.47). For the elementary inequality (5.48) we refer to [21].

Lemma 5.4.3 Let v ∈ N ∩ L2k(Ω). Then∫Ω

v2k ≤ π4k

∫ 2π

0

v2k (5.49)

and ∫Ωα

v2k ≥ 2π[1− 2(1 + 2k)α

] ∫ 2π

0

v2k ≥ π

∫ 2π

0

v2k (5.50)

if 0 ≤ α ≤ 1/4(1 + 2k).

Proof. The convexity inequality

(a− b)2k ≤ 22k−1(a2k + b2k) , ∀ a, b ∈ R

yields ∫Ω

v2k =

∫Ω

(v+ − v−)2k ≤ 22k−1

∫Ω

v2k(t+ x) + v2k(t− x)

which, using Lemma 2.3.3, proves (5.49).Using (5.48) (5.46) and (5.47)∫

Ωα

v2k =

∫Ωα

(v+ − v−)2k(5.48)

≥∫

Ωα

v2k+ + v2k

− − 2k

∫Ωα

v2k−1+ v− + v+v

2k−1−

= 2π(1− 2α)

∫ 2π

0

v2k − 4k

∫Ωα

v2k−1+ v− . (5.51)


By (5.45) and since v has zero average∣∣∣∣∫Ωα

v2k−1+ v−

∣∣∣∣ =1

2

∣∣∣∣∫ 2πα

−2πα

∫ 2π

0

v2k−1(y)v(y + z)dydz

∣∣∣∣≤ 1

2

∫ 2πα

−2πα

∫ 2π

0

|v(y)|2k−1|v(y + z)|dydz . (5.52)

By Holder inequality with p := 2k/(2k − 1) and q := 2k∫ 2π

0

|v(y)|2k−1|v(y + z)|dy ≤(∫ 2π

0

|v(y)|2kdy

) 2k−12k

(∫ 2π

0

|v(y + z)|2kdy

) 12k

=

∫ 2π

0

|v(y)|2kdy

where, in the equality, we have used the periodicity of v. Inserting the lastinequality in (5.52) ∣∣∣∣∫

Ωα

v2k−1+ v−

∣∣∣∣ ≤ 2πα

∫ 2π

0

|v(y)|2kdy (5.53)

Inserting (5.53) in (5.51) follows (5.50).

Proof of Lemma 5.4.1: Since B > 0 in Ω and using Lemma 5.4.3∫Ω

Bv2k ≥ minΩαk

B

∫Ωαk

v2k(5.50)

≥ πminΩαk

B

∫ 2π

0

v2k(5.49)

≥ 1

4kminΩαk

B

∫Ω

v2k .

5.4.1 Step 1: the L2k-estimate

Insert φ := v in the variational inequality (5.42). φ is an admissible variationsince

〈v, φ〉H1 = ‖v‖2H1 > 0 .

Using the variational inequality (5.42), setting w := w(ε, v),∫Ω

(v +H)2kv =

∫Ω

(v +H + w)2kv +

∫Ω

[(v +H)2k − (v + w +H)2k]v

(5.42)

≤∫

Ω

[(v +H)2k − (v + w +H)2k]v

(5.17)

≤ |ε|C1(R)‖v‖L1 ≤ 1 (5.54)

for |ε| ≤ ε1(R) where 0 < ε1(·) ≤ ε0(·).


Since∫

Ωv2k+1 = 0 by (2.41), and using Lemma 5.4.1

1(5.54)

≥∫

Ω

(v +H)2kv =

∫Ω

[(v +H)2k − v2k

]v (5.55)

=

∫Ω

2kHv2k +2k−2∑j=0

(2k

j

)vj+1H2k−j

(5.44)

≥ 2kck(H)‖v‖2kL2k − κ1‖v‖2k−1

L2k − κ2‖v‖L2k

where ck(H) > 0 is defined in (5.44) (recall that H > 0), and we have usedthe Holder inequality to estimate ‖v‖Li ≤ Ci,k‖v‖L2k (i ≤ 2k − 1).

(5.55) implies

‖v‖L2k ≤ κ3 for |ε| ≤ ε1(R) . (5.56)

5.4.2 Step 2: the L∞-estimate.

Let φ be the same admissible variation like in subsection 5.3.1.Using the variational inequality (5.42), setting w := w(ε, v),∫

Ω

(v +H)2kφ =

∫Ω

(v +H + w)2kφ+

∫Ω

[(v +H)2k − (v + w +H)2k]φ

(5.42)

≤∫

Ω

[(v +H)2k − (v + w +H)2k]φ

(5.17)

≤ |ε|C2(R)‖φ‖L1 ≤ ‖φ‖L1 (5.57)

for |ε| ≤ ε2(R) where 0 < ε2(·) ≤ ε1(·).Now, using that

∫Ωv2kφ = 0 by Lemma 2.3.1,

‖φ‖L1

(5.57)

≥∫

Ω

(v +H)2kφ =

∫Ω

[(v +H)2k − v2k

]φ

≥∫

Ω

2kHv2k−1φ− κ4(‖v‖2k−2L∞ + 1)‖φ‖L1

which implies, since ∫Ω

Hv2k−1φ ≥ κ5

∫Ω1/4

φv2k−1

(because vφ ≥ 0 and minΩ1/4H > 0), that∫

Ω1/4

v2k−1φ ≤ κ6(‖v‖2k−2L∞(Ω) + 1)‖φ‖L1(Ω)

(5.30)

≤ κ7(‖v‖2k−2L∞(Ω) + 1)‖q(ˆv)‖L1(T) . (5.58)


We have to give a lower bound of the positive integral∫Ω1/4

v2k−1φ =

∫Ω1/4

(vφ)v2k−2 =

∫Ω1/4

(vφ)(v+ − v−)2k−2 .

Using (5.48)∫Ω1/4

v2k−1φ ≥∫

Ω1/4

vφ[v2k−2

+ + v2k−2− − (2k − 2)(v2k−3

+ v− + v+v2k−3− )

]= 2

∫Ω1/4

v2k−1+ q+ − v2k−1

+ q− + v2k−2+ v−q− − v2k−2

+ v−q+

+ (2k − 2)[−v2k−2+ v−q+ + v2k−2

+ v−q− − v2k−3+ v2

−q− + v2k−3+ v2

−q+]

≥ 2

∫Ω1/4

v2k−1+ q+ (5.59)

− 2

∫Ω1/4

v2k−1+ q− + (2k − 1)v2k−2

+ v−q+ (5.60)

+ (2k − 2)v2k−3+ v2

−q− (5.61)

where in the equality we have used (5.47) and in the last inequality the factthat v+q+, v−q− ≥ 0 (since λq(λ) ≥ 0) and so v2k−2

+ v−q−, v2k−3+ v2

−q+ ≥ 0.The dominant term (5.59) is estimated by

2

∫Ω1/4

v2k−1+ q+

(5.46)= 2π(1− 2

4)

∫ 2π

0

ˆv2k−1

(s)q(ˆv(s))ds

≥ πM2k−1‖q(ˆv)‖L1(T) (5.62)

because λ2k−1q(λ) ≥M2k−1|q(λ)|.The three terms in (5.60)-(5.61) are estimated by∣∣∣∣∣2

∫Ω1/4

v2k−1+ q−

∣∣∣∣∣ ≤ 2

∫Ω

∣∣v2k−1+

∣∣ |q−| ≤ ‖ˆv‖2k−1L2k−1(T)

‖q(ˆv)‖L1(T)

∣∣∣∣∣2∫

Ω1/4

(v2k−2

+ q+

)v−

∣∣∣∣∣ ≤ ‖ˆv2k−2q(ˆv)‖L1(T)‖ˆv‖L1(T)

≤ (2M)2k−2‖q(ˆv)‖L1(T)‖ˆv‖L1(T)

and ∣∣∣∣∣2∫

Ω1/4

v2k−3+

(v2−q−

)∣∣∣∣∣ ≤ ‖ˆv2k−3‖L1(T)‖ˆv2q(ˆv)‖L1(T)

≤ (2M)2‖ˆv‖2k−3L2k−3(T)

‖q(ˆv)‖L1(T) .


By the previous inequalities, (5.62), Holder inequality4 and using the L2k-estimate (5.56) for v∫

Ω1/4

v2k−1φ ≥ πM2k−1‖q(ˆv)‖L1(T) − κ7(M2k−2 + 1)‖q(ˆv)‖L1(T) . (5.63)

By (5.58) and (5.63)

M2k−1‖q(ˆv)‖L1(T) ≤ κ8

(‖v‖2k−2

L∞(Ω) +M2k−2 + 1)‖q(ˆv)‖L1(T)

≤ κ9

(M2k−2 + 1

)‖q(ˆv)‖L1(T)

and finally M2k−1 ≤ κ9(M2k−2 + 1). By our choice of M := 1

2‖ˆv‖L∞(T) (see

(5.25)) the L∞-estimate follows

‖v‖L∞ ≤ κ10 for |ε| ≤ ε2(R) . (5.64)

5.4.3 Step 3: The H1-estimate.

Inserting the admissible variation φ = −D−hDhv in the variational inequality(5.42)

‖φ‖H1

(5.42)

≥∫

Ω

(H + v + w)2kφ

(5.33)=

∫Ω

Dh(H + v + w)2kDhv(5.35)→

∫Ω

[(H + v + w)2k]tvt(5.65)

as h→ 0 since ∂t[(H + v + w)2k] ∈ L2.Since ‖w‖E ≤ C0(R)|ε|, by (5.64) and the Cauchy-Schwartz inequality∫Ω

[(H + v + w)2k

]tvt = 2k

∫Ω

(H + v + w)2k−1(Ht + vt + wt)vt

≥ 2k

∫Ω

(v +H)2k−1v2t − o(1)‖vt‖2

L2 − κ11‖vt‖L2

and by (5.65) ∫Ω

(v +H)2k−1v2t ≤ κ12‖vt‖L2 + o(1)‖vt‖2

L2 . (5.66)

4To estimate ‖ˆv‖Lj(T) ≤ Cj,k‖ˆv‖L2k(T) for j < 2k.


Since v, vt ∈ N then∫

Ωv2k−1v2

t = 0 by Lemma 2.3.1, using the elementaryinequality5

(a+ b)2k−1 − a2k−1 ≥ 41−k b2k−1 , ∀ a ∈ R , b > 0

gives ∫Ω

(v +H)2k−1v2t =

∫Ω

[(v +H)2k−1 − v2k−1

]v2

t

≥ 41−k

∫Ω

H2k−1v2t ≥ 41−kc1(H

2k−1)

∫Ω

v2t (5.67)

where c1(·) was defined in (5.44). By (5.66) and (5.67)

‖vt‖2L2 ≤ κ13‖vt‖L2 + o(1)‖vt‖2

L2

and we finally deduce that there exists a 0 < ε3(R) ≤ ε2(R) such that

‖v‖H1 < κ14 ∀ |ε| ≤ ε3(R) .

Defining R∗ := κ14 and ε∗ := ε3(R∗) we obtain that

‖v(ε)‖H1 < R∗ ∀ |ε| ≤ ε∗

is an interior minimum of Φ in BR∗ := ‖v‖H1 < R∗. Recalling the changeof variables (5.40),

u := ε(H + v(ε2k) + w(ε2k, v(ε2k))

)∈ E

is a weak solution of (5.1) with f = u2k + h.

We shall not prove the higher regularity of the solution. In view of theregularity result of [41] -which holds just for strictly monotone nonlinearities-it is somewhat surprising that the weak solution u of Theorem 5.4.1 is actuallysmooth.

5.4.4 The “maximum principle”

Theorem 5.4.2 ([21]) Let h ∈ N⊥, h > 0 a.e. in Ω. Then there exists aweak solution H ∈ E of (5.41) satisfying H > 0. In particular

H(t, x) :=1

2

∫ κ

0

∫ t−x+ξ

t−x−ξ

h(τ, ξ) dτ dξ − 1

2

∫ x

κ

∫ t+x−ξ

t−x+ξ

h(τ, ξ) dτ dξ (5.68)

for a suitable κ ∈ (0, π).

5Dividing by b2k−1 and setting x := a/b, we have to prove f(x) := (x+1)2k−1− x2k−1 ≥41−k ∀x ∈ R. An elementary calculus shows that x = −1/2 is the unique minimum pointof f(x) and f(−1/2) = 41−k.


Proof. Step 1: H(t, x) ∈ H1(Ω) ∩ C1/2(Ω) for any κ ∈ (0, π) and

2(∂tH) =

∫ κ

0

(h(t− x+ ξ, ξ)− h(t− x− ξ, ξ)

)dξ

−∫ x

κ

(h(t+ x− ξ, ξ)− h(t− x+ ξ, ξ)

)dξ ∈ L2(Ω) ,(5.69)

2(∂xH) =

∫ κ

0

(− h(t− x+ ξ, ξ) + h(t− x− ξ, ξ)

)dξ

−∫ x

κ

(h(t+ x− ξ, ξ) + h(t− x+ ξ, ξ)

)dξ ∈ L2(Ω) .(5.70)

We shall prove that the first addendum in the r.h.s. of (5.68)

H1(t, x) :=1

2

∫ κ

0

∫ t−x+ξ

t−x−ξ

h(τ, ξ) dτ dξ

belongs to C1/2(Ω)∩H1(Ω), the second addendum being analogous. Defining

T (t, x) :=

(τ, ξ) ∈ Ω∣∣∣ t− x− ξ < τ < t− x+ ξ , 0 < ξ < κ

we can write H1(t, x) := (1/2)

∫T (t,x)

h(τ, ξ) dτ dξ. Since |T (t, x)| = κ2 ≤ π2

|H1(t, x)| ≤1

2

∫Ω

1T (t,x)(τ, ξ)|h(τ, ξ)| dτ dξ ≤π

2‖h‖L2(Ω)

using Cauchy-Schwartz inequality.For i = 1, 2 and (ti, xi) ∈ Ω, let define Ti :=T (ti, xi). It results

|T1 \ T2| = |T2 \ T1| ≤ π(|t1 − t2|+ |x1 − x2|)

and, using again Cauchy-Schwartz inequality,

|H1(t1, x1)−H1(t2, x2)| ≤ 1

2

∫T1\T2

|h|+ 1

2

∫T2\T1

|h|

≤√π(|t1 − t2|+ |x1 − x2|)1/2‖h‖L2(Ω)

and H1 ∈ C1/2(Ω).We now prove that H1 ∈ H1(Ω) and that ∂tH1 = −∂xH1 =f1 where

f1(t, x) :=1

2

∫ κ

0

(h(t− x+ ξ, ξ)− h(t− x− ξ, ξ)

)dξ ∈ L2(Ω) .


We first justify that f1 ∈ L2(Ω). Since

|h(t− x+ ξ, ξ)|2 + |h(t− x− ξ, ξ)|2 ≥ 1

2

∣∣∣h(t− x+ ξ, ξ)

− h(t− x− ξ, ξ)∣∣∣2

by periodicity w.r.t. t we obtain that, ∀x ∈ (0, π),

‖h‖2L2(Ω) ≥

∫ κ

0

∫ 2π

0

|h(t, ξ)|2dt dξ

≥ 1

4

∫ κ

0

∫ 2π

0

|h(t− x+ ξ, ξ)− h(t− x− ξ, ξ)|2dt dξ .

Integrating the previous inequality in the variable x between 0 and π, apply-ing Fubini Theorem and Cauchy-Schwartz inequality, we deduce

π‖h‖2L2(Ω) ≥ 1

4

∫ π

0

∫ κ

0

∫ 2π

0

|h(t− x+ ξ, ξ)− h(t− x− ξ, ξ)|2 dt dξ dx

=1

4

∫Ω

∫ κ

0

|h(t− x+ ξ, ξ)− h(t− x− ξ, ξ)|2 dξ dt dx

≥ 1

4κ

∫Ω

(∫ κ

0

|h(t− x+ ξ, ξ)− h(t− x− ξ, ξ)| dξ)2

dt dx .

≥ 1

κ

∫Ω

f 21 (t, x) dt dx =

1

κ‖f1‖2

L2 .

Finally, we prove that ∂xH1 = −f1, being ∂tH1 = f1 analogous. By FubiniTheorem

2

∫Ω

H1φx =

∫ 2π

0

∫ κ

0

∫ π

0

∫ t−x+ξ

t−x−ξ

h(τ, ξ)φx(t, x) dτ dx dξ dt

=

∫ 2π

0

∫ κ

0

( ∫ t+ξ

t+ξ−π

h

∫ t+ξ−τ

0

φx dx dτ +

∫ t+ξ−π

t−ξ

h

∫ π

0

φx dx dτ

+

∫ t−ξ

t−ξ−π

h

∫ π

t−ξ−τ

φx dx dτ)dξ dt

=

∫ 2π

0

∫ κ

0

( ∫ t+ξ

t+ξ−π

h(τ, ξ)φ(t, t+ ξ − τ) dτ

−∫ t−ξ

t−ξ−π

h(τ, ξ)φ(t, t− ξ − τ) dτ)dξ dt

=

∫ 2π

0

∫ κ

0

(−

∫ 0

π

h(t+ ξ − x, ξ)φ(t, x) dx

+

∫ 0

π

h(t− ξ − x, ξ)φ(t, x) dx)dξ dt = 2

∫Ω

f1φ .


With analogue computations for the second addendum of H in (5.68) wederive (5.69) and (5.70).

Step 2: There exists κ ∈ (0, π) such that H(t, x) verifies the Dirichletboundary conditions H(t, 0) = H(t, π) = 0 ∀ t ∈ T.

By (5.68), the function H satisfies, for any κ ∈ (0, π), H(t, 0) = 0 ∀ t ∈ T.It remains to find κ imposing H(t, π) = 0. Taking x = π in (5.68) we obtain

H(t, π) :=1

2

∫ κ

0

∫ t−π+ξ

t−π−ξ


2

∫ π

κ

∫ t+π−ξ

t−π+ξ

h(τ, ξ) dτ dξ

=1

2

∫ π

0

∫ t−π+ξ

t−π−ξ


2

∫ π

κ

∫ t+π−ξ

t−π−ξ

h(τ, ξ) dτ dξ

=1

2

∫ π

0

∫ ξ

−ξ


2

∫ π

κ

∫ π

−π

h(τ, ξ) dτ dξ

=: c− χ(κ)

where in the last line we have used the periodicity of h(·, ξ). ) χ(κ) is acontinuous function, χ(0) > c > 0, χ(π) = 0 and therefore there existsκ ∈ (0, π) solving χ(κ) = c.

Step 3: H ∈ E is a weak solution of (5.41) namely∫Ω

φtHt − φxHx + φh = 0 , ∀φ ∈ C10(Ω) . (5.71)

By Fubini Theorem and periodicity we get∫Ω

(φt(t, x)

∫ κ

0

h(t− x+ ξ, ξ)dξ + φx(t, x)

∫ κ

0

h(t− x+ ξ, ξ)dξ)dt dx

=

∫ π

0

∫ κ

0

∫ 2π

0

(φt(t, x)h(t− x+ ξ, ξ) + φx(t, x)h(t− x+ ξ, ξ)

)dt dξ dx

=

∫ π

0

∫ κ

0

∫ 2π

0

(φt(t+ x− ξ, x) + φx(t+ x− ξ, x)

)h(t, ξ) dt dξ dx

=

∫ κ

0

∫ 2π

0

h(t, ξ)

∫ π

0

d

dx

(φ(t+ x− ξ, x)

)dx dt dξ = 0 (5.72)

by Dirichlet boundary conditions. Analogously,

−∫

Ω

(φt(t, x)

∫ κ

0

h(t− x− ξ, ξ)dξ+ φx(t, x)

∫ κ

0

h(t− x− ξ, ξ)dξ)dt dx = 0 .

(5.73)


Moreover, again by Fubini Theorem,∫Ω

(− φt(t, x)

∫ x

κ

h(t+ x− ξ, ξ)dξ + φx(t, x)

∫ x

κ

h(t+ x− ξ, ξ)dξ)dt dx

=

∫ π

0

∫ x

κ

∫ 2π

0

(− φt(t, x)h(t+ x− ξ, ξ) + φx(t, x)h(t+ x− ξ, ξ)

)dt dξ dx

=

∫ π

0

∫ x

κ

∫ 2π

0

(− φt(t− x+ ξ, x) + φx(t− x+ ξ, x)

)h(t, ξ) dt dξ dx

=

∫ 2π

0

∫ π

0

h(t, ξ)

∫ π

ξ

d

dx

(φ(t− x+ ξ, x)

)dx dt dξ

−∫ 2π

0

∫ κ

0

h(t, ξ)

∫ π

0

d

dx

(φ(t− x+ ξ, x)

)dx dt dξ = −

∫Ω

hφ (5.74)

and, analogously,∫Ω

(φt(t, x)

∫ x

κ

h(t−x+ξ, ξ)dξ+φx(t, x)

∫ x

κ

h(t−x+ξ, ξ)dξ)dt dx = −

∫Ω

hφ .

(5.75)Summing (5.72), (5.73), (5.74), (5.75) and recalling (5.69), (5.70) we get(5.71).

Step 4: H(t, x) > 0 in Ω.

First case: 0 < x ≤ κ. By (5.68) and geometrical considerations on the do-mains of the integrals, we derive that, for 0 < x < κ,H(t, x) =

∫Θh(τ, ξ) dτdξ

where Θ := Θt,x is the trapezoidal region in Ω with a vertex in (τ, ξ) = (t, x)and delimited by the straight lines τ = t− x + ξ, τ = t + x− ξ, × = κ andτ = t− x− ξ. Since h > 0 a.e. in Ω we conclude that H(t, x) > 0.

Second case: κ < x < π. Since H(t+ π − x, π) = 0 we have, by (5.68),∫ κ

0

∫ t−x+ξ

t−x−ξ

h(τ, ξ) dτdξ =

∫ π

κ

∫ t−x−ξ+2π

t−x+ξ

h(τ, ξ) dτdξ .

Therefore, substituting in (5.68), we get, for κ < x < π, the expressionH(t, x) =

∫Θh(τ, ξ) dτdξ where, now, Θ := Θt,x is the trapezoidal region

in Ω with a vertex in (τ, ξ) = (t, x) and delimited by the straight linesτ = t− x+ ξ, τ = t− x− ξ + 2π, ξ = κ and τ = t+ x− ξ. Since h > 0 a.e.in Ω we conclude also in this case that H(t, x) > 0.

Chapter 6

Appendix

6.1 Hamiltonian PDEs

b) The nonlinear Schrodinger equation

(NLS)

i ∂tu−∆u+ f(x, |u|2)u = 0u(x+ 2πk) = u(x) ∀k ∈ Zd , ∀x ∈ Rd

(also Dirichlet or Newmann boundary conditions can be considered) can bewritten in complex Hamiltonian form

∂tu = J∇uH

as in (1) whereJ := iI

and with Hamiltonian

H :=

∫Td

|∇u|2 + F (x, |u|2) dx

where (∂sF )(x, s) = f(x, s) and ∇uH = −∆u + f(x, |u|2)u is the gradientw.r.t. the complex L2-scalar product.

c) The beam equation utt + uxxxx + f(x, u) = 0u(t, 0) = u(t, π) = 0uxx(t, 0) = uxx(t, π) = 0

with Dirichlet and hinged boundary conditions, is, like the wave equation, asecond order Lagrangian equation with Lagrangian

L(u, ut) :=

∫ π

0

u2t

2− u2

xx

2− F (x, u) dx

149

CHAPTER 6. APPENDIX 150

where ∂uF (x, u) = f(x, u). The Hamiltonian of the beam equation is

H(u, p) :=

∫ π

0

p2

2+u2

xx

2+ F (x, u) dx

with symplectic structure

J =

(0 I−I 0

).

d) The KdV equationut − uux + uxxx = 0u(x+ 2π) = u(x) (resp. u(x) → 0 as |x| → +∞)

can be written in the Hamiltonian form (1) with Hamiltonian

H(u) :=

∫T

(u2x

2+u3

6

)dx

defined on the phase space H1(T) of functions u periodic in x, resp. on thespace H1(R) of functions vanishing at infinity (solitons).

The antisymmetric non-degenerate1 operator is

J := ∂x

and ∇H(u) = −uxx + (u2/2) is the gradient w.r.t the L2-scalar product.Other Hamiltonian structures are possible to express the KdV equation,

see for example [73] and [79].

6.1.1 The Euler equations of hydrodynamics

The Hamiltonian formulation of the gravity-capillarity water waves problemdescribing the evolution of a perfect, incompressible, irrotational fluid underthe action of gravity and surface tension, is more complex and it is due toZakharov [130].

The unknowns of the problem are the free surface y = η(x), which boundsthe variable fluid domain

Sη :=

(x, y) ∈ Rn−1 ×R | − h ≤ y ≤ η(x)

with n = 2, 3, and the velocity potential Φ : Sη → R. i.e. the irrotationalvelocity field u = ∇Φ.

1∂x is only weakly non-degenerate since vanishes on the constants.


The gravity-capillarity water-waves problem, see e.g. [122], can be writtenas

∂tΦ + 12|∇Φ|2 + gη = στ at y = η(x) (Bernoulli condition)

∆Φ = 0 in Sη (incompressibility)∂yΦ = 0 at y = −h (impermaebility)∂tη = ∂yΦ− ∂xη · ∂xΦ at y = η(x) (kinematic condition)

(6.1)where g is the acceleration of gravity, σ is the surface tension,

τ := div∂xη√

1 + |∂xη|2

is the mean curvature (we suppose the density of the fluid ρ = 1). Furtherboundary conditions are

η ,∇Φ → 0 for |x| → +∞

(solitons) or periodic boundary conditions

η(x+ Λ) = η(x) , Φ(x+ Λ, y) = Φ(x, y) , ∀x ∈ Rn−1 ,∀Λ ∈ Γ

where Γ is a lattice (cnoidal waves). To fix the ideas we assume in the sequelperiodic boundary conditions x ∈ Tn−1, i.e. Γ = 2πZn−1.

To write the evolution problem (6.1) as an infinite dimensional Hamil-tonian system note that the profile η(x) of the fluid and the value ξ(x) :=Φ(x, η(x)) of the velocity potential Φ restricted on the free boundary uniquelydetermine the velocity potential Φ in the whole Sη, solving the elliptic prob-lem

∆Φ = 0 in Sη

∂yΦ = 0 at y = −hΦ = ξ at y = η(x)

(6.2)

with Φ periodic in x.We claim that (6.1) can be written, in the variables (ξ, η), as the Hamil-

tonian system ∂tη = δξH∂tξ = −δηH

(6.3)

where the Hamiltonian

H =

∫Sη

1

2|∇Φ|2 dxdy +

∫Sη

gy dxdy + σ

∫Tn−1

√1 + |∂xη|2 dx

:= K + U + σA


is the sum of the kinetic energy, the potential energy and the area surfaceintegral, expressed in the variables (ξ, η) and δη, δξ are the L2-gradients ofH. Hence (ξ, η) are canonically conjugated “Darboux coordinates”.

The potential energy is

U =

∫Sη

gy dxdy =g

2

∫Tn−1

η2(x) dx+ constant .

Using the divergence theorem, ∆Φ = 0, the periodicity and the Neumanncondition at the bottom, the kinetic energy can be expressed as

K(ξ, η) =1

2

∫Sη

|∇Φ|2 dxdy =1

2

∫Tn−1

ξ G(η)[ξ] dx (6.4)

where

N(x) :=(−∂xη(x), 1)√1 + |∂xη(x)|2

is the outer normal, and

G(η)[ξ] := ∇Φ(x, η(x)) ·N(x)√

1 + (∂xη(x))2

= (∂yΦ)(x, η(x))− (∂xΦ)(x, η(x)) · ∂xη(x)

is the so called Dirichlet-Neumann operator, see [47].The operator G is linear in ξ, symmetric w.r.t. the L2 scalar product and

semi-positive definite. The nonlinear map η 7→ G(η) is analytic as a functionfrom η ∈ C1 | ‖η‖C1 < R into L(H1, L2), see e.g. [48].

In conclusion the Hamiltonian is

H(ξ, η) =1

2

∫Tn−1

ξ G(η)[ξ] dx+g

2

∫Tn−1

η2 dx+ σ

∫Tn−1

√1 + |∂xη|2 dx .

Let us prove (6.3). By the symmetry of G(η), the partial derivative δξH withrespect to the L2-scalar product is

δξH(ξ, η) = G(η)[ξ] (6.5)

and the forth equation in (6.1) coincides with the first equation of (6.3)

∂tη = G(η)[ξ] = δξH(ξ, η) .

It’s also easy to get the partial derivatives

δηU = gη , δηA = −τ(η) . (6.6)


It remains to compute the partial derivative of K with respect to the shapedomain η,

(δηK, ν)L2 =∂

∂µ |µ=0

∫Sη+µν

|∇Φη+µν |2

2(6.7)

where Φη+µν is the solution of (6.2) in Sη+µν with Dirichlet datum ξ on thevaried boundary y = η(x) + µν(x).

Differentiating w.r.t. µ in (6.7) the term

1

2

∫Sη+µν

|∇Φη+µν |2 dxdy =1

2

∫Tn−1

( ∫ η(x)+µν(x)

−h

|∇Φη+µν |2 dy)dx

yields

(δηK, ν)L2 =1

2

∫Tn−1

|∇Φη(x, η(x))|2ν(x) dx +1

2

∂P

∂µ |µ=0

(6.8)

where, like in (6.4), we can write

P :=

∫Sη

|∇Φη+µν |2 dx dy =

∫Tn−1

ξµG(η)[ξµ] dx

with2 ξµ(x) := Φη+µν(x, η(x)). To compute

1

2

∂P

∂µ |µ=0

=

∫Tn−1

( ∂

∂µ |µ=0

ξµ

)G(η)[ξ] dx (6.9)

it remains to find

∂

∂µ |µ=0

ξµ(x) = ∂BCΦη(x, η(x))[ν] (6.10)

where ∂BC denotes the derivative of the map f 7→ Φf .Differentiating the identity

Φη+µν(x, η(x) + µν(x)) = ξ(x) ∀µ

at µ = 0, yields

∂BCΦη(x, η(x))[ν] = −∂yΦη(x, η(x))ν(x) (6.11)

and (6.8), (6.9), (6.10), (6.11) give

δηK =1

2|∇Φ(x, η(x))|2 − (∂yΦ)(x, η(x))G(η)[ξ] . (6.12)

2We can suppose ν(x) > 0 decomposing in positive and negative part ν = ν+ − ν−.Hence, for µ > 0, Sη ⊂ Sη+µν .


Differentiating ξ(x, t) = Φ(x, η(x, t), t) with respect to time, thanks to thefirst equation in (6.1) and ∂tη = G(η)[ξ] we get

∂tξ = ∂yΦ ∂tη + ∂tΦ = ∂yΦG(η)[ξ]− 1

2|∇Φ|2 − gη + στ

= −δηH(ξ, η)

by (6.6) and (6.12), namely the second equation in (6.3).

Remark 6.1.1 We could express the right hand side of (6.12) through thequantities (ξ, η) only, yielding

δηK =1

2

1

(1 + |∂xη|2)

[|∂xξ|2 − (G(η)[ξ])2 − 2∂xη · ∂xξ G(η)[ξ]

+ |∂xη|2|∂xξ|2 − (∂xη · ∂xξ)2],

see e.g. [47]. Note that the last term vanishes for n = 2.

6.2 Critical point theory

We find convenient to shortly collect the basic ideas and techniques in criticalpoint theory which are used throughout these Lectures.

6.2.1 Preliminaries

Let Φ : X → R be a C1-functional defined on a real Banach space X withnorm ‖ · ‖.

Let DΦ(x) ∈ L(X,R) denote the derivative of Φ at the point x, namelythe unique linear and continuous operator from X to R such that

|Φ(x+ h)− Φ(x)−DΦ(x)[h]| = o(‖h‖) as ‖h‖ → 0 .

For any L ∈ L(X,R) =: X∗ in the dual space we define the operatorial norm

‖L‖∗ := sup‖x‖≤1

|Lx| .

Definition 6.2.1 The point x ∈ X is called a critical point if DΦ(x) = 0 .The corresponding value Φ(x) ∈ R is called a critical value.


When X is a Hilbert space with scalar product (·, ·), we denote by∇Φ(x) ∈ X the gradient of Φ at the point x, uniquely defined, via theRiesz theorem, by

(∇Φ(x), h) = DΦ(x)[h] , ∀h ∈ X .

We have ‖∇Φ(x)‖ = ‖DΦ(x)‖∗.Given a C1 function g : X → R, the set

M := g−1(0) =x ∈ X | g(x) = 0

is a C1-manifold if Dg(x) 6= 0, ∀x ∈ M (by the implicit function theorem),with tangent space at x

TxM :=h ∈ X | Dg(x)[h] = 0

.

Definition 6.2.2 The point x ∈ M is called a critical point of Φ : M ⊂X → R constrained to M if DΦ(x)[h] = 0, ∀h ∈ TxM .

When X is a Hilbert space and x ∈M is a critical point of Φ constrainedto M , we have

∇Φ(x) = λ∇g(x)

for some λ ∈ R, because ∇g(x) spans the orthogonal space to TxM . This iscalled the Lagrange multiplier rule.

6.2.2 Minima

The simplest critical points to look for are the absolute minima and maxima(as we deal with in Chapter 5).

We first recall the following classical result

Theorem 6.2.1 (Weierstrass) Let Φ : X → R be lower semicontinuous,i.e. ∀x,

∀xn → x =⇒ lim infn

Φ(xn) ≥ Φ(x) .

Then, on any compact set K ⊂ X, Φ is bounded from below and attainsabsolute minimum, i.e. there is x ∈ K such that Φ(x) = infK Φ.

Proof. Take a minimizing sequence xn ∈ K, namely such that

Φ(xn) → infx∈K

Φ(x) .


By the compactness of K, up to subsequence, xn → x for some x ∈ K. Bythe lower semicontinuity property of Φ

infK

Φ = limn

Φ(xn) = lim infn

Φ(xn) ≥ Φ(x)

and, therefore, infK Φ = Φ(x), i.e. infK Φ is finite and x ∈ K is an absoluteminimum of Φ in K.

When X is infinite dimensional, bounded sets are not, in general, precom-pact (≡ set with compact closure) in the strong topology, and it is convenientto introduce the weak topology on X. In this weaker topology bounded setsare precompact, see [39]. We write xn x to indicate that xn converges tox in the weak topology.

The following theorem is the main tool in the so called “direct methods”in the calculus of variations.

Theorem 6.2.2 Let X be a reflexive Banach space and Φ : X → R be afunctional satisfying the following conditions:

(Coercivity) Φ(xn) → +∞ for any sequence ‖xn‖ → +∞.

(Sequential weakly lower semicontinuity) ∀x ∈ X, ∀xn x thenlim infn Φ(xn) ≥ Φ(x).

Then Φ is bounded from below and attains absolute minimum on X.

Proof. Let xn ∈ X be a minimizing sequence, namely Φ(xn) → infX Φ.By the coercivity property, the sequence ‖xn‖ is bounded.

Since X is reflexive, up to subsequence, xn x (see e.g. [39] theoremIII.27). By the weakly lower semicontinuity property

infX

Φ = limn

Φ(xn) = lim infn

Φ(xn) ≥ Φ(x)

and, therefore, Φ(x) = infX Φ.

In several applications it happens that the functional Φ is not boundedbelow (neither above); or that the absolute (or local) minima correspond totrivial solutions of the problem (for example, in Chapter 2 the trivial solutionv = 0 is a local minimum of the functional Φε defined in (2.31)).

It is therefore necessary to develop techniques to find “saddle” criticalpoints of unbounded functionals.


6.2.3 The minimax idea

Critical values c of a function Φ : Rn → R should be found where thetopological nature of the sublevels

Ac := Φ−1(−∞, c]

changes, according to the guiding principle that, if there are no critical pointsin Φ−1([a, b]), the sublevel Ab could be deformed into Aa along the trajectoriesof the negative gradient flow

η(t, x) = −∇Φ(η(t, x))η(0, x) = x

where t→ Φ(η(t, x)) is a decreasing function, since

d

dtΦ(η(t, x)) = −‖∇Φ(η(t, x))‖2 .

However, even in finite dimension, saddle critical points need not in generalexist unless some compactness property hold, as illustrated by the followingelementary function Φ : R2 → R in [123],

Φ(x, y) := e−y − x2 .

The sublevels Ac are disconnected sets for c < 0, Ac are connected for c > 0,but there are no critical points at the level c = 0.

The following strong compactness condition has been introduced by Palaisand Smale [103], and guarantees a critical point in many variational problems.

Definition 6.2.3 (Palais-Smale) (i) A sequence xn ∈ X is called a Palais-Smale sequence at the level c ((PS)c for short), if

f(xn) → c and ‖Df(xn)‖X∗ → 0 .

(ii) The functional Φ : X → R is said to satisfy the Palais-Smale conditionat the level c ((PS)c for short), if every (PS)c sequence xn ∈ X is precompactin X, namely xn possesses a convergent subsequence xnk

→ x.

Remark that, if Φ ∈ C1(X,R) satisfies the (PS)c condition, any accumu-lation point x of a (PS)c sequence xn, is a critical point of Φ at the level c,namely DΦ(x) = 0 and Φ(x) = c.

The minimax principle is highlighted in the following theorem [102].


Theorem 6.2.3 (Minimax) Consider Φ : X → R where X is a Hilbert3

space and suppose η = −∇Φ(η) defines a flow η(t, ·) : X → X, ∀t ≥ 0.

(i) Let F ⊂ X be a family of subsets positively invariant under the flow,i.e. if F ∈ F then η(t, F ) ∈ F , ∀t > 0.

(ii) The “minimax” level c := infF∈F supx∈F Φ(x) is finite.

(iii) Φ satisfies (PS)c.

Then c is a critical value.

Proof. We shall prove that for every µ > 0 there exists x ∈ X such that

c− µ ≤ Φ(x) ≤ c+ µ and ‖∇Φ(x)‖ ≤ µ .

Then, choosing µn := 1/n, we find a Palais-Smale sequence xn at the levelc. By the (PS)c condition, xn converges to some critical point in X at thelevel c.

Arguing by contradiction, there exists µ > 0 such that

c− µ ≤ Φ(x) ≤ c+ µ =⇒ ‖∇Φ(x)‖ > µ . (6.13)

By the definition of c there exists an F ∈ F such that supx∈F Φ(x) ≤ c+ µ.We claim that

η(t∗, F ) ∈ Ac−µ for t∗ :=2

µ. (6.14)

Since, by (i), F ∗ := η(t∗, F ) ∈ F , (6.14) implies the contradiction

c := infF∈F

supx∈F

Φ(x) ≤ supx∈F ∗

Φ(x) ≤ c− µ .

To prove (6.14), pick any x ∈ Ac+µ. If Φ(η(t∗, x)) ≤ c − µ we have nothingto prove. If Φ(η(t∗, x)) > c− µ then c− µ < Φ(η(t, x)) ≤ c + µ, ∀t ∈ [0, t∗],because t → Φ(η(t, x)) is decreasing. Hence, by (6.13), ‖∇Φ(η(t, x))‖ > µ,and

Φ(η(t∗, x)) = Φ(x)−∫ t∗

0

‖∇Φ(η(t, x))‖2 dt < c+ µ− t∗µ2 = c− µ .

This contradiction proves the claim (6.14).

Concerning the verification of the Palais-Smale condition, the first step isoften to prove that a Palais-Smale sequence xn is bounded, i.e. there existsC > 0 such that ‖xn‖ ≤ C, ∀n.

This condition is sufficient for example in the following situation:

3The same statement holds when X is a complete Finsler manifold of class C1,1, seee.g. [123]. In this case, a “rich” topology of X implies existence of critical points.


Lemma 6.2.1 Suppose DΦ = L + K where L : X → X∗ is an invertiblelinear map with bounded inverse, and K : X → X∗ is compact (i.e. mapsbounded sets in X in precompact sets in X∗). Then any bounded (PS)c

sequence xn ∈ X is precompact.

Proof. We have DΦ(xn) = Lxn +K(xn) → 0 whence

xn = L−1DΦ(xn)− L−1K(xn) . (6.15)

Since xn is bounded and L−1K is compact then, up to subsequence, L−1K(xn) →x in X. We deduce by (6.15) that xn → −x.

6.2.4 The Mountain Pass Theorem

There is no general recipe for choosing the minimax family F , the choicehaving to reflect some topological change in the nature of the sublevels of Φ.

An important minimax result with a broad spectrum of applications isthe celebrated Mountain Pass Theorem due to Ambrosetti and Rabinowitz[7].

Theorem 6.2.4 (Mountain Pass) Suppose Φ ∈ C1(X,R) and

(i) Φ(0) = 0;

(ii) ∃ ρ, α > 0 such that Φ(x) ≥ α if ‖x‖ = ρ;

(iii) ∃ v ∈ X with ‖v‖ > ρ such that Φ(v) < 0.

Define the “Mountain Pass” value

c := infγ∈Γ

maxt∈[0,1]

Φ(γ(t)) ≥ α

where Γ is the minimax class

Γ :=γ ∈ C([0, 1], X) | γ(0) = 0 , γ(1) = v

.

Then there exists a Palais-Smale sequence of Φ at the level c.As a consequence, if Φ satisfies the (PS)c condition, then c is a critical

value for Φ: there exists x 6= 0 such that DΦ(x) = 0 and Φ(x) = c.


Geometrical Interpretation: Think the graph of Φ as a landscape with alow spot at x = 0 surrounded by a ring of mountains. Beyond these moun-tains lies another low spot at the point v. Then there must be a MountainPass between 0 and v, containing a critical point. From a topologically pointof view, the sublevel Ac−ε is not arcwise connected.

Proof. For simplicity we shall give the proof in the case4 X is a Hilbertspace and ∇Φ is locally Lipschitz continuous. In remark 6.2.1 we explainhow to deal with the general case.

First note that for any γ ∈ Γ the intersection γ([0, 1]) ∩ Sρ 6= ∅ whereSρ := x ∈ X | ‖x‖ = ρ. Therefore

maxt∈[0,1]

Φ(γ(t)) ≥ minx∈Sρ

Φ(x)(ii)

≥ α > 0

andc := inf

γ∈Γmaxt∈[0,1]

Φ(γ(t)) ≥ α > 0 .

Next, we claim that, ∀µ > 0, there exists x ∈ X such that

c− µ ≤ Φ(x) ≤ c+ µ and ‖∇Φ(x)‖ < 2µ .

Then, choosing µn = 1/n we find a Palais-Smale sequence xn at the level c.

The Deformation Argument. Arguing by contradiction suppose thereexists µ > 0 such that

∀x ∈ c− µ ≤ Φ(x) ≤ c+ µ ⇒ ‖∇Φ(x)‖ ≥ 2µ . (6.16)

Let us consider the sets

N := x ∈ X | |Φ(x)− c| ≤ µ and ‖∇Φ(x)‖ ≥ 2µ

N := x ∈ X | |Φ(x)− c| < 2µ and ‖∇Φ(x)‖ > µ

(N ⊂ N). Therefore N and N c are closed, disjoints sets and there exists alocally Lipschitz non-negative function g : X → [0, 1] such that

g =

1 on N0 on N c ,

4These conditions are met in particular in the applications of these Lecture Notes: inChapters 1 and 2 the space X is a Hilbert space (actually finite dimensional in Chapter1) and ∇Φ ∈ C1, see remark 2.2.4.


for example

g(x) :=d(x,N c)

d(x,N c) + d(x, N)

(here d(x, ·) denotes the distance function).Next consider the locally Lipschitz and bounded vector field

X(x) := −g(x) ∇Φ(x)

‖∇Φ(x)‖(6.17)

(note that X(x) is well defined because if ‖∇Φ(x)‖ < µ then x ∈ N c and sog(x) = 0).

For each x ∈ X, the unique solution of the Cauchy problemddtη(t, x) = X(η(t, x))

η(0, x) = x(6.18)

is defined for all t ∈ R (since X is bounded) and

• (i) η(t, ·) is a homeomorphism of X.

Furthermore, since X(x) ≡ 0 for x ∈ N c

• (ii) η(t, x) = x, ∀t, if |Φ(x)− c| ≥ 2µ or if ‖∇Φ(x)‖ ≤ µ .

Finally

• (iii) ∀x ∈ X, ∀t ∈ R

d

dtΦ(η(t, x)) = −g(η(t, x))‖∇Φ(η(t, x))‖ ≤ 0 . (6.19)

Conclusion of the Mountain Pass Theorem. By the definition of c > 0there exists a path γ ∈ Γ such that

maxt∈[0,1]

Φ(γ(t)) ≤ c+ µ .

Since Φ(0) = 0 and Φ(v) ≤ 0, by (ii), if 2µ < c,

η(t, 0) = 0 , η(t, v) = v , ∀t . (6.20)

Furthermore, by (i), η(t, γ(·)) ∈ C([0, 1]) and therefore η(t, ·) γ ∈ Γ (theminimax family Γ is invariant under the deformations η).

But we claim that

Φ(η(1, γ([0, 1])) ≤ c− µ (6.21)


implying the contradiction

c := infγ∈Γ

maxt∈[0,1]

Φ(γ(t)) ≤ maxt∈[0,1]

Φ(η(1, γ(t))) ≤ c− µ .

Let us prove (6.21). Pick a point x ∈ γ([0, 1]). If Φ(η(t, x)) ≤ c− µ for some0 ≤ t ≤ 1 there is nothing to prove since Φ(η(t, x)) is decreasing. Assume bycontradiction that Φ(η(t, x)) > c− µ for all t ∈ [0, 1]. Therefore, by (6.16),

c− µ ≤ Φ(η(t, x)) ≤ c+ µ and ‖∇Φ(η(t, x))‖ ≥ 2µ ,

i.e. η(t, x) ∈ N , ∀t ∈ [0, 1]. By (6.19), since g ≡ 1 on N ,

Φ(η(1, x)) = Φ(x)−∫ 1

0

g(η(t, x)) ‖∇Φ(η(t, x))‖dt

≤ (c+ µ)−∫ 1

0

‖∇Φ(η(t, x))‖dt ≤ (c+ µ)− 2µ = c− µ .

This concludes the proof of the Theorem.

The following example from [42] shows a function Φ : R2 → R,

Φ(x, y) := x2 − (x− 1)3y2 ,

verifying the assumptions (i)-(ii)-(iii) of the Mountain Pass Theorem butwithout critical points except (0, 0).

Indeed the function Φ does not satisfy the Palais-Smale condition at theMountain Pass level c, which is easily verified to be c = 1. Any Palais-Smalesequence (xn, yn) ∈ R2 of Φ at the level c = 1 diverges to infinity, moreprecisely xn → 1, |yn| → +∞, see [72].

Remark 6.2.1 (Pseudo-gradient) To prove Theorem 6.2.4 in the case Xis a Banach space or ∇Φ is not locally Lipschitz, we can replace, in order toconstruct the deformations η(t, ·), the gradient ∇Φ(x) with a locally Lipschitzpseudo-gradient vector field P : DΦ(x) 6= 0 → X such that

‖P (x)‖ ≤ 2‖DΦ(x)‖∗DΦ(x)[P (x)] ≥ ‖DΦ(x)‖2

∗ .

Note that the second inequality implies also ‖P (x)‖ ≥ ‖DΦ(x)‖∗.Any Φ ∈ C1(X,R) admits a locally Lipschitz pseudo-gradient vector field

P , see e.g. Lemma A.2 in [116].

We now discuss the minor variants of the Mountain Pass Theorem usedin Chapter 1.


Theorem 6.2.5 Suppose Φ ∈ C1(Br,R) where Br := ‖x‖ < r, satisfies(i) Φ(0) = 0;(ii) ∃ 0 < ρ < r, α > 0 such that Φ(x) ≥ α if ‖x‖ = ρ;(iii) Φ(x) < 0, ∀x ∈ ∂Br.

There exists a (PS)c sequence of Φ at the level c := infγ∈Γ maxt∈[0,1] Φ(γ(t))where

Γ :=γ ∈ C([0, 1], Br) | γ(0) = 0 , γ(1) ∈ ∂Br

. (6.22)

Proof. Exactly the same proof given for Theorem 6.2.4 above applies, theonly difference being to show that the minimax class Γ in (6.22) is invariantunder the flow η(t, ·) defined in (6.18), i.e. if γ ∈ Γ then η(t, ·) γ ∈ Γ, ∀t.

For this, observe that, if 0 < 2µ < c, then the vector field X definedin (6.17) vanishes on ∂Br (since ∀x ∈ ∂Br, Φ(x) < 0, then x ∈ N c and sog(x) = 0). Hence the Cauchy problem (6.18) defines a family of deformationsη ∈ C([0, 1] × Br, Br) such that η(t, ·)|∂Br = I, whence the minimax class Γis invariant.

Proof of (1.43) . We follow Lemma 1.19 in [113]. Since V is finite dimen-sional, −∇Φ(1, v) is a pseudo-gradient vector field for Φ(λ, v) for all v ∈ ∂Qand we can define, by interpolation, say, a pseudo-gradient vector field P (v)for Φ(λ, v) on Q coinciding with −∇Φ(1, v) on ∂Q.

To apply the same argument of Theorem 6.2.4 again the main issue isto prove that the minimax class Γ defined in (1.42) is invariant under thepositive flow generated by X(v) := −g(v)P (v)/‖P (v)‖.

This is true by proposition 1.2.1: if v ∈ ∂Q verifies Φ(1, v) = c− < 0 (case(ii)) then X(v) ≡ 0 and therefore η(t, v) = v, ∀t. In particular η(t, ·)|T− = I.If v ∈ ∂Q verifies Φ(1, v) = c+ (case (i)) then Φ(1, η(t, v)) ≤ c+, for t > 0,because P (v) is a pseudo-gradient vector field, and so η(t, v) ∈ Q. Otherwise,proposition 1.2.1-(iii) shows that η(t, v) ∈ ∂Q.

For more material in critical point theory we refer to the monographsby Ambrosetti-Malchiodi [5], Mawhin-Willem [88], Rabinowitz [116], Struwe[123].

6.3 Free vibrations of NLW: a global result

As a further application of the Mountain Pass Theorem we prove the follow-ing theorem, originally due to Rabinowitz [115], following the proof given byBrezis-Coron-Nirenberg [40].


Theorem 6.3.1 The nonlinear wave equationutt − uxx + f(u) = 0u(t, 0) = u(t, π) = 0

where f(u) := |u|p−2u, p > 2, possesses a non-trivial weak 2π-periodic solu-tion u ∈ Lp.

Proof. Let

N :=v = η(t+ x)− η(t− x) | η ∈ L1(T) and

∫ 2π

0

η = 0

be the L1-closure of the classical solutions of the linear equation (2.8).For any w belonging to the weak orthogonal complement

N⊥ :=w ∈ L1(Ω) |

∫Ω

w v dt dx = 0 , ∀v ∈ N ∩ L∞,

where Ω := T × (0, π), there exists a unique weak solution ψ := −1w ∈N⊥ ∩ C(Ω) of the equation ψ = w where := ∂tt − ∂xx, namely∫

Ω

ψϕ =

∫Ω

wϕ , ∀ϕ ∈ C20(Ω) .

Actually, by the explicit integral representation formula for −1w given inremark 5.2.1, we deduce the estimates

‖−1w‖L∞ ≤ C‖w‖L1 (6.23)

and, ∀q > 1 (applying Holder inequality like in lemma 5.2.1)

‖−1w‖Cα ≤ C‖w‖Lq α := 1− 1

q(6.24)

where Cα denotes the space of the α-Holder continuous functions.

Let Φ ∈ C1(W,R) be the functional

Φ(w) :=1

p′

∫Ω

|w|p′ + 1

2

∫Ω

w−1w , ∀w ∈ W

defined on the spaceW := N⊥ ∩ Lp′


where p′ ∈ (1, 2) is the conjugated exponent to p, i.e. 1/p+1/p′ = 1, endowedwith its natural norm

‖w‖Lp′ :=( ∫

Ω

|w|p′ dt dx)1/p′

.

Note that Φ is well defined because −1w ∈ L∞(Ω) by (6.23).We have

DΦ(w)[h] =

∫Ω

|w|p′−2w h+

∫Ω

h−1w, ∀h ∈ W , (6.25)

whence, if w ∈ W is a critical point of Φ,

−1w + |w|p′−2w = v

for some v ∈ N ∩ Lp. Define u := v −−1w, namely u := |w|p′−2w. We getw = |u|p−2u and, since v ∈ N ∩ Lp,

u = −w = −|u|p−2u ,

namely u is a solution of utt − uxx + |u|p−2u = 0.

Therefore, Theorem 6.3.1 will be a consequence of the Mountain Pass Theo-rem 6.2.4 and the following claim.

Claim: Φ verifies the assumptions of the Mountain Pass Theorem 6.2.4.

Step 1: Φ verifies assumptions (i)-(ii)-(iii)

We have Φ(0) = 0 and

Φ(w) ≥ 1

p′‖w‖p′

Lp′ − C‖w‖L1‖−1w‖L∞

(6.23)

≥ 1

p′‖w‖p′

Lp′ − C ′‖w‖2L1 ≥

1

p′‖w‖p′

Lp′ − C ′′‖w‖2Lp′

whence, since 1 < p′ < 2, assumption (ii) of the Mountain Pass Theoremfollows.

To verify assumption (iii), take w ∈ W such that∫

Ωw−1w < 0 (recall that

−1 possesses negative eigenvalues). For t ∈ R large enough, we have

Φ(tw) =tp′

p′‖w‖p′

Lp′ +t2

2

∫Ω

w−1w < 0 .

Step 2: Φ satisfies (PS)c.


Let wn be a (PS)c sequence, namely

Φ(wn) =1

p′

∫Ω

|wn|p′+

1

2

∫Ω

wn −1wn → c (6.26)

and ‖DΦ(wn)‖L(Lp′ ,R) → 0.

By (6.25), using the Hahn-Banach extension theorem, and recalling thatthe dual space of Lp′ is Lp, there exists gn ∈ Lp with

‖gn‖Lp = ‖DΦ(wn)‖L(Lp′ ,R) → 0

such that

DΦ(wn)[h] =

∫Ω

(|wn|p

′−2wn + −1wn

)h =

∫Ω

gn h , ∀h ∈ W ,

whence−1wn + |wn|p

′−2wn = gn + vn (6.27)

with vn ∈ N ∩ Lp.Subtracting (6.26) from

1

2DΦ(wn)[wn] =

1

2

∫Ω

|wn|p′+

1

2

∫Ω

wn −1wn =1

2

∫Ω

gnwn (6.28)

yields1

2DΦ(wn)[wn]− Φ(wn) =

(1

2− 1

p′

) ∫Ω

|wn|p′

whence (1

2− 1

p′

)‖wn‖p′

Lp′ ≤∣∣∣12

∫Ω

gnwn

∣∣∣ + |Φ(wn)|

≤ 1

2‖gn‖Lp‖wn‖Lp′ + c+ 1 .

It follows ‖wn‖Lp′ ≤ C is bounded and, therefore, up to subsequence, wnLp′

w ∈ W weakly.

By the convexity of the function |t|p′ we get

1

p′|w|p′ − 1

p′|wn|p

′ ≥ |wn|p′−2wn(w − wn)

(6.27)= (gn + vn −−1wn)(w − wn)


whence

1

p′

∫Ω

|w|p′ − 1

p′

∫Ω

|wn|p′ ≥

∫Ω

(gn −−1wn)(w − wn) (6.29)

since vn ∈ N ∩ Lp.By (6.24) and the Ascoli-Arzela Theorem the operator

−1 : W := N⊥ ∩ Lp′ → Lp (6.30)

is compact. Hence, since wn−w 0 weakly, gn−−1wn converges stronglyin Lp, and we deduce that the right hand side in (6.29) tends to 0, whence

lim supn

‖wn‖Lp′≤ ‖w‖Lp′ . (6.31)

Since the space Lp′ is uniformly convex, (6.31) and wnLp′

w imply that

wnLp′

→ w (see [39] Prop. III. 30).

Remark 6.3.1 The proof of Theorem 6.3.1 can be generalized for any super-linear5 nonlinearity f(u) satisfying lim|u|→+∞ f(u)/u = +∞ and, for someα > 0, 1

2u f(u)− F (u) ≥ α|f(u)| − C, ∀u, where F (u) :=

∫ u

0f(s) ds.

(Regularity) The weak solution u found in Theorem 6.3.1 is actually inL∞(Ω), see [41]. Furthermore, if f(u) is strictly monotone and of class C∞

then u ∈ C∞, see [41], [115].

Remark 6.3.2 Existence of periodic solutions for any rational frequencyω := 2π/T ∈ Q can be proved along the same lines. On the contrary, ifω /∈ Q, the compactness property (6.30) does not hold any more and theabove proof of the Palais-Smale condition fails. This is the reflect of small“denominators” phenomena discussed in these Lecture Notes.

6.4 Approximation of irrationals by rationals

We collect here some elementary results in number theory used in theseLecture Notes. For a complete presentation we refer to [67], [117].

A real number ξ ∈ R may be uniquely written as

ξ = [ξ] + ξ

where [ξ] ∈ N is the integer part of ξ and ξ ∈ [0, 1) its fractional part.

5Previous results in the simpler case when the nonlinearity f grows at most linearlyhad been obtained in [1], [41], see also the survey [38]. In this case, some existence resultof periodic solutions also with a (suitable) irrational frequency has been proved in [52].


Theorem 6.4.1 (Dirichlet) Let x be an irrational number. Then

i) ∀Q ∈ N there exist integers q, p ∈ Z with 1 ≤ q < Q such that

|qx− p| ≤ 1

Q. (6.32)

ii) There exist infinitely many distinct rational numbers p/q such that∣∣∣x− p

q

∣∣∣ < 1

q2. (6.33)

Proof. i) The Q+ 1 numbers

0, x, . . . , (Q− 1)x, 1 ∈ [0, 1]

are all distinct since x is irrational. Therefore at least 2 of them lye at adistance less or equal to 1/Q, namely there exist integers 0 ≤ r1 < r2 < Q,s1, s2 such that ∣∣∣(r1x− s1)− (r2x− s2)

∣∣∣ ≤ 1

Q.

Setting q := r1 − r2, p := s1 − s2, we have 1 ≤ q < Q and (6.32) holds.

ii) Since x is irrational, qx − p is never zero. Hence, as Q → +∞, we findinfinitely many distinct pairs of relatively prime integers satisfying (6.32) andtherefore (6.33).

Corollary 6.4.1 If x ∈ R\Q the set xn+m | (n,m) ∈ Z2 is dense on R.

Proof. ∀Q ∈ N, by the Dirichlet Theorem 6.4.1-i), the sequence

0, qx, 2qx, 3qx, . . .

forms a 1/Q-net of [0, 1].

Definition 6.4.1 (Diophantine number) An irrational number x is calledDiophantine if there exist constants γ, τ > 0 such that∣∣∣x− p

q

∣∣∣ ≥ γ

qτ, ∀q ∈ N, p ∈ Z .

Note that, in view of the Dirichlet Theorem, the constant τ has to be τ ≥ 2,and, if τ = 2, the constant γ has to satisfy γ ∈ (0, 1).

Theorem 6.4.2 Almost every x ∈ R is Diophantine.


Proof. It is sufficient to prove that almost every x ∈ (0, 1) is Diophantine.The complementary set of the Diophantine numbers in (0, 1) for some γ > 0,τ > 2, is contained in

Cγ :=⋃

p/q∈Q∩(0,1)

(pq− γ

qτ,p

q+γ

qτ

),

whose Lebesgue measure is bounded by

|Cγ| ≤∞∑

q=1

q∑p=1

2γ

qτ≤ 2γ

∞∑q=1

q

qτ

which is convergent because τ > 2. We deduce that |Cγ| → 0 as γ → 0.

A Diophantine number with τ = 2 is called “badly-approximable”. Exis-tence of badly approximable numbers is ensured, for example, by the follow-ing theorem (see also Theorem 6.4.6 below).

Theorem 6.4.3 (Liouville) Let x be an irrational root of a second orderpolynomial P (y) := ay2+by+c with a, b, c ∈ Z, a 6= 0 (quadratic irrational).Then x is badly approximable.

Proof. If |x− p/q| > 1 we have nothing to prove. If |x− p/q| ≤ 1 then

|ap2 + bpq + cq2|q2

=∣∣∣P(p

q

)∣∣∣ =∣∣∣P(p

q

)− P (x)

∣∣∣ ≤M∣∣∣pq− x

∣∣∣ (6.34)

where M := max[x−1,+1] |P ′(y)| > 0. Now

|ap2 + bpq + cq2| ≥ 1 ,

for a, b, c ∈ Z; otherwise, P (p/q) = 0, but P can not have rational solutions.We deduce by (6.34) that ∣∣∣x− p

q

∣∣∣ ≥ M−1

q2.

In conclusion: x is badly approximable with γ := minM−1, 1.


6.4.1 Continued fractions

The classical Euclidean division algorithm allows to write every real numberx in the form of a “continued fraction”.

For every x ∈ R we define

a0 := [x] , x0 := x

(so that x = a0 + x0) and, inductively, for n ≥ 0, if xn 6= 0,

an+1 :=[ 1

xn

]≥ 1 , xn+1 :=

1

xn

(so that x−1

n = an+1 + xn+1) whence

x = a0 + x0 = a0 +1

a1 + x1

= . . . = a0 +1

a1 +1

a2 + . . .+1

an + xn

If x ∈ Q is a rational number this Euclidean division algorithm stops after afinite number of steps (when the remainder xn = 0).

On the contrary, if x ∈ R \Q, the previous procedure never ceases anddefines the so called “continued fraction” expansion of the number x,

[a0, a1, a2, . . . ] := a0 +1

a1 +1

a2 + . . .

.

The integers an are called the “partial quotients” of x.

Example The continuous fraction expansion of the famous “golden ratio”number x := (

√5 + 1)/2 is

[1, 1, 1, . . . ] = 1 +1

1 +1

1 + . . .

because x solves the equation x = 1 + x−1.

Definition 6.4.2 (Convergents) Given an irrational number x, we definethe n-th convergents to x as

pn

qn:= [a0, a1, a2, . . . , an] := a0 +

1

a1 +1

a2 + . . .+1

an

∈ Q

for n = 0, 1, 2, . . ..


Exercise Prove, by induction, that pn, qn are recursively determined bypn = anpn−1 + pn−2

qn = anqn−1 + qn−2∀n ≥ 0 ,

defining p−2 := 0, p−1 := 0, q−2 := 1, q−1 := 0. In particular qn+1 > qn >0 and limn→+∞ qn = +∞ (note that, for the golden ratio, qn, pn are theFibonacci sequence). Furthermore

pn qn+1 − pn+1 qn = (−1)n+1 (6.35)

which implies, in particular, that pn and qn are relatively prime for each n.

By the definition 6.4.2 and the previous exercise, given x ∈ R \ Q, thesequence of its convergents satisfy

p0

q0<p2

q2<p4

q4< . . . < x < . . . <

p5

q5<p3

q3<p1

q1

whence∣∣∣pn

qn− x

∣∣∣ < ∣∣∣pn

qn− pn+1

qn+1

∣∣∣ (6.35)=

1

qn qn+1

<1

q2n

, ∀n ≥ 0 , (6.36)

and, in particular,pn

qn→ x for n→ +∞ .

The following somehow “inverse” result holds (see Theorem 5C in [117]).

Theorem 6.4.4 (Legendre) Suppose p, q are relatively prime integers withq > 0 and ∣∣∣x− p

q

∣∣∣ < 1

2q2.

Then p/q is a convergent to x.

The importance of the convergents is finally highlighted by the followingtheorem due to Lagrange, sometimes called the “law of best approximations”(see Theorem 5E in [117]).

Theorem 6.4.5 (Best approximations) Let x be an irrational number.Then

|xq0 − p0| > |xq1 − p1| > |xq2 − p2| > . . .

and, ∀1 ≤ q ≤ qn, (q, p) 6= (qn, pn), n ≥ 1,

|xq − p| > |xqn − pn| .


We finally recall the following identity used in Lemma 2.2.1.

Lemma 6.4.1 (Theorem 5F in [117]). If x ∈ R \Q then∣∣∣x− pn

qn

∣∣∣ =1

q2n

([an+1, an+2, . . .] +

1

[an, an−1, . . . , a1]

)whence

1

q2n(an+1 + 2)

≤∣∣∣x− pn

qn

∣∣∣ ≤ 1

q2n an+1

. (6.37)

By Theorem 6.4.4 and (6.37), arguing as in Lemma 2.2.1, we deduce:

Theorem 6.4.6 An irrational x is badly approximable if and only if thepartial quotients an in its continued fraction expansion are bounded, namely|an| ≤ K(x), ∀n ≥ 0.

6.5 The Banach algebra property of Xσ,s

Lemma 6.5.1 For s > 1/2 the space Xσ,s is an algebra.

Proof. Let u, v ∈ Xσ,s, u =∑

m∈Z um(x) eimt, v =∑

m∈Z vm(x) eimt. Theproduct uv is

uv =∑

l

( ∑k

ul−kvk

)eilt,

so its Xσ,s-norm is, if converging,

‖uv‖2σ,s =

∑l

∥∥∥∑k

ul−kvk

∥∥∥2

H1(1 + l2s) e2|l|σ .

Let us define

alk :=

[(1 + |l − k|2s) (1 + k2s)

(1 + l2s)

]1/2

. (6.38)

By Holder inequality∥∥∥∑k

ul−kvk

∥∥∥2

H1(1 + l2s) ≤ c2l

∑k

‖ul−kvk‖2H1a2

lk (1 + l2s) (6.39)

(6.38)= c2l

∑k

‖ul−kvk‖2H1 (1 + |l − k|2s) (1 + k2s)


where

c2l :=∑

k

1

a2lk

.

We claim thatc2l ≤ c2 , ∀l ∈ Z (6.40)

with

c2 := 2s∑k∈Z

1

1 + k2s< +∞

which is finite if s > 1/2. Let us prove (6.40). By convexity, for s > 1/2, wehave

1 + l2s ≤ 1 +(|l − k|+ |k|

)2s

≤ 1 + 22s−1(|l − k|2s + k2s

)< 22s−1

(1 + |l − k|2s + 1 + k2s

)and then

1

a2lk

< 22s−1( 1

1 + |l − k|2s+

1

1 + k2s

). (6.41)

By the definition of cl

c2l =∑

k

1

a2lk

(6.41)< 22s−1

( ∑k

1

1 + |l − k|2s+

∑k

1

1 + k2s

)= 22s

∑k∈Z

1

1 + k2s=: c2 < +∞.

Finally, using that H1(0, π) is an algebra,

‖uv‖2σ,s =

∑l

∥∥∥∑k

ul−kvk

∥∥∥2

H1(1 + l2s) e2|l|σ

(6.39)

≤∑

l

c2l∑

k

‖ul−k‖2H1‖vk‖2

H1 (1 + |l − k|2s) (1 + k2s)e2|l|σ

(6.40)

≤ c2∑

l

∑k

‖ul−k‖2H1‖vk‖2

H1 (1 + |l − k|2s) (1 + k2s) e2(|l−k|+|k|)σ

= c2∑

k

( ∑l

‖ul−k‖2H1(1 + |l − k|2s) e2|l−k|σ

)‖vk‖2

H1(1 + k2s) e2|k|σ

= c2‖u‖2σ,s‖v‖2

σ,s

proving the algebra property of Xσ,s. Note that the algebra constant c isindependent of σ.

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