L 2 des inverses des opérateurs linéarisés et des "bonnes separations" des "mauvais sites" sont vérifiées). Contents 4.2 Initialization in the iterative Nash-Moser scheme . . . . . . . .

Résumé en français

Les équations aux dérivées partielles (EDP) permettent d'aborder d'un point de vue mathématique des phénomènes observés dans tous les domaines des sciences. Certaines EDP non-linéaires modélisent des problèmes de mécanique statistique, mécanique des fluides, théories de la gravitation ou des mathématiques financières.

L'objectif de ce travail de thèse est l'étude de certains problèmes d' EDP non-linéaires et hamiltoniennes et la recherche des leurs solutions périodiques et quasi-périodiques. Ces solutions sont des fonctions u(ωt) de fréquences ω = (ω 1 , . . . , ω ν ) ∈ R ν telles que

ω • l � = 0, ∀ l ∈ Z \ {0},
où u(•) est definie sur T ν et prend ses valeurs dans un espace de Hilbert de dimension infinie. On parle de solutions périodiques lorsque ν = 1 et de solutions quasi-périodiques lorsque ν > 1.

Dans cette thèse, on prouve l'existence de :

• Chapitre 2 : Solutions périodiques de petite amplitude pour l'équation des ondes non-linéaire autonome avec un potentiel V (x) sur des variétés de Zoll M

u tt -∆u + V (x)u = f (x, u), x ∈ M (1)
• Chapitre 5 : Solutions périodiques pour des EDP non-linéaire et autonomes du type

ω 2 u tt + Au = εf (ε, u), ω ∈ R + (2)
qui généralisent l'équation [START_REF] Ambrosetti | Dual Variational Methods in Critical Point Theory and Applications[END_REF], sous des hypothèses appropriées sur l'opérateur linéaire A, sur la non-linéarité f et á valeurs dans certains espaces de Sobolev

• Chapitre 6 : Solutions quasi-périodiques pour l'équation des ondes non-linéaire forcée avec un potentiel V (x) sur des variétés de Zoll M

u tt -∆u + V (x)u = εf (ωt, x, u), x ∈ M, ω ∈ T ν . (3)
Ces travaux vont étendre les résultats obtenus par Berti, Bolle, Procesi dans l'article [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF].

Une des principales difficultés qu'on rencontre dans l'étude des EDP du type

(1), ( 2), (3) est le phénomène de "petits diviseurs" : la présence de petits dénominateurs dans les inverses des opérateurs linéarisés qui produit une perte de régularité et empêche l'utilisation du Théorème des Fonctions Implicites.

On utilise la méthode de Nash-Moser pour résoudre le problème de petits diviseurs. Cette méthode demande certaines hypothèses de non-résonance sur les fréquences.

Le schéma de Nash-Moser est une méthode plus souple de la Théorie KAM pour EDP (qui est une extension de la Théorie KAM des systèmes dynamiques de dimension finie), mais il ne donne pas d' informations sur la stabilité linéaire des solutions obtenues.

Les autres outils mathémathiques utilisés dans ce travail de thése sont :

• La décomposition de Lyapunov-Schmidt qui nous amène à resoudre deux équations : l' équation de (Q) (sur le noyau E 0 de l'équation linéarisée avec un fréquence fixée ω) résolue par des méthodes variationelles ou de continuation et l'équation (P ) (sur l'espace orthogonal au noyau E 0 par rapport au produit scalaire en L 2 ) résolue via Nash-Moser

• Une analyse multi-échelle pour l'inversion des opérateurs linéarisés, qui permet d'avoir des "bonnes estimations" en grande norme de Sobolev pour les inverses des opérateurs linéarisés, estimations qui sont nécessaires pour la convergence du schéma de Nash-Moser.

Cette analyse multi-échelle utilise des "bonnes separation" des "mauvais sites" (les sites correspondent aux petits diviseurs) et des "bonne estimations" en norm L 2 des inverses des opérateurs linéarisés

• Arguments sur les variations de valeurs propres pour des estimations de mesure de l'ensemble des paramètres pour lesquels on a des solutions (qui sont les paramètres pour lesquels des conditions adéquates de non-résonance sur les fréquences, des "bonnes estimations" en norme

Aim of the thesis

The works of this thesis start from the study of results obtained by Berti, Bolle and Procesi in [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF] ( '10) where they prove an abstract Nash-Moser theorem with parameters and they apply it to prove the existence of periodic solutions for the forced NLW on Zoll manifolds.

The aim of this thesis is to extend the results of [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF] to the research of quasiperiodic solutions for forced NLW equation on a d-dimensional Zoll manifold

M u tt -∆u + V (x)u = εf (ωt, x, u), x ∈ M, ω ∈ T ν (1.1)
and to the research of small amplitude periodic solutions for the autonomous case

u tt -∆u + V (x)u = f (x, u), x ∈ M. (1.2)
Moreover, we generalize this last result to prove the existence of Cantor families of periodic solutions for non-linear autonomous PDEs like

ω 2 u tt + Au = εf (ε, u), ω ∈ R + (1.3)
under suitable assumptions on the linear operator A, on non-linearity f and in appropriate Sobolev spaces.

We recall that periodic and quasi-periodic solutions of the above non-linear equations are solutions u(ωt) with a frequency-vector ω := (ω 1 , ω 2 , . . . , ω ν ) ∈ R ν of rationally independent coordinates

ω • l � = 0, ∀ l ∈ Z ν \ {0},
where u(•) is defined on T ν with values in some infinite dimensional Hilbert space H. We have periodic solutions if ν = 1 and quasi-periodic solutions if Chapter 1. Introduction

ν > 1.
The main tools that we will use in this thesis are: the Lyapunov-Schmidt decomposition (for the autonomous equations (1.2), (1.3)), iterative Nash-Moser schemes in appropriate Sobolev spaces and multiscale inductive arguments (for the key off-diagonal decay estimates of the inverse of linearized operator in the forced equation (1.1)).

We will give later in this introduction more details about our results and the tools used.

In order to understand the main difficulties of these type of equations and the approaches typically used to solve them, we recall a short history of the study of periodic and quasi-periodic solutions for PDEs, in particular for the NLW from which equations (1.1), (1.2) and (1.3) arise.

This historical preface does not pretend to be exhaustive at all.

Historical preface

Non-linear partial differential equations (NL PDEs) are used to describe a wide variety of physical phenomena. Non-linear models appear in statistical mechanics, fluids dynamics, quantum mechanics, etc. Most real physical processes in these different fields can be formalized similarly in terms of PDEs.

A very important role is played by the class of non-linear Hamiltonian PDEs, i.e. a non linear PDEs which can be seen as a Hamiltonian system

∂ t u = J(∇ u H)(t, u), u ∈ H
where the Hamiltonian function

H : R × H → R
is defined on an infinite dimensional Hilbert Space H and J is a non-degenerate antisymmetric operator.

Classical examples are given by the non-linear wave and non-linear Schrödinger equations (NLW) u tt -∆u = f (u), (NLS) ı∂ t u -∆u + f (|u| 2 )u = 0 or the membrane equation

u tt + ∆ 2 u + f (u) = 0.
The Hamiltonian feature of the wawe equation was exploited by Rabinowitz [START_REF] Rabinowitz | Free vibrations for a semi-linear wave equation[END_REF] ('78) and Brezis-Coron-Nirenberg [START_REF] Brezis | Free vibrations for a nonlinear wave equation and a Theorem of P. Rabinowitz[END_REF] ('80): they proved the existence of T -periodic solutions via variational methods for the 1-dimensional NLW

u tt -u xx = f (u), x ∈ [0, π] (1.4)
Chapter 1. Introduction with non-linearity f = u|u| p-2 , p > 2, (see also Theorem 6.3.1 in [START_REF] Berti | Nonlinear Oscillations in Hamiltonian PDEs[END_REF]) under Dirichlet boundary conditions u(t, 0) = u(t, π) = 0 and when T /π is an integer.

In this setting, the "small denominators" problem does not appear (see later for details about the small denominators problem) and the difficulty is due to the "complete resonance" of the frequencies: all the solutions of the linear equation u ttu xx = 0 which satisfy the above Dirichlet boundary conditions are 2π periodic.

When T /π is not rational, small divisors generally appear.

In [START_REF] Bambusi | Families of periodic solutions of resonant PDEs[END_REF] ('01), Bambusi and Paleari avoided the small divisors difficulty for equation (1.4) with f (u) = u 3 + O(u 5 ) imposing on the frequency ω = 2π/T a strong non-resonance condition |ωl -j| ≥ γ l , ∀ l � = j which for 0 < γ < 1/6 is verified by a frequencies set W γ of zero measure.

For the same set W γ of strongly non resonant frequencies, Berti and Bolle proved in [START_REF] Berti | Periodic solutions of nonlinear wave equations with general nonlinearities[END_REF] ('03) and [START_REF] Berti | Multiplicity of periodic solutions of nonlinear wave equations[END_REF] ('04) existence and multiplicity of periodic solutions of (1.4) for any non-linearity f (u).

For non-linear PDEs, one can look for periodic solutions as non-linear continuation of the linear modes, having frequency close to the frequency of the solutions of linearized equation.

It is a generalization in infinite dimension of the Lyapunov Center Theorem for periodic solutions for finite dynamical system.

The Lyapunov Center Theorem assures that close to an elliptic equilibrium of a Hamiltonian system with n degrees of freedom, there exist typically n families of periodic solutions. More precisely, if the frequencies associated to the linearized operator verify the non-resonance condition lω 1ω j � = 0, ∀ l ≥ 1, j = 2, . . . n (1.5) then there exists one continuous family of periodic solutions of frequency ω that are close to the first linear mode, i.e. with ω close to ω 1 . The other families are obtained by replacing the index 1 in (1.5) with any other index j ≥ 2 and consist in periodic solutions of frequency ω, with ω close to ω j .

Chapter 1. Introduction

The main reason is that, in finite dimension, condition (1.5) assures that for any ω sufficiently close to ω 1 , the same non-resonance condition lωω j � = 0, ∀ l ≥ 1, j = 2, . . . n (1.6) holds on ω. The quantities lωω j appear in the Fourier coefficients of the linearized operators: under condition (1.6) we can invert them and apply the Implicit Function Theorem to the problem.

It is easy to realize that in the infinite dimensional case, where

ω j → ∞ as j → ∞,
the quantities lωω j mat accumulate to zero.

The condition (1.6) is no more sufficient: the presence of arbitrarily small Fourier coefficients lωω j of the linearized operator at zero makes its inverse unbounded and the standard Implicit Function Theorem cannot be applied.

This difficulty is known as "small denominators" problem and this name is due to the fact that the arbitrarily small quantities lωω j enter at the denominator of the Fourier coefficients of the inverse of linearized operators.

The first step to overcome this problem is to impose to the expected frequency ω of the solution, close to a fixed frequency ωj, the first order Melnikov diophantine condition: for some appropriate γ ∈ (0, 1), τ > 0

|lω -ω j | ≥ γ 1 + |l| τ +1 ,
∀ l ∈ Z, j ≥ j.

(1.7)

Condition (1.7) allows to invert the linearized operators at the elliptic equilibrium but with a loss of regularity: the standard Implicit Function Theorem is not yet applicable.

Of course in the search of quasi-periodic solutions, one also has to overcome the small denominators problem, even in finite dimension.

In the 1950s, KAM methods were developed by Kolmogorov, Arnold and Moser to deal with this problem in finite dimension.

The KAM procedure consists in an iterative rapidly convergent scheme, which, by infinitely many canonical changes of coordinates, brings the Hamiltonian associated to the equation into another one which has an invariant torus.

This method can provides the existence and the linear stability of the solutions under the first order Melnikon condition and the second order Melnikov condition in which appear not only frequencies ω j , j � = j, but also the differences (ω jω k ) and the sums (ω j + ω k ), j, k � = j.

Chapter 1. Introduction

For tori of lower dimension, these methods was developed by Melnikov, Elliasson [START_REF] Eliasson | Perturbations of stable invariant tori for Hamiltonian systems[END_REF] and Pöschel [START_REF] Pöschel | On elliptic lower dimensional tori in Hamiltonian systems[END_REF].

From the end of the 1980s, KAM Theory was succesfully applied to Hamiltonian PDEs with the works of Kuksin [START_REF] Kuksin | Hamiltonian pertubations of infinite dimensional linear systems with imaginary spectrum[END_REF] ('87), Kuksin-Pöschel [START_REF] Kuksin | Invariant Cantor manifolds of quasi periodic oscillations for a non linear Schrödinger equation[END_REF] ('96) for the NLS equation and Wayne [START_REF] Wayne | Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory[END_REF] ('90), Pöschel [START_REF] Pöschel | A KAM Theorem for some nonlinear partial differential equations[END_REF], [START_REF] Pöschel | Quasi periodic solutions for a nonlienar wawe equations[END_REF] ('96) for the NLW equation. See also [START_REF] Kuksin | Analysis of Hamiltonian PDEs, Oxford Lecture in Mathematics and its application 19[END_REF].

In these results, the spectrum of the space operator -∂ xx + m was assumed to be simple to avoid resonances. In particular they were limited to the 1-spacedimensional case.

The first result dealing with multiple (here double) eigenvalues was obtained by Chierchia and You in [START_REF] Chierchia | KAM tori for 1D non-linear wawe equations with periodic boundary conditions[END_REF] ('00) for 1-dimensional NLW equation (on T).

The first complete KAM result for multi-dimensional PDEs was obtained by

Eliasson and Kuksin in [START_REF] Eliasson | KAM for nonlinear Schrödinger equation[END_REF] ( '10) and applied to the NLS equation on T d .

Eliasson-Kuksin's methods was used and adapted by several authors, see

Eliasson-Grébert-Kuksin [START_REF] Eliasson | KAM for nonlinear beam equation 1: small amplitude solutions[END_REF] ( '14) for the beam equation on T d and Grébert-Paturel [START_REF] Grébert | KAM for the Klein-Gordon equation on S d[END_REF] ('16) for the Klein-Gordon equation on S d .

The KAM results provide not only the existence of periodic and quasi-periodic solutions but also the reducibility of the linearized equation at these solutions as well as linear stability results.

The mere existence of periodic and quasi-periodic solutions in a multidimensional setting had been obtained much earlier by Bourgain for quasiperiodic solutions of NLW and NLS on the higher spatial dimensional torus T d , see [START_REF] Bourgain | Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schrödinger equations[END_REF] ('98), [START_REF] Bourgain | Periodic solutions of nonlinear wave equations[END_REF] ('99), [START_REF] Bourgain | Green's function estimates for lattice Schrödinger operators and applications[END_REF] ('04). He used a method inspirated by Craig and Wayne [START_REF] Craig | Newton's method and periodic solutions of nonlinear wave equations[END_REF] ('93), [START_REF] Craig | Nonlinear waves and the 1 : 1 : 2 resonance" in singular limits of dispersive waves[END_REF] ('94), based on a Nash-Moser implicit function iterative scheme and on the Lyapunov-Schmidt decomposition.

In the Nash-Moser Implicit function scheme, the research of periodic and quasi-periodic solutions is reduced to find zeroes of a nonlinear operator

F (u) = 0
by an iterative convergent Newton-type scheme with regularizations S n in scales of Banach spaces

u 0 := 0, u n+1 := u n + h n+1 h n+1 := -S n (D u F (u n )) -1 (F (u n )).
This method requires only the first order Melnikov condition, which is essentially the minimal assumption possible, but does not give us information about the Chapter 1. Introduction reducibility and the linear stability of the solutions (for which in the KAM theory is required the second order Melnikov condition). The main difficulty is that at each step of iteration we have to invert linear operators L n := D u F (u n )

with variable coefficients represented by matrices which are small perturbations of diagonal matrices with arbitrarily small eigenvalues. For these operators, it is hard to estimate the inverses in high Sobolev norms (small denominators problem). This method was used by Craig and Wayne [START_REF] Craig | Newton's method and periodic solutions of nonlinear wave equations[END_REF] ('93) which introduced for Hamiltonian PDEs the Lyapunov-Schmidt reduction that splits the problem into two equations: the range equation, solved with a Nash-Moser Implicit function theorem, and the bifucration equation, solved via continuation arguments.

The same kind of methods was applied by Berti and Bolle in [START_REF] Berti | Cantor families of periodic solutions for completely resonant nonlinear wave equations[END_REF] ('06) for the existence of Cantor families of periodic solutions for completely resonant 1-dimensional NLW and in [START_REF] Berti | Cantor families of periodic solutions of wave equations with C k nonlinearities[END_REF] for NLW with C k nonlinearities.

In [START_REF] Berti | Sobolev periodic solutions of nonlinear wave equations in higher spatial dimensions[END_REF] ('09), Berti and Bolle proved Sobolev periodic solutions for higher spatial dimensional wave equation

u tt -∆u + mu = f (ωt, x, u), x ∈ T d
with periodic boundary conditions and C k non-linearity and in [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF] they solved

u tt -∆u + V (x)u = f (ωt, x, u), x ∈ M
with a C ∞ multiplicative potential V (x) and on a d-dimensional Zoll manifold M. It is from here that our works start.

Finally we cite recent works of Berti, Corsi and Procesi [START_REF] Berti | An abstract Nash-Moser theorem and quasi-periodic solutions for NLW and NLS on compact Lie groups and homogeneous manifolds[END_REF] ('10) for quasiperiodic solutions of NLW and NLS on Lie groups proved via Nash-Moser.

Main tools used in the thesis

In order to prove periodic and quasi-periodic solutions for equations (1.1),

(1.2) and (1.3), we use the following tools.

(We explain here only the main idea of these tools. All details follow in the description of our results for each particular case).

• The Lyapunov-Schmidt decomposition

In order to solve the autonomous equations (1.2) and (1.3)

in appropriate

Sobolev spaces H s1 , we decompose H s1 as the sum of the kernel E 0 of the linearized equation for a fixed frequency ω = ωj and of its complement Chapter 1. Introduction E s1,⊥ 0 in the whole space

H s1 = E 0 ⊕ E s1,⊥ 0 .
According to this decomposition, we project the whole equation on E 0 and E s1,⊥ 0 obtaining two equations: the range equation in E s1,⊥ 0 and the bifurcation equation in E 0 , which depend on the same parameter λ (or ε)

properly chose with respect to the whole equation.

We will obtain solutions u ∈ H s1 written as

u = h + v with h ∈ E 0 , v ∈ E s1,⊥ 0 .
Fix h ∈ E 0 , we firstly solve the range equation via Nash-Moser finding solutions v ∈ E s1,⊥ 0 in a particular Cantor set of parameters λ; these solutions v are C 1 with respect to (λ, h).

Then, we solve the bifurcation equation writing its variational form and using the Implicit function Theorem: we prove the existence of a smooth path λ � → h(λ) of solutions which intersects transversally, for a positive measure set of parameters λ, the Cantor set where also the range equation has been solved.

• Iterative abstract Nash-Moser schemes

We solve the range equations associated to (1.2) and (1.3) and the whole equation (1.1) using iterative Nash-Moser schemes.

The key point to apply these Nash-Moser schemes is to prove the validity of "good" bounds in high Sobolev H s -norm for the inverse of the linearized operators L N at each step of the Nash-Moser scheme (which are "good" in the sense of convergence of the scheme).

We will prove that "good bounds" in L 2 -norm � • � 0 for the inverse of the linearized operators L N imply these "good bounds" also in H s -norm � • � s . The L 2 -bounds define the set of parameters A ∞ on which we have solutions and the choice to use the L 2 -bounds is in the proof of appropriate measure estimates for the set A ∞ .

We remark that having L 2 -bounds for L -1 N of type �L -1 N � 0 ≤ O(N µ0 ), for some µ 0 > 0 (which is what we will have), implies directly (see Lemma

3.1.16) the following bounds in appropriate

| • | s -norm |L -1 N | s ≤ N s+1/2 �L -1 N � 0,0 ≤ O(N s+1/2+µ0 ).
Using these type of estimates in | • | s -norm, we deduce (see Lemma 3.1.12) the following bounds in Sobolev norm � • � s �L -1 N w� s ≤ C(s)N µ (�w� s + N s �w� s0 ) which are NOT sufficient for the convergence of the Nash Moser-Sceme.

We need to have estimates of type �L -1 N w� s ≤ C(s)N µ (�w� s + N δs �w� s0 ) for some δ < 1, which allows the convergence of the Nash-Moser scheme.

The key property to have these "good bounds" in H s -norm from "good bounds" in L 2 -norm, is the distribution of frequencies ω j which imply a "good separation" of the "singular sites" (in the autonoumous equation (1.2) and (1.3)) or of the "bad sites" (in the forced equation (1.1)).

The singular sites are the Fourier indices which correspond to small denomitators. The definition of the bad sites is more intricate (see chapter 6).

• Multiscale argument

In the case of forced equation (1.1), it is not possible to prove the separation of the singular sites but we can prove the separation of a new classe of "bad sites".

We prove the validity of "good" bounds in high Sobolev norm from "good" bounds on L 2 -norm using a multiscale argument.

The Multiscale scheme says that if the linearized operator L has a sufficient "off-diagonal decay" and if the N -bad sites are sufficiently separated, then the L 2 -bounds for the "large" matrix L -1 N � of size N � = N χ (with some χ large enough) imply bounds in H s -norm too for L -1 N � . We overcome the difficulty to have "good" bounds in � • � s -norm for the inverse of a "larger" matrix with an iterative argument.

• Measure estimates arguments

In order to prove that we have a Cantor family of solutions for our equations, we need to estimate the measure of the sets of parameters A ∞ , given by the Nash-Moser scheme.

In these sets A ∞ , there are parameters for which the frequency-vector ω verifies some appropriate non-resonance conditions (as the first order Melnikov diophantine condition) and for which we have "good" bounds on the L 2 -norm for the inverse of the operators L N .

We use the fact that the operators are self-adjoint in L 2 and appropriate properties of distributions of frequencies ω j . Moreover, we use that the variation of the eigenvalues of the linear operator L N with respect to the parameter gives us imformation on the measure estimates for the set of parameters where L N is not invertible or its inverse L -1 N does not satisfies Chapter 1. Introduction some L 2 -bounds. Here we use the fact that the derivatives of L N with respect to parameters are positive or negative definite.

Follows now a description of the main results that we prove in this thesis.

Main results of the thesis

Chapter 2: Small amplitude periodic solutions of the autonomous NLW equation with potential on Zoll manifolds

In the first result of this thesis, we have extended the works of [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF] to the reasearch of small amplitude periodic solutions for the autonomous NLW (1.2) on a d-dimensional Zoll manifold M

u tt -∆u + V (x)u = f (x, u), x ∈ M
where V (x) ≥ c > 0 is in C ∞ (M, R) and f (x, u) = a p (x)u p + O(u q ) with p ≥ 2, q > p, near u = 0.

A d-dimensional Zoll manifold M is a connected, compact, Riemannian and C ∞ -manifold without boundary, whose the geodesic flow on the unit tangent bundle is periodic of minimal period T > 0.

The sphere S d endowed with its natural metric is the simplest example of a Zoll manifold.

The key property which allows us to solve the equation (1.2) on Zoll manifolds is the well known distribution of the eigeinvalues ω j,k of the operator (-∆ + V (x)) 1/2 on the space L 2 (M, C)

ω j,k → ∞ as j → ∞.
By Lemma 2.1.1, we have that ω j,k are confined in disjoint compact intervals I j centered at c 1 j +c 2 , for some constants c 1 > 0, c 2 ∈ R, whose length decreases as j → ∞ and where in each interval there is a finite number of ω j,k , counted with multiplicity by index k ≤ C 0 j d-1 , which increases as j → ∞. Moreover, there is an orthonormal basis of L 2 (M, C) composed of corresponding eigenvectors ϕ j,k associated to each eigenavalue ω j,k .

We will see in the following where this property of distribution of ω j,k is used.

We fix a particular ω = ω j, k and we impose on ω the first order Melnikov diophantine condition (2.11) and the non-resonance condition (2.12)

� � � � ωl - 2π T p � � � � ≥ γ 1 + |l| τ0+1 ,
∀ (l, p) ∈ Z 2 \ {(0, 0)}.

Chapter 1. Introduction

We look for small amplitude solutions u of (1.2) which are 2π ω -periodic in time, for apropriate ω close to ω. This means looking for 2π-periodic solutions of ω 2 u tt -∆u + V (x)u = εf (ε, x, u),

x ∈ M (1.8) where f (ε, x, u) = a p (x)u p (x) + O(ε q-p p-1 u q ), in appropriate scale of Sobolev spaces (H s (T, H p(M)), � • � s, p) for indices s ≥ s 1 > 1/2, defined in (2.6).

If u is a solution of (1.8) then the function ũ = ε 1/p-1 u is a small amplitude 2π ω -periodic solution of NLW equation (1.2).

For any function u in the Sobolev space H s (T, H p(M)), we performe a Fourier decomposition with respect to the time variable t ∈ T u(t, x) = � l∈Z e ılt u l (x), u l (x) = � j,k u l,j,k ϕ j,k (x) ∈ H p(M) ⊂ L 2 (M), unlike to the case where the spatial variable x lies on the torus, x ∈ T, for which one perfomes a Fourier decomposition both in time and in space u(t, x) = � l,j∈Z e ı(lt+jx) u l,j , u l,j ∈ C. The first result of this thesis is Thorem 2.1.4: fix ε 0 small enough, under appropriate non-degenerate conditions on a p (x) and for V (x) ≥ c > 0 in C ∞ (M, R), we prove the existence of a function u ∈ C 1 ([0, ε 0 ); H s1 (T, H p(M)))

and of a Cantor-like set A ⊂ [0, ε 0 ) of positive measure such that, for all ε ∈ A, u(ε) is a 2π-periodic solution of (1.8) for a set of frequencies ω = ω(ε) close to ω (such that verify condition (1.9)), where the frequency-amplitude relation beetwen ε and ω is given by

ω 2 = ±ε + ω2
depending on the parity of the exponent p and on a p (x). More precisely, using the decomposition of Lyapunov-Schmidt, we obtain a 2π-periodic solution of (1.8) of the form u(t, x) = h(t, x) + v(t, x) with h(t, x) solution of the bifurcation equation and v(t, x) solution of the range equation such that �v� s1, p → 0 as ε → 0. Chapter 1. Introduction Follow here the main ideas of the proof of Theorem 2.1.4.

Fix h, we firstly solve the range equation using a Nash-Moser scheme proved in [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF]. In Theorem 2.3.5, we prove the existence of a function v(ε, ω, h), smooth with respect to parameters (ε, ω, h), with v(0, ω, h) = 0, �v(ε, ω, h)� s1, p → 0 as ε → 0, such that v(ε, ω, h) is solution of the range equation only for parameters (ε, ω, h) in an appropriate set A ∞ .

In the resolution of the range equation, we impose on ω the condition

� � � � ωl - 2π T p � � � � ≥ γ 1 + |l| τ +1 , ∀ (l, p) ∈ Z 2 \ {(0, 0)}.
(1.9)

Under condition (1.9), we have that "good bounds" in L 2 (T, L 2 (M))-norm for the inverse of the linearized operators imply "good bounds" also in H s1 (T, H p(M))norm (see Lemmas 2.3.3 and 2.3.4), required to be verified at each step of the Nash-Moser scheme (and are "good" in the sense of convergence of the scheme).

The L 2 (T, L 2 (M))-bounds define the set of parameters A ∞ on which v(ε, ω, h) is a solution of the range equation.

In this first result for autonomous NLW, we use Lemma 2.3.3 which is already proved in Proposition 3.1 of [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF], using condition (1.9) and the property of distribution of eigenvalues ω j,k which imply a "good separation" of the "singular sites". The "singular sites" are the indices of the Fourier decomposition which correspond to small denominators. In this case, two different singular sites l, l � are "separated" with a distance which grows with the Fourier indices

|l -l � | ≥ c(γ)(|l| + |l � |) δ0 ,
for some δ 0 ∈ (0, 1) (1.10) (see Lemma 3.6 of [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF]).

Then, in order to solve the bifurcation equation is convenient to write its variational formulation. We find the solutions of the bifurcation equation as the critical points of a restricted lagrangian funtional

h ∈ E 0 � → Φ ε (ω, h + v(ε, ω, h)) ∈ R,
(see definition (2.19) and Remark 2.70)

Using the Mountain Pass Theorem 2.4.6, we prove the existence of critical point of the lagrangian functional and so solutions h of the bifurcation equation.

Moreover, under appropriate non-degenerate conditions on a p (x) (see Definition 2.112), we deduce by the Implicit Function Theorem the existence of solutions h = h(ε) of the range equation which are smooth with respect to ε. We need Chapter 1. Introduction the ε-smoothness of the solutions h in order to obtain appropriate measure estimates for the set of parameters.

In fact, the main difficulty for the autonoumous equation (1.8) with respect to the result obtained in [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF] for the forced case, is in the proof of a positive measure of the set

A = {ε : (ε, ω(ε), h(ε)) ∈ A ∞ }.
(We remark also the difference that in the forced case, the presence of ω in the non-linear term f (ωt, x, u) as an external parameter, avoids the Lyapunov-Schmidt decomposition of the equation. In the forced case of [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF], the Nash-Moser scheme is applied directly to the whole NLW equation and not on the range equation as in the autonomous case.)

We prove measure estimates for A using the property of distribution of ω j,k

(again), the first order Melnikov diophantine condition on ω (2.11), arguments of variation of the eigenvalues with respect to ε (see Lemma 2.6.3) and the non-resonance condition (2.12) (see Lemma 2.6.5).

Chapter 5: Periodic solutions for some non-linear autonomous PDEs in infinite dimension

In the second result of this thesis, we try to generalize results of chapter 2 in a non-linear autonomous PDE of the form (1.3)

ω 2 u tt + Au = εf (ε, u), ω ∈ R +
with f (ε, 0) = 0 and D u f (ε, 0) = 0.

The unknown function t � → u(t, •) takes its values in an appropriate separable real Hilbert space (H 0 , � • � 0 ), A is an unbounded, selfadjoint, positive operator with respect to the scalar product in H 0 and there exists an orthonormal Hilbert basis of H 0 composed of the eigenvectors {ϕ j } j≥1 of the operator A with corresponding eigenvalues {ω 2 j } j≥1 , listed in non-decreasing order, such that ∀ j ≥ 1, ω j > 0 and ω j → ∞ for j → ∞.

We solve equation (1.3) imposing on operator A an appropriate hypothesis of distribution of its eigenvalues ω 2 j (see Assumption 5.6 which recall the property of distribution of eigenvalues of the operator (-∆ + V (x)) 1/2 in the first result for the autonomous NLW) and imposing on the non-linearity f some hypotheses of regularity. Our assumption is about the growth of the eigenvalues and not about their separation: it does not exclude the presence of very large clusters of equal or very close eingenvalues. This case is more general not only for the presence of a generic linear operator Chapter 1. Introduction A (exemples of A are operators -∆, -∆ + V (x), ∆ 2 , . . . on appropriate spaces) but also for the generality of the non-linearity

f (ε, •) : H 0 → H 0 g � → f (g).
In fact, take for example H 0 = L 2 (M) for some d-dimensional manifold M, the function f has not to be necessarily a function defined as f (g)(x) := w(g(x))

for some w : R → R, but it could be also an integral function or a convolution.

As done for the autonomous NLW, we fix a frequency ω = ωj on which we impose the first order Melnikov diophantine condition and we look for solutions of (1.3) with ω = ω(λ) given by

ω = (1 -ελ)ω, λ ∈ Λ ⊂ R.
This setting is slightly different with respect to chapter 2: we want a family of solutions of equation (1.3) for all ε fixed. This family of solutions is parametrized by λ (in the autonomous NLW, the parameter of the family of solutions for the initial equation (1.2) is ε, which does not appear in the equation).

Heuristically, the change -ελω of the frequency is created by the non-linearity εf (ε, u) which can be written as

εf (ε, u) = εf (ε, h) + O(ε)
where h is the solution of the bifurcation equation. A change of parameter λ corresponds to a change of h (see also Remark 5.1.4 for the choice of this particular "frequency-amplitude" relation).

Introduce appropriate Sobolev spaces H s,p := H s (T, H p ), with H p ⊂ H 0 , (see definition (5.2)), the main result about this problem is Theorem 5.1.6: under appropriate assumption on operator A, hypotheses of regularity of non-linearity f and some non-degeneracy hypothesis (see Definition 5.4.4), we prove that for all ε small enough being fixed, there exist a Cantor family of solution u ε (λ) ∈ H s1, p, smooth with respect to λ, and a Cantor-like set

A ε ⊂ Λ := [a, b]
of asymptotically full Lebesgue measure, such that for all λ ∈ A ε , u ε (λ) is a solution of (1.3) with ω = ω(λ).

More precisely, using the decomposition of Lyapunov-Schmidt, we obtain a 2π-periodic solution of the form

u ε (λ) = h ε (λ) + v( ε (λ)
with h ε (λ) solution of the bifurcation equation and v ε (λ) solution of the range equation such that �v ε (λ)� s1, p → 0 as ε → 0.

Chapter 1. Introduction

Fix h, we firstly solve the range equation using the Nash-Moser Theorem 2.3.1 proved in chapter 4. This scheme presents some differences with respect to the Nash-Moser Theorem used in the autonomous NLW and proved in [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF].

The reason is in the final measure estimate for the set A ε ⊂ Λ.

In fact, in section 5.5, we obtain measure estimates of the form

meas{(A ε ) c } ≤ CN -σ1 0 ε
for some σ 1 > 0, where N 0 = N 0 (ε) is the initialization scale in the Nash-Moser scheme, and this bound is small if

CN -σ1 0 ε << 1 ⇐⇒ εN σ1 0 >> 1. (1.11)
In the Nash-Moser sheme proved in [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF], N 0 = N 0 (ε) has to verify εN σ2 0 < 1 2 , for some σ 2 > 0 and we could have a contradiction with estimate (1.11).

In the Nash-Moser Theorem 2.3.1, we prove the result imposing to N 0 = N 0 (ε)

ε q N σ2 0 < 1 2 (1.12)
for q > 1 being fixed. The choice of q large enough allows us to avoid the contradiction with (1.11).

The exponent q appears in the hypotheses of our Nash-Moser result: given q > 1, we suppose to have for all ε small enough, an approximate solution ṽ(ε, λ) of the equation F (ε, λ, v) = 0 with an error O(ε q ).

We remark also that we prove Nash-Moser Theorem 2.3.1 between two different scales of Banach spaces and we can apply it directly to the range equation in relation to the loss of regularity of the operator A.

Similar Nash-Moser results are proved in [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF] and [START_REF] Berti | An abstract Nash-Moser theorem and quasi-periodic solutions for NLW and NLS on compact Lie groups and homogeneous manifolds[END_REF].

As said in previous considerations, the key point in the application of the Nash-Moser scheme is the proof of "good" high Sobolev norm estimates for the inverse of any finite dimensionnal restriction L N of linearized operator using "good" bounds in their L 2 -norm.

The result which allows us to prove these "good" bounds in high Sobolev norm (with δ < 1) using bounds in L 2 (T, H 0 )-norm is Proposition 5.3.4 (particular case of the more general result proved in Proposition 3.2.2).

The main hypothesis is a "good separation in blocks" of the "singular site", which are the Fourier indices corresponding to the small divisors (see (5.62)).

In this case (which generalizes the autonomous NLW equation of chapter 2), we cannot prove a lower bound of distance between two different singular sites as (1.10) (which is proved using the particular distribution of the eigenvalues ω j of the linear operator -∆ + V on Zoll manifolds, contained in intervals I j Chapter 1. Introduction centered at c 1 j + c 2 ). We cannot prove that the singular sites are "isolated" but we can prove that the set of singular sites in any interval [-N, N ] is divided in blocks Γ N,r well separated.

If we have this "good separation in blocks" and if L N has a sufficient off-diagonal

decay in | • | s -norm |L N -diag(L N )| s1 ≤ C(s 1 ),
then "good" L 2 -bounds for L -1 N imply "good" bounds in Sobolev norm � • � s,p . See chapter 3 for details about the introduction of | • | s -norm and definitions of regular and singular sites and chapter 5 for the application to the resolution of the range equation.

Then, in Theorem 2.5.5, we prove for all ε small enough, the existence of a Finally, we conclude the proof of main Theorem 5.1.6 with the measure estimates for the set A ε

function v(ε, λ, h), with v(ε, •, •) smooth with respect to parameters (λ, h), with v(0, λ, h) = 0, �v(ε, λ, h)� s1, p → 0 as ε → 0, such that v(ε, λ, h) is
A ε = A 1 ε + A 2 ε
where A 1 ε corresponds to parameters λ for which we have good bounds for the L 2 -norm of the inverse of each L N and A 2 ε corresponds to parameters λ for which there is a "good separation" in blocks for the set of singular sites in each interval [-N, N ].

We prove the measure estimates of A 1 ε proceeding as done in the autonomous NLW using the assumption of the property of distribution of ω j and the first order Melnikov diophantine condition on ω.

The last step is to prove that for almost all parameters λ we have a "good separation" in blocks of the singular sites of L N , and it is possible using again the property of distribution and the first order Melnikov condition. The last result of this thesis is the existence of quasi-periodic solutions of the forced NLW equation (1.1) on Zoll manifold

u tt -∆u + V (x)u = εf (ωt, x, u), x ∈ M, ω ∈ T ν where V (x) ≥ 0, V (x) � ≡ 0, V (x) ∈ C ∞ (M) and the non-linear term f (ϕ, x, u) is in C ∞ (T ν × M × R). Fix a pre-assigned diophantine direction ω ∈ R ν , |ω| ≤ 1 |ω • l| ≥ γ 0 |l| τ0 , ∀ l ∈ Z ν \ {0},
we consider frequencies ω ∈ Λ, with Λ defined in (6.14), which are colinear to ω

ω = λω, λ ∈ � 1 2 , 3 2
� .
Looking for quasi-periodic in time solutions of (1.1) with frequencies ω = λω means looking for (2π) ν -periodic solutions of

λ 2 (ω • ∂ ϕ ) 2 u -∆u + V (x)u = εf (ϕ, x, u) ϕ ∈ T ν . (1.13)
The main result is Theorem 6.1.3: we prove that, for all ε small enough, there exist a map

u ε ∈ C 1 (Λ, H s1 (T ν , H p(M))) with �u ε (λ)� s1, p → 0 as ε → 0
and a Cantor-like set A ε ⊂ Λ with asymptotically full Lebesgue measure, such that, for all λ ∈ A ε , u ε (λ) is a solution of (1.13) with ω = λω.

The forced nature of the non-linearity avoids the Lyapunov-Schmidt decomposition and we apply directly the Nash-Moser scheme 2.3.1 to the whole equation.

It is not possible to prove the separation of the singular sites. We prove that "good" bounds in L 2 (T ν , L 2 (M))-norm for the inverse of the linearized operators imply "good" bounds also in H s1 (T ν , H p(M))-norm using an iterative multiscale argument which uses the separation in blocks at different scales of a new classe of "bad sites" (see Definition 6.2.7), first introduced by Bourgain.

The Multiscale result which we use is the Multiscale Theorem 6.2.16 proved in Appendix C as a consequence of the multiscale result in [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF]. It says that if the linearized operator L has a sufficient "off-diagonal decay" and if the N -bad sites are sufficiently separated, then the L 2 (T ν , L 2 (M))-bounds for the "large" matrix L -1 N � of size N � = N χ (with some χ large enough) imply bounds in H s1 (T ν , H p(M))-norm too for L -1 N � .

Chapter 1. Introduction

We overcome the difficulty to have "good" bounds in � • � s, p-norm for the inverse of a "larger" matrix with an iterative argument (see Proposition 6.2.18).

Similar arguments as the previous results (a little more intricate) are used to have measure estimates for the complementary of the set A ε which conclude he proof of the main theorem.

One future goal could be to extend the existence of quasi-periodic solutions for the aunonomous force NLW equation, using as in the periodic case a Lyapunov-Schmidt decomposition.

Chapter 2

Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds

Main result: general idea and functional setting

We consider the non-linear wave equation

u tt -∆u + V (x)u = f (x, u), x ∈ M (2.1)
where M is a d-dimensional connected, compact, Riemannian C ∞ -manifold without boundary, of Zoll type, namely the geodesic flow on the unit tangent bundle is periodic of minimal period T > 0.

Fix p ∈ N, p > d/2 + 2, for all t ∈ T, the unknown function u(t, •) is in the Sobolev space H p(M, C) defined on M (we refer to section A.1 of Appendix A for definitions of Sobolev spaces

H p (M, R) ⊂ L 2 (M, R), p ∈ N, and their complexifications H p (M, C) ⊂ L 2 (M, C)). The non-linear term f (x, u) is a C ∞ (M × R, R) function such that f (x, 0) = 0, f (x, u) = a p (x)u p + r(x, u) with p ≥ 2
where r(x, u) = O(u q ), q > p, near u = 0.

The potential V verifies

V (x) ≥ 0, V (x) � ≡ 0, V ∈ C ∞ (M, R)
which implies the property

-∆ + V (x) ≥ β 0 I, for some β 0 > 0 (2.2)
Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds where I is the identity map in L 2 (M, C). We recall that (2.2) means

�-∆g, g� L 2 (M,C) + �V g, g� L 2 (M,C) ≥ β 0 �g, g� L 2 (M,C) , ∀g ∈ L 2 (M, C)
i.e. integrating by parts

�∇g� 2 L 2 (M,C) + �V g, g� L 2 (M,C) ≥ β 0 �g� 2 L 2 (M,C) , ∀g ∈ L 2 (M, C).
We look for small amplitude solutions of (2.1).

After the rescaling u → δu, δ > 0, equation (2.1) takes the form

u tt -∆u + V (x)u = δ p-1 � a p (x)u p (x) + r(x, δu) δ p � .
Imposing ε ∈ [0, ε 0 ) for a fixed ε 0 small enough and calling δ p-1 = ε, we study the equation

u tt -∆u + V (x)u = εf (ε, x, u), x ∈ M (2.3)
where the non linear term is

f (ε, x, u) = a p (x)u p (x) + r(ε, x, u) with r(ε, x, u) := r(x, δu) δ p = O((δu) q ) δ p = O(δ q-p u q ) = O(ε q-p p-1 u q ).
We have

f ∈ C 0 ([0, ε 0 ) × M × R, R), εf ∈ C 1 ([0, ε 0 ) × M × R, R) ∂ x f ∈ C 0 ([0, ε 0 ) × M × R, R), ∂ u f ∈ C 0 ([0, ε 0 ) × M × R, R)
and for all ε ∈ [0, ε 0 ),

f (ε, •, •) ∈ C ∞ (M × R, R).
Moreover, the nonlinearity f satisfies f (ε, x, 0) = 0 and

a p ∈ C ∞ (M, R), with p ≥ 2 r(ε, x, 0) = (∂ u r)(ε, x, 0) = • • • = (∂ p u r)(ε, x, 0) = 0. (H p )
Of course, u = 0 is a solution of (2.3).

By (2.2), we can consider the unbounded, linear, self-adjoint operator

P := � -∆ + V (x)
densely defined in L 2 (M, C). The spectrum σ(P ) of P is discrete, real and every eigenvalue of P has a finite multiplicity. The following Lemma, due to Colin de Verdière [START_REF] De Verdière | Sur le spectre des opérateurs ellipriques à bicaractéristiques toutes périodiques[END_REF] and taken from [START_REF] Bambusi | Almost global existence for Hamiltonian semilinear Klein-Gordon equations with small Cauchy data on Zoll manifolds[END_REF], describes the asymptotic distribution of the eigenvalues of P when M is a Zoll manifold.

Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds Lemma 2.1.1. If M is a Zoll manifold, there are constants α ∈ R, c 0 > 0, β ∈ (0, 1), C 0 > 0, and disjoint compact intervals (I j ) j≥1 with I 1 at the left of I 2 , and

I j := � 2π T j + α - c 0 j β , 2π T j + α + c 0 j β � , j ≥ 2
such that the spectrum of P satisfies

spect(P ) ⊂ � j≥1 I j with cardinality card(spect(P ) ∩ I j ) ≤ C 0 j d-1
counted with multiplicity.

We call

ω j,k , 1 ≤ k ≤ d j , d j ≤ C 0 j d-1
, the eigenvalues of P in each I j counted with multiplicity. There is an orthonormal basis of L 2 (M, C) composed of corresponding eigenvectors ϕ j,k .

Since the manifold M has no boundary, the Sobolev norms in H p(M, C) defined in (A.4) of Appendix A, can be written as

�g� 2 H p (M,C) = � � � � � � � � � j≥1 1≤k≤dj g j,k ϕ j,k � � � � � � � � 2 H p (M,C) := � j≥1 1≤k≤dj ω 2 p j,k |g j,k | 2
for all g ∈ H p(M, C), see [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF].

In the basis {ϕ j,k } j,k , every function h(t,

•) ∈ H p(M, C) ⊂ L 2 (M, C) has the form h(t, x) = � j,k a j,k (t)ϕ j,k (x)
and the linearized equation at zero of (2.3)

h tt -∆h + V (x)h = 0 (2.4) is reduced to the equation of harmonic oscillators äj,k (t) + ω 2 j,k a j,k (t) = 0.
We choose a particular ω = ω j, k and we consider solutions h(t, x) of the linear equation (2.4) which are 2π ω -periodic. Performing a decomposition of the space (which we will specify in the following), we want a solution u ε of (2.3) of the form

u ε (t, x) = h ε (t, x) + v ε (t, x)
Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds such that u ε is 2π ω -periodic in time with

ω → ω, h ε (t, x) → h(t, x), v ε (t, x) → 0 as ε → 0.
After a rescaling in time, looking for small amplitude 2π ω -periodic solutions of (2.1) means looking for 2π-periodic solutions of

ω 2 u tt -∆u + V (x)u = εf (ε, x, u) (2.5)
in the Sobolev scale

H s, p := H s (T, H p(M, R)) = � u(t, x) = � l∈Z e ılt u l (x), u l ∈ H p(M, C) | u -l = ūl , �u� 2 s := � l∈Z �l� 2s �u l � 2 H p (M,C) < ∞ � (2.6)
with s ≥ 0 and �l� := max{1, |l|}.

Note that ∀ l ∈ Z, u -l = ūl ∈ H p(M, C) implies u(t, x) ∈ R.
H s, p is the space of 2π-periodic in time functions with values in the Sobolev space H p(M, C), where p > max{2, d/2}.

As said in the section A.1 of Appendix A, for p ≥ d/2, the Sobolev space

H p(M, C) ⊂ C 0 (M, C
) is a Banach algebra with respect to multiplication of functions. Thanks to this property, for s ≥ s 1 > 1/2, H s, p is a Banach algebra too (see e.g. [START_REF] Berti | Nonlinear Oscillations in Hamiltonian PDEs[END_REF]) and there is the continuous embedding

H s (T, H p(M, R)) �→ C 0 (T, C 0 (M, R)) � C 0 (T × M, R) (2.7)
where C 0 (T × M, R) is endowed with the sup-norm

�u� L ∞ (T,L ∞ (M)) := sup t∈T �u(t, x)� L ∞ (M) . (2.8)
See section A.2 of Appendix A for details on the definition of the Sobolev space H s, p and for the proof of the imbedding (2.7).

Moreover, ∀ s ≥ s 1 , ∀ u 1 , u 2 ∈ H s, p �u 1 u 2 � s ≤ C 1 (s 1 )�u 1 � s1 �u 2 � s + C 2 (s 1 , s)�u 1 � s �u 2 � s1 . (2.9)
The proof of (2.9) is given for example in Appendix of [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF] (see also [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF]).

With respect to the standard Moser-Nirenberg interpolation estimate in Sobolev spaces (see e.g. [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF]), in property (2.9) one of the constants is independent of s. We impose on frequency ω to verifies:

Let us define the set

S := � (l, j, k) ∈ N × N × N : ω j,k = lω, k ∈ [0, d j ] � . ( 2
• the first order Melnikov diophantine condition: there is τ 0 > 0 and

γ ∈ (0, 1) such that |ωl-ω j,k | ≥ γ 1 + |l| τ0+1 , ∀ l ∈ Z, j ∈ N, k ∈ [1, d j ], (l, j, k) � ∈ S (2.11)
• for the same τ 0 > 0 and γ ∈ (0, 1) of (2.11), the condition:

� � � � ωl - 2π T p � � � � ≥ 2γ 1 + |l| τ0+1 , ∀ (l, p) ∈ Z 2 \ {(0, 0)}. (2.
12)

The first order Melnikov diophantine condition (2.11) implies that fequency ω is not in resonance with the normal mode frequencies ω j,k :

|ω 2 l 2 -ω 2 j,k | = |ωl -ω j,k |(ωl + ω j,k ) ≥ Cγ 1 + |l| τ0 (2.13) for some constant C > 0, ∀ l ∈ Z, j ∈ N, k ∈ [1, d j ], (l, j, k) � ∈ S.
We will impose on frequency ω a similar condition as (2.12): there is τ > 0 (which we will take large enough with respect to τ 0 , see Lemma 2.6.5) such that

� � � � ωl - 2π T p � � � � ≥ γ 1 + |l| τ +1 , ∀ (l, p) ∈ Z 2 \ {(0, 0)}. (2.14)
Fix a bounded interval (ω 1 , ω 2 ) ⊂ R, we define

Ω := {ω ∈ (ω 1 , ω 2 ) : ω verifies condition (2.14)}. (2.15) Remark 2.1.2. For τ > 0, meas(Ω c ∩ (ω 1 , ω 2 )) = O(γ). (2.16) Proof. Of course, Ω = Ω γ τ . Call R γ τ := (Ω γ τ ) c , fix l ∈ Z and define R γ τ,l := � ω : ∃ p ∈ Z \ {0} | � � � � ωl - 2π T p � � � � ≤ γ 1 + |l| τ +1 � , we have R γ τ ⊂ � l∈Z R γ τ,l . Moreover, fix p ∈ Z \ {0}, meas(R γ τ ∩ (ω 1 , ω 2 )) ≤ � l∈Z � p∈Z\{0} meas(R γ τ,l,p ∩ (ω 1 , ω 2 )),
Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds where

R γ τ,l,p := � ω : � � � � ωl - 2π T p � � � � ≤ γ 1 + |l| τ +1 � .
Define the set

S l := � p ∈ Z \ {0} : ∃ ω ∈ (ω 1 , ω 2 ), � � � � ωl - 2π T p � � � � ≤ 1 � . If p � ∈ S l , then ∀ ω ∈ (ω 1 , ω 2 ), � � ωl -2π T p � � ≥ 1 (and in particular > γ 1+|l| τ +1 ) and we have R γ τ,l,p ∩ (ω 1 , ω 2 ) = ∅.
The integers p which are in S l are only in finite number n = O(|l|).

Moreover, for ω ∈ R γ τ,l,p , the condition

� � ωl -2π T p � � ≤ γ 1+|l| τ +1 implies ω ∈ � 2π T p |l| - Cγ 1 + |l| τ +2 , 2π T p |l| + Cγ 1 + |l| τ +2 � .
By all previous considerations, we conclude

meas(R γ τ ∩ (ω 1 , ω 2 )) ≤ � l∈Z � p∈Z\{0} meas(R γ τ,l,p ∩ (ω 1 , ω 2 )) ≤ � l∈Z � p∈S l meas(R γ τ,l,p ∩ (ω 1 , ω 2 )) ≤ � l∈Z � p∈S l Cγ |l| τ +2 ≤ � l∈Z C|l| γ |l| τ +2 ≤ � l∈Z C γ |l| τ +1 ≤ Cγ if τ + 1 > 1, and it is true because τ > 0. Fix an index 1/2 < s 1 ∈ N. Define E 0 the space of 2π-periodic solutions h ∈ H s1 (T, H p(M, R)) of the linear equation ω2 h tt -∆h + V (x)h = 0.
(2.17)

We have

E 0 := Ker(ω 2 ∂ tt -∆ + V (x)) = � h(t, x) ∈ H s1, p : h(t, x) = � (l,j,k)∈S (α ljk cos(lt)ϕ j,k + β ljk sin(lt)ϕ j,k ) � Chapter 2.

Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds

Since card(S) < +∞, the space E 0 is finite dimensional and E 0 ⊂ H s, p for all s ≥ 0 (in this sense, it does not depend on s).

We decompose the space

H s1 (T, H p(M, R)) = E 0 ⊕ E s1,⊥ 0 where E s1,⊥ 0 = � h(t, x) ∈ H s1, p : h(t, x) = � (l,j,k)∈S c (α ljk cos(lt)ϕ j,k +β ljk sin(lt)ϕ j,k ) � .
We look for 2π-periodic solutions of (2.5) in H s1 (T, H p(M, R)) of the form

u(t, x) = h(t, x) + v(t, x) with h ∈ E 0 and v ∈ E s1,⊥ 0 , 2π-periodic in time too such that �v� s1 → 0 as ε → 0.
Let us consider ∀ a ∈ T the translation in time

T a u : (t, x) → u(t + a, x). (2.18)
As a consequence of the autonomous nature of the equation (2.5), it is clear that the following result holds. In the variational formulation, solving equation (2.5)

ω 2 u tt -∆u + V (x)u = εf (ε, x, u)
is equivalent to finding the critical points in the Sobolev space H s1, p of the functional

Φ ε (ω, u) = � � T×M - ω 2 2 |∂ t u| 2 + 1 2 |∇ x u| 2 + 1 2 V (x)u 2 -εF (ε, x, u) dt dx (2.19)
where

∂ u F (ε, x, u) = f (ε, x, u). (2.20) It is easy to check that for u ∈ H s1, p u is solution of equation (2.5) ⇐⇒ DΦ ε (ω, u) = 0. (2.21)
Since the non linear term f (ε, x, u) does not depend on time, for u ∈ H s1, p we have

Φ ε (ω, T a u) = Φ ε (ω, u) ∀ a ∈ T (2.22) on Zoll manifolds whence DΦ ε (ω, T a u)[T a h] = DΦ ε (ω, u)[h]. (2.23)
Hence, we have

DΦ ε (ω, u) = 0 ⇒ DΦ ε (ω, T a u) = 0 (2.24)
and by (2.21), we have that T a u is solution of (2.5) (we find what yet said in Lemma 2.1.3).

Let us introduce the functionals defined on E 0 , Ψ+

0 , Ψ- 0 ∈ C 2 (E 0 , R) Ψ± 0 (h) := � � T×M h 2 t 2 ± a p (x) p + 1 h p+1 dt dx
which will be studied in subsection 2.4.1.

We can now insert the main result of this chapter.

In what follows, we generally assume Sobolev indices s ∈ N, even if not specified.

Theorem 2.1.4. Let M be a Zoll manifold and let us consider the wave equation (2.5)

ω 2 u tt -∆u + V (x)u = εf (ε, x, u) with potential V (x) ≥ 0, V (x) � ≡ 0, V (x) ∈ C ∞ (M, R
) and the non-linear term

f (ε, x, u) f (ε, x, u) = a p (x)u p + O(ε q-p p-1 u q ) such that f ∈ C 0 ([0, ε 0 ) × M × R, R), εf ∈ C 1 ([0, ε 0 ) × M × R, R) ∂ x f ∈ C 0 ([0, ε 0 ) × M × R, R), ∂ u f ∈ C 0 ([0, ε 0 ) × M × R, R)
and for all ε ∈ [0, ε 0 )

f (ε, •, •) ∈ C ∞ (M × R, R).
We suppose there exists a function 0

� = h 0 ∈ E 0 which verifies    D Ψ+ 0 (h 0 ) = 0, dim(KerD 2 Ψ+ 0 (h 0 )) = 1 condition (P + ) or    D Ψ- 0 (h 0 ) = 0, dim(KerD 2 Ψ- 0 (h 0 )) = 1 condition (P -).

Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds

Then, given 1/2 < s 1 ∈ N and fixed a frequency ω which verifies conditions (2.11) and (2.12), there exist ε ∈ [0, ε 0 ) small enough (ε << γ), a map

u ∈ C 1 ([0, ε); H s1, p) with �u(ε) -h 0 � s1 → 0 as ε → 0
and a Cantor-like set A ⊂ [0, ε) with asymptotically full Lebesgue measure

meas{(A c ) ∩ [0, ε)} ε → 0 as ε → 0 such that, for all ε ∈ A, u(ε) is a solution of (2.5) with ω = ω(ε) ∈ Ω,
where Ω is defined in (2.15) for τ > 2(τ 0 + 2) and the frequency-amplitude relation between ε and ω is given by

ω 2 = s * ε + ω2 with s * =    -1 if h 0 verifies condition (P + ) 1 if h 0 verifies condition (P -).
Moreover, the function ũ = δu, δ = ε 1/p-1 , is a small amplitude 2π ω -periodic solution of wave equation (2.1).

We will prove Theorem 2.1.4 in the next sections.

The Lyapunov-Schmidt decomposition

According to the decomposition of the Sobolev space

H s1 (T, H p(M, R)) = E 0 ⊕ E s1,⊥ 0 ,
we recall that we look for 2π-periodic solutions of (2.5) of form

u(t, x) = h(t, x) + v(t, x) with h(t, x) ∈ E 0 and v(t, x) ∈ E s1,⊥ 0 , 2π-periodic in time too such that �v� s1 → 0 as ε → 0.
In this decomposition, equation (2.5) is equivalent to the system

   ω 2 h tt -∆h + V (x)h = εΠ E0 f (ε, x, h + v) ω 2 v tt -∆v + V (x)v = εΠ E s 1 ,⊥ 0 f (ε, x, h + v) (2.25)
Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds and using that h verified (2.17), we have to solve

   (ω 2 -ω2 )h tt = εΠ E0 f (ε, x, h + v) (Q) ω 2 v tt -∆v + V (x)v = εΠ E s 1 ,⊥ 0 f (ε, x, h + v) (P).
(2.26)

The (Q)-equation is called the bifurcation equation and the (P )-equation is the range equation.

Firstly we fix h ∈ E 0 and we solve the range equation finding a function with respect to ε. Hence, we have a complete solution u = h + v of (2.5) for parameters (ε, ω, h) in a particular set.

v h = v(ε, ω, h) ∈ E s1,⊥ 0 , C 1 with respect to (ε, ω, h),
We need to introduce for the following some useful definitions.

For all s ≥ 0 and 0 ≤ p ≤ p, we introduce the scale of Sobolev spaces For s = 0 = p, we shall denote H 0 (T,

H s,p := H s (T, H p (M, R)),
H 0 (M, R)) = L 2 (T, L 2 (M, R)), with L 2 (T, L 2 (M, R)) � L 2 (T × M, R),
in accordance with the Parseval's identity for which � • � 0,0 is equal to the

standard norm on L 2 (T, L 2 (M, R)) � l∈Z �u l � 2 L 2 (M,C) = 1 2π � T �u(t)� 2 L 2 (M,C) dt = 1 2π � T×M |u(t, x)| 2 dt dx.
L : D(L) ⊂ L 2 (T, L 2 (M, C)) → L 2 (T, L 2 (M, C)) defined by: ∀ u 1 , u 2 ∈ L 2 (T, L 2 (M, R)) L(u 1 + ıu 2 ) = L(u 1 ) + ıL(u 2

The range equation

Let us consider the (P )-equation

ω 2 v tt -∆v + V (x)v = εΠ E s 1 ,⊥ 0 f (ε, x, h + v) in the space E s1,⊥ 0 .
Given h ∈ E 0 such that �h� 0,0 < R, with a suitable R which we will define later (see (2.106)), this is equivalent to solve

F (ε, ω, h, v) = 0 (2.29)
where

F (ε, ω, h, v) := ω 2 Qv tt + v -Qv -εQΠ E s 1 ,⊥ 0 f (ε, x, h + v) (2.30)
and Q is the operator (introduced in [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF]) defined by

Q := (-∆ + V (x) + I) -1 . (2.31)
By classical elliptic estimates the operator Q is regularizing of order 2 in the spatial variable: more precisely, we have

∀ v ∈ H s,p �Qv� s,p = �(-∆ + V (x) + I) -1 v� s,p ≤ �v� s,max{0,p-2} . (2.32)
Note that with the definition of the function F using the operator Q, we have no more the loss of regularity in the spatial variable caused by the operator

-∆ + V (x).
To solve (2.29) we apply a particular Nash-Moser theorem proved in [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF] which follows.

Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds

The Nash-Moser scheme

We consider a scale of Banach spaces (X s , � � s ) s≥0 such that

∀ s ≤ s � , X s � ⊆ X s , �v� s ≤ �v� s � , ∀v ∈ X s � , (2.33)
and we define X := � s≥0 X s . We assume that there are an increasing family (

E (N ) ) N ≥0 of closed subspaces of X such that � N ≥0 E (N )
is dense in X s for every s ≥ 0, and that there are projectors

Π (N ) : X 0 → E (N ) of range E (N ) satisfying, ∀ s ≥ 0, ∀ d ≥ 0, • (S 1 ) �Π (N ) v� s+d ≤ C(s, d)N d �v� s , ∀ v ∈ X s ; • (S 2 ) �(I -Π (N ) )v� s ≤ C(s, d)N -d �v� s+d , ∀ v ∈ X s+d
where C(s, d) are positive constants. The projectors Π (N ) can be seen as smoothing operators.

We consider a map

F : [0, ε 0 ) × Λ × X s1+ν → X s1 (2.34)
where s 1 ≥ 0, ν > 0, ε 0 > 0 and Λ is a bounded open domain of a Banach space (Y, � • � Y ) of finite dimension m > 0. We assume:

• (F 0) F ∈ C 1 ([0, ε 0 ) × Λ × X s1+ν ) and for all (ε, λ) ∈ [0, ε 0 ) × Λ, F (ε, λ, •) ∈ C 2 (X s1+ν ), • (F 1) F (0, λ, 0) = 0, ∀ λ ∈ Λ,
and the "tame" properties:

∀s ≥ s 1 , ∀ v ∈ X s+ν with �v� s1 ≤ 2, ∀ (ε, λ) ∈ [0, ε 0 ) × Λ, • (F 2) �∂ (ε,λ) F (ε, λ, v)� s ≤ C(s)(1 + �v� s+ν ), �D v F (ε, λ, 0)[g]� s ≤ C(s)�g� s+ν ,
where the symbol ∂ (ε,λ) denotes either the partial derivate ∂ ε or ∂ λi , i = 1, . . . , m;

• (F 3) �D 2 v F (ε, λ, v)[w, g]� s ≤ C(s)(�v� s+ν �w� s1 �g� s1 + �g� s+ν �w� s1 +�w� s+ν �g� s1 ); • (F 4) �∂ (ε,λ) D v F (ε, λ, v)[g]� s ≤ C(s)(�g� s+ν + �v� s+ν �g� s1 ).

on Zoll manifolds

Using Taylor formula, one can prove from (F 1) -(F 3) the useful following properties: ∀ s ≥ s 1 , there is C(s) > 0 such that ∀ �v� s1 ≤ 2, �g� s1 ≤ 1,

• (F 5) �D v F (ε, λ, v)[g]� s ≤ C(s)(�v� s+ν �g� s1 + �g� s+ν ), • (F 6) �F (ε, λ, v)� s ≤ C(s)(ε + �v� s+ν ), • (F 7) �F (ε, λ, v + g) -F (ε, λ, v) -D v F (ε, λ, v)[g]� s ≤ C(s)(�v� s+ν �g� 2 s1 +�g� s+ν �g� s1 ).
We consider two parameters µ ≥ 0, σ ≥ 0, such that

σ > 4(µ + ν) (2.35)
and we introduce an index s such that s ≥ s 1 + 4(µ + ν + 1) + 2σ.

(2.36)

We denote by L (N ) (ε, λ, v) the linear operators

L (N ) (ε, λ, v) := Π (N ) D v F (ε, λ, v) |E (N ) ,
and given a non-decreasing function K : [0, +∞) → [1, +∞), for all γ > 0, we assume defined appropriate subsets J

(N ) γ,µ,K such that J (N ) γ,µ,K ⊆ � (ε, λ, v) ∈ [0, ε 0 ) × Λ × E (N ) | L (N ) (ε, λ, v) is invertible and ∀s ≥ s 1 , �(L (N ) (ε, λ, v)) -1 w� s ≤ K(s) N µ γ (�w� s + �v� s �w� s1 ), ∀w ∈ E (N )
� .

(2.37)

Given k > 0, we define

V (N ) k := {v ∈ C 1 ([0, ε 0 ) × Λ, E (N ) ) | �v� s1 ≤ 1, �∂ (ε,λ) v� s1 ≤ k} (2.38) and for all v ∈ V (N ) k , we set G (N ) γ,µ,K (v) := {(ε, λ) ∈ [0, ε 0 ) × Λ | (ε, λ, v(ε, λ)) ∈ J (N ) γ,µ,K }.
We have the following iterative result, where

N n := N 2 n 0 ,
N 0 ∈ N will be chosen large enough (depending on γ) and E n , Π n , J n γ,µ,K are abbreviations for E (Nn) , Π (Nn) , J

(Nn) γ,µ,K respectly. Given a subset A ⊂ [0, ε 1 ) × Λ and η > 0, we denote by N (A, η) the open neigh- borhood of A of width η in ([0, ε 1 )×Λ, |•|+�•� Y ) (which is empty if A is empty).

Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds

We can now insert the following Nash-Moser Theorem taken from [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF].

We remark that in the setting of [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF], one assume

F ∈ C 2 ([0, ε 0 ) × Λ × X s1+ν ).
What is really used in the proof of the Nash-Moser Theorem 3 of [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF] is the C 2 -regularity of F with respect to v ∈ X s1+ν and only the C 1 -regularity with respect to parameters (ε, λ) ∈ [0, ε 0 ) × Λ. Hence, it is enough to suppose (F 0) in order to have the following Nash-Moser result. (2.35) and (2.36). Then, for all γ > 0 there are

Theorem 2.3.1. Assume (F 0) -(F 4),
N 0 := N 0 (γ), K 0 = K 0 (γ) > 0, ε 1 := ε 1 (γ) ∈ (0, ε 0 ] and a sequence (v n ) n≥0 of C 1 maps v n : [0, ε 1 ) × Λ → X s1+ν
with the following properties:

• (P 1 ) n v n (ε, λ) ∈ E n , v n (0, λ) = 0, �v n � s1 ≤ 1, �∂ (ε,λ) v n � s1 ≤ K 0 (γ)N σ/2 0 ; • (P 2 ) n F or 1 ≤ k ≤ n, �v k -v k-1 � s1 ≤ N -σ-1 k , �∂ (ε,λ) (v k -v k-1 )� s1 ≤ N -1-ν k ; • (P 3 ) n Let A n := ∩ n k=0 G N k γ,µ,K (v k-1 ) with v -1 := 0. If (ε, λ) ∈ N (A n , γN -σ/2 n ) then v n (ε, λ) solves the equation (F n ) Π n F (ε, λ, v) = 0; • (P 4 ) n B n := 1 + �v n � s, B � n := 1 + �∂ (ε,λ) v n � s (where s is defined in (2.35)) satisfy (i) B n ≤ 2N µ+ν n+1 (ii) B � n ≤ 2N µ+ν+σ/2 n+1
.

The sequence

(v n ) n≥0 converges uniformly in C 1 ([0, ε 1 ) × Λ, X s1+ν ) (endowed
with the sup-norm of the map and its partial derivatives) to v with v(0, λ) = 0 and

(ε, λ) ∈ A ∞ := ∩ n≥0 A n ⇒ F (ε, λ, v(ε, λ)) = 0.
We want to apply the Nash-Moser scheme: we have to prove that F in (2.30)

verifies the assumptions required in the Theorem 2.3.1.

In our case, the scale of Banach spaces is the following scale of Sobolev spaces

X s = H s (T; H p(M)) ∩ E s1,⊥ 0
and the subspaces E (N ) are Note that E (N ) ⊂ H s,p ∩ E s1,⊥ 0 , for all s ≥ 0.

E (N ) :=    w ∈ E s1,⊥ 0 : w(t, x) = � |l|≤N e ilt w l (x), w l ∈ H p(M, C), w -l = wj    , Chapter
We take as parameter λ the couple (ω, h) with ω in the set Ω defined in (2.15), and h in a open ball B R = B R (0) ⊂ E 0 with center in the origin and radius R,

defining Λ = Ω × B R endowed with norm | • | + � • � 0,0 .
In this setting, we have the following result.

Lemma 2.3.2. For 1/2 < s 1 ∈ N, if the non-linearity f verifies f ∈ C 0 ([0, ε 0 ) × M × R), εf ∈ C 1 ([0, ε 0 ) × M × R) ∂ x f ∈ C 0 ([0, ε 0 ) × M × R), ∂ u f ∈ C 0 ([0, ε 0 ) × M × R)
and for all ε ∈ [0, ε 0 )

f (ε, •, •) ∈ C ∞ (M × R),
then the map F defined in (2.30)

F (ε, ω, h, v) := ω 2 Qv tt + v -Qv -εQΠ E s 1 ,⊥ 0 f (ε, x, h + v) with Q := (-∆ + V (x) + I) -1 , satisfies (2.34) with ν = 2 and (F 0), (F 1) hold F ∈ C 1 ([0, ε 0 ) × Ω × B R (0) × E s1+2,⊥ 0 , E s1,⊥ 0 ), ∀ (ε, ω, h) ∈ [0, ε 0 ) × Ω × B R (0)), F (ε, ω, h, •) ∈ C 2 (E s1+2,⊥ 0 , E s1,⊥ 0 ).
Moreover, the tame properties

(F 2) -(F 4) hold for all s ∈ [s 1 , +∞].
Proof. This Lemma is proved using standard properties of operators of composition in Sobolev spaces (that we prove in section B.3 of Appendix B) for the map F (ε, ω, h, v) for which

D v F (ε, ω, h, v)[w] = ω 2 Qw tt + w -Qw -εQΠ E s 1 ,⊥ 0 (∂ u f )(ε, x, h + v)w.
This kind of result is proved (in a more general setting) in Lemma 5.3.1.

See also see [START_REF] Berti | Cantor families of periodic solutions of wave equations with C k nonlinearities[END_REF] for references.

There remains to define the sets of parameters J

(N ) γ,µ,K for which if (ε, ω, h) ∈ J (N ) γ,µ,K the linear operators L (N ) (ε, ω, h, v) := Π (N ) E ⊥ 0 D v F (ε, ω, h, v) |E (N )
are invertible and the following estimates hold:

∀ w ∈ E (N ) �(L (N ) (ε, ω, h, v)) -1 w� s ≤ K(s) N µ γ (�w� s + �v� s �w� s1 ). (2.39)
Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds Moreover, we have to prove that the set A ∞ has a "large" measure.

We start imposing some conditions in order to have estimates (2.39).

Let us consider in the following, functions v ∈ E s1,⊥ 0 such that �v� s1 ≤ 1.

(2.40)

From the definitions of F and Q, we write

L (N ) (v)[w] := L (N ) (ε, ω, h, v)[w] = ω 2 Qw tt + w -Qw -εQΠ (N ) E ⊥ 0 (b(ε, t, x)w) = QL (N ) (v)[w] (2.41)
where

L (N ) (v)[w] := ω 2 w tt -∆w + V (x)w -εΠ (N ) E ⊥ 0 (b(ε, t, x)w), b(ε, t, x) := (∂ u f )(ε, x, h(t, x) + v(t, x)).
We have the following proposition whose the proof is exactly the same as the one of Proposition 3.1 in [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF], with the different definition of b(ε, t, x).

Proposition 2.3.3. ∀ τ 1 > 0, τ > 0, there exists a non-decreasing function

K : R + → [1, +∞) and ∀ γ > 0 a constant C = C(τ 1 , γ) > 0 such that: if ε(�b� s1 + 1) ≤ C, ω verifies condition (2.14) � � � � ωl - 2π T p � � � � ≥ γ 1 + |l| τ +1 , ∀ (l, p) ∈ Z 2 \ {(0, 0)},
and

∀ 1 ≤ K ≤ N, �(L (K) (v)) -1 � 0,0 ≤ 4K τ1 γ , (2.42
)

then, ∀ s ≥ s 1 , ∀ w ∈ E (N ) �(L (N ) (v)) -1 w� s,0 ≤ K(s) γ N µ0 (�w� s,0 + �b� s �w� s1,0 ) (2.43)
where µ 0 = 2τ 1 + 2.

In our case, the assumption

ε(�b� s1 + 1) ≤ C (2.44)
can be easily verified using the following tame estimates (obtained by (F 5))

∀ s ≥ s 1 , �b� s ≤ C(s)(1 + �h + v� s ).
(2.45)

In fact, since E 0 is finite dimensional, all norm in E 0 are equivalent and for

h ∈ E 0 �h� 0,0 ≤ R =⇒ �h� s ≤ c(s)R, ∀ s ≥ s 1 .
Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds Then, using also �v� s1 ≤ 1, we estimate

�b� s1 ≤ C(s 1 )(1 + �h + v� s1 ) ≤ C(s 1 )(1 + �h� s1 + �v� s1 ) ≤ C(s 1 , R).
Having a bound on �b� s1 allows us to obtain (2.44) taking ε small enough.

We recall that, more generally, the tame estimates (2.45) are proved using standard properties of operators of composition in Sobolev spaces (that we prove in section B.3 of Appendix B). See also see [START_REF] Berti | Cantor families of periodic solutions of wave equations with C k nonlinearities[END_REF] for references.

By a bootstrap type argument, (2.43) implies a similare estimate for

�(L (N ) (v)) -1 w� s . Lemma 2.3.4. Under the assumptions of Proposition (2.3.3), ∀ s ≥ s 1 , �(L (N ) (v)) -1 w� s ≤ K(s) γ N µ (�w� s + �v� s �w� s1 ) ∀w ∈ E (N ) 0 ,
where µ := µ 0 + p + 2 and K(s) is non-decreasing.

Proof.

Let w := (L (N ) (v))[g] = g + Q(ω 2 g tt -g -εΠ (N ) E ⊥ 0 (bg)),
g := (L (N ) (v)) -1 [w] = (QL (N ) (v)) -1 [w] = (L (N ) (v)) -1 Q -1 [w] = (L (N ) (v)) -1 (-∆ + V (x) + I)[w], by
�g� s ≤ C(s)N p+2 � C(s, R) γ N µ0 (�w� s + �v� s �w� s1 ) + �w� s + εC(s)(1 + �h� s + �v� s ) C(s 1 , R) γ N µ0 (�w� s1 + �v� s1 �w� s1 ) + εC(s)(1 + �h� s + �v� s )�w� s1 � ≤ K(s) γ N µ (�w� s + �v� s �w� s1 ).
with µ := µ 0 + p + 2.

Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds Fix an appropriate τ 1 > τ 0 (it will be τ 1 > 2(τ 0 + 2) + d + 2, see Lemma 2.6.4) and τ > 2(τ 0 + 2) (see Lemma 2.6.5) in condition (2.14).

Define for µ = µ(τ 1 ) = 2τ 1 + p + 4 of Lemma 2.3.4 and �v� s1 ≤ 1, the sets

J (N ) γ,µ,K := � (ε, ω, h, v) ∈ [0, ε 1 ) × Ω × B R (0) × E (N ) : ∀ 1 ≤ K ≤ N, L (K) (v) is invertible and �(L (K) (v)) -1 � 0,0 ≤ 4K τ1 γ �
where we recall the definition of Ω as the set of ω ∈ (ω 1 , ω 2 ) which verify condition (2.14).

By Lemma 2.3.4, we have the inclusion

J (N ) γ,µ,K ⊆ � (ε, ω, h, v) ∈ [0, ε 0 ) × Ω × B R (0) × E (N ) | L (N ) (v) is invertible and ∀s ≥ s 1 , �(L (N ) (v)) -1 w� s ≤ K(s) N µ γ (�w| s + �v� s �w� s1 ), ∀w ∈ E (N )
� .

Finally, for all v ∈ V (N ) 1 defined as in (2.38), i.e. such that

�v(ε, ω, h)� s1 ≤ 1, �∂ (ε,ω,h) v(ε, ω, h)� s1 ≤ 1, (2.52)
we define the set

G (N ) γ,µ,K (v) := {(ε, ω, h) ∈ [0, ε 1 ) × Ω × B R (0) : (ε, ω, h, v(ε, ω, h)) ∈ J (N ) γ,µ,K } = � (ε, ω, h) ∈ [0, ε 1 ) × Ω × B R (0) : ∀ 1 ≤ K ≤ N, L (K) (v(ε, ω, h)) is invertible and �(L (K) (v(ε, ω, h)) -1 � 0,0 ≤ 4K τ1 γ � .
By these definitions, Lemma 2.3.2 and Theorem 2.3.1 give us the following result.

Theorem 2.3.5 (Solutions of the (P )-equation). There exist

ε 1 = ε 1 (γ) ∈ (0, ε 0 ) and a sequence {v n } k≥0 of C 1 maps v n : [0, ε 1 ) × Ω × B R (0) → E s1+2,⊥ 0 which converges uniformly to a function v ∈ C 1 ([0, ε 1 ) × Ω × B R (0), E s1+2,⊥ 0 ) with v(0, ω, h) = 0, �v(ε, ω, h)� s1 → 0 as ε → 0, such that ∀ (ε, ω, h) ∈ A ∞ := ∩ +∞ k=0 G N k γ,µ,K (v k-1 ), v(ε, ω, h) is solution of the range equation ω 2 v tt -∆v + V (x)v = εΠ E s 1 ,⊥ 0 f (ε, x, h + v).
Follows here a useful result which we will use in the next sections to prove measure estimates for the set A ∞ . Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds

Proposition 2.3.6. Let v ∈ C 1 ([0, ε 1 )×Ω×B R (0), E s1+2,⊥ 0 ) be the limit of the sequence {v n } n≥0 in Theorem 2.3.5. Given τ 1 > τ 0 and γ > 0, if ∀ 1 ≤ K ≤ N k , �(L (K) (v(ε, ω, h))) -1 � 0,0 ≤ 2K τ1 γ , (2.53)
then

�(L (K) (v k-1 (ε, ω, h))) -1 � 0,0 ≤ 4K τ1 γ .
Proof. The proof is the same as the one of Proposition 4.5.1.

Define

A ∞ := � N ≥N0 � (ε, ω, h) ∈ [0, ε 1 )×Ω×B R (0) : �(L (N ) (v(ε, ω, h)) -1 � 0,0 ≤ 2N τ1 γ �
by the Proposition 2.3.6, we have the inclusion

A ∞ ⊂ A ∞ . (2.54)
In the next sections we will find a solution h = h(ε) of (Q)-equation, C 1 with respect to parameter ε, and in section (2.6) we shall prove that the set

A A := {ε : (ε, ω(ε), h(ε)) ∈ A ∞ }
is a large set, where the relation ω = ω(ε) will be introduced in the following depending in particular on the parity of the exponent p which appears in the non linear term f (ε, x, u) and on the dimension and the form of E 0 .

Hence, define the set

A := {ε : (ε, ω(ε), h(ε)) ∈ A ∞ }, inclusion (2.54) implies A ⊂ A (2.55)
and if A is large, then A is large too: we will find solutions of the wave equation

(2.
3) for ε in a set of positive measure.

Follow now some considerations which give us a useful property which we will use in the following to prove the existence of smooth solutions of the bifurcation equation.

Recall for all a ∈ T and for all u(t, x) = h(t, x) + v(t, x), the translations in time introduced in (2.18)

T a u : (t, x) � → u(t + a, x),
Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds since the equivariance under the action of the group (T a ) a∈T of f (ε, x, u), we have that the function F defined in (2.30) verifies

F (ε, ω, T a h, T a v) = T a F (ε, ω, h, v).
(2.56) Lemma 2.3.7. In the application of the Nash-Moser scheme to the range equation, if we have

F (ε, ω, T a h, T a v) = T a F (ε, ω, h, v), (2.57
)

then for all (ε, ω, h) ∈ A n := ∩ +n k=0 G N k γ,µ,K (v k-1
), the property

(P a ) n    (ε, ω, T a h) ∈ A n v n (ε, ω, T a h) = T a v n (ε, ω, h)
holds for all n ∈ N and for all a ∈ T.

Proof. The proof is the same as the proof of Lemma 5.3.11, noting that in this Nash-Moser scheme, the initialization point is ṽ = 0.

In the Nash-Moser scheme (see Lemma 2.7 of [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF]), the function v n (and so the solution v) are defined on the whole set of parameters (ε, ω, h) ∈ [0, ε 1 ) × Ω × B R (0) using a sequence of cut-off functions {ψ n } n≥0 .

Since the sets A n are invariant under the action of group (T a ) a∈T , for all n ≥ 0, we take ψ n invariant under (T a ) a∈T too

ψ n = ψ n • T a so that ∀ (ε, ω, h) ∈ [0, ε 1 ) × Ω × B R (0) v n (ε, ω, T a h) = T a v n (ε, ω, h).
Hence, by Lemma 2.3.7 we have

(ε, ω, h) ∈ A ∞ =⇒ (ε, ω, T a h) ∈ A ∞ and ∀ (ε, ω, h) ∈ [0, ε 1 ) × Ω × B R (0) v(ε, ω, T a h) = T a v(ε, ω, h). ( 2

Periodic solutions of the bifurcation equation

There remains to solve the (Q)-equation (called also bifurcation equation)

(ω 2 -ω2 )h tt = εΠ E0 f (ε, x, h + v) (Q)
in the finite dimensional space

E 0 := Ker(ω 2 ∂ tt -∆ + V (x)).
We look for periodic solutions h ∈ E 0 of the (Q)-equation which are C 1 with respect to the parameter ε.

In this section we prove the existence of periodic solutions h = h(ε) ∈ E 0 of the (Q)-equation.

We will discuss the C 1 -regularity with respect to ε of the solutions in section 2.5.

We claim that, for any v ∈ E s1,⊥ 0 , solving the bifurcation equation is equivalent to finding critical points of the reduced Lagrangian action functional

h ∈ E 0 → Φ ε (ω, h + v) ∈ R, where Φ ε is the functional introduced in (2.19).
In fact, we have the following result.

Remark 2.4.1. Let us consider h ∈ E 0 . For any v ∈ E s1,⊥ 0 , we have that

h solves the (Q)-equation ⇐⇒ D h Φ ε (ω, h + v) = 0 (2.59)
where we use the notation

D h Φ ε (ω, h + v) := DΦ ε (ω, h + v) | E 0 . Proof. Recall Φ ε (ω, h + v) = � � T×M - ω 2 2 � � ∂ t (h + v) � � 2 + 1 2 � � ∇ x � h + v �� � 2 + 1 2 V (x)(h + v) 2 -εF (ε, x, h + v) dt dx, (2.60)
for all g ∈ E 0 , we have

DΦ ε (ω, h + v)[g] = � � T×M -ω 2 h t g t + ∇ x h • ∇ x g + V (x)hg -ω 2 v t g t + ∇ x v • ∇ x g + V (x)vg -εf (ε, x, h + v)g dt dx.
(2.61)

We write

DΦ ε (ω, h + v)[g] = � � T×M -ω 2 h t g t + ∇ x h • ∇ x g + V (x)hg -ω 2 v t g t + ∇ x v • ∇ x g + V (x)vg -εΠ E0 f (ε, x, h + v)g dt dx (2.62)
Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds using

f (ε, x, h + v) = Π E0 f (ε, x, h + v) + Π E s 1 ,⊥ 0 f (ε, x, h + v) (2.63) and �� T×M Π E s 1 ,⊥ 0 f (ε, x, h + v)g dt dx = 0 by orthogonality, because g ∈ E 0 .
Integrating by parts, we have by orthogonality

� � T×M -ω 2 v t g t + ∇ x v • ∇ x g + V (x)vg dt dx = � � T×M (ω 2 g tt -∆g + V (x)g)v dt dx = 0 (2.64) because if g ∈ E 0 then � ω 2 g tt + (-∆ + V (x))g � ∈ E 0 and v ∈ E s1,⊥ 0 .
For h ∈ E 0 , integrating by parts the linear equation (2.17), we obtain

∀ g ∈ E 0 0 = � � T×M (ω 2 h tt -∆h + V (x)h)g dt dx = � � T×M -ω 2 h t g t + ∇ x h • ∇ x g + V (x)hg dt dx, which implies ∀ g ∈ E 0 � � T×M ω2 h t g t dt dx = � � T×M ∇ x h • ∇ x g + V (x)hg dt dx. ( 2
DΦ ε (ω, h + v)[g] = � � T×M -ω 2 h t g t + ∇ x h • ∇ x g + V (x)hg -εΠ E0 f (ε, x, h + v)g dt dx = � � T×M -ω 2 h t g t + ω2 h t g t -εΠ E0 f (ε, x, h + v)g dt dx = � � T×M � (ω 2 -ω2 )h tt -εΠ E0 f (ε, x, h + v) � g dt dx (2.66)
from which it is clear that

DΦ ε (ω, h + v)[g] = 0 ∀g ∈ E 0 if
D v Φ ε (ω, h + v) = 0
where we use the notation

D v Φ ε (ω, h + v) := DΦ ε (ω, h + v) |E s 1 ,⊥ 0 .
Proof. For all w ∈ E s1,⊥ 0 , we write

DΦ ε (ω, h + v)[w] = � � T×M -ω 2 h t w t + ∇ x h • ∇ x w + V (x)hw -ω 2 v t w t + ∇ x v • ∇ x w + V (x)vw -εf (ε, x, h + v)w dt dx = � � T×M -ω 2 h t w t + ∇ x h • ∇ x w + V (x)hw -ω 2 v t w t + ∇ x v • ∇ x w + V (x)vw -εΠ E s 1 ,⊥ 0 f (ε, x, h + v)w dt dx
where we have used (2.63) and

�� T×M Π E0 f (ε, x, h + v)w dt dx = 0 by orthogo- nality, because w ∈ E s1,⊥ 0 .
Integrating by parts, we have by orthogonality

� � T×M -ω 2 h t w t + ∇ x h • ∇ x w + V (x)hw = � � T×M (ω 2 h tt -∆h + V (x)h)w dt dx = 0 because if h ∈ E 0 then � ω 2 h tt + (-∆ + V (x))h � ∈ E 0 and w ∈ E s1,⊥ 0 . Hence, we have ∀ w ∈ E s1,⊥ 0 DΦ ε (ω, h + v)[w] = � � T×M -ω 2 v t w t + ∇ x v • ∇ x w + V (x)vw -εΠ E s 1 ,⊥ 0 f (ε, x, h + v)w dt dx.
(2.67)

Integrating by parts, we conclude

DΦ ε (ω, h + v)[w] = � � T×M � ω 2 v tt -∆v + V (x)v -εΠ E s 1 ,⊥ 0 f (ε, x, h + v) � w dt dx from which it is clear that DΦ ε (ω, h + v)[w] = 0 ∀ w ∈ E s1,⊥ 0 (2.68)
if and only if v verifies the range equation.

Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds

For any h ∈ B R (0) ⊂ E 0 , let v(ε, ω, h) be the function given by Theorem

2.3.5, v(•, •, •) ∈ C 1 ([0, ε 1 ) × Ω × B R (0), E s1,⊥ 0 ), with v(0, ω, h) = 0, �v(ε, ω, h)� s1 → 0 as ε → 0 (2.69)
which is solution of the range equation for all parameters (ε, ω, h) ∈ A ∞ .

We want to find periodic solutions h = h(ε) of the (Q)-equation written for

v = v(ε, ω, h) (ω 2 -ω2 )h tt = εΠ E0 f (ε, x, h + v(ε, ω, h)) (2.70)
which by Remark 2.4.1, is equivalent to

(D h Φ ε )(ω, h + v(ε, ω, h)) = 0.
Define the functional

Ψ ε : Ω × B R (0) → R Ψ ε (ω, h) := Φ ε (ω, h + v(ε, ω, h)), (2.71) using f (ε, •, •) ∈ C ∞ (M × R), we have that Ψ ε (ω, •) ∈ C 1 (B R (0), R).
Remarks 2.4.1 and 2.4.2 imply the following result. Proof. Using the definition

Ψ ε (ω, h) := Φ ε (ω, h + v(ε, ω, h)),
for all g ∈ B R (0) ⊂ E 0 , we write

DΨ ε (ω, h)[g] = DΦ ε (ω, h + v(ε, ω, h)) � g + Dv(ε, ω, h)[g] � = DΦ ε (ω, h + v(ε, ω, h))[g] + DΦ ε (ω, h + v(ε, ω, h))[Dv(ε, ω, h)[g]].
(2.73) Using (2.66), we have

DΦ ε (ω, h+v(ε, ω, h))[g] = � � T×M � (ω 2 -ω 2 )h tt -εΠ E0 f (ε, x, h+v(ε, ω, h)) � g
(ε, ω, h) ∈ A ∞ DΨ ε (ω, h)[g] = � � T×M � (ω 2 -ω2 )h tt -εΠ E0 f (ε, x, h + v(ε, ω, h)) � g dt dx (2.76)
from which it is clear that

DΨ ε (ω, h)[g] = 0 ∀g ∈ B R (0) ⊂ E 0
if and only if h verifies the bifurcation equation.

We will prove the existence of non-trivial critical points of the functional

Ψ ε (ω, •) ∈ C 1 (B R (
0), R) (and so for (ε, ω, h) ∈ A ∞ of non-trivial solutions of the bifurcation equation) via the Mountain Pass Theorem of Ambrosetti and

Rabinowitz [START_REF] Ambrosetti | Dual Variational Methods in Critical Point Theory and Applications[END_REF] that we insert in the following.

The Mountain Pass argument

Let Φ : X → R be a smooth functional defined on a Banach space (X, � • �),

Φ ∈ C 1 (X, R).
Let us introduce the dual space of X X � := {L : X → R, L is linear and continuous} endowed with the norm � • � X � defined by

�L� X � := sup 0� =x∈X |Lx| �x� . Definition 2.4.4. (i) A sequence {x n } n≥0 ∈ X is called a Palais-Smale se- quence at the level c ∈ R if Φ(x n ) → c and �DΦ(x n )� X � → 0.
In this case, we say that {x n } n≥0 is a (P S) c sequence for short.

(ii) The functional Φ : X → R is said to satisfy the Palais-Smale condition at the level c if every (P S) c sequence {x n } n≥0 ∈ X possesses a convergent subsequence x n k → x.

In this case, we say that Φ verifies the (P S) c condition for short.

(iii) Finally, we say that Φ verifies the (P S) condition if it verifies the (P S) c condition for all c. on Zoll manifolds Note that, if Φ ∈ C 1 (X, R) satisfies (P S) c condition, any accumulation point x of a (P S) c sequence x n , is a critical point of Φ at the level c, namely DΦ(x) = 0 and Φ(x) = c.

Remark 2.4.5. In the finite dimensional case, if we want to prove the (P S) c condition, it is enough to prove that a (P S) c sequence x n is bounded: by the Bolzano-Weierstrass Theorem, a bounded sequence possesses a convergent subsequence x n k → x.

We can now insert the following result of Ambrosetti and Rabinowitz, taken from [START_REF] Ambrosetti | Dual Variational Methods in Critical Point Theory and Applications[END_REF] (see also [START_REF] Berti | Nonlinear Oscillations in Hamiltonian PDEs[END_REF] for the proof).

Theorem 2.4.6. (Mountain Pass) Suppose φ ∈ C 1 (X, R) and

• Φ(0) = 0; • ∃ ρ, α > 0 such that Φ(x) > α, ∀ x such that �x� = ρ; • ∃ v ∈ X with �v� > ρ, such that Φ(v) < 0.
Define the "Mountain Pass" value

c := inf γ∈Γ max t∈[0,1] Φ(γ(t))
where Γ is the minmax class

Γ := � γ ∈ C([0, 1], X) : γ(0) = 0, γ(1) = v � . Since �v� > ρ, then ∀ γ ∈ Γ, γ([0, 1]) ∩ {x ∈ X : �x� = ρ} � = ∅ and so c ≥ α.
Then there exists a Palais-Smale sequence for Φ at the level c.

As a consequence, if Φ satisfies the (P S) c condition, then c is a critical value for Φ: there exists x � = 0 such that DΦ(x) = 0 and Φ(x) = c.

Geometrical interpretation: Think the graph of Φ as a landscape with a low spot at x = 0 surrounded by a ring of mountains. Beyond these mountains lies another low spot at the point v. Then there must be a Mountain Pass between 0 and v, corresponding to a critical point. on Zoll manifolds Using definition (2.71) and (2.60), we write

Ψ ε (ω, h) = � � T×M - ω 2 2 � � ∂ t � h + v(ε, ω, h) �� � 2 + 1 2 � � ∇ x � h + v(ε, ω, h) �� � 2 + 1 2 V (x)(h + v(ε, ω, h)) 2 -εF (ε, x, h + v(ε, ω, h)) dt dx = � � T×M - ω 2 2 � h 2 t + (v t (ε, ω, h)) 2 + 2h t vt (ε, ω, h) � + 1 2 � |∇ x h| 2 + |∇ x v(ε, ω, h)| 2 + 2∇ x h • ∇ x v(ε, ω, h) � + 1 2 V (x) � h 2 + (v(ε, ω, h)) 2 + 2hv(ε, ω, h) � -εF (ε, x, h + v(ε, ω, h)) dt dx (2.77)
Integrating by parts, by orthogonality we have

� � T×M -ω 2 h t vt (ε, ω, h) + ∇ x h • ∇ x v(ε, ω, h) + V (x)hv(ε, ω, h) dt dx = = � � T×M � ω 2 h tt -∆h + V (x)h � v(ε, ω, h) dt dx = 0 (2.78) because if h ∈ B R (0) ⊂ E 0 then � ω 2 h tt + (∆ + V (x))h � ∈ E 0 and v ∈ E s1,⊥ 0 . For (ε, ω, h) ∈ A ∞ , v(ε, ω, h) is
� � T×M - ω 2 2 vt (ε, ω, h)) 2 + 1 2 (∇ x v(ε, ω, h)) 2 + 1 2 V (x)(v(ε, ω, h)) 2 dt dx = = � � T×M ε 2 f (ε, x, h + v(ε, ω, h))v(ε, ω, h) dt dx.
(2.79) By (2.78) and (2.79), we write (2.77) as

Ψ ε (ω, h) = � � T×M - ω 2 2 h 2 t + 1 2 (∇ x h) 2 + 1 2 V (x)h 2 -εF (ε, x, h + v(ε, ω, h)) + ε 2 f (ε, x, h + v(ε, ω, h))v(ε, ω, h) dt dx
Then, using (2.65) written for h = g ∈ E 0 , we have

Ψ ε (ω, h) = � � T×M � - ω 2 2 + ω2 2 � h 2 t -εF (ε, x, h + v(ε, ω, h)) + ε 2 f (ε, x, h + v(ε, ω, h))v(ε, ω, h) dt dx (2.80)
which we write as

Ψ ε (ω, h) = � � T×M � - ω 2 2 + ω2 2 � h 2 t + ε � - a p (x) p + 1 h p+1 � dt dx + εR ε (h, v(ε, ω, h)) (2.81)
Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds

with remainder R ε R ε (h, v(ε, ω, h)) := � � T×M - 1 p + 1 a p (x)(h + v(ε, ω, h)) p+1 + 1 p + 1 a p (x)h p+1 + 1 2 a p (x)(h + v(ε, ω, h)) p v(ε, ω, h) + O(ε q-p p-1 h q ) dt dx, (2.82)
where we have used that f (ε, x, u) = a p (x)u p (x) + O(ε q-p p-1 u q ) and, as a consequence,

F (ε, x, u) = 1 p+1 a p (x)u p+1 (x) + O(ε q-p p-1 u q+1 ). Using the regularity of functions f (ε, •, •), F (ε, •, •) ∈ C ∞ (M × R), we write f (ε, x,h + v(ε, ω, h)) = f (ε, x, h) + ρ(ε, h) with �ρ(ε, h)� L ∞ → 0 as ε → 0 (2.83)
and

F (ε, x,h + v(ε, ω, h)) = F (ε, x, h) + ρ(ε, h), with �ρ(ε, h)� L ∞ → 0 as ε → 0 (2.84) where the � • � L ∞ = � • � L ∞ (T,L ∞ (M)) is introduced in (2.8).
Indeed, write

f (ε, x, h + v(ε, ω, h)) = f (ε, x, h) + � 1 0 D u f (ε, x, h + τ v(ε, ω, h))v(ε, ω, h) dτ,
we define the remainder

ρ(ε, h) = ρ(ε, h, v(ε, ω, h)) := � 1 0 D u f (ε, x, h + τ v(ε, ω, h))v(ε, ω, h) dτ.
Using (2.7), we estimate

�ρ(ε, h)� L ∞ = � � � � 1 0 D u f (ε, x, h + τ v(ε, ω, h))v(ε, ω, h) dτ � � � L ∞ ≤ � 1 0 � � �Duf (ε, x, h + τ v(ε, ω, h))v(ε, ω, h)) � � � L ∞ dτ ≤ C � 1 0 �v(ε, ω, h)� s1 dτ
which implies by estimates (2.69)

�ρ(ε, h)� L ∞ → 0 as ε → 0 (2.85)
The proof of (2.84) is similar.

By definition (2.82), using (2.83) and (2.84), we can easily estimate

�R ε (h, v(ε, ω, h))� L ∞ → 0 as ε → 0 (2.86)
Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds

�DR ε (h, v(ε, ω, h))� L ∞ → 0 as ε → 0 (2.87)
Before to apply the Mountain Pass Theorem, we extend the functional

Ψ ε (ω, •) to whole space E 0 (recall that until now, Ψ ε (ω, •) is defined only in B R (0) ⊂ E 0 , since v(•, •, •) ∈ C 1 ([0, ε 0 ) × Λ × B R (0))).
Step 1. Extension of Ψ ε (ω, •).

Using (2.81), we define the extended action functional Ψε (ω,

•) ∈ C 1 (E 0 , R) as Ψε (ω, h) = � � T×M � - ω 2 2 + ω2 2
� h 2 t + ε � - a p (x) p + 1 h p+1 � dt dx + ε Rε (h, v(ε, ω, h)) (2.88) where Rε (h, v(ε, ω, h)) : E 0 → R is defined by Rε (h, v(ε, ω, h)) := λ(�h� 0,0 )R ε (h, v(ε, ω, h)) (2.89) and λ : [0, +∞] → [0, 1] is a smooth, non-increasing, cut-off function such that λ(y) =    1 for |y| ≤ R -1 0 for |y| ≥ R and |λ � (y)| < 1. By definition Ψε (ω, h) =    Ψ ε (ω, h) for �h� 0,0 ≤ R -1 �� T×M � -ω 2 2 + ω2 2 � h 2 t -ε � ap(x) p+1 h p+1 � dt dx for �h� 0,0 ≥ R.
Furthermore by (2.86), (2.87) and the definition of λ

� Rε (h, v(ε, ω, h))� L ∞ → 0 as ε → 0, �D Rε (h, v(ε, ω, h))� L ∞ → 0 as ε → 0. (2.90)
In order to apply the Mountain Pass Theorem and to find non-trivial critical points of the functional Ψε , we suppose that

∃ h ∈ E 0 : � � T×M a p (x)h p+1 dt dx � = 0. (2.91)
This condition depends in particular on the parity of p, the exponent which appears in the non-linear term f , and on the form of the space E 0 .

Moreover, we impose the following relation between ω and ε

ω 2 = s * ε + ω2 (2.92) with s * =    -1 if ∃ h 1 ∈ E 0 : �� T×M a p (x)h p+1 1 dt dx > 0 1 if ∀ h ∈ E 0 : �� T×M a p (x)h p+1 dt dx ≤ 0.
Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds Just to fix ideas, we suppose in the following that

∃ h 1 ∈ E 0 : � � T×M a p (x)h p+1 1 dt dx > 0 (2.93)
and we impose the relation

-(ω 2 -ω2 ) = ε. ( 2
Ψε (ω(ε), h) = ε � � � T×M h 2 t 2 - a p (x) p + 1 h p+1 dt dx + Rε (h, v(ε, ω(ε), h)) � .
Let us denote in the following v(ε, h) := v(ε, ω(ε), h).

Let us define

Ψε (h) :=

� � T×M h 2 t 2 - a p (x) p + 1 h p+1 dt dx + Rε (h, v(ε, h)) (2.95) with Ψε ∈ C 1 (E 0 , R).
Recall (2.89) and define

Ψ ε (h) := � � T×M h 2 t 2 - a p (x) p + 1 h p+1 dt dx + R ε (h, v(ε, h)),
we have

Ψε (h) =    Ψ ε (h) for �h� 0,0 ≤ R -1 �� T×M h 2 t 2 - ap(x)
p+1 h p+1 dt dx for �h� 0,0 ≥ R.

(2.96)

Of course, (2.96) and Remark 2.4.3 imply the following result.

Remark 2.4.7.

Let (ε, ω, h) ∈ A ∞ ⊂ [0, ε 1 ) × Ω × B R (0). The function h, �h� 0,0 ≤ R -1, solves the (Q)-equation with ω = ω(ε) given by (2.94) and v = v(ε, ω(ε), h) if and only if DΨ ε (h) = D Ψε (h) = 0.
Since E 0 is finite dimensional and so all norm are equivalent, we shall use in E 0 the norm

�h� 2 := � � T×M h 2 t dt dx (2.97)
and by (2.95), we wrire

Ψε (h) = �h� 2 2 - � � T×M a p (x) p + 1 h p+1 dt dx + Rε (h, v(ε, h)).
(2.98)

We claim that the functional Ψε ∈ C 1 (E 0 , R), verifies all the hypotheses of the Mountain Pass Theorem.

Step 2. Ψε verifies the geometrical hypotheses of the Mountain Pass Theorem.

Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds

• Ψε (0) = 0.

The function u = 0 is a solution of the whole equation (2.5). For h = 0, we take v = 0 as solution of the range equation. Using definition (2.82) and (2.89), we have Rε (0, v(ε, 0)) = Rε (ε, 0) = 0 which implies Ψε (0) = 0.

• ∃ ρ, α > 0 such that Ψε (h) > α, ∀ h such that �h� = ρ.

Let ρ > 0. For all �h� = ρ, using (2.90), we have

Ψε (h) = �h� 2 2 - � � T×M a p (x) p + 1 h p+1 dt dx + Rε (h, v(ε, h)) ≥ ρ 2 2 -C(a p )ρ p+1 -α(a p , ε)
for an appropriate α(a p , ε) > 0 such that α(a p , ε) → 0 as ε → 0.

Fix ρ such that ρ 2 2 -C(a p )ρ p+1 > ρ 2 4 and take ε small enough such that α(a p , ε) < ρ 2 8 , we get Ψε (h) ≥ ρ 2 8 > 0.

(2.99)

• ∃ h with � h� > ρ, such that Ψε ( h) < 0.
For �h 1 � = 1 satisfying (2.93) (we can always normalize it), ∃ τ0 > ρ large enough such that

Ψε ( τ0 h 1 ) = � τ0 h 1 � 2 2 - � � T×M a p (x) p + 1 ( τ0 h 1 ) p+1 dt dx + Rε ( τ0 h 1 , v(ε, τ0 h 1 )) = τ0 2 2 - τ0 p+1 p + 1 � � T×M a p (x)h p+1 1 dt dx + Rε ( τ0 h 1 , v(ε, τ0 h 1 )) < 0.
Call h = τ0 h 1 , we have Ψε ( h) < 0.

We define the minmax class

Γ := � γ ∈ C([0, 1], E 0 ) : γ(0) = 0, γ(1) = h�
and the Mountain Pass level

c ε := inf γ∈Γ max s∈[0,1]
Ψε (γ(s)).

(2.100)

Since any path γ ∈ Γ intersects the sphere {h ∈ E 0 : �h� = ρ}, by (2.99)

c ε ≥ min �h�=ρ Ψε (h) ≥ ρ 2 8 > 0.
By the Mountain Pass Theorem, we deduce the existence of a Palais-Smale

sequence for Ψε at level c ε > 0: there exists a sequence {h n } ∈ E 0 such that where Ψε and Ψ ε coincide.

Ψε (h n ) → c ε , ∇ Ψε (h n ) → 0. ( 2
By (2.96), it is enough to prove that {h n } is in B R (0) for an appropriate choice of R that we will define in the next step.

Step 3. Confinement of the Palais-Smale sequence:

{h n } is bounded.
By the definition (2.100) and estimates (2.90),

c ε ≤ max s∈[0,1]
Ψε (γ(sh 1 ))

≤ max s∈[0,1] � s 2 �h 1 � 2 2 - s p+1 p + 1 � � T×M a p (x)h p+1 1 dt dx � +1 =: k (2.102)
for ε small enough such that � Rε (γ(sh 1 ), v(ε, γ(sh

1 )))� L ∞ < 1.
By (2.98), we compute

D Ψε (h n )[h n ] = �h n � 2 - � � T×M a p (x)h p+1 n dt dx + D Rε (h n )[h n ] (2.103)
and, by (2.90)

Ψε (h n ) - D Ψε (h n )[h n ] p + 1 = � 1 2 - 1 p + 1 � �h n � 2 + Rε (h n ) - D Rε (h n )[h n ] p + 1 ≥ � 1 2 - 1 p + 1 � �h n � 2 -1, (2.104) for ε small enough such that � Rε (h n )� L ∞ + �D Rε (h n )� L ∞ < 1.
Moreover, by (2.101) and (2.102) we have that, for n large

Ψε (h n ) - D Ψε (h n )[h n ] p + 1 ≤ c ε + 1 + �h n � ≤ k + 1 + �h n � (2.105)
and by (2.104), we obtain

� 1 2 - 1 p + 1 � �h n � 2 -1 ≤ k + 1 + �h n � whence �h n � ≤ R * ,
for a suitable constant R * independent of ε and depending only on E 0 , p and a p .

We take R such that

R ≥ R * + 1. ( 2

Smooth dependence of periodic solutions of the bifurcation equation with respect to ε

Recall that for any h ∈ B R (0) ⊂ E 0 , we denote by v(ε, ω, h) the function given by Theorem 2.

3.5, v(•, •, •) ∈ C 1 ([0, ε 1 ) × Ω × B R (0), E s1,⊥ 0 ), with v(0, ω, h) = 0, �v(ε, ω, h)� s1 → 0 as ε → 0
which is solution of the range equation for all parameters (ε, ω, h) ∈ A ∞ .

Moreover, we recall equality (2.58)

v(ε, ω, T a h) = T a v(ε, ω, h)
for which if v(ε, ω, h) is solution of the range equation associated to parameters (ε, ω, h) ∈ A ∞ , then T a v(ε, ω, h) is still a solution of the range equation associ-

ated to parameters (ε, ω, T a h) ∈ A ∞ .
Recall the relation ω = ω(ε) introduced in (2.92) and suppose to be in the case where we impose (2.94).

Recall we denote v(ε, h) := v(ε, ω(ε), h) and let us introduce the notation

(D h Φ ε )(h + v(ε, h)) := (DΦ ε )(ω(ε), h + v(ε, h)) |E 0 .
We want to find periodic solutions h ε , smooth with respect to ε, of the (Q)equation written for ω = ω(ε) given by (2.94) and v = v(ε, h)

P ε (h) := h tt + Π E0 f (ε, x, h + v(ε, h)) = 0 (2.107)
(obtained after a division by ε) which by Remark 2.4.1, is equivalent for

ε � = 0 to ( Dh Φ ε )(h + v(ε, h)) = 0
where we have defined for ε � = 0,

( Dh Φ ε )(h + v(ε, h)) := ε -1 (D h Φ ε )(h + v(ε, h)).
Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds Note that, by definition of ( Dh Φ ε )(h + v(ε, h)), we have for ε � = 0

( Dh Φ ε )(h + v(ε, h)) = 0 ⇐⇒ (D h Φ ε )(h + v(ε, h)) = 0.
Recall the functional Ψε ∈ C 1 (E 0 , R) defined in (2.95)

Ψε (h) = � � T×M h 2 t 2 - a p (x) p + 1 h p+1 dt dx + Rε (h, v(ε, h)).
Properties v(ε, T a h) = T a v(ε, h) and �h� 0,0 = �T a h� 0,0 (the action (T a ) a∈T preserves the L 2 -norm) imply the following result.

Remark 2.5.1. The functional Ψε is invariant under the action of the time

translations (T a ) a∈T Ψε (h) = Ψε (T a h) (2.108)
and

(D h Φ ε )(h + v(ε, h)) = 0 ⇒ (D h Φ ε )(T a h + v(ε, T a h)) = 0. (2.109)
Proof. The proof is similar as the one of Remark 5.4.3.

By (2.109) of Remark 2.5.1, the zeroes of (D h Φ ε )(h+ v(ε, h)) are not isolated and they are included in circles of zeroes: if we have

(D h Φ ε )(h + v(ε, h)) = 0, define S h = {g ∈ E 0 : ∃ a ∈ R | g = T a h, },
we have

S h ⊂ {g ∈ E 0 : (D h Φ ε )(g + v(ε, g)) = 0}.
We can write (2.107) as a perturbation of a "simple" (Q 0 )-equation

P 0 (h) = 0 (2.110) with P 0 (h) = h tt + Π E0 f (0, x, h) = h tt + Π E0 a p (x)h p , equivalent to ( Dh Φ 0 )(h) = 0
where, as a consequence of the division by ε of the (Q)-equation, we define Let us introduce the following definition of non-degenerate solution for the (Q) 0 -equation, which it will be justified in the following (see (2.114) and

( Dh Φ 0 )(h) := lim ε→0 ε -1 (D h Φ ε )(h + v(ε, h)). On B R (0), we have Dh (Φ 0 )(h) = D h Ψ0 (h). ( 2
(2.115)).

Definition 2.5.2 (Non-degenerate solutions of the

(Q 0 )-equation). A function h 0 ∈ E 0 is a non-trivial (h 0 � = 0)
and non-degenerate solution of the

(Q 0 )-equation if and only if    D Ψ0 (h 0 ) = 0, dim(KerD 2 Ψ0 (h 0 )) = 1. (2.112)
We suppose there exist a function h 0 ∈ E 0 which is a non-trivial and nondegenerate solution of (Q 0 ) with respect to the Definition 2.5.2.

We choose R such that

R ≥ �h 0 � 0,0 + 2 ⇐⇒ �h 0 � 0,0 ≤ R -2.
(2.113) Using (2.111), we write

dim(KerD 2 Ψ0 (h 0 )) = 1 ⇔ KerD 2 Ψ0 (h 0 ) = span( ḣ0 ) (2.114)
where ḣ0 = d da T a h 0 is the tangent vector in h 0 to the cercle S h0 , we consider the vector space

Z h0 := � span( ḣ0 ) � ⊥ = {H ∈ E 0 : �H, ḣ0 � L 2 = 0}
where we denote for simplicity L 2 := L 2 (T, L 2 (M)) and the affine space

G h0 := h 0 + Z h0 = {h ∈ E 0 : ∃ H ∈ Z h0 | h = h 0 + H} = {h ∈ E 0 : �h -h 0 , ḣ0 � L 2 = 0}.
Let us define Ψ0 := ( Ψ0 ) |G h 0 .

Remark 2.5.3. The following equivalence holds:

KerD 2 Ψ0 (h 0 ) = span( ḣ0 ) ⇐⇒ KerD 2 Ψ0 (h 0 ) = {0}. on Zoll manifolds
Proof. The proof is the same as the one of Remark 5.4.5.

Hence, the condition (2.112) is the usual non-degeneracy condition for D Ψ0

   D Ψ0 (h 0 ) = 0, KerD 2 Ψ0 (h 0 ) = {0}. (2.115) Let us denote F ∈ C 1 ([0, ε 1 ) × Z h0 , L(Z h0 , R)) the function defined by F(ε, H) := ( DH Φ ε )(h 0 + H + v(ε, h 0 + H)) | Z h 0 ∈ L(Z h0 , R).
By (2.111) and (2.115), we have

F(0, 0) = ( DH Φ 0 )(h 0 ) | Z h 0 = D Ψ0 (h 0 ) = 0 and since E 0 is finite dimensonal, condition Ker(D H F(0, 0)) = Ker � ( D2 H Φ 0 )(h 0 ) | Z h 0 ×Z h 0 � = KerD 2 Ψ0 (h 0 ) = {0} implies that D H F(0, 0) = D2 H Φ 0 (h 0 ) ∈ L(Z h0 , L(Z h0 , R)) is invertible. Hence, we can apply the Implicit function theorem to function F ∈ C 1 ([0, ε 1 ) × Z h0 , L(Z h0 , R)) in order to solve F(ε, H) = ( DH Φ ε )(h 0 + H + v(ε, h 0 + H)) |Z h 0 = 0.
The Implicit function Theorem gives us the existence of ε < ε 1 small enough such that ∀ ε ∈ [0, ε) there is a non-trivial H(ε) close to 0, which depends in a C 1 -way on the parameter ε and such that

( DH Φ ε )(h 0 + H(ε) + v(ε, h 0 + H(ε))) |Z h 0 = 0. Define h(ε) := h 0 + H(ε), we have h(ε) close to h 0 , with ε → h(ε) in C 1 ([0, ε)) such that ( Dh Φ ε )(h(ε) + v(ε, h(ε))) | Z h 0 = 0. (2.116)
Note that for ε small enough, using (2.113) for which

h 0 ∈ B R-2 (0), we can obtain h(ε) ∈ B R-1 (0).
We have the following result.

Remark 2.5.4. If (ε, ω(ε), h(ε)) ∈ A ∞ , for ε � = 0, the function h(ε), with h ∈ C 1 ([0, ε), B R (0)), verifies also ( Dh Φ ε )(h(ε) + v(ε, h(ε))) = 0
which means that h(ε) solves the bifurcation equation (2.107).

Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds

Proof. The proof is similar as the one of Remark 5.4.8.

Follow now some considerations about the definition of the relation ω = ω(ε)

given by (2.92)

ω 2 = s * ε + ω2 with s * =    -1 if ∃ h 1 ∈ E 0 : �� T×M a p (x)h p+1 1 dt dx > 0 1 if ∀ h ∈ E 0 : �� T×M a p (x)h p+1 dt dx ≤ 0. Under hypothesis ∃ h 1 ∈ E 0 : � � T×M a p (x)h p+1 1 dt dx > 0 and so for -(ω 2 -ω2 ) = ε, we have defined the functional Ψε ∈ C 1 (E 0 , R) as Ψε (h) := � � T×M h 2 t 2 - a p (x) p + 1 h p+1 dt dx + Rε (h, v(ε, h))
and we have proved that it verifies all the hypotheses of the Mountain Pass Theorem.

In order to prove all the hypotheses of the Mountain Pass Theorem also in the case

∀ h ∈ E 0 : � � T×M a p (x)h p+1 dt dx ≤ 0 for (ω 2 -ω2 ) = ε, it is enough to define Ψ- ε (h) := � � T×M h 2 t 2 + a p (x) p + 1 h p+1 dt dx + Rε (h, v(ε, h)).
Denote by Ψ+ ε (h) := Ψε (h), we have obtained a solution h(ε) of the bifurcation equation depending smoothly on ε, under hypothesis of the existence of a non-degenerate solution for Ψ+ 0 (h) for -(ω 2 -ω2 ) = ε. If we suppose the existence of a non-trivial and non-degenerate solution for Ψ-0 (h), we can obtain a solution h(ε) of the bifurcation equation depending smoothly on ε for (ω 2 -ω2 ) = ε.

Hence, we can now write the following final result for the bifurcation equation.

Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds

Theorem 2.5.5 (Solutions of the (Q)-equation). Suppose there exists a

function 0 � = h 0 ∈ E 0 which verifies    D Ψ+ 0 (h 0 ) = 0, dim(KerD 2 Ψ+ 0 (h 0 )) = 1 condition (P + ) or    D Ψ- 0 (h 0 ) = 0, dim(KerD 2 Ψ- 0 (h 0 )) = 1 condition (P -).
Then there exists ε ∈ (0, ε 1 ) such that ∀ ε ∈ [0, ε) there exists a function

ε → h(ε), h ∈ C 1 ([0, ε), B R (0)), which for (ε, ω, h(ε)) ∈ A ∞ , is a non-trivial
solution of the bifurcation equation

(ω 2 -ω2 )h tt = εΠ E0 f (ε, x, h + v)
for ω = ω(ε) given by

ω 2 = s * ε + ω2 with s * =    -1 if h 0 verifies condition (P + ) 1 if h 0 verifies condition (P -).
Follow in the next subsections particular cases where we can find explicitly non-degenerate solutions of the (Q) 0 -equation under hyphotesis (2.93)

∃ h ∈ E 0 : � � T×M a p (x)h p+1 dt dx � = 0.

The Non-Resonant case

Recall {ω j,k } j,k the sequence of eigenvalues of the operator

� -∆ + V (x)
with corresponding sequence of eigenfunctions {ϕ j,k } j,k ⊂ L 2 (M).

We consider the case where the fixed frequency ω := ω j, k verifies

ω j,k � = lω ∀ l ∈ N, ∀ (j, k) � = ( j, k). ( 2

.117)

The set S defined in (2.10) is reduced here to the singleton

S = {(1, j, k)}.
This condition implies in particular that the frequency ω is a simple eigenvalue for the operator

� -∆ + V (x): call ϕ ω := ϕ j, k, ker(-∆ + V (x) -ω2 I) = Vect(ϕ ω )
is an 1-dimensional space.

When (2.117) holds, we are in the non-resonant case.

Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds

We look for non-degenerate solutions h(t, x) of the (Q) 0 -equation in the space E 0 which, by definition, is here

E 0 := Ker(ω 2 ∂ tt -∆ + V (x)) = span{cos(t)ϕ ω , sin(t)ϕ ω }.
Hence, a function h(t, x) ∈ E 0 has the form

h(t, x) = (α cos(t) + β sin(t))ϕ ω (x). (2.118)
It is clear that in the non-resonant case, resolving the (Q) 0 -equation means finding the two coefficients α, β of the unknown function h: in the non-resonant case, E 0 has dimension two.

Fix h ∈ E 0 , we introduce the notation h = h (α,β) to explicit the dependency on the coefficients α, β.

Recall the translation in time T a introduced in (2.18), choose β = 0 to simplify,

for h (α,0) (t, x) = α cos(t)ϕ ω (x), we compute T a h (α,0) (t, x) = h (α,0) (t + a, x) = α cos(t + a)ϕ ω (x) = α cos(a) cos tϕ ω (x) -α sin(a) sin(t)ϕ ω (x) (2.119)
and so

T a h (α,0) = h (α cos(a),-α sin(a)) .
More generally, we have that (2.120)

T a h (α,β) = h (α cos(a)
Hence the translation T a on h ∈ E 0 is equivalent to the rotation ρ a of angle -a of the vector (α, β)

ρ a (α, β) = (α cos(a) + β sin(a), -α sin(a) + β cos(a)). (2.121) Recall the reduced Lagrangian functional Ψ0 (•) ∈ C 2 (E 0 , R) defined in (2.95)
under hypotheses (2.91) and (2.93)

Ψ0 (h) := � � T×M h 2 t 2 - a p (x) p + 1 h p+1 dt dx
and Remark 2.4.7 for which

h (α,β) ∈ E 0 solves the (Q) 0 -equation ⇐⇒ D Ψ0 (h (α,β) ) = 0. (2.122) Chapter 2.

Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds

Note that for ε = 0, v(0, h) = 0 and we have not to take h ∈ B R-1 (0) ⊂ E 0 .

Let us define the map ψ0

∈ C 2 (R × R, R) ψ0 : R × R → R (α, β) � → ψ0 (α, β) := Ψ0 (h (α,β) ) (2.123)
Lemma 2.5.6. We have the following results.

(i) The map ψ0 is radial, i.e. it depends only on radius r

:= � α 2 + β 2 : ψ0 (α, β) = ψ0 (r) with ψ0 ∈ C 2 (R, R). (ii) A function h (α,β) ∈ E 0 \ {0} is a solution of the (Q) 0 -equation if and only if ψ� 0 (r) = 0. (iii) A function h (α,β) ∈ E 0 \ {0} is a non-degenerate solution of the (Q) 0 -equation if and only if ψ� 0 (r) = 0 and ψ�� 0 (r) � = 0. Proof. (i) By (2.108) written for ε = 0, we have Ψ0 (h (α,β) ) = Ψ0 (T a h (α,β) ).
Using (2.120) and ( 2

D Ψ0 (h (α,β) ) = 0 ⇐⇒ ψ� 0 (r) = 0
and then (2.122) implies that

h (α,β) ∈ E 0 solves the (Q) 0 -equation ⇐⇒ ψ� 0 (r) = 0. (iii) By Definition 2.5.2, a solution h (α,β) of the (Q 0 )-equation is said non- degenerate if dim(KerD 2 Ψ0 (h (α,β) )) = 1, i.e. if dim(KerD 2 ψ0 (α, β)) = 1. Recall ψ0 (α, β) = ψ0 ( � α 2 + β 2 ), we have for z = (α, β) ∈ (R × R) \ {(0, 0)} and ∀ (z 1 , z 2 ) ∈ (R × R) 2 D 2 ψ0 (z)[z 1 , z 2 ] = ψ� 0 (r) r (z 1 • z 2 ) + � ψ�� 0 (r) r 2 - ψ� 0 (r) r 3 � (z • z 1 )(z • z 2 ).

Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds

Since h (α,β) is a solution of the (Q) 0 -equation, we have by (ii) that ψ� 0 (r) = 0 and we obtain

D 2 ψ0 (z)[z 1 , z 2 ] = ψ�� 0 (r) r 2 (z • z 1 )(z • z 2 ). (2.126) Write KerD 2 ψ0 (z) := {z 1 ∈ R : D 2 ψ0 (z)[z 1 , z 2 ] = 0 ∀ z 2 ∈ R}.
By (2.126) we deduce that

z 1 ∈ KerD 2 ψ0 (z) ⇐⇒ ψ�� 0 (r) r 2 (z • z 1 )(z • z 2 ) = 0. If ψ�� 0 (r) = 0, then KerD 2 ψ0 (z) = R × R and so h (α,β) is a degenerate solution. If ψ�� 0 (r) � = 0, then KerD 2 ψ0 (z) = {z 1 ∈ R × R : z • z 1 = 0} and so dim(KerD 2 ψ0 (z)) = 1 and h (α,β) is a non-degenerate solution of the (Q) 0 - equation.
Using Lemma 2.5.6, we find now explicitly non-degenerate solutions h = h (α,β) ∈ E 0 of the (Q) 0 -equation, which in the non-resonant case has the form h(t, x) = (α cos(t) + β sin(t)ϕ ω (x).

Recall that we have supposed hypothesis (2.91)

∃ h ∈ E 0 : � � T×M a p (x)h p+1 dt dx � = 0
and in particular, we have supposed to be in the case (2.93)

∃ h 1 ∈ E 0 : � � T×M a p (x)h p+1 1 dt dx > 0.
In the non-resonant case, since

E 0 = span{cos(t)ϕ ω , sin(t)ϕ ω }, the condition (2.91) is verified only if p is odd: the integral �� T×M a p (x)h p+1 dt dx is a sum of terms of the form � M a p (x)ϕ ω (x) p+1 dx � T (α cos(t)) m (β sin(t)) n dt m, n ∈ N, m + n = p + 1 and � T (α cos(t)) m (β sin(t)) n dt � = 0 ⇐⇒ m + n = p + 1 is even, i.e. p is odd.

Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds

Let us consider in the following only exponents p odd.

Since Ψ0 is radial, we look for solutions h = h (α,0) , with β = 0 of the form

h(t, x) = α cos(t)ϕ ω (x).
By (2.95), we write

Ψ0 (h α,0) ) = � � T×M α 2 sin 2 (t)ϕ 2 ω (x) 2 - a p (x) p + 1 (α cos(t)ϕ ω (x)) p+1 dt dx = π 2 α 2 - c(p) p + 1 α p+1 with c(p) := � � T×M a p (x)(cos(t)ϕ ω (x)) p+1 dt dx
and where we have used � T sin 2 (t) dt = π and the fact that

� M ϕ 2 ω (x) dx = 1. Suppose to have �� M a p (x)ϕ ω (x) p+1 dt dx > 0 which implies c(p) > 0. Hence, we obtain ψ0 (r) = π 2 r 2 -c(p) r p+1 p + 1 . It is easy to check that r = � π c(p) � 1 p-1 is a non-degenerate solution for the equation Ψ� 0 (r) = 0: ψ�� 0 (r) = π-pc(p)r p-1 ⇒ ψ�� 0 � � π c(p) � 1 p-1 � = π(1-p) � = 0 because p ≥ 2.
Hence, by Lemma 2.5.6, the function

h(t, x) = � π c(p) � 1 p-1 cos(t)ϕ ω (x)
is a non-degenerate solution of the (Q) 0 -equation in the non-resonant case (under hypotheses (2.91) and (2.93)).

Genericity of Non-resonance conditions

In this subsection, we prove some results of genericity of the non-resonant case and of non-resonance conditions (2.12) and (2.13).

We start proving that "for almost all" potential V (x) ∈ C ∞ (M) we are in the non-resonant case.

In order to simplify the notations, for any potential V (x), we will denote by {µ j (V )} j≥1 , µ j (V ) = ω 2 j (V ), the sequence of eigenvalues of the operator -∆ + V (x) repeated with multiplicity and listed in increasing order

0 < µ 1 (V ) ≤ µ 2 (V ) ≤ • • • ≤ µ j (V ) ≤ . . .

Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds

{µ j (V )} j≥1 → ∞.
Let us take ω = ω 1 just to fix ideas.

We have the following result.

Lemma 2.5.7. The set

G := � V ∈ C ∞ (M) : ∀ j > 1, ∀ l ≥ 1, ω j (V ) � = lω 1 (V ) � = � V ∈ C ∞ (M) : ∀ j > 1, ∀ l ≥ 1, µ j (V ) � = l 2 µ 1 (V ) � (2.127) contains a countable intersection of open dense subsets of C ∞ (M). By Baire Theorem, G is dense in C ∞ (M).
In order to prove Lemma 2.5.7, we need several results which follow.

We write

G = � j>1,l≥1 G j,l (2.128)
where

G j,l := � V ∈ C ∞ (M) : ω j (V ) � = lω 1 (V ) � = � V ∈ C ∞ (M) : µ j (V ) � = l 2 µ 1 (V ) � .
It is enough to show that ∀ j > 1, ∀ l ≥ 1 , G j,l is a open and dense subset of

C ∞ (M).
We start proving that G j,l is open in C ∞ (M). We will see that it is a consequence of the continuity for all j ≥ 1 of the function

µ j (•) : C ∞ (M) → R + V � → µ j (V ).
In order to prove the continuity of µ j , we insert now four Lemmas.

The first one is a classical result about the operator -∆ + V (x).

Lemma 2.5.8. For a potential V ∈ C ∞ (M), V (x) ≥ 0, V (x) � ≡ 0, let us
consider the unbounded, linear, self-adjoint operator

P 2 := -∆ + V (x)
defined on L 2 (M). The operator P 2 is invertible and the inverse

(P 2 ) -1 is compact in L 2 (M).
The second one is a classical result known as Courant min-max principle (see [START_REF] Kato | Perturbation Theory for Linear Operators[END_REF] for references).

Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds Lemma 2.5.9. Let A be a linear self-adjoint operator defined on D(A) ⊂ H, with (H, � • � H ) a Hilbert space, such that A is invertible and its inverse

A -1 is compact. Let λ 1 ≤ λ 2 ≤ . . ., λ k → +∞, be the eigenvalues of A.
For all k ∈ N + , we have

λ k = min E k ∈S k max x∈E k �x� H =1 �Ax, x� H and λ k = max E k-1 ∈S k-1 min x∈E ⊥ k-1 ∩D(A) �x� H =1 �Ax, x� H , where �•, •� H is the scalar product in H and S k is the set of all linear k- dimensional subspace of D(A) ⊂ H.
Apply Lemma 2.5.9 to the unbounded, linear, self-adjoint operator

P 2 = -∆ + V (x), with D(P 2 ) = D(-∆) ⊂ L 2 (M), we write µ j (V ) = min Ej ∈Sj max g∈Ej �g� L 2 (M) =1 �(-∆ + V (x))g, g� L 2 (M) .
(2.129)

Lemma 2.5.10. For all g ∈ D(-∆), �g� L 2 (M) = 1, the function V ∈ C ∞ (M) � → �(-∆ + V (x))g, g� L 2 (M) ∈ R + is Lipschitz with constant 1. Proof. For V 1 (x), V 2 (x) ∈ C ∞ (M), we estimate directly � � ��(-∆ + V 1 (x))g, g� L 2 (M) -�(-∆ + V 2 (x))g, g� L 2 (M) � � � = � � ��(V1(x) -V 2 (x))g, g� L 2 (M) � � � ≤ �(V 1 (x) -V 2 (x))g� L 2 (M) �g� L 2 (M) ≤ �V 1 -V 2 � L ∞ (M) �g� 2 L 2 (M) ≤ �V 1 -V 2 � L ∞ (M) ≤ �V 1 -V 2 � C ∞ (M) .
Lemma 2.5.11. Let {f i (x)} i∈I be a family of 1-Lipschitz real functions defined on a normed vector space (X, � • � X ).

Suppose there exist x 1 , x 2 ∈ X such that sup i∈I f i (x 1 ) < +∞, inf i∈I f i (x 2 ) > -∞.
Then, the functions

g(x) := sup i∈I f i (x), h(x) := inf i∈I f i (x)
are well defined and 1-Lipschitz. on Zoll manifolds

Proof. The existence of x 1 ∈ X for which sup i∈I f i (x 1 ) < +∞ implies that g(x)

is well defined:

∀ x ∈ X, ∀ i ∈ I f i (x) ≤ f i (x 1 ) + �x -x 1 � X whence sup i∈I f i (x) ≤ sup i∈I f i (x 1 ) + �x -x 1 � X < ∞
where we have used that

∀ i ∈ I, f i is 1-Lipschitz.
By definition, fix ε > 0 there exists an index

i 0 ∈ I such that g(x) = sup i∈I f i (x) verifies ∀ i ∈ I f i (x) ≤ g(x) ≤ f i0 (x) + ε
and so

g(y) ≥ f i0 (y) ≥ � f i0 (y) -f i0 (x) � + f i0 (x) ≥ -�y -x� + g(x) -ε.
Hence, for ε arbitrarily small

g(y) -g(x) ≥ -�y -x� -ε ≥ -�y -x� whence g(x) -g(y) ≤ �y -x�.
Similary, we can prove by symmetry g(y)g(x) ≤ �y -x� and we obtain that g is 1-Lipschitz.

The proof for h(x) = inf i∈I f i (x) is similar.

By (2.129), Lemmas 2.5.10 and 2.5.11, for all j ≥ 1, the function µ j

µ j (•) : C ∞ (M) → R + V � → µ j (V ) is 1-Lipschitz, hence continuous.
Lemma 2.5.12. For all l ≥ 1 and j > 1, the set

G j,l is open in C ∞ (M).
Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds

Proof. Recall for j > 1, l ≥ 1 G j,l := � V ∈ C ∞ (M) : ω j (V ) � = lω 1 (V ) � = � V ∈ C ∞ (M) : µ j (V ) � = l 2 µ 1 (V ) � ,
we can see

G j,l = g -1 � (R \ {0}) � with g(•) : C ∞ (M) → R g(V ) := � µ j (V ) -l 2 µ 1 (V ) � .
Since the function g is continuous, the set

G j,l is open in C ∞ (M).
We prove now that G j,l is dense in

C ∞ (M): ∀ V 0 ∈ C ∞ (M), there exists a sequence {w m } ∈ C ∞ (M) with w m → 0 in C ∞ (M) such that V 0 + w m ∈ G j,l .
We distinguish two cases:

• Case 1: the eigenvalue µ 0 1 of (-∆ + V 0 (x)) is simple. Define V mn := V 0 + m n , m n → 0 for n → +∞, it is clear V mn → V 0 in C ∞ (M) and, call µ 0 1 < µ 0 2 ≤ µ 0 3 ≤ . . . the eigenvalues of (-∆ + V 0 (x)), then µ mn 1 := µ 0 1 + m n , µ mn 2 := µ 0 2 + m n , . . . are the eigenvalues of (-∆ + V mn (x)).
Of course, for all j > 1, µ mn j � = µ mn 1 since µ 0 1 � = µ 0 j (µ 0 1 is simple). Moreover, for all j > 1 and l > 1

µ mn j = lµ mn 1 ⇐⇒ µ 0 j + m n = l(µ 0 1 + m n ) ⇐⇒ m n = µ 0 j -µ 0 1 l -1 . If m n � = µ 0 j -µ 0 1 l-1 , then µ mn j � = lµ mn 1 and V mn ∈ G j,l . • Case 2: the eigenvalue µ 0 1 of (-∆ + V 0 (x)) is multiple. We call E 0 1 ⊂ L 2 (M) the eigenspace associated to µ 0 1 of dimension d ≥ 2: � -∆ + V 0 (x) � u = µ 0 1 u, ∀ u ∈ E 0 1 ⊂ L 2 (M).
We use the following Lemma.

Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds Lemma 2.5.13. Let be

L ε := -∆ + V 0 (x) + εw(x), with w(•) ∈ C ∞ (M)
such that for ε = 0, µ 0 1 is a eigenvalue of multiplicity d ≥ 2. Let be λ 1 , . . . , λ d the eigenvalues of the operator

Π E 0 1 w | E 0 1 : u ∈ E 0 1 → Π E 0 1 (wu) ∈ E 0 1
and suppose they are all simple.

Then, for ε

� = 0, µ 0 1 splits into d eigenvalues of L ε µ 0 1 + ελ 1 + o(ε), µ 0 1 + ελ 2 + o(ε), . . . µ 0 1 + ελ d + o(ε),
which are simple too for ε small enough.

This Lemma is a consequence of a general result taken from [START_REF] Kato | Perturbation Theory for Linear Operators[END_REF].

Lemma 2.5.14. Let A be a selfadjoint operator defined on a Hilbert space H which has λ as an isolate eigenvalue of finite multiplicity m λ = d ≥ 2

and be E := ker(A -λI) of dimension d. Let B be a continuous and selfadjoint operator defined on H and B E be the selfadjoint operator defined by

B E := Π |E B E .
Let be ε 0 such that

(λ -ε 0 , λ + ε 0 ) ∩ Sp(A) = {λ}
where Sp(A) is the spectrum of operator A.

If the eigenvalues of B E are all simple, then for ε small enough the eigenvalues λ 1 , . . . , λ d of (A + εB) close to λ

(λ -ε 0 , λ + ε 0 ) ∩ Sp(A + εB) = {λ 1 , . . . , λ d } are simple. Let U ⊂ C ∞ (M) be a open neighborhood of V 0 with respect to the C ∞ (M)-topology. Suppose we have a function w(•) ∈ C ∞ (M) as in
Lemma 2.5.13. For ε small enough, V 0 + εw ∈ U and the first eigenvalue of the operator -∆ + V 0 (x) + εw(x) is simple.

For the Case 1, there exists a sequence {m n } n → 0 as n → ∞ such that

V mn = V 0 + εw(x) + m n ∈ G j,l . on Zoll manifolds
This concludes the proof of the density of G j,l in L ∞ (M).

There remains to prove that there is w(•) ∈ C ∞ (M) that verifies hypothesis of Lemma 2.5.13.

We write the eigenspace E 0 1 = span{ψ 1 , ψ 2 , . . . , ψ d }, with {ψ i } ⊂ {ϕ j,k } the Hilbert basis in L 2 (M) formed by eigenvectors of the operator P .

For the functions w ∈ C ∞ (M), u ∈ E 0 1 , we have

Π E 0 1 (wu) = d � i=1 �wu, ψ i � L 2 (M) ψ i (x)
and the d × d real matrix which represent the operator

Π E 0 1 w | E 0 1 in the basis {ψ 1 , . . . , ψ d } is A d (w) : =     �wψ 1 , ψ 1 � L 2 (M) �wψ 2 , ψ 1 � L 2 (M) . . . �wψ d , ψ 1 � L 2 (M) . . . . . . . . . �wψ 1 , ψ d � L 2 (M) �wψ 2 , ψ d � L 2 (M) . . . �wψ d , ψ d � L 2 (M)     =     �w, ψ 2 1 � L 2 (M) �w, ψ 1 ψ 2 � L 2 (M) . . . �w, ψ 1 ψ d � L 2 (M) . . . . . . . . . �w, ψ 1 ψ d � L 2 (M) �w, ψ 2 ψ d � L 2 (M) . . . �w, ψ 2 d � L 2 (M)     .
Take a point x ∈ M, we consider the corresponding Dirac measure δ x defined for all functions f ∈ C 0 (M) by

�δ x , f � L 2 (M) = f (x).
Lemma 2.5.15. Take (d -1) distinct points x i ∈ M and a vector

(λ 1 , λ 2 , . . . , λ d-1 ) ∈ R d-1 \ {0}, let us consider the Dirac measure D := d-1 � i=1 λ i δ xi .
Suppose the real matrix

A d (D) =     �D, ψ 2 1 � L 2 (M) �D, ψ 1 ψ 2 � L 2 (M) . . . �D, ψ 1 ψ d � L 2 (M) . . . . . . . . . �D, ψ 1 ψ d � L 2 (M) �D, ψ 2 ψ d � L 2 (M) . . . �D, ψ 2 d � L 2 (M)    
has only simple eigenvalues.

Then there exists w ∈ C ∞ (M) such that the corresponding matrix

A d (w) =     �w, ψ 2 1 � L 2 (M) �w, ψ 1 ψ 2 � L 2 (M) . . . �w, ψ 1 ψ d � L 2 (M) . . . . . . . . . �w, ψ 1 ψ d � L 2 (M) �w, ψ 2 ψ d � L 2 (M) . . . �w, ψ 2 d � L 2 (M)    
has only simple eigenvalues too.

Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds

Proof. We consider a family of positive functions

{f µ,xi } ⊂ C ∞ (M) such that    suppf µ,xi ⊂ B(x i , µ) � M f µ.xi (x) dx = 1. We have that f µ,xi → δ xi for µ → 0 in the sense that �f µ,xi , g� L 2 (M) → �δ xi , g� L 2 (M) = g(x i ) ∀ g ∈ C 0 (M).
Define for μ small enough

w(x) := w μ(x) = d-1 � i=1 λ i f μ,xi ∈ C ∞ (M).
By the continuity of eigenvalues of a symmetric matrix with respect to its coefficients, we have that for µ ≤ μ small enough, the spectrum of the

≤ n ≤ d dimensional subspace E ⊂ R d , there exist (n -1) different points in M x 1 , x 2 , . . . , x n-1 ∈ M and a vector (λ 1 , λ 2 , . . . , λ n-1 ) ∈ R n-1 \ {0} such that the operator Π E n-1 � i=1 λ i φ(δ xi ) |E has only simple eigenvalues, where for all i ∈ [1, . . . , n -1], φ(δ xi ) : R d → R d is the endomorfism corresponding to the real matrix A d (δ xi ) A d (δ xi ) =     ψ 2 1 (x i ) ψ 1 (x i )ψ 2 (x i ) . . . ψ 1 (x i )ψ d (x i ) . . . . . . . . . ψ 1 (x i )ψ d (x i ) ψ 2 (x i )ψ d (x i ) . . . ψ 2 d (x i )     .
Proof. We prove this result by induction on n.

It is always possible to fix a point x 1 ∈ M such that

Ψ(x 1 ) := (ψ 1 (x 1 ), . . . , ψ d (x 1 )) � ∈ E ⊥ .
In fact, suppose by contradiction that ∀ x ∈ M we have

Ψ(x) ∈ E ⊥ .
Then, take α := (α 1 , . . . , α d ) ∈ E \ {0}, we have

α 1 ψ 1 (x) + • • • + α d ψ d (x) = 0, ∀ x ∈ M i.e. α 1 ψ 1 + • • • + α d ψ d = 0
and it is impossible because the vectors ψ 1 , . . . , ψ d are independent.

Moreover, it is easy to see that for all

V ∈ E Π E φ(δ x1 )[V ] = �Ψ(x 1 ), V � R d Π E Ψ(x 1 ).
Hence the operator Π E φ(δ x1 ) |E has rank 1.

In the case n = 2, by Rank Nullity Theorem,

dim(Ker(Π E φ(δ x1 ) |E )) = 1
and we obtain that 0 is a simple eigenvalues for Π E φ(δ x1 ) |E . Hence there is an other simple eigenvalue different from 0.

We proceed now to prove the inductive step.

Consider a (n + 1)-dimensional subspace E, since the operator A :=

Π E φ(δ x1 ) |E has rank 1, the Rank Nullity Theorem implies that dim(Ker(A)) = n
which means that 0 is a multiple eigenvalue for A of multiplicity n.

By induction hypothesis, for the n-dimensional subspace

F := Ker(A) ⊂ E,
we have the existence of (n -1) different points x 2 , . . . , x n and of the vector ( λ2 , . . . , λn ) ∈ R n-1 \ {0} for which the operator

Π F n � i=2 λi φ(δ xi ) |F
has only simple eigenvalues.

Then, define the linear combination Hence, write

B := Π E n � i=2 λi φ(δ xi ) |E
A + εB = Π E φ(δ x1 ) |E + εΠ E n � i=2 λi φ(δ xi ) |E = Π E � φ(δ x1 ) + n � i=2 ε λi φ(δ xi ) � |E = Π E n � i=1 λ i φ(δ xi ) |E with λ 1 = 1, λ 2 := ε λ2 , . . . , λ n := ε λn , we have that Π E � n i=1 λ i φ(δ xi
) |E has all simple eigenvalues and this concludes the proof.

Similar arguments are studied in [START_REF] Kappeler | Multiplicities of the Eigenvalues of the Schrödinger Equation in any dimension[END_REF] and [START_REF] Kappeler | Strong nonresonance of the Schrödinger operators and an averaging theorem[END_REF].

By cases 1 and 2, we have so proved the following result.

Lemma 2.5.17. For all l ≥ 1 and j > 1, the set G j,l is dense in C ∞ (M). By (2.128), Lemmas 2.5.12 and 2.5.17 imply Lemma 2.5.7.

We prove now two results about the genericity of non-resonance conditions (2.12) and (2.13).

Let us introduce the following definition: we say that a potential

V (x) ∈ C ∞ (M) satisfies the non-resonance diophantine property if |µ 1 l 2 -µ j | ≥ Cγ 1 + |l| τ0 ∀ l ∈ Z, j > 1 (2.130)
where for all j ≥ 1, µ j is an eigenvalues of the operator -∆ + V in L 2 (M)

µ j := ω 2 j .
Of course, if a potential V satisfies the non-resonance diophantine property then

µ j � = l 2 µ 1 ∀ l ∈ Z, j > 1
Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds i.e. we are in the non-resonant case (see (2.117)) and in particular µ 1 is a simple eigenvalue of -∆ + V .

Let us define the set

V := {V ∈ C ∞ , V (x) ≥ 0, V (x) � ≡ 0 : µ 1 is simple} = {V ∈ C ∞ , V (x) ≥ 0, V (x) � ≡ 0 : µ 1 � = µ j ∀j > 1}.
We have the following result.

Lemma 2.5.18. For all potential V ∈ V, there exists a subset

B V ⊂ [0, 1] with meas(B V ) = O(γ) such that for all m ∈ [0, 1] \ B V , the potential V + m
satisfies the non-resonance diophantine property (2.130).

Proof. For any potential V ∈ V, let us define

G V := {m ∈ [0, 1] : V +m satisfies the non-resonance diophantine property (2.130)}.
We call µ m j , j ≥ 1, the eigenvalues of

-∆ + V + m µ m j = µ j + m = ω 2 j + m. If m ∈ G V , then ∀ l ∈ Z, ∀ j > 1 |µ m 1 l 2 -µ m j | ≥ Cγ 1 + |l| τ0 ⇐⇒ |(ω 2 1 + m)l 2 -(ω 2 j + m)| ≥ Cγ 1 + |l| τ0 ⇐⇒ |(ω 2 1 l 2 -ω 2 j ) + m(l 2 -1)| ≥ Cγ 1 + |l| τ0 .
Hence, we have

B V := G c V = � l∈Z B V,l
where

B V,l := � m ∈ [0, 1] : ∃ j > 1, |(ω 2 1 l 2 -ω 2 j ) + m(l 2 -1)| < Cγ 1 + |l| τ0 � .
For l = ±1, since µ 1 = ω 2 1 is a simple eigenvalue, we have that

|ω 2 1 -ω 2 j | < Cγ 2 is impossible for γ ∈ (0, 1) small enough. Hence B V,±1 = ∅. Fix ±1 � = l ∈ Z. If m ∈ B V,l then � � �m + (ω 2 1 l 2 -ω 2 j ) (l 2 -1) � � � < Cγ �l� τ0+2
(2.131)

where �l� := max{1, |l|}, i.e. Hence, we estimate

m ∈ � - (ω 2 1 l 2 -ω 2 j ) (l 2 -1) - Cγ �l� τ0+2 , - (ω 2 1 l 2 -ω 2 j ) (l 2 -1) + Cγ �l� τ0+2 � .
meas(B V ) ≤ � l∈Z c|l| d Cγ �l� τ0+2 ≤ � l∈Z Cγ �l� τ0+2-d = O(γ) if we take τ 0 + 2 -d > 1, i.e. τ 0 > d -1.
We can prove with similar argument also the following result.

Lemma 2.5.19. For all potential V ∈ V, there exists a subset

B � V ⊂ [0, 1] with meas(B � V ) = O(γ) such that for all m ∈ [0, 1] \ B � V , the first eigenvalue µ m 1 = µ 1 + m = ω 2 1 + m of the operator (-∆ + V (x) + m) verifies � � � � ( � µ m 1 )l - 2π T p � � � � ≥ 2γ 1 + |l| τ0+1 , ∀ (l, p) ∈ Z 2 \ {(0, 0)}.

Periodic solutions of the whole equation

This final section is devoted to the proof of Theorem 2.1.4.

Using a Nash-Moser scheme, we have proved Theorem 2.3.5 which gives us the existence of

ε 1 = ε 1 (γ) ∈ (0, ε 0 ) and of a function v ∈ C 1 ([0, ε 1 ) × Ω × B R (0), E s1+2,⊥ 0 ) with v(0, ω, h) = 0, �v(ε, ω, h)� s1 → 0 as ε → 0, such that v(ε, ω, h) is solutions of the range equation for all (ε, ω, h) ∈ A ∞ .
Moreover, we have proved the inclusion

A ∞ ⊂ A ∞ with A ∞ := � N ≥N0 � (ε, ω, h) ∈ [0, ε 1 ) × Ω × B R : �(L (N ) (v(ε, ω, h)) -1 � 0,0 ≤ 2N τ1 γ � .
Moreover in Theorem 2.5.5, we have proved that there exists ε ∈ (0, ε 1 )

and a function h ∈ C 1 ([0, ε), B R (0)) such that for (ε, ω, h(ε)) ∈ A ∞ h(ε) is a
non-trivial solution of the bifurcation equation for ω = ω(ε) given by

ω 2 = ±ε + ω2 .

Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds

With the C 1 -dependence on ε of functions h(ε) and v(ε

) := v(ε, ω(ε), h(ε)),
we can obtain an appropriate estimate of measure for the complement set A c of

A := {ε ∈ [0, ε) : (ε, ω(ε), h(ε)) ∈ A ∞ }
and have, as a consequence, a solution h(ε) + v(ε) of the whole equation (2.5)

for a large set of parameter ε.

Suppose that we are in the case where we impose the following relation

ω 2 = -ε + ω2
for defining A. We write

A = A 1 ∩ A 2 (2.132)
where

A 1 = � N ≥N0 � ε ∈ [0, ε) : ∃ (L (N ) (v(ε))) -1 and �(L (N ) (v(ε))) -1 � 0,0 ≤ 2 N τ1 γ � and A 2 := � ε ∈ [0, ε) : ω(ε) ∈ Ω � = � ε ∈ [0, ε) : � � � � ω(ε)l - 2π T p � � � � ≥ γ 1 + |l| τ +1 , ∀ (l, p) ∈ Z 2 \ {(0, 0)} � .
The next two subsections are devoted to prove appropriate measure estimates for A c 1 and A c 2 for appropriate choices of τ 1 and τ .

Measure estimate for the set A 1

Let us recall that for all w ∈ E (N ) , we have

L (N ) (v(ε))[w] = ω 2 w tt -∆w + V (x)w -εΠ (N ) E ⊥ 0 (b(ε, t, x)w) with b(ε, t, x) := (∂ u f )(ε, x, h(ε)(t, x) + v(ε)(t, x)). The operator -∆ + V (x) is self-adjoint in L 2 (M). Moreover, the fact that ∀ (ε, t, x) we have f (ε, x, u(t, x)) ∈ R and (2.20) ∂ u F (ε, x, u) = f (ε, x, u) imply that b(ε, t, x) is self-adjoint in L 2 (T ν , L 2 (M, C)).
Hence, the operators L (N ) (v(ε)) are self-adjoint with respect to the scalar

product in L 2 (T ν , L 2 (M, C)).
We write ) , with D (N ) the diagonal part

L (N ) (v(ε)) = D (N ) + R (N
D (N ) w = ω 2 w tt -∆w + V (x)w
Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds and

R (N ) w = -εΠ (N ) E ⊥ 0 (b(ε, t, x)w).
In the basis {e ılt ϕ j,k } l,j,k of L 2 (T × M, C), where ϕ j,k are the eigenvectors of the operator � -∆ + V (x), we write

D (N ) = (ω 2 ∂ tt -∆ + V ) | E (N ) = diag |l|≤N,j,k {-ω 2 l 2 + ω 2 j,k }. ( 2
�R (N ) � 0,0 = sup 0� =w∈E N �R (N ) w� 0,0 �w� ≤ sup 0� =w∈E N ε�bw� 0,0 �w� 0,0 ≤ sup 0� =w∈E N ε�b� L ∞ (T×M) �w� 0,0 �w� 0,0 ≤ ε�b� L ∞ (T×M) ≤ εC�b� s1 ≤ εC(s 1 )(1 + �h(ε)� s1 + �v(ε)� s1 ) ≤ εC 1 (2.134)
for some constant C 1 > 0, where in the last inequality we have used that �h(ε)� s1 ≤ cR and �v(ε)� s1 → 0 as ε → 0.

Recall the first order Melnikov diophantine condition (2.11) on ω which implies the non-resonance diophantine property (2.13)

|ω 2 l 2 -ω 2 j,k | ≥ Cγ 1 + |l| τ0 , ∀ l ∈ Z, j ∈ N, k ∈ [1, d j ], (l, j, k) � ∈ S and the relation ω 2 -ω2 = -ε.
We have the following result. Lemma 2.6.1. The non-resonance diophantine property (2.13) on ω implies that there exists

N 1 = N 1 (γ, ε) N 1 ≤ C � γ ε � 1 τ 0 +2 large enough N 1 ≥ N 0 if we take ε << γ, such that ∀ K ≤ N 1 and ∀ ε ∈ [0, ε) �(L (K) (v(ε))) -1 � 0,0 ≤ 2 K τ1 γ
for an appropriate τ 1 > τ 0 . on Zoll manifolds Proof. Using (2.13), (2.161) and ω 2 -ω2 = -ε, we write

|ω 2 l 2 -ω 2 j,k | ≥ |ω 2 l -ω 2 j,k | -|(ω 2 -ω2 )l 2 | ≥ Cγ 1 + |l| τ0 -ε|l| 2 . (2.135) If l ∈ Z verifies ε|l| 2 < Cγ 2(1 + |l| τ0 ) ⇐⇒ |l| ≤ C � γ ε � 1 τ 0 +2 then (2.135) implies ∀ |l| ≤ C � γ ε � 1 τ 0 +2 |ω 2 l 2 -ω 2 j,k | ≥ Cγ 2(1 + |l| τ0 ) , ∀ j ∈ N, k ∈ [1, d j ], (l, j, k) � ∈ S. (2.136) Let us choose N 1 = N (γ, ε) such that N 1 ≤ C � γ ε � 1 τ 0 +2 . Recall L (K) (v(ε)) = D (K) + R (K) , by (2.133) we write (D (K) ) -1 = diag |l|≤K,j,k {(-ω 2 l 2 + ω 2 j,k ) -1 }, (2.137)
and by (2.136), we have that

∀ K ≤ N 1 , �(D (K) ) -1 � 0,0 ≤ C K τ0 γ .
Using (2.134), we estimate

�(D (K) ) -1 R (K) � 0,0 ≤ �(D (K) ) -1 � 0,0 �R (K) � 0,0 ≤ C K τ0 γ C 1 ε and if we take ε = ε(γ) small enough such that ∀ K ≤ N 1 C K τ0 γ C 1 ε ≤ Cγ 1 τ 0 +2 -1 (ε) -τ 0 τ 0 +2 ε < C (ε) 2 τ 0 +2 γ τ 0 +1 τ 0 +2 < 1 2 , then Lemma B.1.1 in Appendix B implies that L (K) (v(ε)) is invertible with ∀ K ≤ N 1 , �L (K) (v(ε))� 0,0 ≤ 2C K τ0 γ ≤ 2 K τ1 γ for an appropriate τ 1 > τ 0 . Lemma 2.6.1 implies that ∀ K ≤ N 1 � ε ∈ [0, ε) : ∃ (L (K) (v(ε))) -1 and �(L (K) (v(ε))) -1 � 0,0 ≤ 2 K τ1 γ � = [0, ε)
Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds and we can write

A 1 = � N ≥N1 � ε ∈ [0, ε) : ∃ (L (N ) (v(ε))) -1 and �(L (N ) (v(ε))) -1 � 0,0 ≤ 2 N τ1 γ � .
Let us consider now the sets

E N 1 := � (l, j, k) : |l| ≤ N ; l 2 < 2 C or j > 2 c 1 ω|l|, 1 ≤ k ≤ d j � , E N 2 := � (l, j, k) : |l| ≤ N ; l 2 ≥ 2 C and j ≤ 2 c 1 ω|l|, 1 ≤ k ≤ d j �
with an appropriate choice of constants C and c 1 which will be specified later and d j is defined in Lemma 2.1.1.

Recall that we can take N ≥ N 0 , with N 0 the term of initialization in the Nash-Moser scheme sufficiently large such that 2 C < N 2 0 and E N 2 is never an empty set. Moreover, by Lemma 2.1.1 and by definition of

E N 2 , for (l, j, k) ∈ E N 2 , we have card(E N 2 ) ≤ � |l|≤N � j≤ 2 c 1 ω|l| C 0 j d-1 ≤ � |l|≤N C|l| d ≤ CN d+1 .
(2.138)

Let E N 1 , E N 2 be the the subspaces of E (N ) corresponding respectively to the sets of indices E N

1 and E N 2 and let Π E1 ,Π E2 be the L 2 -projections respectively on E N

1 and E N 2 . We write

L (N ) (v(ε)) = D (N ) + R (N ) as L (N ) (v(ε)) = � B (N ) (ε) R (N ) 1 (ε) R (N ) 2 (ε) C (N ) (ε) � (2.139)
where

B (N ) (ε) := Π E1 L (N ) (v(ε)) | E 1 , R (N ) 1 (ε) := Π E1 L (N ) (v(ε)) | E 2 = Π E1 (R (N ) ) | E 2 , C (N ) (ε) := Π E2 L (N ) (v(ε)) | E 2 , R (N ) 2 (ε) := Π E2 L (N ) (v(ε)) | E 1 = Π E2 (R (N ) ) | E 1 .
By (2.134) we have for ε small enough and for i = 1, 2

�R (N ) i (ε)� 0,0 ≤ �R (N ) � 0,0 ≤ εC 1 .
(2.140) on Zoll manifolds

The matrices B (N ) (ε) and

C (N ) (ε) are self-adjoint and R (N ) 2 (ε) = (R (N )
1 (ε)) T . In order to prove that

A 1 = � N ≥N1 � ε ∈ [0, ε) : ∃ (L (N ) (v(ε))) -1 and �(L (N ) (v(ε))) -1 � 0,0 ≤ 2 N τ1 γ �
is a large set, we will apply the following Lemma.

Lemma 2.6.2. Let L(ε) be a selfadjoint operator of the form

L(ε) = � B(ε) R 1 (ε) R 2 (ε) C(ε) � with C(ε) defined on a M -finite dimension space.
If we assume that B(ε) is invertible and that there are constants

c 0 > 0, C 1 > 0, C 2 > 0, C 3 > 0 and C 4 > 0 such that (i) �R i (ε)� 0,0 ≤ εC 1 , (ii)
d dε C(ε) ≥ c 0 I, (iii) �B -1 (ε)� 0,0 ≤ 2 C2 , (iv) � d dε R i (ε)� 0,0 ≤ C 3 , (v) � d dε (B -1 )(ε)� 0,0 ≤ C 4
, then, there is ε � > 0 small enough (depending on all constants c 0 , C i ) and a constant K > 0 (depending only on C 1 , C 2 , C 3 ) such that for δ small enough meas

� ε ∈ (0, ε � ) :� ∃ L -1 (ε) or �L -1 (ε)� 0,0 ≥ 1 δ � ≤ K δM c 0 .
Proof. The proof is similar to the proof of a more general result proved in Appendix B, see Lemma B.2.3.

We show now that we have all the hypotheses required in the Lemma 2.6.2 and so the measure estimate for A will follow for ε small enough.

We write in the following B(ε) and C(ε) instead of B (N ) (ε) and C (N ) (ε) and

� • � = � • � 0,
0 in order to simplify the notation.

Lemma 2.6.3. The linear operator L (N ) (v(ε)) written as (2.139) verifies hy-

potheses (i) -(v) of Lemma 2.6.2 with C(ε) defined on the subspace E N 2 of finite dimension M ≤ CN d+1 . Proof. Hypothesis (i) is satisfied by (2.140). Since d dε L (N ) (v(ε))[w] = -w tt -Π (N ) E ⊥ 0 (b(ε, t, x)w) -εΠ (N ) E ⊥ 0 (∂ ε b(ε, t, x)w) (2.141)
Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds for

w(t, x) = � |l|≤N l 2 ≥2
C e ılt w l (x), we have

�-∂ tt w, w� L 2 (T,L 2 (M)) = � |l|≤N l 2 ≥2 C(-ı 2 l 2 )�w l � 2 L 2 (M) ≥ 2 C � |l|≤N l 2 ≥2 C �w l � 2 L 2 (M) ≥ 2 C�w� 2 L 2 (T,L 2 (M)) , (2.142
)

�Π (N ) E ⊥ 0 (b(ε, t, x)w), w� L 2 (T,L 2 (M)) ≤ �b� L ∞ (T×M) �w� 2 L 2 (T,L 2 (M)) ≤ C�b� s1 �w� 2 L 2 (T,L 2 (M)) ≤ C(s 1 )(1 + �h(ε)� s1 + �v(ε)� s1 )�w� 2 L 2 (T,L 2 (M)) ≤ K�w� 2 L 2 (T,L 2 (M)) (2.143)
for a constant K > 0 and similarly, using also (2.52)

�εΠ (N ) E ⊥ 0 (∂ ε b(ε, t, x)w), w� L 2 (T,L 2 (M)) ≤ ε K�w� 2 L 2 (T,L 2 (M)) .
(2.144) By (2.142), (2.143), (2.144), taking 2 K ≤ C, we have also hypothesis (ii).

We want to prove (iii). We show first that there exists a constant

C 2 > 0 such that (l, j, k) ∈ E N 1 ⇒ | -ω 2 l 2 + ω 2 j,k | ≥ C 2 ω 2 �l� 2 (2.145)
where �l� := max{1, |l|}.

Using the non-resonance property of ω (2.13) and the relation

ω 2 = -ε + ω2 , we have ∀ l ∈ Z, j ∈ N, k ∈ [1, d j ], (l, j, k) � ∈ S |ω 2 l 2 -ω 2 j,k | = |(-ε + ω2 )l 2 -ω 2 j,k | ≥ |ω 2 l 2 -ω 2 j,k | -εl 2 ≥ Cγ 1 + |l| τ0 -εl 2 .
(2.146) By the definition of E N 1 , we have to distinguish two different cases for (l, j, k).

Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds

• First case: |l| ≤ N and l 2 < 2 C.
By (2.146), we estimate easily for

| -ω 2 l 2 + ω 2 j,k | �l� 2 ≥ Cγ �l� 2 + �l� 2τ0 -ε ≥ Cγ 2 C + 2 Cτ0 -ε > C
for ε small enough and some C > 0.

Then in this case we have | -

ω 2 l 2 + ω 2 j,k | ≥ C�l� 2 .
• Second case: |l| < N and j > 2 c1 ω|l|. By Lemma 2.1.1,

ω j,k ∈ � 2π T j + α - c 0 j β , 2π T j + α + c 0 j β � and so ω j,k > 2π T j + α - c 0 j β > 2π T 2 c 1 ω|l| + α - c 0 j β > 2π T 2 c 1 ω|l| + α -c 0 > 2 ω|l| + 1
if we choose c 1 small enough such that the last inequality holds.

This implies that

| -ω 2 l 2 + ω 2 j,k | > -ω 2 l 2 + 4 ω 2 l 2 + 1 = 3 ω 2 l 2 + 1.
The constant C 2 in the (2.145) is chosen properly.

By (2.145), B(ε) is invertible and an estimate like (iii) holds.

In fact, we write

B(ε) = D B (ε) + R B (ε) (2.147) with D B (ε) = diag (l,j,k)∈E N 1 {-ω 2 l 2 + ω 2 j,k } (2.148)
and

�R B (ε)� ≤ εC R . (2.149) Since D -1 B (ε) = diag (l,j,k)∈E N 1 {(-ω 2 l 2 + ω 2 j,k ) -1 }, ( 2
�D -1 B (ε)R B (ε)� ≤ �D -1 B (ε)��R B (ε)� ≤ εC R C 2 < 1 2 ,
which implies that B(ε) is invertible and

B -1 (ε) = (D B (ε) + R B (ε)) -1 = +∞ � k=0 (-1) k (D -1 B (ε)R B (ε)) k D -1 B (ε).
with

�B -1 (ε)� ≤ 2 C 2 .
(2.152)

In order to prove (iv), it is enough to show that there exists a constant

C 3 > 0 such that � � � � d dε (L (N ) (v(ε)) -D (N ) )(ε, v) � � � � 0,0 < C 3 .
(2.153) By (2.141), proceding as done to prove (2.140)

� � � � d dε (L (N ) (v(ε)) -D (N ) )w � � � � = �Π (N ) E ⊥ 0 (b(ε, t, x)w) + εΠ (N ) E ⊥ 0 (∂ ε b(ε, t, x)w)� ≤ | K + ε K|�w� ≤ (1 + ε) K�w� ≤ C 3 �w�
if we take a constant C 3 sufficiently large.

Finally, we want to prove (v).

Applying Lemma B.1.2, we write

d dε (B -1 )(ε) = -B -1 (ε) � d dε B(ε) � B -1 (ε) (2.154)
where by (2.147)

d dε B(ε) = d dε D B (ε) + d dε R B (ε) (2.155)
and by (2.153)

� � � d dε R B (ε) � � � ≤ C 3 . (2.156)
Write on the one hand

B -1 (ε) = (D B (ε) + R B (ε)) -1 = (D B (ε)(I + D B (ε) -1 R B (ε))) -1 = (I + D -1 B (ε)R B (ε)) -1 D -1 B (ε) (2.157)
Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds and on the other hand

B -1 (ε) = (D B (ε) + R B (ε)) -1 = ((I + R B (ε)D -1 B (ε))D B (ε)) -1 = D -1 B (ε)(I + R B (ε)D -1 B (ε)) -1 ,
(2.158) using (2.154), (2.155), we obtain

� � � d dε (B -1 )(ε) � � � = � � �B -1 (ε) � d dε B(ε) � B -1 (ε) � � � = �(I + D -1 B (ε)R B (ε)) -1 D -1 B (ε) � d dε D B (ε) + d dε R B (ε) � D B (ε) -1 (I + R B (ε)D -1 B (ε)) -1 � ≤ �(I + D -1 B (ε)R B (ε)) -1 �× �� � �D -1 B (ε) � d dε D B (ε) � D -1 B (ε) � � � + � � �D -1 B (ε) � d dε R B (ε) � D -1 B (ε) � � � � × �(I + R B (ε)D -1 B (ε)) -1 �.
We have to estimate

� � �D -1 B (ε) � d dε D B (ε) � D -1 B (ε) � � �, � � �D -1 B (ε) � d dε R B (ε) � D -1 B (ε) � � �.
By (2.148),

d dε D B (ε) = diag � d dε (-ω 2 l 2 + ω 2 j,k ) � = diag � d dε ((ε -ω2 )l 2 + ω 2 j,k ) � = diag{l 2 }
and by (2.150), we have

D -1 B (ε) � d dε D B (ε) � D -1 B (ε) = diag � l 2 (-ω 2 l 2 + ω 2 j,k ) 2 � .
By (2.145) and (2.156), we estimate

� � �D -1 B (ε) � d dε D B (ε) � D -1 B (ε) � � � ≤ 1 C 2 and � � �D -1 B (ε) � d dε R B (ε) � D -1 B (ε) � � � ≤ 1 C 2 Now, applying Lemma B.1.1, we obtain �(I + D -1 B (ε)R B (ε)) -1 � ≤ 2
and

�(I + R B (ε)D -1 B (ε)) -1 � ≤ 2 on Zoll manifolds if ε is such that εC R C2 < 1 2 .
We conclude the proof of (v) with

� � � d dε (B -1 )(ε) � � � ≤ 4 C 2 + 4 C 2 = 8 C 2 =: C 4 .
We conclude this subsection with the following Lemma.

Lemma 2.6.4. For τ 1 > 2(τ 0 + 2) + d + 2 and ε << γ, we have

meas(A c 1 ) < C ε2 γ << ε (2.159)
and in particular

meas(A c 1 ) ε → 0 as ε → 0. (2.160) Proof. Recall A 1 = � N ≥N1 � ε ∈ [0, ε) : ∃ (L (N ) (v(ε))) -1 and �(L (N ) (v(ε))) -1 � 0,0 ≤ 2N τ1 γ � ,
we have

A c 1 = � N ≥N1 � ε ∈ [0, ε) :� ∃ (L (N ) (v(ε))) -1 or �(L (N ) (v(ε))) -1 � 0,0 > 2N τ1 γ � .
By Lemmas 2.6.2 and 2.6.3 we deduce

meas(A c 1 ) ≤ � N ≥N1 meas � ε ∈ [0, ε) :� ∃ (L (N ) (v(ε))) -1 or �(L (N ) (v(ε))) -1 � 0,0 > 2N τ1 γ � ≤ � N ≥N1 C γ c 0 2N τ1 N d+1 = � N ≥N1 C γ c 0 N d+1-τ1 ≤ C γ c 0 N d+2-τ1 1 ≤ C γ c 0 � ε γ � τ 1 -(d+2) τ 0 +2
which is small if we take

τ 1 -(d + 2) τ 0 + 2 > 2 ⇐⇒ τ 1 > 2(τ 0 + 2) + d + 2 such that if ε = ε(γ) << γ meas(A c 1 ) ≤ C γ c 0 � ε γ � τ 1 -(d+2) τ 0 +1 < C c 0 ε2 γ << ε.
In particular, we have Let us consider the set

meas(A c 1 ) ε → 0 as ε → 0. Chapter 2.
A 2 := � ε ∈ [0, ε) : ω(ε) ∈ Ω � = � ε ∈ [0, ε) : � � � � ω(ε)l - 2π T p � � � � ≥ γ 1 + |l| τ +1 , ∀ (l, p) ∈ Z 2 \ {(0, 0)} � .
We prove measure estimates for A c 2 using similar argument as in Remark 2.1.2. Recall the frequency-amplitude relation ω = ω(ε)

(ω(ε)) 2 = ω2 -ε which implies for ω ∈ (ω 1 , ω 2 ) |ω(ε) -ω| = ε |ω(ε) + ω| ≤ cε. (2.161)
Recall also the condition (2.12) on ω

� � � � ωl - 2π T p � � � � ≥ 2γ 1 + |l| τ0+1 , ∀ (l, p) ∈ Z 2 \ {(0, 0)}.
We have the following result.

Lemma 2.6.5. For τ > 2(τ 0 + 2) and ε << γ, we have

meas(A c 2 ) ≤ c ε2 γ << ε and in particular meas(A c 2 ) ε → 0 as ε → 0. (2.162) Proof. We write A 2 = � 0� =l∈Z G l where G l := � ε ∈ [0, ε) : ∀ p ∈ Z, � � � � ω(ε)l - 2π T p � � � � ≥ γ 1 + |l| τ +1 � from which we have A c 2 = � 0� =l∈Z B l with B l := G c l = � ε ∈ [0, ε) : ∃ p ∈ Z, � � � � ω(ε)l - 2π T p � � � � < γ 1 + |l| τ +1 � .
Fix l ∈ Z \ {0}, using (2.161) and (2.12), we obtain for all p ∈ Z

� � � � ω(ε)l - 2π T p � � � � ≥ � � � � ωl - 2π T p � � � � -|(ω(ε) -ω)l| ≥ 2γ 1 + |l| τ0+1 -cε|l|. (2.163) If l ∈ Z \ {0} verifies cε|l| < γ 1 + |l| τ0+1 ⇐⇒ |l| ≤ C � γ ε � 1 τ 0 +2 ,
Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds then (2.163) implies for all ε ∈ [0, ε), for all p ∈ Z and for τ > 2(τ 0 + 2)

� � � � ω(ε)l - 2π T p � � � � ≥ γ 1 + |l| τ0+1 > γ 1 + |l| τ +1
and so B l = ∅. Hence, define

l 0 := C � γ ε � 1 τ 0 +2
, we can write

A c 2 = � 0� =l∈Z |l|≥l0 B l . (2.164)
Moreover, fix l ∈ Z \ {0}, we write

B l = � p∈Z B l,p (2.165)
where

B l,p := � ε ∈ [0, ε) : � � � � ω(ε)l - 2π T p � � � � < γ 1 + |l| τ +1 � . If ε ∈ B l,p , then the condition � � � � ω(ε)l - 2π T p � � � � ≤ γ 1 + |l| τ +1 ⇐⇒ � � � � � ω2 -ε - 2π |l|T p � � � � ≤ γ 1 + |l| τ +2 (2.166) implies � ω2 -ε ∈ � 2π T p |l| - Cγ 1 + |l| τ +2 , 2π T p |l| + Cγ 1 + |l| τ +2 � and 1 meas(B l,p ) ≤ meas � ε ∈ [0, ε) : � ω2 -ε ∈ � 2π T p |l| - Cγ 1 + |l| τ +2 , 2π T p |l| + Cγ 1 + |l| τ +2 � � ≤ Cγ 1 + |l| τ +2 .
Moreover, the integers p which may verify (2.166) for some ε ∈ [0, ε) are only in finite number n = O(|l|). 1 Let us consider a real function f : R → R with f � (ε) ≤ -c 1 , for some c 1 > 0. Then, for any I ⊂ R we have

meas{ε : f (ε) ∈ I} ≤ meas(I) c 1 .
We apply this result with f (ε) := √ ω2ε and

I := � 2π T p |l| - Cγ 1+|l| τ +2 , 2π T p |l| + Cγ 1+|l| τ +2 � .

Chapter 2. Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds

By all previous considerations, we conclude

meas(A c 2 ) ≤ � 0� =l∈Z |l|≥l0 meas(B l ) ≤ � 0� =l∈Z |l|≥l0 c|l| Cγ |l| τ +2 ≤ � 0� =l∈Z |l|≥l0 Cγ |l| τ +1 ≤ C γ |l 0 | τ ≤ Cγ � ε γ � τ τ 0 +2 .
Hence, if we take τ > 2(τ 0 + 2), we obtain for ε = ε(γ) << γ small enough

meas(A c 2 ) < Cγ ε2 γ 2 < C ε2 γ << ε and in particular meas(A c 2 ) ε → 0 as ε → 0. Recall A = A 1 ∩ A 2 .
The measure estimates for A c 1 in Lemma 2.6.4 and the measure estimates for A c 2 in Lemma 2.6.5 imply that for ε = ε(γ) << γ small enough, A is a large set with asymptotically full Lebesgue measure:

meas(A c ) < C ε2 γ << ε with meas(A c ) ε → 0 as ε → 0.
By inclusion (2.55), A is a large set with asymptotically full Lebesgue measure too.

The main Theorem 2.1.4 is finally proved.

Chapter 3

An abstract analysis for some linear invertible operators

In this chapter, we develop an abstract analysis for studying linear operators with off-diagonal decay that we want to invert.

We consider linear operators in Sobolev spaces H s defined in (3.7).

For these operators, we introduce the useful | • | s -norms (see Definition 3.5

taken from [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF]), their tame estimates (3.14) and their properties of algebra (3.15).

This setting is convenient to estimate the decay of inverse operators via Neumann series because products (and then powers) of operators with finite | • | snorm exhibit the same off-diagonal decay.

The main result of this chapter is Proposition 3.2.2: if an operator L has a sufficient off-diagonal decay and if its singular sites are sufficiently separated (see Definition 3.2.1 for singular sites), then "good" L 2 -bounds for L -1 imply "good" bounds in Sobolev operator norm � • � s .

Preliminaries

Let (H, � • � H ) be a separable Hilbert space.

With notation L(H) we always identify the set of all linear and continuous operators defined from H into itself L(H) := {A : H → H, A linear and continuous}.

The space L(H) is a Banach space with operator norm

�A� L(H) := sup 0� =h∈H �Ah� H �h� H .
For operators A 1 and A 2 in L(H), we have

�A 1 A 2 � L(H) ≤ �A 1 � L(H) �A 2 � L(H) . (3.1)
We define for T := R/2πZ, the space

L 2 (T, H) := {u = � k∈Z e ikt u k , u k ∈ H : �u� 2 0 := � k∈Z �u k � 2 H < +∞}
in accordance with the Parseval's identity for which �•� 0 is equal to the standard

norm on L 2 (T, H) � k∈Z �u k � 2 H = 1 2π � T �u(t)� 2 H dt.
The Fourier coefficients u k ∈ H of u ∈ L 2 (T, H) are defined by For a real linear operator L : L 2 (T, H) → L 2 (T, H), we consider the C-linear complexification of L that we still denote by L defined by: for

u k := 1 2π � T e -ıkt u(t) dt.
u 1 , u 2 ∈ L 2 (T, H) L(u 1 + ıu 2 ) := Lu 1 + ıLu 2 .
We associate to each linear and continuous operator L ∈ L(L 2 (T, H)) a matrix of linear and continuous coefficients (

L k l ) k,l∈Z , L k l ∈ L(H), defined by: ∀ h ∈ H, L k l (h) := 1 2π � T e -ilt L(e ikt h) dt.
Chapter 3. An abstract analysis for some linear invertible operators Remark 3.1.1. Let us consider L ∈ L(L 2 (T, H)). For all u ∈ L 2 (T, H),

define v := L(u) ∈ L 2 (T, H), we have that the Fourier coefficients v l of v verify v l = � k∈Z L k l u k . (3.3)
Proof. By definition of the Fourier coefficients of u ∈ L 2 (T, H)

u k := 1 2π � T e -ıkt u(t) dt, u k ∈ H,
we have that

N � k=-N e ıkt u k → u as N → ∞, in � • � 0 -norm, i.e. lim N →∞ N � k=-N e ıkt u k = u in L 2 (T, H). Since L ∈ L(L 2 (T, H)), we write L(u) = lim N →∞ L � N � k=-N e ıkt u k � = lim N →∞ N � k=-N L(e ıkt u k ) in L 2 (T, H).
Hence, the Fourier coefficients v l of v := L(u) verify

v l := 1 2π � T e -ılt v(t) dt = 1 2π � T e -ılt L(u)(t) dt = 1 2π � T e -ılt � lim N →∞ N � k=-N L(e ıkt u k ) � dt = 1 2π lim N →∞ � � T e -ılt N � k=-N L(e ıkt u k ) dt � = lim N →∞ N � k=-N � 1 2π � T e -ılt L(e ıkt u k ) dt � = lim N →∞ � N � k=-N L k l u k � = � k∈Z L k l u k . Remark 3.1.2.
Let us consider a continuous map

P : T → L(H) t � → P (t).
We define the Fourier coefficients P k ∈ L(H) of P as

P k := 1 2π � T e -ıkt P (t) dt.
Consider the operator L P L P : u � → P u defined on L 2 (T, H) by

(P u)(t) := P (t)u(t), ∀ t ∈ T.
Then, L P ∈ L(L 2 (T, H)) and its corresponding matrix coefficients are

(L P ) k l = P l-k .
Proof. By definition, L P is linear in u ∈ L 2 (T, H) and since P is continuous, we estimate

�L P u� 0 = �P u� 0 = � 1 2π � T �P (t)u(t)� 2 H dt � 1 2 ≤ � max t∈T �P (t)� L(H) �� 1 2π � T �u(t)� 2 H dt � 1 2 ≤ � max t∈T �P (t)� L(H) � �u� 0 .
Hence, the operator L P maps L 2 (T, H) into itself and it is continuous.

Using he definitions, we compute the matrix coefficients of L P : ∀ h ∈ H

(L P ) k l (h) := 1 2π � T e -ilt L P (e ikt h) dt = 1 2π � T e -ilt P (t)(e ikt h) dt = 1 2π � T e -ilt e ikt P (t)(h) dt = 1 2π � T e -i(l-k)t P (t)(h) dt =: P l-k (h).
By (3.3), we conclude that, define v(t) := P (t)u(t)

v l = � k∈Z P l-k u k .
Chapter 3. An abstract analysis for some linear invertible operators Given L ∈ L(L 2 (T, H)), we introduce the following useful definitions taken from [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF] :

fix n ∈ Z, we define the semi-norm [ • ] as [L(n)] := max l-k=n �L k l � L(H) , (3.4)
and fix s ∈ R, the norms | • | s as

|L| 2 s := C � n∈Z �n� 2s [L(n)] 2 (3.5)
where �n� := max{1, |n|}.

The role of the constant C > 0 (independent of s) is minor: it is chosen large enough so that Lemma 3.1.9 holds with C(s 0 ) = 1. We shall often omit it when we estimate the | • | s -norm.

Recall the continuous map P : T → L(H) and the definitions of the operator

L P ∈ L(L 2 (T, H)) introduced in Remark 3.1.2.
Define the following norm

|||P ||| 2 s,L(H) := � k∈Z �k� 2s �P k � 2 L(H) (3.6)
where �k� := max{1, k}, we have the following result. Proof. By Remark 3.1.2, we estimate directly

[L P (n)] := max l-k=n �(L P ) k l � L(H) = max l-k=n �P l-k � L(H) = �P n � L(H)
which implies by definition (3.6)

|L P | 2 s = � n∈Z �n� 2s [L(n)] 2 = � n∈Z �n� 2s �P n � 2 L(H) = |||P ||| 2 s,L(H) .
Let us define for all s ≥ 0, the Sobolev spaces For all s > 1/2, there is the continuous embedding

H s (T, H) := � u ∈ L 2 (T, H) : �u� 2 s := C � k∈Z �k� 2s �u k � 2 H < +∞ � . ( 3
H s (T, H) �→ C 0 (T, H) (3.8)
where C 0 (T, H) is the set of all continuous functions on T into H endowed with the � • � L ∞ -norm defined as

�u� L ∞ := sup t∈T �u(t)� H .
See Remark A.2.1 for the proof of (3.8).

Moreover, for α ∈ N+, let us denote u (α) the α-th derivative of u with respect to t ∈ T. Then, for all s > 1/2 + k, we have

H s (T, H) �→ C k (T, H)
where C k (T, H) is the set of all continuous functions u on T into H such that all its derivatives u (α) , for α = 1, . . . k, are continuous, endowed with the

� • � C k -norm defined as �u� C k := �u� L ∞ + k � α=1 �u (α) � L ∞ .
fixed s 0 > 1/2 �u� L ∞ ≤ �u� s0 (3.9)
without a presence of a constant.

We shall often omit it when we estimate the � • � s -norm.

Let B, C be subsets of Z. We introduce the subspace

H s B := {u ∈ H s (T, H) : u k = 0 if k � ∈ B}. (3.10) Remark 3.1.5. If B is finite, H s B ⊂ H s (T, H), ∀ s ≥ 0.
In this sense it does not depend on s and will be denoted H B .

If L : H s B → H s C is a linear and continuous operator, its associated matrix M is in the set M B C of matrices of coefficients (M k l ) k∈B, l∈C , with M k l : H → H linear and continuous. In the following, we will identify each operator L with its matrix M .

For L : H s B → H s C linear and continuous, the definition (3.4) becomes

[L(n)] :=        max l-k=n k∈B, l∈C �L k l � L(H) if n ∈ C -B 0 if n / ∈ C -B. (3.11)
Follow now some classical properties of the | • | s -norms. We do not insert here the proof of all results: many Lemmas are taken from [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF] where the setting is a little different (the Hilbert space H considered in [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF] is finite dimensional) but it does not change the heart of the proofs.

We need the definitions of the following useful space

H + (T, R) := {h ∈ L 2 (T, R) : h = � k∈Z e ikt h k , h k ∈ R +
and h k � = 0 only for a finite number of indices k}.

Note that the sum and the product of two functions in H + (T, R) remain in

H + (T, R) and H + (T, R) ⊂ H s (T, R) for all s ≥ 0. Definition 3.1.6. Let B, C be finite subsets of Z. Given L ∈ M B C , h ∈ H + (T, R), we say that L is dominated by h, and we write L ≺ h, if [L(n)] ≤ h n , ∀ n ∈ Z, i.e. if �L k l � L(H) ≤ h l-k for all k ∈ B, l ∈ C.
Remark 3.1.7. Since B and C are finite subsets of Z, we have that for any

L ∈ M B C ∃ h ∈ H + (T, R), L ≺ h such that ∀ s ≥ 0, |L| s = �h� s , (3.12)
in particular

|L| s = min{�h� s : h ∈ H + (T, R), L ≺ h}. (3.13) Proof. Take h ∈ H + (T, R) such that L ≺ h, by all previous definitions |L| 2 s := � n∈Z �n� 2s [L(n)] 2 ≤ � n∈Z �n� 2s h 2 n = �h� 2 s .
Moreover, define

h n := [L(n)], the function h = � k∈Z e ikt h k = � k∈Z e ikt [L(k)] is in H + (T, R) because B and C are finite, L ≺ h and verifies |L| s = � h� s , ∀ s ≥ 0.
Chapter 3. An abstract analysis for some linear invertible operators Lemma 3.1.8. Let B, C, D be finite subsets of Z.

For L 1 ∈ M C D , L 2 ∈ M B C , L 3 ∈ M C D , we have that L 1 ≺ h 1 , L 2 ≺ h 2 , L 3 ≺ h 3 imply L 1 + L 3 ≺ h 1 + h 3 and L 1 L 2 ≺ h 1 h 2 .
Proof. The property

L 1 + L 3 ≺ h 1 + h 3 is straightforward. For k ∈ B, l ∈ C, using (3.1), �(L 1 L 2 ) k l � L(H) = � � q∈C (L 1 ) q l (L 2 ) k q � L(H) ≤ � q∈C �(L 1 ) q l (L 2 ) k q � L(H) ≤ � q∈C �(L 1 ) q l � L(H) �(L 2 ) k q � L(H) ≤ � q∈Z (h 1 ) l-q (h 2 ) q-k ≤ (h 1 h 2 ) l-k implies L 1 L 2 ≺ h 1 h 2 by Definition 3.1.6.
The following results are consequences of Lemma 3.1.8.

Lemma 3.1.9. For all s ≥ s 0 > 1 2 there is a constant C(s) ≥ 1, with C(s 0 ) = 1, such that for any finite subsets B, C, D ⊂ Z and all

L 1 ∈ M B C , L 2 ∈ M C D L 1 : H B → H C , L 2 : H C → H D , L 2 L 1 ∈ M B D and |L 2 L 1 | s ≤ 1 2 |L 2 | s0 |L 1 | s + C(s) 2 |L 2 | s |L 1 | s0 . (3.14)
In particular,

|L 2 L 1 | s0 ≤ |L 1 | s0 |L 2 | s0 , |L 2 L 1 | s ≤ C(s)|L 1 | s |L 2 | s . (3.15) Corollary 3.1.10. Let B be a finite subset of Z and L ∈ M B B . For n ≥ 1, we consider the composition map L n = L • L • • • • • L, n times. Then the following estimates hold |L n | s0 ≤ |L| n s0 , |L n | s ≤ C(s)|L| n-1 s0 |L| s . Lemma 3.1.11. Let L ∈ M B C . If for all k ∈ B, l ∈ C, we have that |k -l| > N ⇒ L k l = 0, then for all s � ≥ s ≥ 0 |L| s � ≤ N s � -s |L| s . Moreover, if for all k ∈ B, l ∈ C, |k -l| < N ⇒ L k l = 0, then |L| s ≤ N -(s � -s) |L| s � .
For B a finite set of indices, we identify H B with the space M {0} B of column matrices and the Sobolev norm

� • � s is equal to the s-norm | • | s , i.e. ∀ u ∈ H B , �u� s = |u| s , ∀ s ≥ 0. (3.16)
We will denote this identification with

H B ≡ M {0} B .
Note that by (3.16), the following Lemma is a particular case of Lemma 3.1.9.

Lemma 3.1.12. For all s ≥ s 0 > 1 2 there is a constant C(s) ≥ 1, with C(s 0 ) = 1, such that for any finite subsets B, C ⊂ Z,

�Lw� s ≤ 1 2 |L| s0 �w� s + C(s) 2 |L| s �w� s0 , ∀ L ∈ M B C , w ∈ H B . Given subsets B ⊂ B � ⊂ Z, C ⊂ C � ⊂ Z and L ∈ M B �
C � , we define the "restricted" operators

L C := Π C L, L B := L | H s B , L B C := Π C L | H s B where Π C is the L 2 -orthogonal projector onto H s C .
Lemma 3.1.13 (Decay along lines). Let L ∈ M B C , with B, C finite subsets of Z. For all s ≥ 0 and b > 1/2, we have

|L| s ≤ C 0 max l∈C |L {l} | s+b for a positive constant C 0 = C 0 (b) > 0.
Proof. Fix l ∈ C. By definitions (3.5) and (3.11), we write for all s ≥ 0

|L {l} | 2 s+b = � n∈l-B �n� 2(s+b) � max l-k=n k∈B �L k l � L(H) � 2 = � k∈B �l -k� 2(s+b) �L k l � 2 L(H) .
Chapter 3. An abstract analysis for some linear invertible operators

Then, for all l ∈ C, k ∈ B

|L {l} | 2 s+b ≥ �l -k� 2(s+b) �L k l � 2 L(H)
and so

�L k l � L(H) ≤ |L {l} | s+b �l -k� (s+b) ≤ M �l -k� (s+b) (3.17)
where

M := max l∈C |L {l} | s+b .
As a consequence, we can conclude

|L| s = � � n∈C-B �n� 2s � max l-k=n l∈C,k∈B �L k l � L(H) � 2 � 1 2 ≤ � � n∈C-B �n� 2s � M �n� (s+b) � 2 � 1 2 ≤ M � � n∈Z �n� -2b � 1 2 ≤ C 0 M provided if � n∈Z �n� -2b = C 0 < ∞, which is the case if b > 1/2.
We are interested now in the relation between the operator norm � • � 0 of L and its | • | s -norm. We have the following results. Proof. Let h 1 ∈ H + (T, R) be such that L ≺ h 1 and |L| s = �h 1 � s for all s ≥ 0, see (3.12). Also for U ∈ M {0} B , there is h 2 ∈ H + (T, R) such that U ≺ h 2 and |U | s = �h 2 � s for all s ≥ 0. Then, from Lemma 3.1.8, LU ≺ h 1 h 2 and using the continuous embedding (3.9), we have

|LU | 0 ≤ �h 1 h 2 � 0 ≤ �h 1 � L ∞ (T,R) �h 2 � 0 ≤ �h 1 � s0 �h 2 � 0 = |L| s0 |U | 0 . ( 3
�L� 0 := sup 0� =U ∈H B �LU � 0 �U � 0 = sup 0� =U ∈M {0} B |LU | 0 |U | 0 .
Using (3.18), we have

�L� 0 ≤ |L| s0 .
Chapter 3. An abstract analysis for some linear invertible operators Using the completeness of the Banach space L(L 2 (T, H)), Lemma 3.1.14 implies the following Lemma.

Lemma 3.1.15. Let L ∈ L(L 2 (T, H)). If s 0 > 1 2 , then �L� 0 ≤ |L| s0 . Hence, ∀ s ≥ s 0 �L� 0 ≤ |L| s . Lemma 3.1.16. Let be L ∈ M B C . If ∀ k ∈ B, ∀ l ∈ C, |k -l| > N =⇒ L k l = 0, then for s > 0 |L| s ≤ CN s+ 1 2 �L� 0 . Proof. We start proving that ∀ l, k ∈ Z, �L k l � L(H) ≤ �L� 0 . By definition �L� 0 = sup 0� =u∈L 2 (T,H) �Lu� 0 �u� 0 ,
and so

∀ g ∈ H, ∀ k ∈ B, ∀ l ∈ C, we have �L� 0 ≥ �L(e ikt g)� 0 �g� H ≥ �L k l g� H �g� H which implies �L� 0 ≥ sup 0� =g∈H �L k l g� H �g� H = �L k l � L(H) .
We compute now

|L| 2 0 = � n∈Z [L(n)] 2 = � |n|≤N � sup l-k=n �L k l � 2 L(H) � ≤ (2N + 1)�L� 2 0 since sup l-k=n �L k l � L(H) ≤ �L� 0 . Hence, using Lemma 3.1.11, we conclude |L| s ≤ N s |L| 0 ≤ CN s+ 1 2 �L� 0 .
We insert now a useful perturbation result.

Chapter 3. An abstract analysis for some linear invertible operators Lemma 3.1.17. Let L be an operator of the form

L : D(L) ⊂ L 2 (T, H) → L 2 (T, H) L = D + T.
We suppose that

D : D(D) ⊂ L 2 (T, H) → L 2 (T, H)
is linear (possibly unbounded, i.e. not continuous), invertible with continuous inverse

D -1 ∈ L(L 2 (T, H)) and T ∈ L(L 2 (T, H)).
Of course, D(L) = D(D).

If we have

|D -1 | s0 |T | s0 < 1,
then L is invertible and verifies estimates

|L -1 | s0 ≤ � |D -1 | s0 1 -|D -1 | s0 |T | s0 � (3.19)
and ∀ s ≥ s 0

|L -1 | s ≤ C(s) � |D -1 | s + |D -1 | 2 s0 |T | s 1 -|D -1 | s0 |T | s0 � . (3.20)
As a consequence, if

|D -1 | s0 |T | s0 ≤ 1 2 , then |L -1 | s0 ≤ 2|D -1 | s0 and ∀ s ≥ s 0 |L -1 | s ≤ C(s)(|D -1 | s + |D -1 | 2 s0 |T | s ).
Proof. The fact that L = D + T is invertible is standart: it is enough to write

L = D + T = D(I + D -1 T )
and to notice that

I + D -1 T is invertible in the Banach space L(L 2 (T, H))
since, by Lemma 3.1.15,

�D -1 T � 0 ≤ |D -1 T | s0 ≤ |D -1 | s0 |T | s0 < 1. (3.21)
By completeness of L(L 2 (T, H)), we write directly

L -1 = (D(I + D -1 T )) -1 = (I + D -1 T ) -1 D -1 (3.22)
and by (3.21), we can use the Neumann series formula to write

(I + D -1 T ) -1 = +∞ � n=0 (-1) n (D -1 T ) n .
Chapter 3. An abstract analysis for some linear invertible operators By Corollary 3.1.10, we obtain the estimates

|(I + D -1 T ) -1 | s0 ≤ � n≥0 |(D -1 T ) n | s0 ≤ � n≥0 |(D -1 T )| n s0 ≤ 1 1 -|D -1 | s0 |T | s0 and ∀ s ≥ s 0 |(I + D -1 T ) -1 | s ≤ |I| s + � n≥1 |(D -1 T ) n | s ≤ c + C(s) � n≥1 |(D -1 T )| n-1 s0 |(D -1 T )| s ≤ c + C(s) � |D -1 T | s 1 -|D -1 | s0 |T | s0 � .
Hence, by (3.22) and Lemma 3.1.9 we conclude

|L -1 | s0 = |(I + D -1 T ) -1 D -1 | s0 ≤ |(I + D -1 T ) -1 | s0 |D -1 | s0 ≤ |D -1 | s0 1 -|D -1 | s0 |T | s0 and |L -1 | s = |(I + D -1 T ) -1 D -1 | s ≤ 1 2 |(I + D -1 T ) -1 | s |D -1 | s0 + C(s) 2 |(I + D -1 T ) -1 | s0 |D -1 | s ≤ c 2 |D -1 | s0 + C(s) � |D -1 T | s |D -1 | s0 1 -|D -1 | s0 |T | s0 � + C(s) 2 � |D -1 | s 1 -|D -1 | s0 |T | s0 � ≤ C(s) � |D -1 | s + |D -1 | 2 s0 |T | s 1 -|D -1 | s0 |T | s0 � .
Remark 3.1.18. All results of this section hold also if the time variable lies on the ν-dimensional torus T ν .

In this case, we define the Sobolev spaces H s (T ν , H) and all norms in accordance with the fact that the Fourier indices are in Z ν .

The only difference in the results is that we impose to the Sobolev index s to be s ≥ s 0 > ν/2 (and not only > 1/2) in order to have the continuous embedding

H s (T ν , H) �→ C 0 (T ν , H)
that we use in several lemmas.

Regular and Singular sites

We consider a linear operator of the form

L : D(L) ⊂ L 2 (T, H) → L 2 (T, H) L = D + T
where D is the diagonal part with coefficients defined by D k l := d l δ l k , with d l : H → H linear (possibly unbouded), and T ∈ L(L 2 (T, H)) is the nondiagonal term.

Using the notation in (3.10), we define

E (N ) := H [-N,N ] =    w ∈ L 2 (T, H) : w(t) = � |l|≤N e ilt w l (x), w l ∈ H    , (3.23)
with the corresponding L 2 -orthogonal projector Π (N ) .

Of course, E (N ) ⊂ H s (T, H) for all s ≥ 0, see Remark 3.1.5.

For a linear and continuous operator A : H s (T, H) → H s (T, H) we consider the operator norm

�A� s := sup 0� =u∈H s (T,H) �Au� s �u� s .
We want to study the invertibility, ∀ N ∈ N, of the linear operators L (N )

L (N ) := Π (N ) L |E (N )
and find some estimates of �(L (N ) ) -1 w� s , for w ∈ E (N ) .

We introduce the following definition.

Definition 3.2.1. Fix θ > 0. An index l ∈ Z is said to be a "regular site" for the operator L if

�d -1 l � L(H) ≤ θ -1 .
An index which is not regular is said a "singular site".

Let be R N the set of all regular sites of L in the interval [-N, N ] and let be S N := R c N the set of the singular sites in [-N, N ].

We have the following Proposition:

Proposition 3.2.2. Suppose there exist constants 0 < δ < 1, 0 < β < 1 such that in the interval [-N, N ], the singular sites of L are divided in blocks Γ r

S N = ∪ r Γ r such that    diam(Γ r ) ≤ N δ ∀ r, dist(Γ r , Γ r � ) ≥ N β if r � = r � (3.24)
where dist(Γ r , Γ r � ) := inf l∈Γr, k∈Γ r � |l -k|.

Fix µ 0 > 0, let s 1 = s 1 (µ 0 ) be such that

β(s 1 -s 0 ) ≥ µ 0 + δ 2 + δs 0 . (3.25)
Fix γ ∈ (0, 1), there exists ε 0 = ε 0 (s 1 ) (independent of N ), a constant µ independent of s µ := 2(µ 0 + δs 0 ) + δ and a non-decreasing function

K : R + → [1, ∞) such that if L (N ) is invertible with �(L (N ) ) -1 � 0 ≤ N µ0 γ (3.26)
and if

|T | s1 ≤ ε 0 , (3.27)
then ∀ s ≥ s 0 , ∀ w ∈ E (N ) �(L (N ) ) -1 w� s ≤ K(s) γ 2 N µ � (N δs + |T | s )�w� s0 + �w� s � . (3.28)
For proving Proposition 3.2.2, we need some considerations which follow.

Let us define the sets

E R := {u ∈ E (N ) : u = � l∈R N e ilt u l , u l ∈ H} E S := {u ∈ E (N ) : u = � l∈S N e ilt u l , u l ∈ H}
with the corresponding L 2 -orthogonal projectors Π ER and Π ES .

According to the orthogonal decomposition ) as the block matrix

E (N ) := E R � E S , we represent L (N
L (N ) = � L R L S R L R S L S � (3.29)
where

L R := Π E R L (N ) | E R , L S R := Π E R L (N ) | E S , L R S := Π E S L (N ) | E R , L S := Π E S L (N ) | E S .
Chapter 3. An abstract analysis for some linear invertible operators Lemma 3.2.3.

For |T | s1 ≤ θ 2 , L R is invertible with |(L R ) -1 | s0 ≤ 2θ -1 (3.30)
and ∀ s > s 0

|(L R ) -1 | s ≤ C(s)(θ -1 + θ -2 |T | s ). (3.31) Proof. Write L R = D R + T R , with (D R ) k l = d l δ l k for l, k ∈ R N ,
by definition of regular sites and of the norm (3.11), it is clear that

[D -1 R (n)] := max l-k=n �(D -1 R ) k l � L(H) = max l-k=n �(d l ) -1 δ l k � L(H)
is different to zero only for n = 0. Hence, ∀ s ≥ s 0

|D -1 R | 2 s := � n �n� 2s [D -1 R (n)] 2 = [D -1 R (0)] 2 = � max l∈R N �d -1 l � L(H) � 2 ≤ max l∈R N �d -1 l � 2 L(H) ≤ θ -2 . Since |T R | s0 ≤ |T | s0 ≤ |T | s1 ≤ θ 2 , we have |D -1 R | s0 |T R | s0 ≤ θ -1 θ 2 = 1 2
and we can apply Lemma 3.1.17 obtaining

|(L R ) -1 | s0 ≤ 2θ -1 and ∀ s > s 0 |(L R ) -1 | s ≤ C(s)(|D -1 R | s + |D -1 R | 2 s0 |T R | s ) ≤ C(s)(θ -1 + θ -2 |T | s ).
By Lemma 3.1.12, estimates (3.30) and (3.31) imply

�(L R ) -1 � s0 ≤ 2θ -1 (3.32)
Chapter 3. An abstract analysis for some linear invertible operators and ∀ s > s 0

�(L R ) -1 w� s ≤ θ -1 �w� s + C(s) 2 (θ -1 + θ -2 |T | s )�w� s0 . (3.33)
The invertibility of L (N ) is then reduced to the invertibility of the operator

U : E S → E S U := L S -L R S (L R ) -1 L S R
by the "resolvent identity"

(L (N ) ) -1 = � I -(L R ) -1 L S R 0 I � � (L R ) -1 0 0 U -1 � � I 0 -L R S (L R ) -1 I � .
According to the separation property of the singular sites, we can see the matrix U as divided in blocks associated to each Γ r and we write

U = U � + T � (3.34)
where U � is the "block-diagonal" part:

(U � ) k l =    U k l if l, k ∈ Γ r , for some r 0 otherwise, (3.35)
and

(T � ) k l =    U k l if l ∈ Γ r , k ∈ Γ r � , r � = r � 0 otherwise. (3.36) Lemma 3.2.4. Assume |T | s1 ≤ θ 2 .
According to the definition of T � in (3.36), we have

|T � | s0 ≤ 2|T | s0 (3.37)
and for all s > s 0

|T � | s ≤ C(s)|T | s . (3.38) Proof. Since U = L S -L R S (L R ) -1 L S R
, we can write

T � = T � 1 -T � 2 where for l ∈ Γ r , k ∈ Γ r � , r � = r � , (T � 1 ) k l := (L S ) k l = T k l , (T � 2 ) k l := (L R S (L R ) -1 L S R ) k l and for l, k ∈ Γ r (T � 1 ) k l = (T � 2 ) k l = 0.
Of course, by definition of norm | • | s , for all s ≥ s 0

|T � 1 | s ≤ |T | s . (3.39)
By Lemma 3.1.9, by estimate (3.30) and since

|T | s0 < |T | s1 ≤ θ 2 |T � 2 | s0 ≤ |L R S | s0 |(L R ) -1 | s0 |L S R | s0 ≤ 2θ -1 |T | 2 s0 ≤ |T | s0
and the estimate (3.37) comes easily using (3.39)

|T � | s0 = |T � 1 -T � 2 | s0 ≤ |T � 1 | s0 + |T � 2 | s0 ≤ 2|T | s0 .
For s > s 0 , Lemma 3.1.9 implies

|T � 2 | s = |L R S (L R ) -1 L S R | s ≤ 1 2 |L R S | s0 |(L R ) -1 L S R | s + C(s) 2 |L R S | s |(L R ) -1 L S R | s0 ≤ 1 2 |L R S | s0 � 1 2 |(L R ) -1 | s0 |L S R | s + C(s) 2 |(L R ) -1 | s |L S R | s0 � + C(s) 2 |L R S | s |(L R ) -1 | s0 |L S R | s0 ≤ C(s) � |L R S | s0 |(L R ) -1 | s0 |L S R | s + |L R S | s0 |(L R ) -1 | s |L S R | s0 + |L R S | s |(L R ) -1 | s0 |L S R | s0 � . By estimate (3.30), for |T | s0 ≤ |T | s1 ≤ θ 2 , we have |L R S | s0 |(L R ) -1 | s0 |L S R | s ≤ 2θ -1 |T | s0 |T | s ≤ |T | s , |L R S | s |(L R ) -1 | s0 |L S R | s0 ≤ 2θ -1 |T | s0 |T | s ≤ |T | s
and, using also (3.31)

|L R S | s0 |(L R ) -1 | s |L S R | s0 ≤ |T | 2 s0 � C(s)(θ -1 + θ -2 |T | s ) � ≤ C(s)(|T | s0 |T | s0 θ -1 + θ -2 |T | 2 s0 |T | s ) ≤ C(s)(|T | s0 + |T | s ) = C(s)|T | s
Chapter 3. An abstract analysis for some linear invertible operators which imply

|T � 2 | s ≤ C(s)|T | s .
Then, by (3.39), we have estimate (3.38)

|T � | s = |T � 1 -T � 2 | s ≤ |T � 1 | s + |T � 2 | s ≤ |T | s + C(s)|T | s ≤ C � (s)|T | s .
Follow here four Lemmas that will prove Proposition 3.2.2.

Lemma 3.2.5. Let U = U � + T � be as in (3.34). If L (N ) is invertible then U is invertible too and

�U -1 � 0 ≤ �(L (N ) ) -1 � 0 .
Proof. For proving that the operator

U : E S → E S U := L S -L R S (L R ) -1 L S R
is invertible, i.e. there exists U -1 , we first have to show that

∀ g � ∈ E S , ∃! h � ∈ E S such that U -1 g � = h � , i.e. U h � = g � . Fix g � ∈ E S . Since L (N ) is invertible, we can define h := (L (N ) ) -1 g � .
According to the decomposition

E (N ) := E R � E S , we write any function h ∈ E (N ) as h = h R + h S with (h R ) k =    h k if k ∈ R N 0 if k � ∈ R N , (h S ) k =    h k if k ∈ S N 0 if k � ∈ S N . By (3.29), since g � ∈ E S , h = (L (N ) ) -1 g � ⇐⇒ L (N ) h = g � ⇐⇒    L R h R + L S R h S = 0 L R S h R + L S h S = g � . The first equation implies h R = -(L R ) -1 (L S R h S )
and put it in the second equation, we obtain

g � = (L S -L R S (L R ) -1 L S R )h S = U h S .
Chapter 3. An abstract analysis for some linear invertible operators Hence, for any g � ∈ E S , there exists a function h � = h S ∈ E S , such that

U h � = g � . Moreover �U -1 g � � 0 = �h S � 0 ≤ �h� 0 ≤ �(L (N ) ) -1 g � � 0 ≤ �(L (N ) ) -1 � 0 �g � � 0 which implies �U -1 � 0 ≤ �(L (N ) ) -1 � 0 .
Lemma 3.2.6. Let U = U � + T � be as in (3.34). Under the assumptions of Proposition 3.2.2, for ε 0 small enough, U � is invertible and

�U -1 � 0 ≤ N µ0 γ =⇒ �(U � ) -1 � 0 ≤ 2N µ0 γ . (3.40)
Proof. We want to apply Lemma B.1.1 to the matrix

U � = U -T � .
Recall the definition of T � in (3.36)

(T � ) k l =    U k l if l ∈ Γ r , k ∈ Γ r � , r � = r � 0 otherwise,
and the hypothesis of separation of the singular sites in (3.24) for which

dist(Γ r , Γ r � ) ≥ N β if r � = r � ,
we can apply Lemma 3.1.11 to

T � |T � | s0 ≤ N -β(s1-s0) |T � | s1
and using Lemma 3.1.14 and (3.38), we estimate for

s 1 > s 0 > 1 2 �T � � 0 ≤ C|T � | s0 ≤ CN -β(s1-s0) |T � | s1 ≤ C(s 1 )N -β(s1-s0) |T | s1 .
For ε 0 ≤ C(s 1 ) γ 2 small enough and s 1 satisfying (3.25) (it is enough here that β(s 1s 0 ) > µ 0 ), we have

�U -1 � 0 �T � � 0 ≤ C(s 1 ) ε 0 N β(s1-s0) �U -1 � 0 ≤ C(s 1 ) ε 0 N β(s1-s0) N µ0 γ ≤ 1 2
and by Lemma B.1.1 applied to U � = U -T � , we obtain

�(U � ) -1 � 0 ≤ 2N µ0 γ .
Chapter 3. An abstract analysis for some linear invertible operators Lemma 3.2.7. Under the assumptions of Proposition 3.2.2, the estimate

�(U � ) -1 � 0 ≤ 2N µ0 γ implies |U -1 | s0 ≤ CN μ+δs0 γ (3.41)
and ∀ s > s 0

|U -1 | s ≤ C(s) � N 2μ+2δs0 γ 2 |T | s + N μ+δs γ � (3.42)
with μ := µ 0 + δ 2 . As a consequence, we have

�U -1 � s0 ≤ CN μ+δs0 γ (3.43)
and ∀ s > s 0

�U -1 w� s ≤ C(s) N µ γ 2 � (N δs + |T | s )�w� s0 + �w� s � (3.44)
with µ := 2(µ 0 + δs 0 ) + δ.

Proof. Recall U = U � + T � , the definition of U � in (3.35) (U � ) k l =    U k l if l, k ∈ Γ r ,
|(U � ) -1 | s0 ≤ C(N δ ) s0+ 1 2 �(U � ) -1 � 0 ≤ CN μ+δs0 γ (3.45) with μ := µ 0 + δ 2 and similarly ∀ s > s 0 |(U � ) -1 | s ≤ CN μ+δs γ . ( 3
|(U � ) -1 | s0 |T � | s0 ≤ CN μ+δs0 γ |T � | s1 N β(s1-s0) ≤ C(s 1 ) N μ+δs0 γ |T | s1 N β(s1-s0) ≤ C(s 1 ) γ ε 0 N β(s1-s0)-μ-δs0 ≤ 1 2 .
(3.47) Chapter 3. An abstract analysis for some linear invertible operators Then, Lemma 3.1.17 implies that U is invertible with

|U -1 | s0 ≤ 2|(U � ) -1 | s0
and for s > s 0

|U -1 | s ≤ C(s)(|(U � ) -1 | s + |(U � ) -1 | 2 s0 |T � | s ).
By (3.45) we obtain estimate (3.41)

|U -1 | s0 ≤ CN μ+δs0 γ
and using also (3.38) and (3.46), we have

|U -1 | s ≤ C(s) � N μ+δs γ + N 2μ+2δs0 γ 2 |T | s � .
By Lemma 3.1.12 and the previous estimates, we conclude

�U -1 w� s0 ≤ |U -1 | s0 �w� s0 ≤ CN μ+δs0 γ �w� s0
which means

�U -1 � s0 ≤ CN μ+δs0 γ and ∀ s > s 0 �U -1 w� s ≤ 1 2 |U -1 | s0 �w� s + C(s) 2 |U -1 | s �w� s0 ≤ CN μ+δs0 γ �w� s + C(s) 2 � N 2μ+2δs0 γ 2 |T | s + N μ+δs γ � �w� s0 ≤ C(s) N µ γ 2 � �w� s + (N δs + |T | s )�w� s0 � with µ := 2(μ + δs 0 ) = 2(µ 0 + δs 0 ) + δ.
Lemma 3.2.8. Under the assumptions of Proposition 3.2.2, we have for all s ≥ s 0 the estimate (3.28)

�(L (N ) ) -1 w� s ≤ K(s) N µ γ 2 � �w� s + (N δs + |T | s )�w� s0 � .
Proof. Let h, w ∈ E (N ) be such that h = (L (N ) ) -1 w. According to the decom-

position h = h R + h S , w = w R + w S ,
�h S � s0 = �U -1 (w S -L R S (L R ) -1 w R )� s0 ≤ 4N μ+δs0 γ 2 (�w S � s0 + �L R S (L -1 R )w R � s0 )
and ∀ s > s 0

�h S � s = �U -1 (w S -L R S (L R ) -1 w R )� s ≤ C(s) N µ γ 2 � �w S -L R S (L R ) -1 w R � s + (N δs + |T | s )�w S -L R S (L R ) -1 w R � s0 � ≤ C(s) N µ γ 2 � (�w S � s + �L R S (L R ) -1 w R � s ) + (N δs + |T | s )(�w S � s0 + �L R S (L R ) -1 w R � s0 ) � .
By Lemma 3.1.12, estimate (3.49) and hypothesis (3.27) for which

|L R S | s0 ≤ |T | s0 ≤ |T | s1 ≤ ε 0 < 1, we have �L R S (L R ) -1 w R � s0 ≤ |L R S | s0 �(L R ) -1 w R � s0 ≤ �(L R ) -1 w R � s0 and �L R S (L R ) -1 w R � s ≤ 1 2 |L R S | s0 �(L R ) -1 w R � s + C(s) 2 |L R S | s �(L R ) -1 w R � s0 ≤ �(L R ) -1 w R � s + C(s)|T | s �(L R ) -1 w R � s0 .
By the previous estimates, using also (3.32) and (3.33), for θ fixed, we estimate

�h S � s0 ≤ CN μ+δs0 γ 2 (�w S � s0 + �L R S (L -1 R )w R � s0 ) ≤ CN μ+δs0 γ 2 (�w S � s0 + �(L R ) -1 w R � s0 ) ≤ CN μ+δs0 γ 2 (�w S � s0 + 2θ -1 �w R � s0 ) ≤ CN μ+δs0 γ 2 �w� s0
and

�h S � s ≤ C(s) N µ γ 2 � �w S � s + �(L R ) -1 w R � s + (N δs + |T | s )(�w S � s0 + �(L R ) -1 w R � s0 ) � ≤ C(s) N µ γ 2 � �w S � s + θ -1 �w R � s + (θ -1 + θ -2 |T | s )�w R � s0 + (N δs + |T | s )(�w S � s0 + θ -1 �w R � s0 ) � ≤ C(s) N µ γ 2 � �w� s + (N δs + |T | s )�w� s0 � .
Similarly, we have

�h R � s = �(L R ) -1 (w R -L S R h S )� s ≤ 2θ -1 �w R -L S R h S � s + C(s)(θ -1 + θ -2 |T | s )�w R -L S R h S � s0 ≤ 2θ -1 (�w R � s + �L S R h S � s ) + C(s)(θ -1 + θ -2 |T | s )(�w R � s0 + �L S R h S � s0 ) ≤ 2θ -1 (�w R � s + 1 2 |L S R | s0 �h S � s + C(s) 2 |L S R | s �h S � s0 ) + C(s)(θ -1 + θ -2 |T | s )(�w R � s0 + |L S R | s0 �h S � s0 ) ≤ C(s) � (�w R � s + �h S � s + |T | s �h S � s0 ) + |T | s (�w R � s0 + �h S � s0 ) � ≤ C(s) � �w� s + �h S � s + |T | s (�w� s0 + �h S � s0 )
� .

Hence, we can easily conclude for µ = 2(μ + δs 0 )

�(L (N ) (v)) -1 w� s = �h� s ≤ �h R � s + �h S � s ≤ K(s) N µ γ 2 � �w� s + (N δs + |T | s )�w� s0 � .
By Lemmas 3.2.5, 3.2.6, 3.2.7 and 3.2.8 we have proved Proposition 3.2.2.

Chapter 4

An abstract Nash-Moser

Theorem

In this chapter we prove an abstract Nash-Moser implicit function theorem with parameters (ε, λ) which we will apply in chapter 5 to solve the range equation associated to the non-linear autonomous PDE (5.1) and in chapter 6

to the forced NLW equation (6.1).

The Nash-Moser Theorem 4.1.1 presents some differences with respect to the Nash-Moser scheme 2.3.1 used in chapter 2 and taken from [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF].

The main differences concern:

• the use of two different scales of Banach spaces, X s and X � s . In the applications, this allows to apply this scheme directly to equations with operators which could present loss of regularity • the existence, for all ε small enough, of an approximate solution ṽ(ε, λ) of the equation F (ε, λ, v) = 0 with an error ε q , for some q > 1.

With this hypothesis, the condition which connects the intialization term N 0 of the scheme and the parameter ε is

N 2(µ+δs1)+ν+2σ 0 γ 4 ε q < 1 2 .
This is the key point for the final measure estimates of the set of parameters in the applications (see subsections 5.5.1 and 5.5.2).

In the Nash-Moser Theorem 2.3.1, we have ṽ(ε, λ) = 0

• At each step of the scheme, we require that the inverse of the linearized operator L N satisfies the following "tame" estimates

�(L (N ) (ε, λ, v)) -1 w� s ≤ K(s) N µ γ 2 (|w| s + (N δs + �v� s )|w| s1 )
Chapter 4. An abstract Nash-Moser Theorem for some δ ∈ (0, 1).

These tame estimates assure the convergence of the scheme.

In the Nash-Moser Theorem 2.3.1, we have δ = 0

• We do not consider the variation of the approximate solutions v n with respect to parameter ε. We consider only derivatives with respect to parameter λ. As a consequence, at the end of the scheme, we prove only that the approximate solutions v n and the limit v are continuous functions with respect to parameter λ.

Similar Nash-Moser schemes are proved in [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF] and [START_REF] Berti | An abstract Nash-Moser theorem and quasi-periodic solutions for NLW and NLS on compact Lie groups and homogeneous manifolds[END_REF].

Functional setting and main result

We consider two different scales of Banach spaces (X s , | • | s ) s≥0 and

(X � s , � • � s ) s≥0 such that ∀ s ≤ s � X s � ⊆ X s , |v| s ≤ |v| s � ∀ v ∈ X s � X � s � ⊆ X � s , �v� s ≤ �v� s � ∀ v ∈ X � s �
and we define the Fréchet spaces

X := � s≥0 X s , X � := � s≥0 X � s .
We endow X and X � with the topologies induced respectively by the families of norms | • | s and � • � s .

We assume that there are an increasing family (E (N ) ) N ≥0 of closed subspaces of

X 0 such that ∀ N ∈ N, E (N ) ⊂ X, � N ≥0 E (N )
is dense in X s for every s ≥ 0, and that there are projectors

Π (N ) : X 0 → E (N ) of range E (N )
satisfying the following properties:

∀ s ≥ 0, ∀ d ≥ 0 • (S 1 ) |Π (N ) v| s+d ≤ C(s, d)N d |v| s , ∀ v ∈ X s ; • (S 2 ) �(I -Π (N ) )v| s ≤ C(s, d)N -d |v| s+d , ∀ v ∈ X s+d .
We assume that similar properties hold for an increasing family (E �(N ) ) N ≥0 of closed subspaces of X � 0 with projectors

Π �(N ) : X � 0 → E �(N ) of range E �(N ) .
Chapter 4. An abstract Nash-Moser Theorem

We consider a C 2 map

F : [0, ε 0 ) × Λ × X � s1+ν → X s1 (4.1)
where s 1 ≥ 0, ν ≥ 0, ε 0 > 0 and Λ is a bounded open subset of a finite dimensional Banach space (Y, � • � Y ). We assume

• (F 1) F (0, λ, 0) = 0, ∀ λ ∈ Λ
and the "tame" properties:

∀s ≥ s 1 , ∀ v ∈ X � s+ν with �v� s1 ≤ 2, ∀ (ε, λ) ∈ [0, ε 0 ) × Λ, • (F 2) |F (ε, λ, 0)| s ≤ C(s), |∂ λ F (ε, λ, v)| s ≤ C(s)(1 + �v� s+ν ), |D v F (ε, λ, 0)[g]| s ≤ C(s)�g� s+ν
where the symbol ∂ λ denotes partial derivatives with respect to any component of λ

• (F 3) |D 2 v F (ε, λ, v)[w, g]| s ≤ C(s)(�v� s+ν �w� s1 �g� s1 + �g� s+ν �w� s1 +�w� s+ν �g� s1 ) • (F 4) |∂ λ D v F (ε, λ, v)[g]| s ≤ C(s)(�g� s+ν + �v� s+ν �g� s1 ).
Using Taylor integral formula, we deduce from (F 1) -(F 3) the useful following properties : ∀ s ≥ s 1 , there is C(s) such that ∀ �v� s1 ≤ 2, �g� s1 ≤ 1,

• (F 5) |D v F (ε, λ, v)[g]| s ≤ C(s)(�v� s+ν �g� s1 + �g� s+ν ) • (F 6) |F (ε, λ, v)| s ≤ C(s)(1 + �v� s+ν ) • (F 7) |F (ε, λ, v + g) -F (ε, λ, v) -D v F (ε, λ, v)[g]| s ≤ C(s)(�v� s+ν �g� 2 s1 +�g� s+ν �g� s1 ).
We consider parameters 0

< δ < 1, χ 0 > 1, µ ≥ 0, σ ≥ 0, s ≥ s 1 such that χ 0 δ < 1, σ > max{4(µ + δs 1 + ν), 2(µ + 1)}, (4.2)
s ≥

s 1 + ν + 3χ 0 (σ + 1) + 2σχ0 χ0-1 1 -χ 0 δ . (4.3)
We suppose there exists q > 1 and ṽ = ṽ(ε, λ) ∈ X � such that for ε small enough

• ( Ṽ 1) (i) �ṽ(ε, λ, h)� s+ν ≤ √ ε (ii) ∀ S � > s, �ṽ(ε, λ)� S � ≤ C(S � )
Chapter 4. An abstract Nash-Moser Theorem

• ( Ṽ 2) |F (ε, λ, ṽ(ε, λ))| s1 ≤ C(s 1 )ε q • ( Ṽ 3) �∂ λ ṽ(ε, λ)� s+ν ≤ C(s)ε 1 4
• ( Ṽ 4)

� � �∂λ � (F (ε, λ, ṽ(ε, λ)) � � � � s1 ≤ C(s 1 )ε q .
For ε small enough, ṽ(ε, λ) is an approximate solution of the equation F (ε, λ, v) = 0 with an error ε q which is smaller than ε for q > 1. Of course, larger is q, closer is ṽ to be a solution.

We denote by L (N ) (ε, λ, v) the linear operators

L (N ) (ε, λ, v) : E �(N ) → E (N ) L (N ) (ε, λ, v) := Π (N ) D v F (ε, λ, v) |E �(N ) ,
and given a non-decreasing function K : [0, +∞) → [1, +∞), for all γ ∈ (0, 1],

we assume defined appropriate subsets J

(N ) γ,µ,K such that J (N ) γ,µ,K ⊆ � (ε, λ, v) ∈ [0, ε 0 ) × Λ × E �(N ) | L (N ) (ε, λ, v) is invertible and ∀s ≥ s 1 �(L (N ) (ε, λ, v)) -1 w� s ≤ K(s) N µ γ 2 (|w| s + (N δs + �v� s )|w| s1 ), ∀w ∈ E (N ) � . (4.4)
We assume the following property:

• (L1) If (ε, λ, v) ∈ J (N )
γ,µ,K and

�λ � -λ� Y + �v � -v� s1 ≤ 2γN -(µ+1) then (ε, λ � , v � ) ∈ J (N ) γ,µ,2K , which implies ∀ w ∈ E (N ) �(L (N ) (ε, λ � , v � )) -1 w� s ≤ 2K(s) N µ γ 2 (|w| s + (N δs + �v � � s )|w| s1 ). (4.5)
We remark that property (L1) is not a consequence of previous hypotheses and of a perturbative result: in estimate (4.5), the term �v� s does not appear.

Given k > 0, we define

V (N ) k := {v ∈ C 1 ([0, ε 0 ) × Λ, E �(N ) ) | �v� s1 ≤ 1, �∂ λ v� s1 ≤ k} (4.6)
and for all v ∈ V (N )

k , we set

G (N ) γ,µ,K (v) := {(ε, λ) ∈ [0, ε 0 ) × Λ | (ε, λ, v(ε, λ)) ∈ J (N )
γ,µ,K }.

Chapter 4. An abstract Nash-Moser Theorem

We have the following iterative result, with

N n := [N χ n 0 0 ], (4.7)
where [ • ] denotes the interger part of a real number and where the initialization term N 0 ∈ N will be chosen large enough (depending on parameters ε, γ, s 1 , s).

We denote by E n , E � n , Π n , Π � n , J n γ,µ,K the abbreviations for E (Nn) , E �(Nn) , Π (Nn) , Π �(Nn) J (Nn) γ,µ,K respectively. Given a set Λ ⊂ Λ and η > 0, we denote by N ( Λ, η) the open neighborhood of Λ of width η in the topological space (Λ, � • � Y ). We define N ( Λ, η) as the emptyset if Λ is empty.

Theorem 4.1.1 (The Nash-Moser Theorem). Assume (F 1)-(F 4), ( Ṽ 1)-( Ṽ 4), (L1), (4.2) and (4.3). Then, for all γ ∈ (0, 1], there is

ε 1 := ε 1 (γ) ∈ (0, ε 0 ] such that ∀ ε ∈ [0, ε 1 ) and ∀ N 0 := N 0 (ε, γ, s 1 , s) large enough satisfying N 2(µ+δs1)+ν+2σ 0 γ 4 ε q < 1 2 , (4.8)
there is a sequence {v n } n≥0 of functions v n : [0, ε 1 ) × Λ → X � s1+ν , which are C 1 with respect to λ, with the following properties:

• (P 1 ) n (v n (ε, λ) -ṽ(ε, λ)) ∈ E � n , v n (0, λ) = 0, �v n -ṽ� s1 ≤ N -σ 0 , �∂ λ (v n -ṽ)� s1 ≤ N -σ/2 0 ; • (P 2 ) n F or 1 ≤ k ≤ n, �v k -v k-1 � s1 ≤ N -σ-1 k , �∂ λ (v k -v k-1 )� s1 ≤ N -σ-1 k ; • (P 3 ) n Let A n := ∩ n k=0 G N k γ,µ,K (v k-1 ) with v -1 := ṽ and Λ ε,n := {λ ∈ Λ : (ε, λ) ∈ A n }. For all ε ∈ [0, ε 1 ), if λ ∈ N (Λ ε,n , γN -σ/2 n ) then v n (ε, λ) solves the equation (F n ) Π n F (ε, λ, v) = 0; • (P 4 ) n B n := 1 + �v n � s, B � n := 1 + �∂ λ v n � s satisfy (i) B n ≤ N µ+ν χ 0 -1 n+1 N δs 0 (ii) B � n ≤ N 2(µ+ν)+σ χ 0 -1 n+1 N δs 0 .
For all ε ∈ [0, ε 1 ), the sequence {v n (ε, •)} n≥0 converges uniformly in C 1 (Λ, X � s1+ν ) (endowed with the sup-norm of the map and its partial derivatives) to v with v(0, λ) = 0 and

(ε, λ) ∈ A ∞ := ∩ n≥0 A n ⇒ F (ε, λ, v(ε, λ)) = 0. Moreover, v(ε, λ) is in X � s for any s, i.e. v(ε, λ) ∈ X � .
We will prove the Nash-Moser Theorem 4.1.1 by induction. What follows is devoted to the construction of the sequence {v k }.

Throughout this construction we shall take N 0 := N 0 (γ, ε, s 1 , s) going to infinity as ε tends to zero, but satisfying (4.8).

Initialization in the iterative Nash-Moser scheme

We want to find a function g ∈ E � 0 such that v 0 := ṽ +g verifies (P 1 ) 0 -(P 4 ) 0 . Let A 0 := G (N0) γ,µ,K (ṽ). By definition, the parameters (ε, λ) are in A 0 if and only if (ε, λ, ṽ(ε, λ)) ∈ J

(N0) γ,µ,K . For all ε ∈ [0, ε 0 ), let us define Λ ε,0 := {λ : (ε, λ) ∈ A 0 }. Now, ∀ λ ∈ N (Λ ε,0 , 2γN -σ/2 0
), by (4.2) for which σ > 2(µ+1) and by hypothesis (L1), the operator L (N0) (ε, λ, ṽ) is invertible and using ( Ṽ 1), we have

�(L (N0) (ε, λ, ṽ)) -1 w� s1 ≤ C(s 1 ) N µ+δs1 0 γ 2 |w| s1 (4.9)
and

�(L (N0) (ε, λ, ṽ)) -1 w� s ≤ 2K(s) N µ 0 γ 2 (|w| s + (N δs 0 + �ṽ� s)|w| s1 ) ≤ C(s) N µ 0 γ 2 (|w| s + N δs 0 |w| s1 ). (4.10)
Let us introduce

L 0 := L (N0) (ε, λ, ṽ), r -1 := Π 0 F (ε, λ, ṽ) R -1 (g) := Π 0 (F (ε, λ, ṽ + g) -F (ε, λ, ṽ) -D v F (ε, λ, ṽ)[g]).
A fixed point of

G 0 : E � 0 → E � 0 G 0 (g) := -L -1 0 (r -1 + R -1 (g))
is a solution of equation

Π 0 F (ε, λ, ṽ + g) = 0.
B 0 := � g ∈ E � 0 : �g� s1 ≤ ρ 0 := ε q γ -2 C 0 N µ+δs1 0 �
defined for some appropriate C 0 = C 0 (s 1 ) and endowed with the norm � • � s1 .

Chapter 4. An abstract Nash-Moser Theorem

Proof. By (4.9), ∀ �g� s1 ≤ ρ

0 := ε q γ -2 C 0 N µ+δs1 0 , �G 0 (g)� s1 ≤ C(s 1 )γ -2 N µ+δs1 0 (|r -1 | s1 + |R -1 (g)| s1 ) (4.11) with |r -1 | s1 = |Π 0 F (ε, λ, ṽ)| s1 ≤ |F (ε, λ, ṽ)| s1 ≤ ( Ṽ 2) C(s 1 )ε q (4.12)
and

|R -1 (g)| s1 = |Π 0 (F (ε, λ, ṽ + g) -F (ε, λ, ṽ) -D v F (ε, λ, ṽ)[g])| s1 ≤ |(F (ε, λ, ṽ + g) -F (ε, λ, ṽ) -D v F (ε, λ, ṽ)[g])| s1 ≤ (F 7),( Ṽ 1) C(s 1 )�g� s1 �g� s1+ν ≤ (S1) C(s 1 )N ν 0 �g� 2 s1 . (4.13)
Take C 0 := 4C(s 1 ), where C(s 1 ) is the constant of (4.11), for which

C(s 1 )γ -2 ε q N µ+δs1 0 ≤ ε q γ -2 C 0 2 N µ+δs1 0 = ρ 0 2 .
For ε small enough such that by (4.8)

C(s 1 )γ -2 N µ+δs1+ν 0 ρ 0 = C(s 1 )γ -2 N µ+δs1+ν 0 ε q γ -2 4C(s 1 )N µ+δs1 0 ≤ C(s 1 ) 1 2 N -2σ 0 << 1 2 , (4.14)
we obtain

�G 0 (g)� s1 ≤ C(s 1 )γ -2 N µ+δs1 0 (ε q + N ν 0 �g� 2 s1 ) ≤ C(s 1 )γ -2 N µ+δs1 0 (ε q + N ν 0 ρ 2 0 ) ≤ ρ 0 2 + ρ 0 2 = ρ 0 .
In the same way, if ε is small enough, we have by (F 3), ∀ g ∈ B 0 ,

�DG 0 (g)[w]� s1 ≤ �w� s1 2 .
Hence, for λ ∈ N (Λ ε,0 , 2γN

-σ/2 0 ), G 0 is a contraction on (B 0 , � • � s1 ).
Note that, by (4.14), we have the estimate

ρ 0 ≤ C(s 1 )γ 2 N -(µ+δs1+ν)-2σ 0 (4.15)
which we will often use in the following.

Chapter 4. An abstract Nash-Moser Theorem

The only difference between the proofs in this first step and those of the iteration step is that the term r -1 is small thanks to the smallness of ε (see estimate (4.12)).

Since G 0 is a contraction in the complete metric space (B 0 , � • � s1 ), it has a unique fixed point in this set.

For all ε ∈ [0, ε 1 ) and for all λ ∈ N (Λ ε,0 , 2γN

-σ/2 0
), let g0 (ε, λ) ∈ B 0 denote this unique fixed point which is solution of

Π 0 F (ε, λ, ṽ + g) = 0.
By ( Ṽ 1), ṽ(0, λ) = 0 and by (F 1), if (0, λ) ∈ N (A 0 , 2γN

-σ/2 0
) the uniqueness gives us g0 (0, λ) = 0.

For λ ∈ N (Λ ε,0 , 2γN -σ/2 0
) with σ > 2(µ + 1) and �g 0 � s1 ≤ ρ 0 , using (4.15), hypothesis (L1) give us that L (N0) (ε, λ, ṽ + g0 ) is invertible with, using Ṽ ( 1)

�(L (N0) (ε, λ, ṽ + g0 )) -1 w� s1 ≤ C(s 1 ) N µ+δs1 0 γ 2 |w| s1 (4.16)
and

�(L (N0) (ε, λ, ṽ + g0 )) -1 w� s ≤ C(s) N µ 0 γ 2 (|w| s + (N δs 0 + �g 0 � s)|w| s1 ). (4.17)
Hence, by the Implicit Function Theorem, g0 ∈ C 1 (N (Λ ε,0 , 2γN

-σ/2 0
); B 0 ) and

∂ λ g0 = -(L (N0) (ε, λ, ṽ + g0 )) -1 [Π 0 (∂ λ F )(ε, λ, ṽ + g0 ) + Π 0 D v F (ε, λ, ṽ + g0 )[∂ λ ṽ]]. (4.18)
Call

H(g 0 ) := Π 0 (∂ λ F )(ε, λ, ṽ + g0 ) + Π 0 D v F (ε, λ, ṽ + g0 )[∂ λ ṽ], (4.19)
we write

H(g 0 ) = Π 0 (∂ λ F )(ε, λ, ṽ) + Π 0 D v F (ε, λ, ṽ)[∂ λ ṽ] + Π 0 (∂ λ F )(ε, λ, ṽ + g0 ) -Π 0 (∂ λ F )(ε, λ, ṽ) + Π 0 D v F (ε, λ, ṽ + g0 )[∂ λ ṽ] -Π 0 D v F (ε, λ, ṽ)[∂ λ ṽ] =: H 1 + H 2 (g 0 ) + H 3 (g 0 ). (4.20)
We estimate by ( Ṽ 4)

|H 1 | s1 = � � �Π0(∂λF )(ε, λ, ṽ) + Π 0 D v F (ε, λ, ṽ)[∂ λ ṽ] � � � s1 = � � �∂λΠ0 � F (ε, λ, ṽ(ε, λ)) � � � � s1 ≤ C(s 1 )ε q , (4.21)
Chapter 4. An abstract Nash-Moser Theorem by (F 4), ( Ṽ 1),

|H 2 (g 0 )| s1 = � � � � 1 0 Π 0 D v (∂ λ F )(ε, λ, ṽ + tg 0 )[g 0 ] dt � � � s1 ≤ � 1 0 � � �Π0Dv(∂λF )(ε, λ, ṽ + tg 0 )[g 0 ] � � � s1 dt ≤ � 1 0 � � �(∂λDvF )(ε, λ, ṽ + tg 0 )[g 0 ] � � � s1 dt ≤ C(s 1 )(�g 0 � s1+ν + (�ṽ� s1+ν + �g 0 � s1+ν )�g 0 � s1 ) ≤ C(s 1 )(N ν 0 ρ 0 + N ν 0 ρ 2 0 ) ≤ C(s 1 )N ν 0 ρ 0 (4.22)
and by ( Ṽ 1), ( Ṽ 3), (F 3)

|H 3 (g 0 )| s1 = � � � � 1 0 Π 0 D 2 v F (ε, λ, ṽ + tg 0 )[g 0 , ∂ λ ṽ] dt � � � s1 ≤ � 1 0 � � �D 2 v F (ε, λ, ṽ + tg 0 )[g 0 , ∂ λ ṽ] � � � s1 dt ≤ C(s 1 ) � (�ṽ� s1+ν + �g 0 � s1+ν )�g 0 � s1 �∂ λ ṽ� s1 + �g 0 � s1+ν �∂ λ ṽ� s1 + �g 0 � s1 �∂ λ ṽ� s1+ν � ≤ C(s 1 )(N ν 0 ρ 2 0 ε 1 4 + N ν 0 ρ 0 ε 1 4 ) ≤ C(s 1 )N ν 0 ρ 0 ε 1 4 (4.23)
where we have used ρ 2 o ≤ ρ 0 by (4.15). By estimates (4.18), (4.16), (4.21), (4.22), (4.23) and for N 0 = N 0 (γ) large enough we have

�∂ λ g0 � s1 ≤ C(s 1 ) N µ+δs1 0 γ 2 |H(g 0 )| s1 ≤ C(s 1 ) N µ+δs1 0 γ 2 (|H 1 | s1 + |H 2 (g 0 )| s1 + |H 3 (g 0 )| s1 ) ≤ C(s 1 ) N µ+δs1 0 γ 2 (ε q + N ν 0 ρ 0 ) ≤ C(s 1 ) � N µ+δs1 0 γ 2 ε q + N -2σ 0 � ≤ C(s 1 ) � N 2(µ+δs1)+ν+2σ 0 γ 4 N -2σ 0 ε q + N -2σ 0 � ≤ C(s 1 )N -2σ 0 (4.24)
where in the last inequalities we have used also (4.8) and (4.15).

We define the map g 0 := ψ 0 g0 : [0, ε 1 ) × Λ → E � 0 where the cut-off ) and 0 outside N (Λ ε,0 , 2γN -σ/2 0

), and

|∂ λ ψ 0 | ≤ CN σ/2 0 γ -1 .
The function g 0 is C 1 with respect to parameter λ.

Let us define v 0 := ṽ + g 0 .

Lemma 4.2.2. For ε ∈ [0, ε 1 ) and N 0 = N 0 (ε, γ, s 1 , s) satisfying (4.8), the map v 0 verifies properties (P 1 ) 0 , (P 2 ) 0 , (P 3 ) 0 and (P 4 ) 0 .

Proof. Property (P 3 ) 0 is satisfied by the definition of v 0 .

Using ṽ(0, λ) = 0 and g0 (0, λ) = 0, we have v 0 (0, λ) = 0.

Moreover, by (4.15) we have

�v 0 -ṽ� s1 = �g 0 � s1 = �ψ 0 g0 � s1 ≤ |ψ 0 |�g 0 � s1 ≤ ρ 0 ≤ N -2σ
0 and using also (4.24) and N 0 = N 0 (ε, γ, s 1 )

�∂ λ (v 0 -ṽ)� s1 = �∂ λ g 0 � s1 ≤ �∂ λ (ψ 0 g0 )� s1 ≤ �∂ λ (ψ 0 )g 0 + ψ 0 ∂ λ (g 0 )� s1 ≤ Cγ -1 N σ/2 0 � g0 � s1 + C(s 1 )N -2σ 0 ≤ N -σ-1 0
which imply properties (P 1 ) 0 and (P 2 ) 0 (which in this case coincide).

There remains to show (P 4 ) 0 .

By (4.10), for ε ∈ (0, ε 1 ) and for all λ ∈ N (Λ ε,0 , γN -σ/2 0

), we estimate

�g 0 � s = �G 0 (g 0 )� s ≤ C(s)N µ 0 γ -2 � |r -1 | s + |R -1 (g 0 )| s + N δs 0 (|r -1 | s1 + |R -1 (g 0 )| s1 ) �
where, by (F 6), ( Ṽ 1)

|r -1 | s = |Π 0 F (ε, λ, ṽ)| s ≤ |F (ε, λ, ṽ)| s ≤ C(s)(1 + �ṽ� s+ν ) ≤ C(s)(1 + √ ε)
and proceeding as in (4.13), using (F 7),

|R -1 (g 0 )| s ≤ C(s)(�ṽ� s+ν �g� 2 s1 + N ν 0 �g 0 � s1 �g 0 � s) ≤ C(s)N ν 0 �g 0 � s1 �g 0 � s.
123 Chapter 4. An abstract Nash-Moser Theorem Then, by (4.12), (4.13), (4.15), we have

�g 0 � s ≤ C(s)N µ 0 γ -2 � 1 + √ ε + N ν 0 �g 0 � s1 �g 0 � s + N δs 0 (ε q + N ν 0 �g 0 � 2 s1 ) � ≤ C(s)N µ 0 γ -2 � 1 + N ν 0 ρ 0 �g 0 � s + N δs 0 ε q + N δs+ν 0 ρ 2 0 � ≤ C(s)γ -2 N µ+δs 0 + 1 2 �g 0 � s taking N 0 = N 0 (ε, s 1 ) large enough (i.e. ε small enough) such that N µ 0 γ -2 N ν 0 ρ 0 ≤ N µ+ν 0 γ -2 C(s 1 )γ 2 N -(µ+δs1+ν)-2σ 0 ≤ C(s 1 )N -δs1-2σ 0 ≤ 1 2 .
Hence

�g 0 � s ≤ C(s, γ)N µ+δs 0 . (4.25)
We want to estimate now �∂ λ g0 � s.

Recall (4.18)

∂ λ g0 = -(L (N0) (ε, λ, ṽ + g0 )) -1 [Π 0 (∂ λ F )(ε, λ, ṽ + g0 ) + Π 0 D v F (ε, λ, ṽ + g0 )[∂ λ ṽ]],
using (4.25), we estimate

|Π 0 (∂ λ F )(ε, λ, ṽ + g0 )| s ≤ (F 2) C(s)(1 + �ṽ + g0 � s+ν ) ≤ C(s)(1 + �ṽ� s+ν + �g 0 )� s+ν ) ≤ ( Ṽ 1) C(s)(1 + √ ε + N ν 0 �g 0 � s) ≤ C(s)N µ+δs+ν 0 (4.26)
and

|Π 0 D v F (ε, λ, ṽ + g0 )[∂ λ ṽ]| s ≤ (F 5) C(s)(�ṽ + g0 � s+ν �∂ λ ṽ� s1 + �∂ λ ṽ� s+ν ) ≤ C(s)(�ṽ)� s+ν �∂ λ ṽ� s1 + �g 0 � s+ν �∂ λ ṽ� s1 + �∂ λ ṽ� s+ν ) ≤ ( Ṽ 1),( Ṽ 3) C(s)N µ+δs+ν 0 . ( 4
|Π 0 (∂ λ F )(ε, λ, ṽ+g 0 )+Π 0 D v F (ε, λ, ṽ+g 0 )[∂ λ ṽ]| s1 ≤ C(s 1 )(ε q +N ν 0 ρ 0 ). (4.28)
Hence, by (4.17), (4.26), (4.27) and (4.28) we have

�∂ λ g0 � s = �(L (N0) (ε, λ, ṽ + g0 )) -1 [Π 0 (∂ λ F )(ε, λ, ṽ + g0 ) + Π 0 D v F (ε, λ, ṽ + g0 )[∂ λ ṽ]]� s ≤ C(s) N µ 0 γ 2 � |Π 0 (∂ λ F )(ε, λ, ṽ + g0 ) + Π 0 D v F (ε, λ, ṽ + g0 )[∂ λ ṽ]| s + (N δs 0 + �g 0 � s)|Π 0 (∂ λ F )(ε, λ, ṽ + g0 ) + Π 0 D v F (ε, λ, ṽ + g0 )[∂ λ ṽ]| s1 � ≤ C(s) N µ 0 γ 2 � N µ+δs+ν 0 + (N δs 0 + �g 0 � s)(ε q + N ν 0 ρ 0 ) � ≤ (4.25) C(s, γ)N 2µ+δs+ν 0 .
Then, we have

�v 0 � s ≤ �ṽ� s + �ψ 0 g0 � s ≤ �ṽ� s + |ψ 0 |�g 0 � s ≤ (4.25) C(s) √ ε + C(s, γ)N µ+δs 0 ≤ C(s, γ)N µ+δs 0
and using also σ > 4(µ + δs 1 + ν),

�∂ λ v 0 � s ≤ �∂ λ ṽ� s + �∂ λ (ψ 0 g0 )� s ≤ �∂ λ ṽ� s + �∂ λ (ψ 0 )g 0 + ψ 0 ∂ λ (g 0 )� s ≤ C(s)ε 1 4 + Cγ -1 N σ/2 0 � g0 � s + C(s, γ)N 2µ+δs+ν 0 ≤ C(s, γ)N µ+δs+σ/2 0 . Define now B 0 := 1 + �v 0 � s and B � 0 := 1 + �∂ λ v 0 � s. For N 1 := N χ0 0 , χ 0 > 1 which implies χ 0 χ 0 -1 > 1
and provided N 0 ≥ N 0 (γ, s) is large enough, we have

B 0 ≤ C(s, γ)N µ+δs 0 ≤ N µ+ν 0 N δs 0 ≤ N χ 0 χ 0 -1 (µ+ν) 0 N δs 0 ≤ N µ+ν χ 0 -1 1 N δs 0 and B � 0 ≤ C(s, γ)N µ+δs+σ/2 0 ≤ N 2(µ+ν)+σ 0 N δs 0 ≤ N χ 0 χ 0 -1 (2(µ+ν)+σ) 0 N δs 0 ≤ N 2(µ+ν)+σ χ 0 -1 1 N δs 0 . (4.29)
Chapter 4. An abstract Nash-Moser Theorem

Iteration in the Nash-Moser scheme

Assume for all ε ∈ [0, ε 1 ) that v n (ε, •) ∈ C 1 (Λ, E � n ) satisfies the properties (P 1 ) n -(P 4 ) n . We want to find now g n+1 ∈ E � n+1 such that

v n+1 := v n + g n+1 verifies (P 1 ) n+1 -(P 4 ) n+1 . We write ∀ g ∈ E � n+1 Π n+1 F (ε, λ, v n (ε, λ) + g) = r n + L n+1 [g] + R n (g)
where

r n := Π n+1 F (ε, λ, v n ), L n+1 := L n+1 (ε, λ) := L (Nn+1) (ε, λ, v n ), R n (g) := Π n+1 (F (ε, λ, v n + g) -F (ε, λ, v n ) -D v F (ε, λ, v n )[g]).
The "quadratic" term R n (g) is estimated, by (F 7), as

|R n (g)| s ≤ C(s)(�v n � s+ν �g� 2 s1 + �g� s+ν �g� s1 ). (4.30)
By (P 3 ) n , for all ε ∈ [0, ε 1 ), if λ ∈ N (Λ ε,n , γN -σ/2 n
) then v n solves equation (F n ) and so

r n := Π n+1 F (ε, λ, v n ) = Π n+1 F (ε, λ, v n ) -Π n F (ε, λ, v n ) = Π n+1 (I -Π n )(F (ε, λ, v n )). (4.31)
By the definition of σ and J n γ,µ,K , the operator L n+1 (ε, λ) is invertible on the set

A n+1 = A n ∩ G (Nn+1) γ,µ,K (v n ). If A n+1 = ∅, we define v k = v n , ∀ k > n.
Otherwise we continue the iteration.

Note that for all ε ∈ [0, ε 1 ), by N i := N χ i 0 0 , for N 0 large enough, we have the inclusion

N (Λ ε,n+1 , 2γN -σ/2 n+1 ) ⊂ N (Λ ε,n , γN -σ/2 n ). (4.32) For all λ ∈ N (Λ ε,n+1 , 2γN -σ/2 n+1 ), there is λ � ∈ Λ ε,n+1 (i.e. (ε, λ � , v n (ε, λ � )) ∈ J (Nn+1) γ,µ,K ) such that �λ -λ � � Y ≤ 2γN -σ/2
n+1 and then for N 0 = N 0 (γ, s 1 ) large enough

�λ � -λ� Y + �v n (ε, λ � ) -v n (ε, λ)� s1 ≤ (P1)n 2γN -σ/2 n+1 + N -σ/2 0 �λ -λ � � Y ≤ 2γN -(µ+1) n+1 .
Hence, hypothesis (L1) implies that for all ε ∈ (0, ε 1 ) and for all λ ∈ N (Λ ε,n+1 , 2γN

-σ/2 n+1 ), the operator L n+1 (ε, λ) is invertible with �L -1 n+1 [w]� s1 ≤ C(s 1 )γ -2 N µ+δs1 n+1 |w| s1 (4.33)
and

�L -1 n+1 [w]� s ≤ C(s) N µ n+1 γ 2 � |w| s + (N δs n+1 + �v n � s)|w| s1 � ≤ C(s) N µ n+1 γ 2 � |w| s + (N δs n+1 + B n )|w| s1 � . (4.34)
Defining for all ε ∈ (0, ε 1 ) and for all λ ∈ N (Λ ε,n+1 , 2γN

-σ/2 n+1 ) the map G n+1 : E � n+1 → E � n+1 , G n+1 (g) := -L -1 n+1 [r n + R n (g)], (4.35)
equation (F n+1 ) is equivalent to the fixed point problem g = G n+1 (g). Before

we prove the next lemma, we note that ∀s ∈ [s 1 , s]

�v n � s+ν = �v n -ṽ + ṽ� s+ν ≤ �v n -ṽ� s+ν + �ṽ� s+ν ≤ (S1) N ν n (�v n � s + �ṽ� s ) + �ṽ� s+ν ≤ ( Ṽ 1) C(s)N ν n (�v n � s + 1) (4.36)
where we remark that we have used the property (S 1 ) because

v n -ṽ ∈ E � n . Lemma 4.3.1. (Contraction, iterative step) For all ε ∈ [0, ε 1 ), let λ ∈ N (Λ ε,n+1 , 2γN -σ/2 n+1
). For N 0 ≥ N 0 (γ, s 1 , s) large enough, G n+1 is a contraction in the complete metric space

B n+1 := {g ∈ E � n+1 : �g� s1 ≤ ρ n+1 := N -2σ-1 n+1 } endowed with the norm � • � s1 .
Proof. For all ε ∈ [0, ε 1 ) and λ ∈ N (Λ ε,n+1 , 2γN -σ/2 n+1 ), by (4.33) and by (4.35), we have

�G n+1 (g)� s1 ≤ C(s 1 )γ -2 N µ+δs1 n+1 (|r n | s1 + |R n (g)| s1 ) (4.37)
and r n has the form (4.31) because of (4.32). Now, using (4.36), we estimate

|r n | s1 + |R n (g)| s1 ≤ (S2),(4.30) C(s) � N -(s-s1) n |F (ε, λ, v n )| s+ �v n � s1+ν �g� 2 s1 + N ν n+1 �g� 2 s1 � ≤ (F 6),( Ṽ 1) C(s) � N -(s-s1) n (1 + �v n � s+ν ) + N ν n+1 �g� 2 s1 � ≤ (P1)n C(s) � N -(s-s1)+ν n B n + N ν n+1 �g� 2 s1 � (4.38) from which if �g� s1 ≤ ρ n+1 := N -2σ-1 n+1 |r n | s1 + |R n (g)| s1 ≤ C(s)ρ n+1 � N -(s-s1)+ν n N 2σ+1 n+1 B n + N ν n+1 ρ n+1 � . (4.39)
Chapter 4. An abstract Nash-Moser Theorem For s defined in (4.2), we can easy estimate s ≥ s 1 + ν + χ 0 (2σ + 1) + χ0 χ0-1 (µ + ν) + χ 0 (µ + δs 1 + 1) 1δ , which implies

N -(s-s1)+ν n N 2σ+1 n+1 B n ≤ (P4)n N -(s-s1)+ν n N 2σ+1+ µ+ν χ 0 -1 + δs χ 0 n+1 ≤ N -(s-s1)+ν n N χ0(2σ+1)+ χ 0 (µ+ν) χ 0 -1 +δs n ≤ N χ0(-µ-δs1-1)
n and then

|r n | s1 + |R n (g)| s1 ≤ C(s)ρ n+1 � N χ0(-µ-δs1-1) n + N ν n+1 N -2σ-1 n+1 � ≤ ρ n+1 N -µ-δs1 n+1 γ 2 C(s 1 ) (4.40)
for N 0 (γ, s) large enough. Hence by (4.37), G n+1 (B n+1 ) ⊂ B n+1 .

Next, differentiating (4.35) with respect to g we get, ∀ g ∈ B n+1 ,

D g G n+1 (g)[w] = -L -1 n+1 [Π n+1 (D v F (ε, λ, v n + g)[w] -D v F (ε, λ, v n )[w])]
and by (4.33)

�D g G n+1 (g)[w]� s1 ≤ C(s 1 )γ -2 N µ+δs1 n+1 |Π n+1 (D v F (ε, λ, v n + g)[w] -D v F (ε, λ, v n )[w])| s1 ≤ (F 3) C(s 1 )γ -2 N µ+δs1 n+1 (�v n � s1+ν �g� s1 �w� s1 + �g� s1+ν �w� s1 + �g� s1 �w� s1+ν ) ≤ (P1)n C(s 1 , s)γ -2 N µ+δs1+ν n+1 ρ n+1 �w� s1 ≤ C(s 1 , s)γ -2 N µ+δs1+ν-2σ-1 n+1 �w� s1 ≤ C(s 1 , s)γ -2 N -1 n+1 �w� s1 ≤ �w� s1 2 for N 0 (γ, s 1 , s) large enough. Hence G n+1 is a contraction in (B n+1 , � • � s1 ). For all ε ∈ [0, ε 1 ) and λ ∈ N (Λ ε,n+1 , 2γN -σ/2 n+1 ), let gn+1 := gn+1 (ε, λ) ∈ E �
n+1 be the unique fixed point of G n+1 in the complete metric space B n+1 . We already have that

�g n+1 � s1 ≤ ρ n+1 = N -2σ-1 n+1 . Moreover, since gn+1 = G n+1 (g n+1 ), G n+1 (0) = -L -1 n+1 [r n ], Lemma 4.3.1 give us �G n+1 (g n+1 ) -G n+1 (0)� s1 ≤ 1 2 �g n+1 -0� s1 ⇐⇒ �g n+1 + L -1 n+1 [r n ]� s1 ≤ 1 2 �g n+1 � s1
from which, using (4.31), (4.33) and (4.36)

�g n+1 � s1 ≤ �L -1 n+1 [r n ]� s1 ≤ C(s 1 )γ -2 N µ+δs1 n+1 |r n | s1 ≤ C(s 1 )γ -2 N µ+δs1 n+1 |Π n+1 (I -Π n )F (ε, λ, v n )| s1 ≤ C(s)γ -2 N µ+δs1 n+1 N -(s-s1) n (1 + �v n � s+ν ) ≤ C(s)γ -2 N µ+δs1 n+1 N -(s-s1) n N ν n B n . (4.41)
Estimate (4.41) will be useful in the following because the index s appears with a negative sign in the power of N n , unlike to the estimate �g n+1 � s1 ≤ ρ n+1 where s does not appear.

Since gn+1 solves

U n+1 (g) := Π n+1 F (ε, λ, v n (ε, λ) + g) = 0 (4.
�g n+1 � s ≤ C(s)γ -2 N µ+ ν χ 0 n+1 B n . (4.44)
Proof. By gn+1 = G n+1 (g n+1 ) and (4.34), we write

�g n+1 � s ≤ C(s) N µ n+1 γ 2 � |r n | s + |R n (g n+1 )| s + (N δs n+1 +B n )(|r n | s1 + |R n (g n+1 )| s1 ) � .
By (F 6) and (4.36)

|r n | s ≤ C(s)(1 + �v n � s+ν ) ≤ C(s)(1 + N ν n B n ) ≤ C(s)N ν χ 0 n+1 B n
and by (4.30), (S 1 ), recall

�g n+1 � s1 ≤ ρ n+1 := N -2σ-1 n+1 and σ > 4(µ + δs 1 + ν), |R n (g n+1 )| s ≤ C(s)(�v n � s+ν �g n+1 � 2 s1 + N ν n+1 �g n+1 � s�g n+1 � s1 ) ≤ C(s)(N ν n B n �g n+1 � 2 s1 + N ν n+1 �g n+1 � s�g n+1 � s1 ) ≤ C(s)(N ν χ 0 n+1 N 2(-2σ-1) n+1 B n + N ν n+1 N -2σ-1 n+1 �g n+1 � s) ≤ C(s)(N ν χ 0 n+1 B n + N -2σ-1+ν n+1 �g n+1 � s).
�g n+1 � s ≤ C(s)γ -2 N µ n+1 � N ν χ 0 n+1 B n + N -2σ-1+ν n+1 �g n+1 � s + (N δs n+1 (N -(s-s1)+ν n B n + N ν n+1 �g n+1 � 2 s1 ) + B n � ≤ C(s)γ -2 N µ n+1 � N ν χ 0 n+1 B n + N -2σ-1+ν n+1 �g n+1 � s + N δs n+1 N -(s-s1)+ν n B n + N δs+ν n+1 �g n+1 � 2 s1 + B n � . (4.47) Using χ 0 > 1, χ 0 δ < 1, σ > 4(µ + δs 1 + ν) and s in (4.3) from which s > s 1 + ν 1 -χ 0 δ , we have N δs n+1 N -(s-s1)+ν n ≤ N χ0δs-(s-s1)+ν n << 1. (4.48) By �g n+1 � s1 ≤ ρ n+1 ≤ N -2σ-1 n+1
, we have

�g n+1 � 2 s1 ≤ �g n+1 � s1 (4.49)
and using the bound (4.41), we obtain

N δs+ν n+1 �g n+1 � 2 s1 ≤ N δs+ν n+1 �g n+1 � s1 ≤ C(s)γ -2 N δs+ν n+1 N µ+δs1 n+1 N -(s-s1)+ν n B n ≤ C(s)γ -2 N χ0δs+χ0ν n N χ0(µ+δs1) n N -(s-s1)+ν n B n ≤ C(s)B n (4.50)
if we use as usually χ 0 δ < 1 and s in (4.3).

Hence, by (4.47), (4.48) and (4.50) we conclude

�g n+1 � s ≤ C(s)γ -2 N µ n+1 N ν χ 0 n+1 B n + 1 2 �g n+1 � s (4.51)
taking N 0 (γ, s) large enough such that

C(s)γ 2 N µ n+1 N -2σ-1+ν n+1 < 1 2 .
Estimate (4.44) follows from (4.51).

Chapter 4. An abstract Nash-Moser Theorem Lemma 4.3.3. (Estimates of the derivatives) For all ε ∈ [0, ε 1 ), the map

gn+1 (ε, •) is in C 1 (N (Λ ε,n+1 , 2γN -σ/2
n+1 ); B n+1 ) and we have

(i) �∂ λ gn+1 � s1 ≤ 1 2 N -σ-1 n+1 , (ii) �∂ λ gn+1 � s ≤ C(s, γ)N 2(µ+ν) n+1 (B � n + B n ).
Proof. Recall that

U n+1 (ε, λ, gn+1 ) = Π n+1 F (ε, λ, v n (ε, λ) + gn+1 ) = 0.
Arguing as in the proof of estimates (4.16) and (4.17), if for all ε ∈ [0, ε 1 ),

λ ∈ N (Λ ε,n+1 , 2γN -σ/2
n+1 ) with σ > 2(µ + 1) and

�g n+1 � s1 ≤ ρ n+1 = N -2σ-1 n+1
, hypothesis (L1) gives us that the partial derivative

D g U n+1 (ε, λ, gn+1 ) = L (Nn+1) (ε, λ, v n (ε, λ) + gn+1 ) is invertible with ∀ w ∈ E n+1 �(D g U n+1 (ε, λ, gn+1 )) -1 [w]� s1 ≤ C(s 1 )γ -2 N µ+δs1 n+1 |w| s1 , (4.52)
and using (4.44)

�(D g U n+1 (ε, λ, gn+1 )) -1 [w]� s ≤ C(s) N µ n+1 γ 2 � |w| s + (N δs n+1 + �v n + gn+1 � s)|w| s1 � ≤ C(s) N µ n+1 γ 2 � |w| s + (N δs n+1 + B n + γ -2 N µ+ ν χ 0 n+1 B n )|w| s1 � ≤ C(s, γ)N µ n+1 � |w| s + (N δs n+1 + N µ+ ν χ 0 n+1 B n )|w| s1 � . (4.53)
Then for all ε ∈ [0, ε 1 ), by the Implicit Function Theorem,

gn+1 (ε, •) ∈ C 1 (N (Λ ε,n+1 , 2γN -σ/2 n+1 ); B n+1 )
and

∂ λ gn+1 = - � (D g U n+1 (ε, λ, gn+1 ) � -1 ∂ λ U n+1 (ε, λ, gn+1 ). (4.54)
Now, using that for all ε ∈ [0, ε 1 ) and for all λ ∈ N (Λ ε,n , 2γN

-σ/2 n ), v n (ε, λ)
solves (F n ), we get by the inclusion (4.32)

∂ λ U n+1 (ε, λ, g) = Π n+1 ((∂ λ F )(ε, λ, v n (ε, λ) + g) + D v F (ε, λ, v n + g)[∂ λ v n ]) = Π n+1 ((∂ λ F )(ε, λ, v n (ε, λ) + g) + D v F (ε, λ, v n + g)[∂ λ v n ]) -Π n ((∂ λ F )(ε, λ, v n (ε, λ)) -Π n D v F (ε, λ, v n )[∂ λ v n ]) = Π n+1 ((∂ λ F )(ε, λ, v n (z) + g) -(∂ λ F )(ε, λ, v n (ε, λ))) + Π n+1 (D v F (ε, λ, v n (ε, λ) + g)[∂ λ v n ] -D v F (ε, λ, v n )[∂ λ v n ]) + Π n+1 (I -Π n )((∂ λ F )(ε, λ, v n (ε, λ)) + D v F (ε, λ, v n )[∂ λ v n ]) =: A(ε, λ, g) + B(ε, λ, g) + C(ε, λ).

Chapter 4. An abstract Nash-Moser Theorem

Using (F 3), (F 4), (P 1 ) n and (4.49), we get

|A(ε, λ, gn+1 )| s1 ≤ � � � � 1 0 D v ∂ λ F (ε, λ, v n + tg n+1 )[g n+1 ] dt � � � s1 ≤ � 1 0 � � �Dv∂λF (ε, λ, v n + tg n+1 )[g n+1 ] � � � s1 dt ≤ (F 4) C(s 1 )(�g n+1 � s1+ν + (�v n � s1+ν + �g n+1 � s1+ν )�g n+1 � s1 ) ≤ (P1)n C(s 1 )N ν n+1 �g n+1 � s1 , (4.55) |B(ε, λ, gn+1 )| s1 ≤ � � � � 1 0 D 2 v F (ε, λ, v n + tg n+1 )[g n+1 , ∂ λ v n ] dt � � � s1 ≤ � 1 0 � � �D 2 v F (ε, λ, v n + tg n+1 )[g n+1 , ∂ λ v n ] � � � s1 dt ≤ (F 3) C(s 1 ) � (�v n � s1+ν + �g n+1 � s1+ν )�g n+1 � s1 �∂ λ v n � s1 + �g n+1 � s1+ν �∂ λ v n � s1 + �g n+1 � s1 �∂ λ v n � s1+ν � ≤ (P1)n C(s 1 )N ν n+1 �g n+1 � s1 , (4.56)
and by (S 2 ), (F 2), (F 5), (P 1 ) n , (P 4 ) n and (4.36)

|C(ε, λ)| s1 ≤ (S2) N -(s-s1) n � � �(∂λF )(ε, λ, v n (ε, λ)) + D v F (ε, λ, v n )[∂ λ v n ] � � � s ≤ (F 2) N -(s-s1) n � C(s)(1 + �v n � s+ν ) + |D v F (ε, λ, v n )[∂ λ v n ]| s� ≤ (F 5) C(s)N -(s-s1) n (�v n � s+ν + �v n � s+ν �∂ λ v n � s1 + �∂ λ v n � s+ν ) ≤ (4.36) C(s)N -(s-s1) n (N ν n (�v n � s + 1) + N ν n (�v n � s + 1)�∂ λ v n � s1 + N ν n (�∂ λ v n � s + 1)) ≤ (P1)n C(s)N -(s-s1) n (N ν n B n + N ν n B n + N ν n B � n ) ≤ C(s)N -(s-s1)+ν n (B n + B � n ) (4.57)
where we have used also

�∂ λ v n � s+ν ≤ C(s)N ν n (�∂ λ v n � s + 1) (4.58)
which we can prove as (4.36).

Chapter 4. An abstract Nash-Moser Theorem

Then, by (4.52), we deduce estimate (i) for N 0 (γ, s) large enough:

�∂ λ gn+1 � s1 ≤ C(s 1 )γ -2 N µ+δs1 n+1 |∂ λ U n+1 (z, gn+1 )| s1 ≤ C(s 1 )γ -2 N µ+δs1 n+1 (|A(z, gn+1 )| s1 + |B(z, gn+1 )| s1 + |C(z)| s1 ) ≤ C(s 1 )γ -2 N µ+δs1 n+1 (N ν n+1 �g n+1 � s1 + N -(s-s1)+ν n (B n + B � n )) ≤ (P4)n C(s 1 )γ -2 N µ+δs1 n+1 � N ν n+1 N -2σ-1 n+1 + N -(s-s1)+ν n N 2(µ+ν)+σ χ 0 -1 n+1 N δs 0 � ≤ C(s 1 )γ -2 N µ+δs1 n+1 � N ν-2σ-1 n+1 + N -(s-s 1 )+ν χ 0 n+1 N 2(µ+ν)+σ χ 0 -1 + δs χ 0 n+1 � ≤ 1 2 N -σ-1 n+1
where in the last inequality we have used σ > 4(µ + δs 1 + ν) and s defined in

(4.3) from which s > χ 0 (µ + δs 1 ) + s 1 + ν + 2χ(µ+ν)+χ0σ χ0-1 + χ(σ + 1) 1 -δ that implies µ + δs 1 + -(s -s 1 ) + ν χ 0 + 2(µ + ν) + σ χ 0 -1 + δs χ 0 ≤ -σ -1.
To obtain (ii), we use (4.53) and (4.54) whence

�∂ λ gn+1 � s ≤ C(s, γ)N µ n+1 � |∂ λ U n+1 (z, gn+1 )| s + (N δs n+1 + N µ+ν n+1 B n )|∂ λ U n+1 (z, gn+1 )| s1
� and using (F 2), (F 5), (P 1 ) n , (4.36) and (4.58), we estimate

|∂ λ U n+1 (ε, λ, gn+1 )| s ≤ |Π n+1 ((∂ λ F )(ε, λ, v n + gn+1 ) + D v F (ε, λ, v n + gn+1 )[∂ λ v n ])| s ≤ |(∂ λ F )(ε, λ, v n + gn+1 )| s + |D v F (ε, λ, v n + gn+1 )[∂ λ v n ]| s ≤ C(s) � (1 + �v n � s+ν + �g n+1 � s+ν ) + (�v n � s+ν + �g n+1 � s+ν )�∂ λ v n � s1 + �∂ λ v n � s+ν � ≤ C(s) � N ν n+1 (B n + �g n+1 � s) + N ν n B � n � .
By (4.53), (4.54), (4.55), (4.56), (4.57) and (4.44), we write

�∂ λ gn+1 � s ≤ C(s, γ)N µ n+1 � N ν n+1 (B n + �g n+1 � s) + N ν n B � n + (N δs n+1 + N µ+ ν χ 0 n+1 B n )(N ν n+1 �g n+1 � s1 + N -(s-s1)+ν n (B n + B � n )) � ≤ C(s, γ)N µ n+1 � N ν n+1 (B n + N µ+ ν χ 0 n+1 B n ) + N ν n B � n + (N δs n+1 + N µ+ ν χ 0 n+1 B n )(N ν n+1 �g n+1 � s1 + N -(s-s1)+ν n (B n + B � n ))
� .

�g n+1 � s1 ≤ C(s)γ -2 N µ+δs1-(s-s1)+ν n+1
B n , we estimate

N δs n+1 N ν n+1 �g n+1 � s1 ≤ N δs n+1 N ν n+1 C(s)γ -2 N µ+δs1-(s-s1)+ν n+1 B n ≤ C(s)B n , N µ+ ν χ 0 n+1 B n N ν n+1 �g n+1 � s1 ≤ N µ+ ν χ 0 n+1 B n N ν n+1 N -2σ-1 n+1 ≤ B n , N δs+ν n+1 N -(s-s1)+ν n (B n + B � n ) ≤ N χ0δs n N -(s-s1)+ν n (B n + B � n ) ≤ B n + B � n and N µ+ ν χ 0 n+1 B n N -(s-s1)+ν n+1 (B n + B � n ) ≤ N µ+ ν χ 0 n+1 N µ+ν χ 0 -1 n+1 N δs 0 N -(s-s1)+ν n+1 (B n + B � n ) ≤ N µ+ ν χ 0 n+1 N µ+ν χ 0 -1 n+1 N χ0δs n+1 N -(s-s1)+ν n+1 (B n + B � n ) ≤ B n + B � n ,
Hence, we conclude

�∂ λ gn+1 � s ≤ C(s, γ)N 2(µ+ν) n+1 (B n + B � n ). Lemma 4.3.4. (Extension) For all ε ∈ [0, ε 1 ), there is g n+1 (ε, •) ∈ C 1 (Λ; B n+1 ) satisfying g n+1 (0, λ) = 0, �g n+1 � s1 ≤ N -2σ-1 n+1 , �∂ λ g n+1 � s1 ≤ N -σ-1 n+1
and that is equal to gn+1 (ε,

•) on N (Λ ε,n+1 , γN -σ/2 n+1 ).
Proof. For all ε ∈ [0, ε 1 ), we define

g n+1 (ε, λ) =    ψ n+1 gn+1 (ε, λ) if λ ∈ N (Λ ε,n+1 , 2γN -σ/2 n+1 ) 0 if λ � ∈ N (Λ ε,n+1 , 2γN -σ/2 n+1 ) (4.59)
where

ψ n+1 (ε, •) is a C ∞ cut-off function satisfying 0 ≤ ψ n+1 (ε, •) ≤ 1, ψ n+1 (ε, •) = 1 on N (Λ ε,n+1 , γN -σ/2 n+1 ), ψ n+1 (ε, •) = 0 outside N (Λ ε,n+1 , 2γN -σ/2
n+1 ) and

∂ λ ψ n+1 ≤ Cγ -1 N σ/2 n+1 .

Chapter 4. An abstract Nash-Moser Theorem

By definition, in N (Λ ε,n+1 , γN -σ/2 n+1 ), we have

�g n+1 � s1 = �g n+1 � s1 ≤ N -2σ-1 n+1
and by Lemma 4.3.3, for N 0 = N 0 (γ, s 1 ) large enough

�∂ λ g n+1 � s1 ≤ �(∂ λ ψ n+1 )g n+1 � s1 + �ψ n+1 ∂ z gn+1 � s1 ≤ Cγ -1 N σ/2 n+1 N -2σ-1 n+1 + 1 2 N -σ-1 n+1 ≤ N -σ-1 n+1 .
Finally, for all ε ∈ [0, ε 1 ), we define v n+1 (ε,

•) ∈ C 1 (Λ; E � n+1 ) as v n+1 := v n + g n+1 . By Lemma 4.3.4, on N (Λ ε,n+1 , γN -σ/2 n+1 ) we have g n+1 (ε, •) = gn+1 (ε, •)
which verifies U n+1 (ε, λ, g) = 0 and so v n+1 solves (F n+1 ). Hence property (P 3 ) n+1 holds. By Lemma 4.3.4 property (P 2 ) n+1 holds. Using the estimates computed in the initialitation of the scheme and (P 2 ) n+1 , for

N 0 = N 0 (ε, γ, s 1 , s) large enough �v n+1 -ṽ� s1 ≤ �v 0 -ṽ� s1 + n � k=0 �g k+1 � s1 ≤ N -2σ 0 + n � k=0 �g k+1 � s1 ≤ N -2σ 0 + n � k=0 N -2σ-1 k+1 ≤ N -2σ 0 + N -2σ 1 ≤ N -σ 0 and �∂ λ (v n+1 -ṽ)� s1 ≤ �∂ λ (v 0 -ṽ)� s1 + n � k=0 �∂ λ g k+1 � s1 ≤ N -σ-1 0 + n � k=0 N -σ-1 k+1 ≤ N -σ-1 0 + N -σ 1 ≤ N -σ/2 0 .
Moreover, still by Lemma 4.3.4 we have v n+1 (0, λ) = 0, and also property (P 1 ) n is verified. We conclude the induction of the Nash-Moser scheme with the following result. Lemma 4.3.5. For N 0 (γ) large, property (P 4 ) n+1 holds.

Proof. By definition of g n+1 and by (4.44), we have

�g n+1 � s ≤ C(s)γ -2 N µ+ ν χ 0 n+1 B n .
By (P 4 ) n , using that χ 0 > 1 which implies ν -ν χ0 > 0, for N 0 (γ, s) large enough, we estimate

B n+1 ≤ B n + �g n+1 � s ≤ (4.44) B n + C(s)γ -2 N µ+ ν χ 0 n+1 B n ≤ B n + C(s)γ -2 N µ+ν n+1 N -(ν-ν χ 0 ) n+1 B n ≤ N µ+ν n+1 B n ≤ (P4)n N µ+ν n+1 N µ+ν χ 0 -1 n+1 N δs 0 ≤ N (χ 0 -1)(µ+ν)+µ+ν χ 0 -1 n+1 N δs 0 ≤ N χ 0 (µ+ν) χ 0 -1 n+1 N δs 0 ≤ N µ+ν χ 0 -1
n+2 N δs 0 and using Lemma 4.3.3

B � n+1 ≤ B � n + �∂ λ g n+1 � s ≤ B � n + |∂ λ ψ n+1 |�g n+1 � s + |ψ n+1 |�∂ λ gn+1 � s ≤ (4.44) B � n + C(s, γ)(N σ/2 n+1 N µ+ ν χ 0 n+1 B n + N 2(µ+ν) n+1 (B n + B � n )) ≤ C(s, γ)N 2(µ+ν)+σ/2 n+1 (B n + B � n ) ≤ (P4)n N 2(µ+ν)+σ n+1 N 2(µ+ν)+σ χ 0 -1 n+1 N δs 0 ≤ N 2χ(µ+ν)+σ χ 0 -1 n+1 N δs 0 ≤ N 2(µ+ν)+σ χ 0 -1 n+2 N δs 0 .

Nash-Moser Theorem completed

For all ε ∈ [0, ε 1 ), the sequence v n (ε,

•) ∈ C 1 (Λ; E � n+1 ) converges in C 1 (Λ; X s1+ν ) to v because X �
s1+ν is a Banach space and we have

� n≥0 �v n -v n-1 � s1+ν ≤ (S1) K � n≥0 N ν n �v n -v n-1 � s1 ≤ (P2)n K � n≥0 N ν-σ-1 n < ∞
and similarly

� n≥0 �∂ λ v n -∂ λ v n-1 � s1+ν ≤ K � � n≥0 N -1 n < ∞.

Chapter 4. An abstract Nash-Moser Theorem

Finally, if (ε, λ) ∈ A ∞ := ∩ n≥0 A n , the Nash-Moser scheme implies that

∀ n ∈ N, Π n F (ε, λ, v n ) = 0
from which we obtain

F (ε, λ, v) = 0 since |F (ε, λ, v)| s1 = |(I -Π n )(F (ε, λ, v)) + Π n F (ε, λ, v) -Π n F (ε, λ, v n )| s1 ≤ |(I -Π n )(F (ε, λ, v))| s1 + |F (ε, λ, v) -F (ε, λ, v n )| s1 → 0 for n → ∞.
There remains to prove that v(ε, λ) is in X � s for any s. This result is a consequence of the following Lemma. Lemma 4.4.1. For all S � ≥ s, there exists a constant C(S � ) (which may go to

∞ as S � tends to ∞) such that �v n � S � ≤ C(S � )N µ+ν χ 0 -1 n . (4.60)
Proof. Using the same argument as in Lemma 4.3.2, we write

�g n+1 � S � ≤ C(S � ) N µ n+1 γ 2 � |r n | S � + |R n (g n+1 )| S � + (N δS � n+1 +�v n � S � )(|r n | s1 + |R n (g n+1 )| s1 ) � with |r n | S � = |Π n+1 F (ε, λ, v n )| S � ≤ (F 6) C(S � )(1 + �v n � S � +ν ) ≤ C(S � )N ν n (�v n � S � + 1) ≤ C(S � )N ν χ 0 n+1 (�v n � S � + 1)
where we have used as usually

�v n � S � +ν = �v n -ṽ + ṽ� S � +ν ≤ �v n -ṽ� S � +ν + �ṽ� S � +ν ≤ N ν n �v n -ṽ� S � + �ṽ� S � +ν ≤ ( Ṽ 1)(ii) C(S � )N ν n (�v n � S � + 1) (4.61)
and

|R n (g n+1 )| S � ≤ C(S � )(�v n � S � +ν + N -2σ-1+ν n+1 �g n+1 � S � ) ≤ (4.61) C(S � )(N ν χ 0 n+1 (�v n � S � + 1) + N -2σ-1+ν n+1 �g n+1 � S � ).

Chapter 4. An abstract Nash-Moser Theorem

We already know by (4.45) that

|r n | s1 + |R n (g n+1 )| s1 ≤ C(s 1 )
from which we write

�v n � S � � |r n | s1 + |R n (g n+1 )| s1 � ≤ C(s 1 )�v n � S � .
Moreover, proceeding similarly as in (4.38), we have

|r n | s1 + |R n (g n+1 )| s1 ≤ C(S � ) � N -(S � -s1)+ν n �v n � S � + N ν n+1 �g n+1 � 2 s1 � .
By the previous estimates, we write

�g n+1 � S � ≤ C(S � )γ -2 N µ n+1 � N ν χ 0 n+1 �v n � S � + N -2σ-1+ν n+1 �g n+1 � S � + N δS � n+1 (N -(S � -s1)+ν n �v n � S � + N ν n+1 �g n+1 � 2 s1 ) + C(s 1 )�v n � S � � .
Proceeding as in (4.41) we write

�g n+1 � s1 ≤ C(S � )γ -2 N µ+δs1 n+1 N -(S � -s1)+ν n �v n � S �
that allows us to obtain, proceeding as in Lemma 4.3.2, estimates

�g n+1 � S � ≤ C(S � )γ -2 N µ+ ν χ 0 n+1 �v n � S � (4.62)
for n ≥ n 0 (S � ) large enough such that

C(S � )γ -2 N µ n+1 N -2σ-1+ν n+1 < 1 2 .
Of course, g n+1 defined in (4.59) satisfies (4.62) as well.

Therefore, for n ≥ n 0 (S � )

�v n+1 � S � ≤ �v n � S � + �g n+1 � S � ≤ �v n � S � + C(S � )γ -2 N µ+ ν χ 0 n �v n � S � ≤ C(S � )γ -2 N µ+ ν χ 0 n+1 (1 + �v n � S � ) ≤ C(S � )N µ+ ν χ 0 n+1 �v n � S � .
(4.63) By (4.63) we deduce that

∃ β > 0 : ∀ n ∈ N, �v n � S � ≤ C(S � )N β n+1 .
In fact, define α n := �v n � S � N -β n+1 , for some β which we will define later, we write for n ≥ n 0 (S � )

α n+1 = �v n+1 � S � N -β n+2 ≤ (4.63) C(S � )N µ+ ν χ 0 n+1 �v n � S � N -β n+2 ≤ C(S � )N µ+ ν χ 0 n+1 N -χ0β n+1 �v n � S � ≤ C(S � )N µ+ ν χ 0 -χ0β n+1 N β n+1 N -β n+1 �v n � S � ≤ C(S � )N µ+ ν χ 0 -(χ0-1)β n+1 α n .

Chapter 4. An abstract Nash-Moser Theorem

If we impose

µ + ν χ 0 -(χ 0 -1)β < 0 ⇐⇒ β > µ + ν χ0 χ 0 -1 ,
taking for example β := µ+ν χ0-1 , we obtain that the sequence {α n } n is bounded by a constant K(S � )

α n = �v n � S � N -β n+1 ≤ K(S � )
i.e.

�v n � S � ≤ K(S � )N β n+1 ≤ K(S � )N µ+ν χ 0 -1 n+1 .
By (4.60), we deduce

�g n � S � ≤ C(S � )N µ+ν χ 0 -1 n . ( 4
�g n � s ≤ C(s, S � )�g n � 1-θ s1 �g n � θ S � ≤ C(s, S � )N (-2σ-1)(1-θ) n N θ(µ+ν) χ 0 -1 n .
From (4.65)

θ = s -s 1 S � -s 1 , 1 -θ = S � -s S � -s 1
and chosing S � large enough such that

(-2σ -1)(1 -θ) + θ(µ + ν) χ 0 -1 < 0 ⇐⇒ (-2σ -1) � S � -s S � -s 1 � + � s -s 1 S � -s 1 �� µ + ν χ 0 -1 � < 0,
we obtain that � n �g n � s < ∞ and so v(ε, λ) ∈ X � s for any s.

Useful result

Follow in this last subsection, a useful result that we will use in the applications in chapters 5 and 6 for the measure estimates of the sets of parameters of solutions.

Take (X � 0 , � • � 0 ), (X 0 , | • | 0 ) the largest Banach spaces in the Nash-Moser scheme.

For linear and continuous operators

L 1 : X � 0 → X 0 , L 2 : X 0 → X � 0 ,
we define the operator norms (and we use the same notation)

|||L 1 ||| 0 := sup 0� =u∈X � 0 |L 1 u| 0 �u� 0 , |||L 2 ||| 0 := sup 0� =g∈X0 �L 2 g� 0 |g| 0 . (4.66)
We understand from the operator what norm we consider.

Consider the linear and continuous operators

L (N ) (v) : E �(N ) ⊂ X � 0 → E (N ) ⊂ X 0 L (N ) (v) := L (N ) (ε, λ, v) = Π (N ) D v F (ε, λ, v) | E �(N ) . Proposition 4.5.1. For all ε ∈ [0, ε 1 ), let v(ε, •) ∈ C 1 (Λ, X � s1+ν ) be the limit of the sequence {v k (ε, •)} in the Theorem 4.1.1. Given µ 0 > 0 and γ ∈ (0, 1), if |||(L (N k ) (v)) -1 ||| 0 ≤ 1 2 N µ0 k γ (4.67) then |||(L (N k ) (v k-1 )) -1 ||| 0 ≤ N µ0 k γ .
Proof. By Theorem 4.1.1, we have that

�v k -v k-1 � s1 ≤ N -σ-1 k and we can estimate �v -v k � s1 = �v k -v k+1 + v k+1 -... -v� s1 ≤ ∞ � i=k N -σ-1 i+1 = ∞ � i=0 (N χ i 0 k+1 ) -σ-1 ≤ ∞ � i=0 (N -σ-1 k+1 ) ci+1 = N -σ-1 k+1 1 -N c(-σ-1) k+1 ≤ 2N -σ-1 k+1 (4.68)
using, by definition, that

∀ i ∈ N, N i := N χ i 0 0
Chapter 4. An abstract Nash-Moser Theorem which implies N i+1 = N χ0 i , and χ i 0 ≥ ci + 1 for χ 0 > 1 and an appropriate constant c. Write

L (N k ) (v k-1 ) = L (N k ) (v) + R (N k ) k-1 with R (N k ) k-1 := L (N k ) (v k-1 ) -L (N k ) (v),
from (F 3) and (4.68) we have

|||R (N k ) k-1 ||| 0 = |||L (N k ) (v k-1 ) -L (N k ) (v)||| 0 ≤ C�v -v k-1 � s1+ν ≤ CN -σ-1+ν k . (4.69)
This implies that the operator L (N k ) (v k-1 ) is invertible. In fact, we write

L (N k ) (v k-1 ) = L (N k ) (v) + R (N k ) k-1 = L (N k ) (v) � I + (L (N k ) (v)) -1 R (N k ) k-1 � where � I + (L (N k ) (v)) -1 R (N k ) k-1 � is invertible because |||(L (N k ) (v)) -1 R (N k ) k-1 ||| 0 ≤ |||L (N k ) (v) -1 ||| 0 |||R (N k ) k-1 ||| 0 ≤ 2 N µ0 k γ CN -σ-1+ν k ≤ 2C γ N µ0-σ-1+ν k < 1
� � � � � � � � � � I + (L (N k ) (v)) -1 R (N k ) k-1 � -1 � � � � � � � � � 0 ≤ 2 and by (L (N k ) (v k-1 )) -1 = � I + (L (N k ) (v)) -1 R (N k ) k-1 � -1 (L (N k ) (v)) -1 ,
we conclude

|||(L (N k ) (v k-1 )) -1 ||| 0 ≤ 2|||(L (N k ) (v)) -1 ||| 0 ≤ N µ0 k γ .
Chapter 5

Periodic solutions of some non-linear autonomous equations in infinite dimension

Main result and functional setting

We consider the non-linear autonomous equation

ω 2 u tt + Au = εf (ε, u), ω ∈ R + (5.1)
where for all t ∈ T, the unknown function u(t, •) takes its values in an appropriate separable real Hilbert space (H 0 , � • � 0 ), A is an unbounded, selfadjoint, positive operator with respect to the scalar product in H 0 and there exists an orthonormal Hilbert basis of H 0 composed of the eigenvectors {ϕ j } j≥1 of the operator A with corresponding eigenvalues {ω 2 j } j≥1 , listed in non-decreasing order, such that ∀ j ≥ 1, ω j > 0 and ω j → ∞ for j → ∞.

Define for all 0 < p ≤ p, p ≥ 2, the "Sobolev" scales

H p := � g ∈ H 0 , g = � 1≤j∈N g j ϕ j , g j ∈ R, | �g� 2 p := � j �ω j � 2p |g j | 2 < +∞ �
where �ω j � := max{1, ω j }, with the scalar product defined by

∀ g, z ∈ H p , �g, z� H p := � j �ω j � 2p g j z j ,
and their complexifications

H p C := � g = g 1 + ıg 2 , g 1 , g 2 ∈ H p � = � g = � 1≤j∈N g j ϕ j , g j ∈ C, | �g� 2 p := � j �ω j � 2p |g j | 2 < +∞ � Chapter 5.

Periodic solutions of some non-linear autonomous equations in infinite dimension

with the scalar product defined by

∀ g, z ∈ H p C , �g, z� H p C := � j �ω j � 2p g j zj .
Moreover, define for all s ≥ 0, 0 ≤ p ≤ p, the scale of separable Sobolev spaces

with values in H p H s,p := H s (T, H p ) = � u(t) = � l∈Z u l e ılt , u l ∈ H p C , u -l = ūl , | �u� 2 s,p := � l∈Z �l� 2s �u l � 2 p < +∞ � (5.2)
where for l ∈ Z, �l� := max{1, |l|}, the Fourier coefficients u l ∈ H p C are defined by

u l := 1 2π � T e -ılt u(t) dt (5.3)
and the scalar product is defined by

∀ u, v ∈ H s,p , �u, v� H s,p := � l∈Z �l� 2s �u l , v l � H p C .
We have ∀ 0 < p ≤ p,

H p ⊂ H p ⊂ H 0 , � • � p ≤ � • � p and ∀ 0 ≤ s � ≤ s, ∀ 0 ≤ p ≤ p H s, p ⊂ H s,p ⊂ H s � ,p ⊂ H 0,p , � • � 0,p ≤ � • � s � ,p ≤ � • � s,p ≤ � • � s, p.
For s = 0, we will write H 0,p = L 2 (T, H p ) in accordance with Parseval's identity

� l∈Z �u l � 2 p = 1 2π � T �u(t)� 2 p dt
for which � • � 0,p is equal to the standard norm on L 2 (T, H p ).

See section A.2 in Appendix A for details on the definitions of the Sobolev spaces.

For all s > 1/2, 0 ≤ p ≤ p, there is the continuous embedding

H s (T, H p ) �→ C 0 (T, H p ) (5.4)
where C 0 (T, H p ) is the set of all continuous functions on T into H p endowed with the � • � L ∞ (T,H p ) -norm defined as

�u� L ∞ (T,H p ) := sup t∈T �u(t)� p .

infinite dimension

See Remark A.2.1 for the proof of (5.4).

For all s ≥ 0, 0 ≤ p ≤ p, we define also the complexification H s (T, H p C ) of the Sobolev spaces H s,p = H s (T, H p )

H s (T, H p C ) := � u = u 1 + ıu 2 , u 1 , u 2 ∈ H s (T, H p ) � .
With L(H s,p ), we denote the space of all linear and continuous operators from H s,p into itself. Given a linear and continuous operator L ∈ L(H s,p ), �L� s,p denotes the associated operator norm:

�L� s,p := sup 0� =u∈H s,p
�Lu� s,p �u� s,p .

(5.5)

The space (L(H s,p ), � • � s,p ) is a Banach space.

Moreover, take a linear operator L : H s,p → H s,p , we consider its C-linear complexification (that we still denote by L)

L : H s (T, H p C ) → H s (T, H p C )
defined by By definition of the Sobolev spaces H p , the linear operator A is invertible and A -1 is a smoothing operator between the spaces

L(u 1 + ıu 2 ) = Lu 1 + ıLu 2 , ∀ u 1 , u 2 ∈ H s (T, H p ).
A -1 : H p → H p+2 �A -1 g� p+2 ≤ C�g� p , ∀ g ∈ H p . (5.6)
We suppose that the operator A verifies the following assumption.

Assumption 5.1.2 (Distribution of eigenvalues of the operator A).

There exist some positive constants σ

∈ (0, 1], c 1 , c 2 , c 3 , c 4 , d ≥ 1 and r > 1 σ + 1 such that the spectrum of √ A satisfies spect( √ A) ⊂ � j≥1 I j , I j := [p j -δ j , p j + δ j ]
Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension with

p j ≥ c 1 j r -c 2 , |δ j | ≤ c 3 |p j | σ and cardinality card(spect( √ A) ∩ I j ) ≤ c 4 j d (5.7)
where the eigenvalues of √ A are counted with multiplicity.

Hypothesis (5.7) says that in each interval I j there are at most c 4 j d eigenvalues counted with multiplicity.

For every ε small enough, we assume for the non-linear term f the following properties:

• (P 1 ) the function f (ε, •) f (ε, •) : H p → H p g � → f (ε, g) is in C ∞ (H p, H p) with f (ε, 0) = 0, D g f (ε, 0) = 0 (5.8) • (P 2 ) the C ∞ -function D g f (ε, •) : H p → L(H p),
where L(H p) is the space of all linear and continuous operators of H p into itself, satisfies

∀ 1 ≤ k ∈ N, ∀ z 1 , . . . , z k ∈ H p �D k g f (ε, g)[z 1 , . . . , z k ]� p ≤ C(k, �g� p)�z 1 � p . . . �z k � p.
(5.9)

• (P 3 ) the linear operator D g f (ε, g) ∈ L(H p) can be extended as a continous operator of H 0 , sending for all 0 ≤ p ≤ p, H p into H p , i.e.

�D g f (ε, g)z� p ≤ C(�g� p)�z� p , ∀ z ∈ H p .
(5.10)

Moreover D g f (ε, •) is in C ∞ (H p, L(H p )) for all 0 ≤ p ≤ p and ∀ 1 ≤ k ∈ N, ∀ g 1 , . . . , g k ∈ H p, ∀ z ∈ H p �D k+1 g f (ε, g)[g 1 , . . . , g k ; z]� p ≤ C(k, �g� p)�g 1 � p . . . �g k � p�z� p . (5.11)
Finally, we assume that there exists a C ∞ -function

F : [0, ε 0 ) × H p → R such
that for all ε ∈ [0, ε 0 ), D g F(ε, g) can be extended as a linear continous operator of H 0 to R and

∇ g F(ε, g) = f (ε, g) ∀ g ∈ H p (5.12)
Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension where ∇ g is the gradient in H 0 , in the sense that

D g F(ε, g)[g � ] = �f (ε, g), g � � H 0 ∀ g � ∈ H 0 .
For example, we can consider the following wave equation on a d-dimensional,

C ∞ -riemannian manifold M u tt -∆u + V (x)u = εf (ε, x, u), x ∈ M defined in the Sobolev spaces H s (T, H p(M, R)), where V ≥ c > 0 is in C ∞ (M).
The operator (-∆ + V (x)) is an example of operator A: it is invertible with inverse (-∆ + V (x)) -1 regularizing of order 2 in the spatial variables and there exists an orthonormal basis of L 2 (M) composed by its eigenfunctions {ϕ i } with corresponding eigenvalues ω 2 i , ω i → ∞ for i → ∞, which verify the asymptotic Weyl law ω j ∼ cj

f (ε, •) ∈ C ∞ (H p, H p)
which verifies by hypothesis estimates (5.9), we obtain that for all integer index 1/2 < s 0 ≤ s ∈ N, the following map (which we still call f )

f (ε, •) : H s (T, H p) → H s (T, H p) u � → f (ε, u) with f (ε, u) defined by f (ε, u)(t) := f (ε, u(t)), is in C ∞ (H s (T, H p), H s (T, H p)) and ∀ 1 ≤ k ∈ N, ∀ u, v 1 , . . . , v k ∈ H s (T, H p) �f (ε, u)� s, p ≤ C(s, �u� s0, p)(1 + �u� s, p),
(5.13)

�D k u f (ε, u)[v 1 , . . . , v k ]� s, p ≤ C(k, s, �u� s0, p) � k � i=1 �v i � s, pΠ k j=1 j� =i �v j � s0, p
+ �u� s, pΠ k j=i �v j � s0, p� .

(5.14)

Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension

In this application, D u f (ε, u) is seen as a linear (and continuous) operator from

H s (T, H p) into itself, i.e. D u f (ε, u) ∈ L(H s (T, H p)).
By property (P 3 ), it can be extended as a linear and continuous operator from

L 2 (T, H 0 ) into itself, D u f (ε, u) ∈ L(L 2 (T, H 0 )), sending for all 0 ≤ p ≤ p, H s (T, H p ) into itself D u f (ε, u) ∈ L(H s (T, H p ))
and, proceeding as done in Proposition B.3.3 to obtain (5.14), one can prove

∀ 1/2 < s 0 ≤ s ∈ N, ∀ 0 ≤ p ≤ p, ∀ v ∈ H s (T, H p ) �D u f (ε, u)[v]� s,p ≤ C(s, �u� s0, p)(1 + �u� s, p)�v� s,p . (5.15)
For all 0 ≤ p ≤ p, we associate to the linear and continuous operator D u f (ε, u) ∈ L(H s (T, H p )) the matrix of linear and continuous coefficients

(D u f (ε, u)) m l ∈ L(H p ) (D u f (ε, u)) m l : H p → H p g � → (D u f (ε, u)) m l (g)
where

(D u f (ε, u)) m l (g) is defined by (D u f (ε, u)) m l (g) := 1 2π � T e -ılt D u f (ε, u(t)
)(e ımt g) dt.

(5.16)

For all 0 ≤ p ≤ p and u ∈ H s0,p , we will use the notation D u f (ε, u) also for the continuous function

D u f (ε, u) : T → L(H p ) t � → D u f (ε, u)(t) (5.17)
defined by

D u f (ε, u)(t) := D u f (ε, u(t)).
We define the Fourier coefficients

(D u f (ε, u)) k ∈ L(H p ) of the function D u f (ε, u)
in (5.17) as

(D u f (ε, u)) k := 1 2π � T e -ıkt D u f (ε, u(t)) dt. ( 5
(D u f (ε, u)) m l = (D u f (ε, u)) l-m ∈ L(H p ). ( 5

.19)

In order to distinguish D u f (ε, u) as operator with its operator norm in L(H s (T, H p ))

from the function (5.17), we will write infinite dimension

• �D u f (ε, u)� (op) s,p
for refering to the operator norm (5.5)

• |||D u f (ε, u)||| s,L(H p )
for refering to the norm of the function (5.17)

defined ∀ 0 ≤ p ≤ p by |||D u f (ε, u)||| 2 s,L(H p ) := � k∈Z �k� 2s �(D u f (ε, u)) k � 2 L(H p ) .
(5.20)

Estimates (5.15) imply by the definition of the operator norm (5.5)

�D u f (ε, u)� (op) s,p ≤ C(s, �u� s0, p)(1 + �u� s, p).
(5.21)

We conclude this remark with the proof of the following tame estimates: ∀ s ≥

s 0 > 1/2, 0 ≤ p ≤ p |||D u f (ε, u)||| s,L(H p ) ≤ C(s, �u� s0, p)(1 + �u� s+1, p). ( 5

.22)

In order to prove (5.22), we need estimates on

�(D u f (ε, u)) k � L(H p ) . Take v ∈ H p , let us see v as a constant function in H s,p , v : T → v ∈ H p . For D u f (ε, u) ∈ L(H s (T, H p )), we consider D u f (ε, u)[v] ∈ H s (T, H p ).
By definition of norm in H s (T, H p ) (see (5.2)) and estimates (5.15), we deduce

that ∀ k ∈ Z �(D u f (ε, u)[v]) k � p ≤ 1 �k� s �D u f (ε, u)[v]� s,p ≤ 1 �k� s C(s, �u� s0, p)(1 + �u� s, p)�v� p .
(5.23)

By definition (5.3) of Fourier coefficients (D u f (ε, u)[v]) k ∈ H p of the function t � → D u f (ε, u(t))[v] ∈ H s (T, H p )
and by definition (5.18) of Fourier coefficients

(D u f (ε, u)) k ∈ L(H p ) of the function D u f (ε, u) : T → L(H p ), we have for v ∈ H p (D u f (ε, u)[v]) k := 1 2π � T e -ıkt D u f (ε, u(t))v dt = � 1 2π � T e -ıkt D u f (ε, u(t)) dt � v =: (D u f (ε, u)) k v.
(5.24)

Hence, using (5.23) and (5.24), we estimate

∀ 1/2 < s 0 ≤ s ∈ N �(D u f (ε, u)) k � L(H p ) := sup 0� =v∈H p �(D u f (ε, u)) k v� p �v� p = sup 0� =v∈H p �(D u f (ε, u)[v]) k � p �v� p ≤ 1 �k� s C(s, �u� s0, p)(1 + �u� s, p).
(5.25)

Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension

We conclude using (5.25) written for s + 1 to estimate

|||D u f (ε, u)||| 2 s,L(H p ) := � k∈Z �k� 2s �(D u f (ε, u)) k � 2 L(H p ) ≤ � k∈Z �k� 2s 1 �k� 2(s+1) C(s, �u� s0, p)(1 + �u� s+1, p) 2 ≤ C(s, �u� s0, p)(1 + �u� s+1, p) 2 since � k∈Z 1 �k� 2 < ∞, which proves (5.22).
We look for solutions u ∈ H s, p, for some s > 1/2, of equation (5.1).

By (5.8), u = 0 is a solution.

Recall that {ω 2 j } j≥1 are the eigenvalues of the operator A : H 0 → H 0 , we choose a particular ω = ωj and we define

S := � j ∈ N, j ≥ 1 : ∃ l ∈ N | ω j = lω � .
We consider the case where S is finite:

card(S) < +∞.
We impose on ω the first order Melnikov diophantine condition: there exist τ 0 > 1 and γ ∈ (0, 1) such that

|ωl -ω j | ≥ γ 1 + |l| τ0+1 , ∀ l ∈ Z, j ∈ N \ S.
(5.26)

This condition implies that the frequency ω is not in resonance with the normal mode frequencies ω j :

|ω 2 l 2 -ω 2 j | = |ωl -ω j ||ωl + ω j | ≥ Cγ 1 + |l| τ0 , (5.27) for some constant C > 0 and ∀ l ∈ Z, j ∈ N \ S.
Define E 0 the space of 2π-periodic solutions h ∈ L 2 (T, H p) of the linear equation ω2 h tt + Ah = 0.

(5.28) For all j ∈ S, define l j := ωj ω ∈ N, we have

E 0 := Ker(ω 2 ∂ tt + A) = � h ∈ L 2 (T, H p) : h = � j∈S α j cos(l j t)ϕ j + β j sin(l j t)ϕ j � .
(5.29)

Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension

Since card(S) < +∞, the space E 0 is finite dimensional and E 0 ⊂ H s, p, ∀ s ≥ 0 (in this sense, it does not depend on s).

Fix an index s 1 ∈ N, s 1 ≥ s 0 > 1/2, on which we will impose appropriate conditions (see (5.63)).

We write the space

H s1, p = E 0 ⊕ E s1, p⊥ 0
, where E s1, p⊥ 0 is the orthogonal space of E 0 with respect to the L 2 (T, H 0 )-scalar product Note that ω → ω as ε → 0.

E s1, p⊥ 0 = � h ∈ H s1, p : h = � l∈ j≥1 j / ∈S or l� =lj α j cos(lt)ϕ j + β j sin(lt)ϕ j � . ( 5
Using the non-resonance property (5.27) on ω, for a ≤ λ ≤ b, we estimate

|ω 2 l 2 -ω 2 j | = |(1 -ελ) 2 ω2 l 2 -ω 2 j | ≥ |ω 2 l 2 -ω 2 j | -|(ελ) 2 ω2 l 2 -2ελω 2 l 2 | ≥ Cγ 1 + |l| τ0 -εcl 2 , ∀ l ∈ Z, ∀ j ∈ N \ S (5.32)
with an appropriate constant c > 0. Estimate (5.32) is the non-resonance property on ω for lower frequencies l.

We look for 2π-periodic in time solutions u ε (λ) ∈ H s1, p of equation (5.1) for ω = ω(λ) given by (5.31), C 1 with respect to the amplitude parameter λ, of the form

u ε (λ) = h ε (λ) + v ε (λ) (5.33) with h ε (λ) ∈ E 0 , v ε (λ) ∈ E s1,

Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension

Using hypothesis (5.12), equation (5.1) with ω = ω(λ) given by (5.31), is the Euler-Lagrange equation of the Lagrangian action functional Φ defined on

H s1, p by Φ(ε, λ, u) := � T - 1 2 (ω(λ)) 2 �u t , u t � H 0 + 1 2 �Au, u� H 0 -εF(ε, u) dt. (5.34)
In fact, it is easy to check that for u ∈ H s1, p u is solution of equation ( 5.1) ⇐⇒ DΦ(ε, λ, u) = 0.

(5.35)

Let us consider ∀ a ∈ T the translation in time

T a u : t → u(t + a).
(5.36)

As consequence of the autonomous nature of the equation (5.1) which implies

Φ(ε, λ, T a u) = Φ(ε, λ, u), ∀ a ∈ T (5.37)
we have the following result. Let us introduce the reduced functional Ψ(0, λ, •) defined on E 0 by

Ψ(0, λ, h) := � T λω 2 �h t , h t � H 0 -F(0, h) dt.
(5.38)

We can now insert the main result of this chapter.

In what follows, we generally consider all Sobolev indices s in N, even if not specified.

Theorem 5.1.6. Suppose that the operator A verifies Assumption 5.1.2.

Suppose there exist (0, 0

) � = (λ 0 , h 0 ) ∈ Λ × E 0 such that    D h Ψ(0, λ 0 , h 0 ) = 0, dim(KerD 2 h Ψ(0, λ 0 , h 0 )) = 1.
Then, there exist 1/2 < s 1 ∈ N large enough, ε = ε(s 1 , p) and δ 0 > 0 small enough such that for all ε ∈ [0, ε), there exist a map

u ε ∈ C 1 (Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ), H s1, p) with �u ε (λ) -h0 (λ)� s1, p → 0 as ε → 0 infinite dimension for some h0 : Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ) → E 0 , h0 (λ 0 ) = h 0 and a Cantor-like set A ε ⊂ Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ) such that for all λ ∈ A ε , u ε (λ)
is a solution of (5.1) with ω = ω(λ) given by ω = (1λε)ω.

Moreover, the Cantor-like set A ε ⊂ Λ ∩ (λ 0δ 0 , λ 0 + δ 0 ) has asymptotically full Lebesgue measure, i.e.

meas{(Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 )) \ A ε } → 0 as ε → 0 and u ε ∈ C 1 (Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ), H s, p) for all s 1 ≤ s ∈ N.
We will prove Theorem 5.1.6 in the next sections.

The Lyapunov-Schmidt decomposition

According to the decomposition of the space

H s, p = E 0 ⊕ E s, p⊥ 0 ,
we recall that we look for 2π-periodic in time solutions u ε (λ) ∈ H s1, p of equation (5.1) for ω = ω(λ) given by (5.31), C 1 with respect to the amplitude parameter λ, of the form

u ε (λ) = h ε (λ) + v ε (λ) with h ε (λ) ∈ E 0 , v ε (λ) ∈ E s1, p⊥ 0 , 2π-periodic in time functions too, such that �v ε (λ)� s1, p → 0 as ε → 0.
In this decomposition, equation (5.1) is equivalent to the system

   ω 2 h tt + Ah = εΠ E0 f (ε, h + v) ω 2 v tt + Av = εΠ E s 1 , p⊥ 0 f (ε, h + v) (5.39)
and using that h verified (5.28), we have to solve

   (ω 2 -ω2 )h tt = εΠ E0 f (ε, h + v) (Q) ω 2 v tt + Av = εΠ E s 1 , p⊥ 0 f (ε, h + v) (P).
(5.40)

The (Q)-equation is called the bifurcation equation and the (P )-equation is the range equation.

Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension

Recall that ω = ω(λ) is given by (5.31).

For all fixed ε small enough, firstly we fix h ∈ E 0 and we solve the range equation finding a function v(ε, λ, h) ∈ E s1, p⊥ 0 , smooth on λ, which is solution of the range equation only for parameters (ε, λ, h) in a particular set. Then, under an appropriate hypothesis of non-degeneracy (see Definition 5.4.4), we will have a solution h ∈ E 0 of the bifurcation equation where v = v(ε, λ, h), smooth on λ, and so a complete solution u = h + v of (5.1) for a particular set of parameters.

The range equation

We consider the (P )-equation in the space E s1, p⊥ 0

ω 2 v tt + Av = εΠ E s 1 , p⊥ 0 f (ε, h + v)
which for ω = (1λε)ω, we write as

(1 -λε) 2 ω2 v tt + Av = εΠ E s 1 , p⊥ 0 f (ε, h + v). (5.41)
Given h ∈ E 0 such that �h� 0,0 < R, for some R ∈ R + which we will define later (see (5.100)), we look for solutions v = v(ε, λ, h) of (5.41).

Define

F (ε, λ, h, v) := (1 -λε) 2 ω2 v tt + Av -εΠ E s 1 , p⊥ 0 f (ε, h + v), (5.42)
solving equation (5.41) means solving the functional equation

F (ε, λ, h, v) = 0
in the space E s1, p⊥ 0 and to do this we will apply the Nash-Moser Theorem 4.1.1.

The Nash-Moser scheme

We have to prove that the function

F (ε, λ, h, v) F (ε, λ, h, v) = (1 -λε) 2 ω2 v tt + Av -εΠ E s 1 , p⊥ 0 f (ε, h + v)
verifies all the assumptions required in Theorem 4.1.1.

In our case, the scales of Banach spaces are the following Sobolev scales

(X s , | • | s ) = (E s, p-2⊥ 0 , � • � s, p-2 ), (X � s , � • � s ) = (E s, p⊥ 0 , � • � s, p),

Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension

where (as done for s 1 in (5.30)), we introduce

E s, p⊥ 0 = � h ∈ H s, p : h = � l∈N, j≥1 j / ∈S or l� =lj α j cos(lt)ϕ j + β j sin(lt)ϕ j � .
The closed subspaces are

E (N ) = E (N ) p-2,⊥ and E �(N ) = E (N ) p,⊥ where
E (N ) p,⊥ :=    w ∈ L 2 (T, H p ) : w(t) = � |l|≤N e ilt w l , w l ∈ H p C , w -l = wl    , with the corresponding L 2 -orthogonal projectors Π (N ) p-2,⊥ , Π (N ) p,⊥ . Of course, E (N ) p-2,⊥ ⊂ E s, p-2⊥ 0 and E (N ) p,⊥ ⊂ E s, p⊥ 0
, for all s ≥ 0 (in this sense they do not depend on s).

We look for solutions v ∈ E s1+2, p⊥ 0 of the range equation such that �v� s1, p ≤ 1.

(5.43)

Index s 1 will be the lower index in the Nash Moser scheme.

In all this chapter, we consider functions v such that (5.43) holds, even if not specified.

We take as parameter the couple (λ, h) with λ in the bounded interval

Λ = [
= (-∆ + V (x) + I) -1 defined in (2.31)).
Using standard properties of the composition operators in Sobolev spaces, we will prove all hypotheses of the Nash-Moser scheme in the two following Lemmas.

For s 1 being fixed, assume v satisfies (5.43), p ≥ 2 and fix ε 0 = ε 0 (s 1 , p) small enough.

Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension

Lemma 5.3.1. If f (ε, •) is a C ∞ map
from H p into itself which verifies (P 1 ), (P 2 ) and (P 3 ), then the function F defined in (5.42)

F : [0, ε 0 ) × Λ × B R (0) × E s1+2, p⊥ 0 → E s1, p-2⊥ 0 is C ∞ , satisfies (
F (ε, λ, h, v) = (1 -λε) 2 ω2 v tt + Av -εΠ E s 1 , p⊥ 0 f (ε, h + v) is in C ∞ ([0, ε 0 ) × Λ × B R (0) × E s1+2, p⊥ 0 ; E s1, p-2⊥ 0
) and verifies property (F 1)

• (F 1) F (0, λ, h, 0) = 0, ∀ (λ, h) ∈ Λ × B R (0).
We prove now that:

∀ s ≥ s 1 , ∀ (ε, λ, h) ∈ [0, ε 0 ) × Λ × B R (0) and ∀ v ∈ E s+2, p⊥ 0 with �v� s1, p ≤ 1, • (F 2) �F (ε, λ, h, 0)� s, p-2 ≤ C(s), �∂ (λ,h) F (ε, λ, h, v)� s, p-2 ≤ C(s)(1 + �v� s+2, p), �D v F (ε, λ, h, 0)[g]� s, p-2 ≤ C(s)�g� s+2, p.
Using (5.13) and (5.44), we have

�F (ε, λ, h, 0)� s, p-2 ≤ �εΠ E s 1 , p⊥ 0 f (ε, h)� s, p-2 ≤ εC(s, R) ≤ C(s).
For a ≤ λ ≤ b, we estimate

�∂ λ F (ε, λ, h, v)� s, p-2 = � � �∂λ � (1 -λε) 2 ω2 v tt + Av -εΠ E s 1 , p⊥ 0 f (ε, h + v) �� � � s, p-2 = � -2ε(1 -λε)ω 2 v tt � s, p-2 ≤ C�v� s+2, p-2 . Write h ∈ E 0 h(t) = � j∈S (α j cos(l j t) + β j sin(l j t))ϕ j , l j := ω j ω . ( 5

.45)

For all 1 ≤ j ∈ S, using (5.14) for k = 1, we estimate

�∂ αj F (ε, λ, h, v)� s, p-2 = � � �∂α j � (1 -λε) 2 ω2 v tt + Av -εΠ E s 1 , p⊥ 0 f (ε, h + v) �� � � s, p-2 = �εΠ E s 1 , p⊥ 0 D u f (ε, h + v)[cos(l j t)ϕ j ]� s, p-2 ≤ �εD u f (ε, h + v)[cos(l j t)ϕ j ]� s, p ≤ C(s)(� cos(l j t)ϕ j � s, p + �h + v� s, p� cos(l j t)ϕ j � s0, p) ≤ C(s, p)(1 + �v� s, p). infinite dimension
Similar estimates hold for �∂ βj F (ε, λ, h, v)� s, p-2 and conclude the proof of

�∂ (λ,h) F (ε, λ, h, v)� s, p-2 ≤ C(s)(1 + �v� s+2, p).
Finally, using (5.14) for k = 1, we estimate

�D v F (ε, ω, h, 0)[g]� s, p-2 ≤ �(1 -λε) 2 ω2 g tt + Ag -εΠ E s 1 , p⊥ 0 f (ε, h)[g]� s, p-2 ≤ C(s, R)(�g� s+2, p-2 + �g� s, p + ε�g� s, p) ≤ C(s)�g� s+2, p.
Using (5.14) for k = 2 and (5.43), we obtain directly property (F 3)

• (F 3) �D 2 v F (ε, λ, h, v)[w, g]� s, p-2 = �εD 2 v f (ε, h + v)[w, g]� s, p-2 ≤ C(s)(�v� s, p�w� s1, p�g� s1, p+ �g� s, p�w� s1, p+�w� s, p�g� s1, p).
There remains to prove:

• (F 4) �∂ (λ,h) D v F (ε, λ, h, v)[g]� s, p-2 ≤ C(s)(�g� s+2. p+�v� s+2, p�g� s1, p).
For a ≤ λ ≤ b, we estimate

�∂ λ D v F (ε, ω, h, v)[g]� s, p-2 = � � �∂λ � D v � (1 -λε) 2 ω2 v tt + Av -εΠ E s 1 , p⊥ 0 f (ε, h + v) � [g] �� � � s, p-2 = � � �∂λ � (1 -λε) 2 ω2 g tt + Ag -εΠ E s 1 , p⊥ 0 D u f (ε, h + v)[g] �� � � s, p-2 = � -2ε(1 -λε)ω 2 g tt � s, p-2 ≤ C�g� s+2, p-2 .
Write h ∈ E 0 as in (5.45), using �h� s,p ≤ c(s, p)R and estimates (5.14) written for k = 2, for all j ∈ S, we have

�∂ αj D v F (ε, ω, h, v)[g]� s, p-2 = � � �∂α j � (1 -λε) 2 ω2 g tt + Ag -εΠ E s 1 , p⊥ 0 D u f (ε, h + v)[g] �� � � s, p-2 = �εΠ E s 1 , p⊥ 0 D 2 u f (ε, h + v)[g, cos(l j t)ϕ j ]� s, p-2 ≤ �εD 2 u f (ε, h + v)[g, cos(l j t)ϕ j ]� s, p ≤ C(s, p)(�g� s, p� cos(l j t)ϕ j � s1, p + �g� s1, p� cos(l j t)ϕ j � s, p + �h + v� s, p�g� s1, p� cos(l j t)ϕ j � s1, p) ≤ C(s, p)(�g� s, p + �v� s, p�g� s1, p).
Similar estimates hold for �∂ βj D v F (ε, ω, h, v)[g]� s, p-2 and conclude the proof of (F 4).

Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension

Recall the non-resonance property (5.32) on ω for lower frequencies

|ω 2 l 2 -ω 2 j | ≥ Cγ 1 + |l| τ0 -εcl 2 , ∀ l ∈ Z, ∀ 1 ≤ j / ∈ S we take Ñ = Ñ (ε) ∈ N such that    Cγ 1+| Ñ | τ 0 > ε 1/4 cε Ñ 2 << Cγ 2(1+ Ñ τ 0 ) .
(5.46)

Since τ 0 > 1, these condition are satisfied choosing

Ñ < C(γε -1/4 ) 1 τ 0 (5.47)
for an appropriare constant C > 0.

For |l| ≤ Ñ , we estimate

|ω 2 l 2 -ω 2 j | ≥ Cγ 1 + |l| τ0 -εcl 2 ≥ Cγ 1 + | Ñ | τ0 -εc Ñ 2 ≥ Cγ 1 + | Ñ | τ0 - Cγ 2(1 + | Ñ | τ0 ) ≥ Cγ 2(1 + | Ñ | τ0 ) ≥ 1 2 ε 1/4 .
(5.48)

We can now prove the second Lemma for the hypotheses of the Nash-Moser scheme.

Fix an index s > s 1 large enough.

Lemma 5.3.2. Given q > 0, for ε = ε(s, p, q, R) small enough, there exists

ṽ = ṽ(ε, λ, h) ∈ � � s≥0 E s, p⊥ 0 � such that: ∀ (λ, h) ∈ Λ × B R (0) • ( Ṽ 1) (i) �ṽ(ε, λ, h)� s+2, p ≤ √ ε (ii) ∀ S � > s, ∀ ε ≤ ε(s, p, q, R) �ṽ(ε, λ, h)� S � , p ≤ C(S � , p, R) • ( Ṽ 2) �F (ε, λ, h, ṽ(ε, λ, h))� s1, p-2 ≤ ε q • ( Ṽ 3) �∂ (λ,h) ṽ(ε, λ, h)� s+2, p ≤ ε 1 4 • ( Ṽ 4) �∂ (λ,h) � F (ε, λ, h, ṽ(ε, λ, h)) � � s1, p-2 ≤ ε q .
Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension

Proof. We start proving the existence of ṽ.

For Ñ defined in (5.47), call

Π Ñ := Π ( Ñ )
p,⊥ the projector on the subspace

E ( Ñ ) p,⊥
to simplify the notations.

We write

Π Ñ F (ε, λ, h, v) = Π Ñ ((1 -λε) 2 ω2 v tt + Av -εΠ E s 1 , p⊥ 0 f (ε, h + v)) = D Ñ v -εΠ Ñ f (ε, h + v),
where the diagonal operator D Ñ written in Hilbert basis {e ılt ϕ j } l,j of L 2 (T, H 0 C ) as

D Ñ := diag j� ∈S or l� =lj |l|≤ Ñ , j≥1 {-ω 2 l 2 + ω 2 j }, ω = (1 -λε)ω
is invertible by (5.48) and for all s ≥ 0

�D -1 Ñ � s, p = max j� ∈S or l� =lj |l|≤ Ñ , j≥1 � 1 | -ω 2 l 2 + ω 2 j | � ≤ 2ε -1 4 .
In the following, we will use this estimate for s = s + 2

�D -1 Ñ � s+2, p ≤ 2ε -1 4 . (5.49) Write Π Ñ F (ε, λ, h, v) = 0 ⇐⇒ v = εD -1 Ñ (Π Ñ f (ε, h + v)),
we want to find the fixed points of

G(v) := εD -1 Ñ (Π Ñ f (ε, h + v)).
Define the set

B δ := {v ∈ E ( Ñ )
p,⊥ : �v� s+2, p ≤ δ} for an appropriate choice of δ > 0, G maps B δ into itself: using (5.49), (5.13) and �h� s+2, p ≤ c(s, p)R, we estimate

�G(v)� s+2, p ≤ 2 ε ε -1 4 �Π Ñ f (ε, h + v)� s+2, p ≤ C(s, p)ε 3 4 (1 + �h + v� s+2, p) ≤ C(s, p, R)ε 3 4 (1 + �v� s+2, p) ≤ C(s, p, R)ε 3 4 (1 + δ) ≤ δ
taking ε ≤ ε(s, p, R) small enough and δ = 2C(s, p, R)ε

Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension

In the same way, we can prove that ∀ v ∈ B δ ,

�DG(v)[w]� s+2, p ≤ �w� s+2, p 2
and so G is a contration in B δ .

Let ṽ = ṽ(ε, λ, h) be the unique fixed point of G in B δ .

Since

B δ ⊂ E (N ) p,⊥ ⊂ � � s≥0 E s, p⊥ 0 � , we have that ṽ ∈ � � s≥0 E s, p⊥ 0 � . • Estimate ( Ṽ 1) (i) For ε ≤ ε(s, p, R) small enough, we have directly �ṽ(ε, λ, h)� s+2, p ≤ δ ≤ 2C(s, p, R)ε 3/4 ≤ √ ε.
(ii) Take S � > s. We write as usual

�ṽ(ε, λ, h)� S � , p = �εD -1 Ñ (Π Ñ f (ε, h + ṽ(ε, λ, h)))� S � , p ≤ 2 ε ε -1 4 �Π Ñ f (ε, h + ṽ(ε, λ, h))� S � , p ≤ C(S � , p)ε 3 4 (1 + �h + ṽ(ε, λ, h)� S � , p) ≤ C(S � , p, R)ε 3 4 (1 + �ṽ(ε, λ, h)� S � , p). If C(S � , p, R)ε 3/4 ≤ 1/2, we obtain �ṽ(ε, λ, h)� S � , p ≤ C(S � , p, R)ε 3/4 ≤ 1.
Let us consider now the case C(S � , p, R)ε 3/4 > 1/2.

Using the properties of the projectors and estimate ( Ṽ 1)(i), we write

�ṽ(ε, λ, h)� S � , p ≤ Ñ (S � -s) �ṽ(ε, λ, h)� s, p ≤ Ñ (S � -s) √ ε.
We choose Ñ = ε -α satisfying (5.47) for a suitable α > 0 (α = 1/5τ 0 for example).

We have, using

C(S � , p, R)ε 3/4 > 1/2 which implies ε -1 < (2C(S � , p, R)) 4/3 �ṽ(ε, λ, h)� S � , p ≤ Ñ (S � -s) √ ε ≤ ε -α(S � -s)+1/2 ≤ (2C(S � , p, R)) 4(α(S � -s)+1/2)/3
with (2C(S � , p, R)) 4(α(S � -s)+1/2)/3 =: C � (S � , p, R) independent of ε.

• Estimate ( Ṽ 2)

Since Π Ñ F (ε, λ, h, ṽ) = 0, we write

F (ε, λ, h, ṽ) = (I -Π Ñ )F (ε, λ, h, ṽ) = -ε(I -Π Ñ )Π E s 1 , p⊥ 0 f (ε, h + ṽ).
(5.50)

Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension

For Ñ = ε -α , using the properties of the projectors, estimates (5.13) and ( Ṽ 1)(i), we have

�F (ε, λ, h, ṽ)� s1, p-2 = ε�(I -Π Ñ )Π E s 1 , p⊥ 0 f (ε, h + ṽ)� s1, p-2 ≤ ε Ñ -(s-s1) �f (ε, h + ṽ)� s, p-2 ≤ ε Ñ -(s-s1) C(s)(1 + �h� s, p + �ṽ� s, p) ≤ C(s, p)ε α(s-s1)+1
≤ ε q for α(ss 1 ) + 1 > q taking s large enough and ε ≤ ε(s, p, q) small enough.

• Estimate ( Ṽ 3)

We will distinguish between the partial derivative in λ and in h.

Write as usual h ∈ E 0 as in (5.45)

h(t) = � j∈S (α j cos(l j t) + β j sin(l j t))ϕ j . Since Π Ñ F (ε, λ, h, ṽ(ε, λ, h)) = 0,
the Implicit Function Theorem implies

(Π Ñ ∂ λ F )(ε, λ, h, ṽ) + Π Ñ D v F (ε, λ, h, ṽ)∂ λ ṽ = 0 (5.51) and ∀ j ∈ S (Π Ñ ∂ αj F )(ε, λ, h, ṽ) + Π Ñ D v F (ε, λ, h, ṽ)∂ αj ṽ = 0, (5.52) (Π Ñ ∂ βj F )(ε, λ, h, ṽ) + Π Ñ D v F (ε, λ, h, ṽ)∂ βj ṽ = 0. (5.53)
Write the operator

Π Ñ D v F (ε, λ, h, ṽ) = Π Ñ ((1 -λε) 2 ω2 ∂ tt + A -εΠ E s 1 , p⊥ 0 D v f (ε, h + ṽ)) = D Ñ + T Ñ ,
where we consider D v f (ε, h+ ṽ) as operator of H s, p into itself as explained in Remark 5.1.3. Recall that D Ñ verifies (5.49)

�D -1 Ñ � s+2, p ≤ 2ε -1 4 .
Using (5.21), ( Ṽ 1)(i) and �h� s+2, p ≤ c(s, p)R, we have for ε ≤ ε(s, p, R)

small enough �D -1 Ñ � s+2, p�T Ñ � s+2, p ≤ 2ε -1 4 ε�D v f (ε, h + ṽ)� (op) s+2, p ≤ C(s)ε 3 4 (1 + �h + ṽ� s+2, p) ≤ 1 2
Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension and Lemma B.1.1 implies that Π Ñ D v F (ε, λ, h, ṽ) is invertible with

�(Π Ñ D v F (ε, λ, h, ṽ)) -1 � s+2, p ≤ 4ε -1/4 .
Hence, by (5.51), (5.52) and (5.53), we have

∂ λ ṽ = - � Π Ñ D v F (ε, λ, h, ṽ) � -1 � (Π Ñ ∂ λ F )(ε, λ, h, ṽ)
� and for all j ∈ S

∂ αj ṽ = - � Π Ñ D v F (ε, λ, h, ṽ) � -1 � (Π Ñ ∂ αj F )(ε, λ, h, ṽ) � , ∂ βj ṽ = - � Π Ñ D v F (ε, λ, h, ṽ) � -1 � (Π Ñ ∂ βj F )(ε, λ, h, ṽ) � .
We estimate, using �ṽ(ε,

λ, h)� s+2, p ≤ δ ≤ 2C(s, p)ε 3/4 �∂ λ ṽ� s+2, p = � � � � Π Ñ D v F (ε, λ, h, ṽ) � -1 � (Π Ñ ∂ λ F )(ε, λ, h, ṽ) �� � � s+2, p ≤ 4ε -1/4 �(Π Ñ ∂ λ F )(ε, λ, h, ṽ)]� s+2, p ≤ 4ε -1/4 � -2ε(1 -λε)ω 2 ∂ tt ṽ� s+2, p ≤ Cε -1/4 ε Ñ 2 �ṽ� s+2, p ≤ 2C(s, p)ε 3/4 ε 3/4 ε -2α ≤ ε 1/4
if we take 3/4 + 3/4 -2α > 1/4 ⇐⇒ α < 5/8 and ε ≤ ε(s, p) small enough.

Using (5.14) for k = 1, we estimate for all j ∈ S and for ε ≤ ε(s, p, R)

small enough �∂ αj ṽ� s+2, p = � � � � Π Ñ D v F (ε, λ, h, ṽ) � -1 � (Π Ñ ∂ αj F )(ε, λ, h, ṽ) �� � � s+2, p ≤ 4ε -1/4 �(Π Ñ ∂ αj F )(ε, λ, h, ṽ)]� s+2, p ≤ 4ε -1/4 ε�D v f (ε, h + v)[cos(l j t)ϕ j ]� s+2, p ≤ C(s, p)ε -1/4 ε(� cos(l j t)ϕ j � s+2, p + �h + v� s+2, p� cos(l j t)ϕ j � s0+2, p) ≤ C(s, p, R)ε 3/4 ≤ ε 1/4 .
Similar estimates follow for �∂ βj ṽ� s+2, p and conclude the proof of ( Ṽ 3).

• Estimate ( Ṽ 4) Derivating (5.50), we have

∂ λ � F (ε, λ, h, ṽ) � = -ε(I -Π Ñ )Π E s 1 , p⊥ 0 ∂ λ (f (ε, h + ṽ))
Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension and

∂ αj � F (ε, λ, h, ṽ) � = -ε(I -Π Ñ )Π E s 1 , p⊥ 0 ∂ αj f (ε, h + ṽ), ∂ βj (F (ε, λ, h, ṽ)) = -ε(I -Π Ñ )Π E s 1 , p⊥ 0 ∂ βj (f (ε, h + ṽ)).
We estimate, using the properties of projectors, estimates (5.15), ( Ṽ 1)(i)

and ( Ṽ 3)

�∂ λ � F (ε, λ, h, ṽ) � � s1, p-2 = ε�(I -Π Ñ )Π E s 1 , p⊥ 0 ∂ λ (f (ε, h + ṽ))� s1, p-2 ≤ ε Ñ -(s-s1) �∂ λ (f (ε, h + ṽ))� s, p-2 ≤ ε Ñ -(s-s1) �D v f (ε, h + ṽ)∂ λ ṽ� s, p-2 ≤ C(s, p)ε Ñ -(s-s1) (�∂ λ ṽ� s, p + �ṽ� s, p�∂ λ ṽ� s1, p) ≤ C(s, p)ε α(s-s1)+1+1/4
≤ ε q taking s large enough such that α(ss 1 ) + 5/4 > q and ε ≤ ε(s, p, R, q) small enough.

The estimates for

�∂ αj � F (ε, λ, h, ṽ) � � s1, p-2 and �∂ βj � F (ε, λ, h, ṽ) � � s1, p-2
are exactly the same. We have proved also ( Ṽ 4).

There remains to define the sets of parameters J

(N ) γ,µ,K for which if (ε, λ, h) ∈ J (N )
γ,µ,K the linear operators

L (N ) (ε, λ, h, v) := Π (N ) p-2,⊥ D v F (ε, λ, h, v) |E (N ) p,⊥
(5.54) are invertible and the following estimates hold:

∀ w ∈ E (N ) p-2,⊥ �(L (N ) (ε, λ, h, v)) -1 w� s, p ≤ K(s) N µ γ 2 � (N δs + �v� s, p)�w� s1, p-2 + �w� s, p-2 � .
(5.55) Moreover, we have to prove Hypothesis (L1) of the Nash-Moser scheme and that the set of parameters A ∞ has a "large" measure.

We will obtain estimates (5.55) by the application of Proposition 3.2.2 to the linear operator

L(ε, λ, h, v) : D(L) ⊂ L 2 (T, H 0 ) → L 2 (T, H 0 ) L(ε, λ, h, v) = D v F (ε, λ, h, v) = (1 -λε) 2 ω2 ∂ tt + A -εΠ E s 1 , p⊥ 0 D v f (ε, h + v) = D + T (5.56)
Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension where

D := (1 -λε) 2 ω2 ∂ tt + A, T := -εΠ E s 1 , p⊥ 0 D v f (ε, h + v),
and by the application of Lemma 5.3.6.

By hypotheses on operator A and on the map f , we have that the diagonal

operator D is unbounded, D -1 ∈ L(L 2 (T, H 0 )) and T ∈ L(L 2 (T, H 0 )).
We define the "matrix coefficients" associated to the linear and unbouded operator L(ε, λ, h, v) as

L l l := D l l + T l l , L m l := T m l , l � = m (5.57) with D l l := diag j � -(1 -λε) 2 ω2 l 2 + ω 2 j � , T m l := -ε � Π E s 1 , p⊥ 0 D v f (ε, h + v) � m l .
For a linear and continuous operator P ∈ L(L 2 (T, H 0 )), we recall the operator norm (5.5)

�P � 0,0 := sup u� =0
�P u� 0,0 �u� 0,0 ,

for n ∈ Z, the definition of seminorms

[P (n)] := max l-m=n �P m l � L(H 0 ) (5.58)
where P m l are the matrix coefficients associated to the linear and continuous operator P (see chapter 3 for their definition) and for s ∈ R the definition of norms

|P | 2 s := � n∈Z �n� 2s [L(n)] 2 . (5.59)
Let us consider a linear operator L = D + T defined on L 2 (T, H 0 ), with

diagonal part D = d l δ m l , d l : H 0 → H 0 possibly unbounded, d -1 l ∈ L(H 0 ) and T ∈ L(L 2 (T, H 0 )).
We recall the definition of regular sites: l ∈ Z is a regular site for a linear operator L = D + R with respect to a fixed constant θ if and only if

�d -1 l � L(H 0 ) ≤ θ -1 .
Let S N,θ be the set of singular sites of L in the interval [-N, N ] with respect to constant θ.

Let us insert here the definition of "good separation in blocks" for the set S N,θ .

Let us fix two constants 0 < δ < 1, 0 < β < 1, independent of N , which will be specified later.

Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension Definition 5.3.3 (Good separation in blocks of set S N,θ ). We say that there is a "good separation in blocks" (with respect to the fixed constants δ, β) for the set S N,θ if and only if the singular sites l ∈ S N,θ of L are divided in blocks Γ N,r

S N,θ = ∪ r Γ N,r such that    diam(Γ N,r ) ≤ N δ ∀ r, dist(Γ N,r , Γ N,r � ) ≥ N β if r � = r � (5.60)
where

dist(Γ N,r , Γ N,r � ) := inf l∈ΓN,r k∈Γ N,r � |l -k|.
Fix θ = 1, we consider the singular sites l ∈ Z for L(ε, λ, h, v): by definition

�(-(1 -λε) 2 ω2 l 2 I + A) -1 � L(H 0 ) ≥ 1 (5.61)
where I : H 0 → H 0 is the identity map in H 0 or, equivalently,

∃ j ≥ 1 : | -(1 -λε) 2 ω2 l 2 + ω 2 j | < 1. (5.62)
Let S N be the set of singular sites of L(ε, λ, h, v) in the interval [-N, N ] with respect to constant θ = 1.

Fix an index s 0 > 1/2. Proposition 3.2.2 written for L(ε, λ, h, v) gives us the following result.

Proposition 5.3.4. Suppose that the set of singular sites S N of L(ε, λ, h, v)

satisfies the "good separation in blocks" defined in 5.3.3.

Fix µ 0 > 0 and impose to index s 1 introduced in (5.43) to satisfy:

s 1 > s 0 > 1/2 and s 1 = s 1 (µ 0 ) β((s 1 -1) -s 0 ) ≥ µ 0 + δ 2 + δs 0 . (5.63)
Fix γ ∈ (0, 1), there exists a constant µ 1 independent of s µ 1 := 2(µ 0 + δs 0 ) + δ and a non-decreasing function K : R + → [1, ∞) such that if the operator

L (N ) (ε, λ, h, v) defined in (5.54) is invertible with �(L (N ) )(ε, λ, h, v)) -1 � 0,0 ≤ N µ0 γ (5.64)
and if

|T | s1-1 ≤ ν(s 1 ),
for some ν(s 1 ) > 0 small enough Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension

then ∀ s ≥ s 0 , ∀ w ∈ E ⊥(N ) s,0 �(L (N ) (ε, λ, h, v)) -1 w� s,0 ≤ K(s) N µ1 γ 2 � (N δs + �v� s, p)�w� s0,0 + �w� s,0 � . ( 5
|T | 2 s1-1 = � n∈Z �n� 2(s1-1) [T (n)] 2 = � n∈Z �n� 2(s1-1) � max l-m=n �T m l � L(H 0 ) � 2 ≤ ε 2 � n∈Z �n� 2(s1-1) � max l-m=n �(D v f (ε, h + v)) m l � L(H 0 ) � 2 ≤ ε 2 � n∈Z �n� 2(s1-1) � max l-m=n �(D v f (ε, h + v)) l-m � L(H 0 ) � 2 ≤ ε 2 � n∈Z �n� 2(s1-1) �(D v f (ε, h + v)) n � 2 L(H 0 ) ≤ ε 2 |||D v f (ε, h + v)||| 2 s1-1,L(H 0 ) .
Using tame estimates (5.22) and �h� s1,0 ≤ c(s 1 )R, �v� s1, p ≤ 1 for R being fixed, we have

|T | s1-1 ≤ ε|||D v f (h + v)||| s1-1,L(H 0 ) ≤ εC(s 1 )(1 + �h + v� s1,0 ) ≤ εC(s 1 , R)(1 + �v� s1, p)
≤ εC(s 1 ).

(5.66)

Hence, it is sufficient to take ε = ε(s 1 ) small enough in order to have |T | s1-1 ≤ ν(s 1 ), with ν(s 1 ) small enough, as required in Proposition 5.3.4.

Lemma 5.3.6. Under the assumptions of Proposition 5.3.4, for ε = ε(s 1 , s, p)

small enough, we have

∀ s ≥ s 1 , ∀ w ∈ E (N ) p-2,⊥ �(L (N ) (ε, λ, h, v)) -1 w� s, p ≤ K(s) N µ γ 2 � (N δs + �v� s, p)�w� s1, p-2 + �w� s, p-2

�

(5.67)

with µ = µ 1 + p + δs 1 + 2.

Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension

Proof. Write in this proof, Π ⊥ := Π (N ) p-2,⊥ to simplify the notations and recall

ω = ω(λ) with ω = (1 -λε)ω, λ ∈ Λ = [a, b]. Setting for g ∈ E (N ) p,⊥ , w ∈ E (N ) p-2,⊥ , w := L (N ) (ε, λ, h, v)[g] = ω 2 g tt + Ag -εΠ ⊥ D v f (h + v)[g]
we have

L (N ) (ε, λ, h, v) -1 w = g = A -1 (-ω 2 g tt + εΠ ⊥ D v f (h + v)[g] + w).
By (5.6), (5.15) and �h� s,p ≤ c(s, p)R, we estimate

�g� s, p ≤ �A -1 (-ω 2 g tt + εΠ ⊥ D v f (h + v)[g] + w)� s, p ≤ C� -ω 2 g tt + εΠ ⊥ D v f (h + v)[g] + w� s, p-2 ≤ C(N 2 �g� s, p-2 + ε�Π ⊥ D v f (h + v)[g]� s, p-2 + �w� s, p-2 ) ≤ C(N 2 �g� s, p-2 + ε(�g� s, p-2 + �h + v� s, p�g� s1, p-2 ) + �w� s, p-2 ) ≤ C(s, R)(N 2 �g� s, p-2 + ε�v� s, p�g� s1, p-2 + �w� s, p-2 )
(5.68) and using �v� s1, p ≤ 1

�g� s1, p-2 ≤ �A -1 (-ω 2 g tt + εΠ ⊥ D v f (h + v)[g] + w)� s1, p-2 ≤ C� -ω 2 g tt + εΠ ⊥ D v f (h + v)[g] + w� s1,max{0, p-4} ≤ C(N 2 �g� s1,max{0, p-4} + ε�Π ⊥ D v f (h + v)[g]� s1,max{0, p-4} + �w� s1,max{0, p-4} ) ≤ C(s 1 , R)(N 2 �g� s1,max{0, p-4} + �w� s1,max{0, p-4} ).
(5.69) Iterating in (5.68) and (5.69), we have

�g� s1, p ≤ C(s 1 , p)N p+2 (�g� s1,0 + �w� s1, p-2 ) and �g� s, p ≤ C(s, R)N p+2 (�g� s,0 + �v� s, p(�g� s1,0 + �w� s1, p-2 ) + �w� s, p-2 ).
By (5.65), we have

�g� s1,0 ≤ K(s 1 ) N µ1+δs1 γ 2 �w� s1,0 , �g� s,0 ≤ K(s) N µ1 γ 2 � (N δs + �v� s, p)�w� s0,0 + �w� s,0 � .
Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension and we conclude

�g� s, p ≤ CN p+2 � K(s) N µ1 γ 2 � �w� s, p-2 + (N δs + �v� s, p)�w� s1, p-2 � + �v� s, p� K(s) N µ1+δs1 γ 2 �w� s1, p-2 + �w� s1, p-2 � + �w� s, p-2 � ≤ K(s) N µ γ 2 � (N δs + �v� s, p)�w� s1, p-2 + �w� s, p-2 � with µ = µ 1 + p + δs 1 + 2.
For �v� s1, p ≤ 1, we can define now

J (N ) γ,µ0 := � (ε, λ, h, v) ∈ [0, ε 0 ) × Λ × B R (0) × E (N ) p,⊥ :
the set of singular sites S N = ∪ r Γ N,r satisfies (5.60),

∃ (L (N ) ) -1 (ε, λ, h, v) and �(L (N ) ) -1 (ε, λ, h, v)� 0,0 ≤ N µ0 γ � . (5.70)
We remark that condition (5.60) about the separation of the singular sites depends only on parameters (ε, λ) and not on (h, v) (see definition in (5.61)).

By Proposition 5.3.4 and Lemma 5.3.6, for µ := 2(µ 0 + δs 0 ) + δ + δs 1 + p + 2 and ε = ε(s 1 , s, p) small enough, we have the inclusion

J (N ) γ,µ0 ⊆ {(ε, λ, h, v) ∈ [0, ε 0 ) × Λ × B R (0) × E (N ) p,⊥ | ∃ (L (N ) (ε, λ, h, v)) -1 and (5.55) holds ∀ s ≥ s 1 }.
(5.71)

We can now prove the last hypothesis for applying the Nash-Moser Theorem:

we will obtain hypothesis (L1) of the Nash-Moser scheme as a consequence of the following lemma.

Lemma 5.3.7. Let us consider µ := 2(µ 0 + δs 0 ) + δ + δs 1 + p + 2.

Take (ε, λ, h, v) ∈ J (N ) γ,µ0 and (ε, λ � , h � , v � ) ∈ [0, ε 0 ) × Λ × B R (0) × E (N ) p,⊥ such that |λ � -λ| + �h � -h� s1, p + �v � -v� s1, p ≤ 2N -(µ+1) .
(5.72)

Consider the linear operator defined by

L(ε, λ � , h � , v � ) = (1 -λ � ε) 2 ω2 ∂ tt + A -εΠ s, p⊥ 0 D v f (ε, h � + v � ).
Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension

If l ∈ [-N, N ] is a regular sites for L(ε, λ, h, v) with respect to θ = 1, then l is a regular site for L(ε, λ � , h � , v � ) with respect to an appropriate constant θ � < 1 such that θ � → 1 as N → ∞ (or as ε → 0).
Moreover, define

S � N,θ � := � l ∈ [-N, N ], l singular site for L(ε, λ � , h � , v � ) with constant θ � � , we have that (ε, λ � , h � , v � ) ∈ J � (N )
γ,µ0 , with

J � (N ) γ,µ0 := � (ε, λ � , h � , v � ) ∈ [0, ε 0 ) × Λ × B R (0) × E ⊥(N ) s, p : the set of singular sites S � N,θ � = ∪ r Γ N,r satisfies (5.60), ∃ (L (N ) ) -1 (ε, λ � , h � , v � ) and �(L (N ) ) -1 (ε, λ � , h � , v � )� 0,0 ≤ 2N µ0 γ � .
Proof. We will write

Π ⊥ := Π (N )
p-2,⊥ to simplify the notation. Consider the restricted linear operator defined on

E (N ) p,⊥ L (N ) (ε, λ � , h � , v � ) = (1 -λ � ε) 2 ω2 ∂ tt + A -εΠ ⊥ D v f (ε, h � + v � ).
We write

L (N ) (ε, λ � , h � , v � ) = L (N ) (ε, λ, h, v) + R (N )
(5.73) with

R (N ) = L (N ) (ε, λ � , h � , v � ) -L (N ) (ε, λ, h, v) = � ε 2 � (λ � ) 2 -λ 2 � -2ε(λ � -λ) � ω2 ∂ tt -ε(Π ⊥ D v f (ε, h � + v � ) -Π ⊥ D v f (ε, h + v)).
Recall that λ, λ � ∈ Λ = [a, b], using (5.72) we estimate

|(λ � ) 2 -λ 2 | = |λ � -λ|(λ � + λ) ≤ CN -(µ+1) and � � � � ε 2 � (λ � ) 2 -λ 2 � -2ε(λ � -λ) � ω2 � � � ≤ CεN -(µ+1) (5.74) which implies ∀ w ∈ E (N ) p,⊥ � � � � ε 2 � (λ � ) 2 -λ 2 � -2ε(λ � -λ) � ω2 ∂ tt w � � � 0,0 ≤ CεN -(µ+1) N 2 �w� 0,0 ≤ CεN -µ+1 �w� 0,0 .
(5.75) Moreover, by C ∞ -regularity of f and for s 1 > 1/2 by the embedding (5.4)

H s1 (T, H p) �→ C 0 (T, H p),
Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension we estimate

�ε(Π ⊥ D v f (ε, h � + v � ) -Π ⊥ D v f (ε, h + v))w� 0,0 ≤ Cε�w� 0,0 � �h � -h� L ∞ (T,H p ) + �v � -v� L ∞ (T,H p ) � ≤ Cε�w� 0,0 (�h � -h� s1, p + �v � -v� s1, p)
≤ CεN -(µ+1) �w� 0,0 .

(5.76)

Hence by (5.75) and (5.76), we have

�R (N ) � 0,0 ≤ CεN -µ+1 . (5.77) This implies that L (N ) (ε, λ � , h � , v � ) is invertible. In fact, write L (N ) (ε, λ � , h � , v � ) = L (N ) (ε, λ, h, v) + R (N ) = L (N ) (ε, λ, h, v)(I + (L (N ) (ε, λ, h, v)) -1 R (N ) ), by (5.77), we can estimate for (ε, λ, h, v) ∈ J (N ) γ,µ0 �(L (N ) (ε, λ, h, v)) -1 R (N ) � 0,0 ≤ �(L (N ) (ε, λ, u)) -1 � 0,0 �R (N ) � 0,0 ≤ Cε N µ0 γ N -µ+1 ≤ CεN µ0-µ+1 0 ≤ 1 2
(5.78) taking ε small enough and having that µ 0µ + 1 < 0.

Hence by (5.78), Lemma B.1.1 gives us that L (N ) (ε, λ � , h � , v � ) is invertible and

�(L (N ) (ε, λ � , h � , v � )) -1 � 0,0 ≤ 2�(L (N ) (ε, λ, h, v)) -1 � 0,0 ≤ 2N µ0 γ . (5.79)
We prove now that if l ∈ [-N, N ] is a regular sites for L(ε, λ, h, v) with respect to θ = 1 then l is a regular site also for L(ε, λ � , h � , v � ) with respect to an appropriate θ � < 1.

By definition (5.62), l ∈ [-N, N ] is a regular site for L(ε, λ, h, v) if and only if

∀ j ≥ 1 : | -(1 -λε) 2 ω2 l 2 + ω 2 j | > 1.
Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension Using (5.72) and (5.74), l ∈ [-N, N ], we estimate

| -(1 -λ � ε) 2 ω2 l 2 + ω 2 j | = � � � -(1 -λ � ε) 2 ω2 l 2 + (1 -λε) 2 ω2 l 2 -(1 -λε) 2 ω2 l 2 + ω 2 j � � � ≥ - � � � -(1 -λ � ε) 2 ω2 l 2 + (1 -λε) 2 ω2 l 2 � � � + � � � -(1 -λε) 2 ω2 l 2 + ω 2 j � � � ≥ - � � � � ε 2 � λ 2 -(λ � ) 2 � -2ε(λ -λ � ) � ω2 l 2 � � � + 1 ≥ -CεN -(µ+1) N 2 + 1 ≥ -CεN -µ+1 + 1 ≥ θ � for an appropriate θ � < 1 such that, since µ > 1, θ � → 1 as N → ∞, (or as ε → 0).
Hence, we have that l is a regular site for L(ε, λ � , h � , v � ) with constant θ � and this implies that, for

S � N,θ � = � l ∈ [-N, N ], l singular site for L(ε, λ � , h � , v � ) with constant θ � < 1 � ,
we have the inclusion

S � N,θ � ⊂ S N . Since (ε, λ, h, v) ∈ J (N )
γ,µ0 , the good separation (5.60) for S N implies a good separation also for S � N,θ � and the proof of Lemma is completed.

Lemma 5.3.7 implies the validity of hypothesis (L1) proceeding as in the proof of inclusion (5.71).

In fact, for ε ≤ ε(s 1 , s, p) small enough and (ε,

λ � , h � , v � ) ∈ J � (N )
γ,µ0 , we have the same results as in Proposition 5.3.4 and Lemma 5.3.6 for L(ε, λ � , h � , v � ) (with

S �
N,θ � as the set of singular sites) and we obtain the following inclusion

J � (N ) γ,µ0 ⊂ � (ε, λ � , h � , v � ) ∈ [0, ε 0 ) × Λ × E (N ) p,⊥ : ∃ (L (N ) (ε, λ � , h � , v � )) -1 and ∀ s ≥ s 1 �(L (N ) (ε, λ � , h � , v � )) -1 w� s ≤ 2K(s)N µ (�w� s, p-2 + (N δs + �v � � s )�w� s1, p-2 )

�

where µ := 2(µ 0 + δs 0 ) + δ + δs 1 + p + 2, which proves (L1).

Hence, the function F defined in (5.42) verifies all hypotheses of the Nash-Moser Theorem.

Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension

For a map v

: Λ×B R (0) → E (N ) p,⊥ such that �v(λ, h)� s1, p ≤ 1, �∂ λ v(λ, h)� s1, p ≤ 1, �∂ h v(λ, h)� s1, p ≤ 1, we define the set G (N ) γ,µ0 (v) := � (ε, λ, h) ∈ [0, ε 0 ) × Λ × B R (0) : (ε, λ, h, v(ε, λ, h)) ∈ J (N ) γ,µ0 � = � (ε, λ, h) ∈ [0, ε 0 ) × Λ × B R (0) : S N = ∪ r Γ N,r satisfies (5.60), ∃ (L (N ) (ε, λ, h, v(ε, λ, h))) -1 and �(L (N ) (ε, λ, h, v(ε, λ, h)) -1 � 0,0 ≤ N µ0 γ � .
Lemmas 5.3.1, 5.3.2, 5.3.7 and Theorem 4.1.1 give us the following result.

Theorem 5.3.8 (Solutions of the (P )-equation). There exist 1/2 < s 1 ∈ N large enough and a parameter

ε 1 = ε 1 (s 1 , p, q, R, γ) ∈ [0, ε 0 (s 1 , p)) small enough such that ∀ ε ∈ [0, ε 1 ) there exist N 0 = N 0 (ε, s 1 , s, p, γ) ∈ N satisfying N 2(µ+δs1)+2+2σ1 0 γ 4 ε q < 1 2 for σ 1 > max{4(µ + δs 1 + 2), 2(µ + 1)}
and a sequence of functions {v k } k≥0 which converges to a smooth function

v = v(ε, •, •) with respect to parameters (λ, h) v(ε, •, •) ∈ C 1 (Λ × B R (0), E s1+2, p⊥ 0 ), with �v(ε, λ, h)� s1, p → 0 as ε → 0 �∂ λ,h v(ε, λ, h)� s1, p → 0 as ε → 0 (5.80)
which for parameters

(ε, λ, h) ∈ A ∞ := ∩ +∞ k=0 G N k γ,µ0 (v k-1 )
is solution of

F (ε, λ, h, v) = 0
and then of the range equation.

Moreover, ∀ ε ∈ [0, ε 1 ), v(ε, λ, h) ∈ E s, p⊥ 0 for all integers s ≥ s 1 .
We insert now some remarks which we will use in the following for the measure estimates of the set A ∞ .

Remark 5.3.9. Since the definition of singular sites depends only on the parameters (ε, λ), we remark that l is a singular site for L(ε, λ, h, v) if and only if it is a singular site also for L(v(ε, λ, h))) for the same constant θ.

Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension

Remark 5.3.10. Define

A ∞ := � N ≥N0 � (ε, λ, h) ∈ [0, ε 1 ) × Λ × B R (0) : S N = ∪ r Γ N,r satisfies (5.60), ∃ (L (N ) (v(ε, λ, h))) -1 and �(L (N ) (v(ε, λ, h)) -1 � 0,0 ≤ 1 2 N µ0 γ � .
by the Proposition 4.5.1, we have the inclusion

A ∞ ⊂ A ∞ .
(5.81)

We will prove in the next sections that, for an appropriate choice of δ and β (the constants which appear in the "good separation in blocks") the set A ∞ is large: by inclusion (5.81), A ∞ is large too.

Useful results about solutions of the range equation

Follow now some considerations which give us a useful result, used in the next section to suppose the existence of a solution h ∈ E 0 of the bifurcation equation which is smooth with respect to parameters λ.

Take u : T → H 0 , u = h + v with h ∈ E 0 and v ∈ E s1, p⊥ 0 .
Recall for all a ∈ T the translation in time T a defined by

T a u : t � → u(t + a),
since the equivariance of f (ε, u) under the action of the group (T a ) a∈T , we have that the function F defined in (5.42) is (T a )-equivariant too:

F (ε, λ, T a h, T a v) = T a F (ε, λ, h, v). (5.82)
Moreover, we claim that the function ṽ(ε, λ, h) built in Lemma 5.3.2 verifies ṽ(ε, λ, T a h) = T a ṽ(ε, λ, h).

(5.83)

In fact, by construction, ṽ(ε, λ, h) is the unique solution close to zero of

Π Ñ F (ε, λ, h, v) = 0.
Using (5.82) and

T a • Π N = Π N • T a ∀ N ∈ N, (5.84)
we have

Π Ñ F (ε, λ, T a h, T a ṽ(ε, λ, h)) = Π Ñ T a F (ε, λ, h, ṽ(ε, λ, h)) = T a Π Ñ F (ε, λ, h, ṽ(ε, λ, h)) = 0
Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension and then T a ṽ(ε, λ, h) is a solution of

Π Ñ F (ε, λ, T a h, v) = 0
which is close to T a 0 = 0. By Lemma 5.3.2 the equation

Π Ñ F (ε, λ, T a h, v) = 0
has an unique solution ṽ(ε, λ, T a h) close to zero. By uniqueness of this solution, we conclude that ṽ(ε, λ, T a h) = T a ṽ(ε, λ, h).

Recall now that in the Nash-Moser scheme, the solution v(ε, λ, h) is built as the limit of a sequence of functions

{v n (ε, λ, h)} n≥0 . For all n ≥ 0, v n (ε, λ, h) is solution of Π (N ) p-2,⊥ F (ε, λ, h, v) = 0 for parameters (ε, λ, h) ∈ A n .
We have the following result.

Lemma 5.3.11. In the application of the Nash-Moser scheme to the range equation, if we have

F (ε, λ, T a h, T a v) = T a F (ε, λ, h, v), (5.85)
and ṽ(ε, λ, T a h) = T a ṽ(ε, λ, h),

then for all (ε, λ, h) ∈ A n , the property

(P a ) n    (ε, λ, T a h) ∈ A n v n (ε, λ, T a h) = T a v n (ε, λ, h)
holds for all n ∈ N and for all a ∈ T.

Proof. We prove this result by induction on n. Let us consider n = 0.

We start proving that if (ε, λ, h)

∈ A 0 , then (ε, λ, T a h) ∈ A 0 .
Recall the notation using in the Nash-Moser scheme, with v -1 := ṽ, we have

(ε, λ, T a h) ∈ A 0 := G N0 γ,µ0 (ṽ) ⇐⇒ (ε, λ, T a h, ṽ(ε, λ, T a h)) ∈ J N0 γ,µ0 .
By hypothesis (5.86)

(ε, λ, T a h, ṽ(ε, λ, T a h)) ∈ J N0 γ,µ0 ⇐⇒ (ε, λ, T a h, T a ṽ(ε, λ, h)) ∈ J N0 γ,µ0
and by definition (5.70), it means that L N0 (ε, λ, T a h, T a ṽ(ε, λ, h)) is invertible and

�(L N0 ) -1 (ε, λ, T a h, T a ṽ(ε, λ, h))� 0,0 ≤ N µ0 0 γ . infinite dimension
(Recall that the definition of singular sites depends only on the parameters (ε, λ) and not on (h, v) and so the condition on the separation in blocks for the

set S N0 is satisfies if (ε, λ, h) ∈ A 0 ).
By (5.85) and definition of L N0 , for all z ∈ E ⊥(N0) 0,0

L N0 (ε, λ, T a h, T a ṽ(ε, λ, h))[T a z] = T a (L N0 (ε, λ, h, ṽ(ε, λ, h))[z])
i.e.

L N0 (ε, λ, T a h, T a ṽ(ε, λ, h)) • T a = T a • (L N0 (ε, λ, h, ṽ(ε, λ, h))) ⇐⇒ L N0 (ε, λ, T a h, T a ṽ(ε, λ, h)) = T a • (L N0 (ε, λ, h, ṽ(ε, λ, h))) • (T a ) -1 .
Since the action (T a ) a∈T preserves the L 2 -norm, we have

�(L N0 ) -1 (ε, λ, T a h, T a ṽ(ε, λ, h))� 0,0 ≤ N µ0 0 γ ⇐⇒ �T a • (L N0 ) -1 (ε, λ, h, ṽ(ε, λ, h)) • (T a ) -1 � 0,0 ≤ N µ0 0 γ ⇐⇒ �(L N0 ) -1 (ε, λ, h, ṽ(ε, λ, h))� 0,0 ≤ N µ0 0 γ and this is what we have by definition for (ε, λ, h) ∈ A 0 . We prove now that v 0 (ε, λ, T a h) = T a v 0 (ε, λ, h).
In the construction of the Nash-Moser scheme, for parameters (ε, λ, h) ∈ A 0 , v 0 (ε, λ, h) is the unique solution close to ṽ(ε, λ, h) of

Π 0 F (ε, λ, h, v) = 0, Π 0 := Π N0 .
Using (5.84) and (5.85),

Π 0 F (ε, λ, T a h, T a v 0 (ε, λ, h)) = Π 0 T a F (ε, λ, h, v 0 (ε, λ, h)) = T a Π 0 F (ε, λ, h, v 0 (ε, λ, h)) = 0 and then T a v 0 (ε, λ, h) is a solution of Π 0 F (ε, λ, T a h, v) = 0 which is close to T a ṽ(ε, λ, h).
For what proved before, (ε, λ, T a h) ∈ A 0 and so the Theorem 4.1.1 implies that the equation

Π 0 F (ε, λ, T a h, v) = 0
has an unique solution v 0 (ε, λ, T a h) close to ṽ(ε, λ, T a h).

Since ṽ(ε, λ, T a h) = T a ṽ(ε, λ, h), we have by this uniqueness property that

v 0 (ε, λ, T a h) = T a v 0 (ε, λ, h).

Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension

We suppose now that (P a ) n holds and we prove (P a ) n+1 in the same way as in the base case n = 0.

We start proving that if (ε, λ, h) ∈ A n+1 , then (ε, λ, T a h) ∈ A n+1 . We have

(ε, λ, T a h) ∈ A n+1 := ∩ n+1 k=0 G N k γ,µ0 (v k-1 ) ⇐⇒ (ε, λ, T a h, v k-1 (ε, λ, T a h)) ∈ J N k γ,µ0 ∀ k ≤ n + 1. Using the inductive hypothesis v k-1 (ε, λ, T a h) = T a v k-1 (ε, λ, h), ∀ k ≤ n + 1, this is equivalent to (ε, λ, T a h, T a v k-1 (ε, λ, h)) ∈ J N k γ,µ0 ∀ k ≤ n + 1
and by definition (5.70), it means that

L N k (ε, λ, T a h, T a v k-1 (ε, λ, h)) is invert- ible and �(L N k ) -1 (ε, λ, T a h, T a v k-1 (ε, λ, h))� 0,0 ≤ N µ0 k γ .
(Similar considerations for the condition of separation in blocks follow as in the base step).

As done for the base step of induction, we have

�(L N k ) -1 (ε, ω, T a h, T a v k-1 (ε, ω, h))� 0,0 ≤ N µ0 k-1 γ ∀ k ≤ n + 1 ⇐⇒ �(L N k ) -1 (ε, ω, h, v k-1 (ε, ω, h))� 0,0 ≤ N µ0 k γ ∀ k ≤ n + 1
and this is what we have by (ε, ω, h) ∈ A n .

We prove now that v n+1 (ε, ω, T a h) = T a v n+1 (ε, ω, h).

In the construction of the Nash-Moser scheme, for parameters (ε, λ, h) ∈ A n+1 ,

v n+1 (ε, λ, h) is the unique solution close to v n (ε, λ, h) of Π n+1 F (ε, λ, h, v) = 0, Π n+1 := Π Nn+1 .
Then, as done in the base step of induction, we can prove that

T a v n+1 (ε, λ, h) is a solution of Π n+1 F (ε, λ, T a h, v) = 0 which is close to T a v n (ε, λ, h).
For what proved before, (ε, λ, T a h) ∈ A n+1 and so Theorem 4.1.1 implies that the equation

Π n+1 F (ε, λ, T a h, v) = 0 has an unique solution v n+1 (ε, λ, T a h) close to v n (ε, λ, T a h).
By inductive hypothesis, v n (ε, λ, T a h) = T a v n (ε, λ, h) and we conclude by this uniqueness property that

v n+1 (ε, λ, T a h) = T a v n+1 (ε, λ, h).

Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension

Recall that v is the limit of the sequence {v n } n≥0 in the Nash-Moser scheme and that it is solution of the range equation for parameters (ε, λ, h) ∈ A ∞ .

In the Nash-Moser scheme (see Lemma 4.3.4), we define the solution v(ε, •, •)

on the whole set of parameters (ε, λ, h) ∈ [0, ε 1 ) × Λ × B R (0) using a sequence of cut-off functions {ψ n } n≥0 .
Since the sets A n are invariant under the action of group (T a ) a∈T , for all n ≥ 0, we take ψ n invariant under (T a ) a∈T too:

ψ n = ψ n • T a .
This condition and Lemma 5.3.11 imply

(ε, λ, h) ∈ A ∞ =⇒ (ε, λ, T a h) ∈ A ∞ and ∀ (ε, λ, h) ∈ [0, ε 1 ) × Λ × B R (0) v(ε, λ, T a h) = T a v(ε, λ, h).
(5.87)

The bifurcation equation

We consider the (Q)-equation (called also bifurcation equation)

(ω 2 -ω2 )h tt = εΠ E0 f (ε, h + v) (Q)
in the finite dimensional space

E 0 := Ker(ω 2 ∂ tt + A).
Using the frequency-amplitude relation

ω = ω(ε, λ) ω = (1 -ελ)ω, λ ∈ [a, b]
and dividing by ε � = 0, we write the (Q)-equation as

(ελ 2 -2λ)ω 2 h tt -Π E0 f (ε, h + v) = 0. ( 5.88)
We claim that, for any v ∈ E s1, p⊥ 0 , solving the bifurcation equation (5.88) for ε � = 0 is equivalent to finding critical points of the reduced Lagrangian action

functional h ∈ E 0 → ε -1 Φ(ε, λ, h + v) ∈ R,
where Φ is the functional introduced in (5.34).

In fact, using the notation

D h Φ(ε, λ, h + v) := DΦ(ε, λ, h + v) |E 0 ,
we have the following result. where we have defined for

ε � = 0 Dh Φ(ε, λ, h + v) := ε -1 D h Φ(ε, λ, h + v). (5.90)
Proof. The proof is the same as the one of Remark 2.4.1 adapted to our functional setting. Note that, for ε � = 0,

Dh Φ(ε, λ, h + v) = 0 ⇐⇒ D h Φ(ε, λ, h + v) = 0. For any h ∈ B R (0) ⊂ E 0 , we denote by v(ε, λ, h) the function obtained in Theorem 5.3.8 v(ε, •, •) ∈ C 1 (Λ × B R (0), E s1+2, p⊥ 0 ), with v(0, λ, h) = 0, �v(ε, λ, h)� s1, p → 0 as ε → 0, �∂ (λ,h) v(ε, λ, h)� s1, p → 0 as ε → 0
which is solution of the range equation for parameters (ε, λ, h) ∈ A ∞ .

Moreover, in (5.87) we have

v(ε, λ, T a h) = T a v(ε, λ, h)
which means that if v(ε, λ, h) is solution of the range equation associated to parameters (ε, λ, h), T a v(ε, λ, h) is still a solution of the range equation associated to parameters (ε, λ, T a h).

We want to solve the (Q)-equation (5.88

) for v = v(ε, λ, h) (ελ 2 -2λ)ω 2 h tt -Π E0 f (ε, h + v(ε, λ, h)) = 0 (5.91) which by Remark 5.4.1 is equivalent for ε � = 0 to ( Dh Φ)(ε, λ, h + v(ε, λ, h)) = 0. Moreover, introduce the functional Ψ(ε, λ) ∈ C 1 (B R (0), R) Ψ(ε, λ, h) := Φ(ε, λ, h + v(ε, λ, h)), (5.92)
we have the following remark. where we have defined for ε � = 0

DΨ(ε, λ, h) := ε -1 DΨ(ε, λ, h). (5.93)
Proof. The proof is the same as the one of Remark 2.4.3 adapted to our functional setting. Note that, for ε � = 0

DΨ(ε, λ, h) = 0 ⇐⇒ DΨ(ε, λ, h) = 0. Property v(ε, λ, T a h) = T a v(ε, λ, h) implies the following result.
Ψ(ε, λ, h) = Ψ(ε, λ, T a h) (5.94)
and

(D h Φ)(ε, λ, h + v(ε, λ, h)) = 0 ⇒ (D h Φ)(ε, λ, T a h + v(ε, λ, T a h)) = 0. (5.95)
Proof. By definition (5.92), using (5.37) and property v(ε, λ,

T a h) = T a v(ε, λ, h),
we write

Ψ(ε, λ, h) := Φ(ε, λ, h + v(ε, λ, h)) = Φ(ε, λ, T a (h + v(ε, λ, h))) = Φ(ε, λ, T a h + T a v(ε, λ, h))) = Φ(ε, λ, T a h + v(ε, λ, T a h))) =: Ψ(ε, λ, T a h). Moreover, from Φ(ε, λ, h + v) = Φ(ε, λ, T a h + T a v), we deduce D h Φ(ε, ω, h + v)[g] = D h Φ(ε, ω, T a h + T a v)[T a g], ∀ g ∈ E 0 . (5.96) Write (5.96) for v = v(ε, λ, h), we obtain ∀ g ∈ E 0 (D h Φ)(ε, ω, h + v(ε, λ, h))[g] = (D h Φ)(ε, ω, T a h + T a v(ε, λ, h))[T a g] = (D h Φ)(ε, ω, T a h + v(ε, λ, T a h))[T a g]
which implies (5.95) since T a : E 0 → E 0 is an isomorphism.

Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension By (5.95) of Remark 5.4.3, the zeroes of (D h Φ)(ε, λ, h + v(ε, λ, h)) are not isolated and they are included in circles of zeroes: if we have

(D h Φ)(ε, λ, h + v(ε, λ, h)) = 0, define S h = {g ∈ E 0 : ∃ a ∈ R | g = T a h, },
we have

S h ⊂ {g ∈ E 0 : (D h Φ)(ε, λ, g + v(ε, λ, g)) = 0}.
We can see the (Q)-equation ( 5.91) as a perturbation of a "simple" (Q 0 )-equation

-2λω 2 h tt -Π E0 f (0, h) = 0 (5.97) equivalent to Dh Φ(0, λ, h) = 0
where, as a consequence of the division by ε � = 0,

Dh Φ(0, λ, h) := lim ε→0 ε -1 D h Φ(ε, λ, h).
Note that, for ε = 0, v(0, λ, h) = 0 and

Dh Φ(0, λ, h) = DΨ(0, λ, h) (5.98)
where Ψ(0, λ, h) is defined in (5.38)

Ψ(0, λ, h) := � T λω 2 �h t , h t � H 0 -F(0, h) dt.
We introduce the following definition of non-degenerate solution for the (Q) 0 -equation, which it will be justified in the following (see (5.101) and (5.108)).

Definition 5.4.4 ("Non-degenerate" solutions of the (Q) 0 -equation). A couple (λ 0 , h 0 ) ∈ Λ × E 0 is a non-trivial (i.e. (λ 0 , h 0 ) � = (0, 0)) and nondegenerate solution of the (Q) 0 -equation if and only if

   DΨ(0, λ 0 , h 0 ) = 0, dim(KerD 2 Ψ(0, λ 0 , h 0 )) = 1.
(5.99)

We suppose there exists a couple (λ 0 , h 0 ) ∈ E 0 × Λ which is a non-trivial and non-degenerate solution of the (Q) 0 -equation with respect to Definition 5.4.4.

We define R > 0 (the one which defines the ball B R (0) ⊂ E 0 that we consider in the resolution of the range equation) such that

�h 0 � 0,0 ≤ R -1 infinite dimension i.e. h 0 ∈ B R-1 (0) := {h ∈ E 0 : �h� 0,0 < R -1}.
(5.100) Uising (5.98), we write

dim(KerD 2 Ψ(0, λ 0 , h 0 )) = 1 ⇔ KerD 2 Ψ(0, λ 0 , h 0 ) = span( ḣ0 ) (5.101)
where ḣ0 = d da T a h 0 is the tangent vector in h 0 to the cercle S h0 . We consider the vector space

Z h0 := � span( ḣ0 ) � ⊥ = {H ∈ E 0 : �H, ḣ0 � L 2 = 0}
where we denote by simplicity L 2 := L 2 (T, H 0 ) and the affine space

G h0 := h 0 + Z h0 = {h ∈ E 0 : ∃ H ∈ Z h0 | h = h 0 + H} = {h ∈ E 0 : �h -h 0 , ḣ0 � L 2 = 0}.
Define Ψ(0, λ, •) := Ψ(0, λ, •) |G h 0 , we have the following remark.

Remark 5.4.5. The following equivalence holds:

KerD 2 Ψ(0, λ 0 , h 0 ) = span( ḣ0 ) ⇐⇒ KerD 2 Ψ(0, λ 0 , h 0 ) = {0}.
Proof. Let us consider the functions DΨ(0, λ 0 , h 0 ) :

E 0 → R, D 2 Ψ(0, λ 0 , h 0 ) : E 0 × E 0 → R and D Ψ(0, λ 0 , h 0 ) : Z h0 → R, D 2 Ψ(0, λ 0 , h 0 ) : Z h0 × Z h0 → R
and let us start to prove that KerD 2 Ψ(0, λ 0 , h 0 ) ⊂ KerD 2 Ψ(0, λ 0 , h 0 ).

(5.102)

Using the time translation invariance (5.94)

Ψ(0, λ 0 , T a h 0 ) = Ψ(0, λ 0 , h 0 )
and derivating with respect to a DΨ(0, λ 0 , T a h 0 ) d da T a h 0 = 0, we obtain for a = 0 DΨ(0, λ 0 , h 0 ) ḣ0 = 0.

(5.103) Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension Derivating (5.103), we have

DΨ(0, λ 0 , h 0 ) ġ + D 2 Ψ(0, λ 0 , h 0 )[g, ḣ0 ] = 0 ∀ g ∈ E 0 .
(5.104)

Using DΨ(0, λ 0 , h 0 ) = 0 in (5.104) (because we have supposed that (λ 0 , h 0 ) is a non-trivial and non-degenerate solution of the (Q 0 )-equation), we obtain

D 2 Ψ(0, λ 0 , h 0 )[g, ḣ0 ] = 0 ∀ g ∈ E 0 .
(5.105)

Take now H ∈ KerD 2 Ψ(0, λ 0 , h 0 ) ⊂ Z h0 , i.e. by definition of KerD 2 Ψ(0, λ 0 , h 0 )

D 2 Ψ(0, λ 0 , h 0 )[H, G] = 0 ∀ G ∈ Z h0 .
Since Ψ(0, λ 0 ) = Ψ(0, λ 0 ) |G h 0 , where G h0 := h 0 + Z h0 , we have that

D 2 Ψ(0, λ 0 , h 0 ) = D 2 Ψ(0, λ 0 , h 0 ) on Z h0 × Z h0
and so

D 2 Ψ(0, λ 0 , h 0 )[H, G] = D 2 Ψ(0, λ 0 , h 0 )[H, G] = 0 ∀ G ∈ Z h0 . ( 5
D 2 Ψ(0, λ 0 , h 0 )[H, ∆] = 0 ∀ ∆ ∈ E 0 i.e. H ∈ KerD 2 Ψ(0, λ 0 , h 0 ).
The inclusion (5.102) is so proved

KerD 2 Ψ(0, λ 0 , h 0 ) ⊂ KerD 2 Ψ(0, λ 0 , h 0 ).
We conclude noting that KerD 2 Ψ(0, λ 0 , h 0 ) = span( ḣ0 ) implies

Z h0 ∩ KerD 2 Ψ(0, λ 0 , h 0 ) = {0}
and so KerD 2 Ψ(0, λ 0 , h 0 ) ⊂ Z h0 ∩ KerD 2 Ψ(0, λ 0 , h 0 ) = {0}.

F ε ∈ C 1 (U, Z) (5.109) defined on a open subset U ⊂ X × Y of the Banach space (X × Y, � • � X×Y )
with values in the Banach space (Z, � • � Z ) such that Moreover, we suppose that

F ε → F 0 in C 0 (U, Z) as ε → 0 (5.
D x F ε → D x F 0 in C 0 (U, L(X, Z)) as ε → 0 (5.112) D y F ε → D y F 0 in C 0 (U, L(Y, Z)) as ε → 0 (5.113) where the � • � L ∞ (U,L(X,Z)) -norm and � • � L ∞ (U,L(Y,Z))
-norm are defined properly as in (5.111).

Suppose there exists a couple (x 0 , y 0 ) ∈ U ⊂ X × Y such that F 0 (x 0 , y 0 ) = 0 and D y F 0 (x 0 , y 0 ) ∈ L(Y, Z) is invertible.

(5.114)

Then there exist ε ∈ (0, ε 1 ), δ 0 > 0 and a constant L > 0 such that ∀ ε ∈ [0, ε) there exists a C 1 map

y ε : B δ0 (x 0 ) → Y x � → y ε (x)
Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension where

B δ0 (x 0 ) := {x ∈ X : �x -x 0 � X < δ 0 } ⊂ X such that (x, y ε (x)) verifies F ε (x, y ε (x)) = 0, �Dy ε ∆� Y ≤ L�∆� X , ∀ ∆ ∈ X, y ε → ȳ0 in C 0 (B δ0 (x 0 ), Y ) as ε → 0, for some ȳ0 : B δ0 (x 0 ) → Y, ȳ0 (x 0 ) = y 0 .
We want to apply the Implicit function Theorem 5.4.6 to the family of functionals

( Dh Φ)(ε, λ, h + v(ε, λ, h)) | Z h 0 for h ∈ G h0 := h 0 + Z h0
which we write as

F ε (λ, H) := ( DH Φ)(ε, λ, h 0 + H + v(ε, λ, h 0 + H)) | Z h 0 with U = (a, b) × B 1 (0), X = R, Y = E 0 and Z = L(Z h0 , R).
Lemma 5.4.7. The family of functionals It is easy to check that

F ε (λ, H) = ( DH Φ)(ε, λ, h 0 + H + v(ε, λ, h 0 + H)) | Z h 0
( Dh Φ)(ε,λ, h + v(ε, λ, h)) = = � (ελ 2 -2λ)ω 2 h tt -Π E0 f (ε, h + v(ε, λ, h)), • � L 2 (T,H 0 )
from which we define the vector Fε (λ, H) which represents F ε (λ, H)

Fε (λ, H) := (ελ 2 -2λ)ω 2 � (h 0 ) tt + H tt � -Π E0 f (ε, h 0 + H + v(ε, λ, h 0 + H))
.

By regularity of v(ε, •, •) ∈ C 1 (Λ×B R (0), E s1, p⊥ 0 ) and of function f ∈ C ∞ (H s1, p, H s1, p) (see Remark 5.1.3), it is clear that Fε ∈ C 1 ((a, b) × B 1 (0), E 0 ).
Moreover, we compute

∂ λ Fε (λ, H) =(2ελ -2)ω 2 � (h 0 ) tt + H tt � -Π E0 D u f (ε, h 0 + H + v(ε, λ, h 0 + H))∂ λ v(ε, λ, h 0 + H)
Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension and

D H Fε (λ, H)[G] =(ελ 2 -2λ)ω 2 G tt -Π E0 D u f (ε, h 0 + H + v(ε, λ, h 0 + H))[G] -Π E0 D u f (ε, h 0 + H + v(ε, λ, h 0 + H))∂ h v(ε, λ, h 0 + H)[G].
Using that v(0, λ, h) = 0 and

�v(ε, λ, h)� s1, p, �∂ (λ,h) v(ε, λ, h)� s1, p → 0 as ε → 0,
it is clear that also hypothesis (5.110), (5.112) and (5.113) are verified where, by (5.97), we have

D λ F0 (λ, H) = -2ω 2 � (h 0 ) tt + H tt �
and

D H F0 (λ, H)[G] = -2λω 2 G tt -Π E0 D u f (0, h 0 + H)[G].
By regularity of Fε ∈ C 1 ((a, b) × B 1 (0), E 0 ), we have that

F ε = (D H Φ)(ε, •, h 0 + • + v(ε, λ, h 0 + •)) | Z h 0 ∈ C 1 ((a, b) × B 1 (0), L(Z h0 , R)).
There remains to verify hypothesis (5.114).

Since we have supposed to have a couple (λ 0 , h 0 ) which verifies (5.108), using

(5.98) we have that

F ε (λ 0 , 0) = ( DH Φ)(0, λ 0 , h 0 ) | Z h 0 = DΨ(0, λ 0 , h 0 ) | Z h 0 = D Ψ(0, λ 0 , h 0 ) = 0. Moreover, since Z h0 ⊂ E 0 is finite dimensional, condition Ker � ( D2 H Φ(0, λ 0 , h 0 ) | Z h 0 ×Z h 0 � = KerD 2 Ψ(0, λ 0 , h 0 ) = {0}
implies that the differential

DF ε (λ 0 , 0) = D2 H Φ(0, λ 0 , h 0 ) ∈ L(Z h0 , L(Z h0 , R)) is invertible.
Hence, the Implicit function Theorem 5.4.6 and Lemma 5.4.7 give us the existence of ε < ε 1 , δ 0 > 0 and a constant L > 0 such that ∀ ε ∈ [0, ε) there

exists a map H ε ∈ C 1 (Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ), E 0 ), �∂ λ H ε (λ)� 0,0 ≤ L H ε → H0 as ε → 0 infinite dimension for some H0 : Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ) → E 0 , H0 (λ 0 ) = 0, such that ( DH Φ)(ε, λ, h 0 + H ε (λ) + v(ε, λ, h 0 + H ε (λ))) | Z h 0 = 0. Let us define h ε := h 0 + H ε ∈ C 1 (Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ), E 0 ). We have �∂ λ h ε (λ)� 0,0 ≤ L, h ε → h0 as ε → 0 for some h0 : Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ) → E 0 , h0 (λ 0 ) = h 0 and ( Dh Φ)(ε, λ, h ε (λ) + v(ε, λ, h ε (λ))) |Z h 0 = 0. (5.115)
In particular, for ε and δ 0 small enough, using (5.100) for which h 0 ∈ B R-1 (0), we can have

h ε (λ) ∈ B R (0).
The following remark concludes the proof of the existence of a solution of the bifurcation equation (5.91) which depends smoothly on λ.

( Dh Φ)(ε, λ, h ε (λ) + v(ε, λ, h ε (λ)))H = 0, ∀ H ∈ Z h0
and we can write

( Dh Φ)(ε, λ, h ε (λ) + v(ε, λ, h ε (λ)))G = α� ḣ0 , G� L 2 ∀ G ∈ E 0 (5.116)
for some α ∈ R.

DΨ(ε, λ, h ε (λ)) ḣε (λ) = DΦ(ε, λ, h ε (λ) + v(ε, λ, h ε (λ))) � ḣε (λ) + Dv(ε, λ, h ε (λ)) ḣε (λ) � = 0. If (ε, λ, h ε (λ)) ∈ A ∞ , v(ε, λ, h ε (λ))
(λ) + v(ε, λ, h ε (λ))) | E s 1 , p⊥ 0 = 0. Since Dv(ε, λ, h ε (λ)) ḣε (λ) ∈ E s1, p⊥ 0 , we have DΦ(ε, λ, h ε (λ) + v(ε, λ, h ε (λ)))Dv(ε, λ, h ε (λ)) ḣε (λ) = 0. Hence, DΨ(ε, λ, h ε (λ)) ḣε (λ) = DΦ(ε, λ, h ε (λ) + v(ε, λ, h ε (λ))) ḣε (λ) = 0.
Using (5.116), we have DΨ(ε, λ, h ε (λ)) ḣε (λ) = α� ḣ0 , ḣε (λ)� L 2 = 0.

For ε and δ 0 small enough, h ε (λ) is close to h 0 and so � ḣ0 , ḣε (λ)� L 2 � = 0.

This implies that α = 0 and so (5.116) is reduced to

( Dh Φ)(ε, λ, h ε (λ) + v(ε, λ, h ε (λ)))G = 0 ∀ G ∈ E 0 i.e. ( Dh Φ)(ε, λ, h ε (λ) + v(ε, λ, h ε (λ))) = 0.
In this section, we have so proved the following result.

Theorem 5.4.9 (Solutions of the (Q)-equation). Suppose there exists a couple (0, 0) � = (λ 0 , h 0 ) ∈ Λ × E 0 which verifies

   DΨ(0, λ 0 , h 0 ) = 0, dim(KerD 2 Ψ(0, λ 0 , h 0 )) = 1.
Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension

Then there exist ε ∈ (0, ε 1 ), δ 0 > 0 small enough and a constant L > 0 such that ∀ ε ∈ [0, ε) there exists a function

h ε ∈ C 1 (Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ), B R (0)), �∂ λ h ε (λ)� 0,0 ≤ L h ε → h0 as ε → 0 for some h0 : Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ) → E 0 , h0 (λ 0 ) = h 0 such that if (ε, λ, h ε (λ)) ∈ A ∞ , h ε (λ) is a non-trivial solution of the bifurcation equation (ελ 2 -2λ)ω 2 h tt -Π E0 f (ε, h + v(ε, λ, h)) = 0.

Periodic solutions of the whole equation

In Theorem 5.3.8, using a Nash-Moser scheme, we have obtained for all fixed ε ∈ [0, ε 1 ), functions v = v(ε, λ, h), smooth with respect to parameters (λ, h) ∈ Λ × B R (0), which are solutions of the (P )-equation only for (ε, λ, h) ∈ A ∞ .

Moreover, by Remark 5.3.10, we have the inclusion

A ∞ ⊂ A ∞ with A ∞ := � N ≥N0 � (ε, λ, h) ∈ [0, ε 1 ) × Λ × B R (0) : S N = ∪ r Γ N,r satisfies (5.60), ∃ (L (N ) (v(ε, λ, h))) -1 and �(L (N ) (v(ε, λ, h)) -1 � 0,0 ≤ 1 2 N µ0 γ � .
In Theorem 5.4.9, for all fixed ε ∈ [0, ε), ε < ε 1 and under condition of existence of a non-degenerate solution of the (Q) 0 -equation, we have obtained non-trivial

solutions h ε (λ) of the (Q)-equation in C 1 � Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ) � .
With the C 1 -dependences on λ of functions v(ε), h ε , for all ε ∈ [0, ε), let us introduce the set

A ε := {λ ∈ Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ) : (ε, λ, h ε (λ)) ∈ A ∞ }.
The aim of this section is to prove positive measure estimates for the set A ε .

By definition, for v(ε, λ) := v(ε, λ, h ε (λ)),

A ε = � N ≥N0 � λ ∈ Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ) : S N = ∪ r Γ N,r verifies (5.60), ∃ (L (N ) (v(ε, λ))) -1 and �(L (N ) (v(ε, λ))) -1 � 0,0 ≤ 1 2 N µ0 γ � ,
Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension and we write

A ε = A 1 ε ∩ A 2 ε
where

A 1 ε := � N ≥N0 � λ ∈ Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ) : ∃ (L (N ) (v(ε, λ))) -1 and �(L (N ) (v(ε, λ))) -1 � 0,0 ≤ 1 2 N µ0 γ � and A 2 ε := � N ≥N0 � λ ∈ Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ) : S N = ∪ r Γ N,r verifies (5.60), � .
For proving that A 1 ε and A 2 ε are large sets, we will use that the operator A verifies Assumption 5.1.2, that we recall in the following.

L (N ) (v(ε, λ)) = (1-ελ) 2 ω2 ∂ tt +A-εΠ ⊥ D u f (ε, h ε (λ)+ v(ε, λ, h ε (λ))) (5.120)
Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension where we write Π ⊥ = Π (N ) p-2,⊥ to simplify the notations. The operator A is self-adjoint with respect to the scalar product in H 0 . Moreover

f (ε, g) = ∇ g F(ε, g) implies that D g f (ε, g) is self-adjoint too in H 0 �D g f (ε, g)g 1 , g 2 � H 0 = �g 1 , D g f (ε, g)g 2 � H 0 , ∀ g 1 , g 2 ∈ H 0 .
Hence the operator L (N ) is self-adjoint as an operator from

D(L (N ) ) ⊂ L 2 (T, H 0 ) into L 2 (T, H 0 ).
For a linear continuous operator P in L(L 2 (T, H 0 )), we recall the operator norm (5.5)

�P � 0,0 := sup u� =0
�P u� 0,0 �u� 0,0 .

As done in Remark 5.1.3, (only) for the operator

D u f (ε, h + v) ∈ L(L 2 (T, H 0 )),
we will use the notation �D u f (ε, h + v)�

(op) 0,0 for its operator norm.

We define the sets

E N 1 := {(l, j, k) : |l| ≤ N ; l 2 < K 1 or |ω j,k | > K 2 ω|l|, 1 ≤ k ≤ d j }, E N 2 := {(l, j, k) : |l| ≤ N ; l 2 ≥ K 1 and |ω j,k | ≤ K 2 ω|l|, 1 ≤ k ≤ d j }
with an appropriate choice of constants K 1 and K 2 which will be specified later.

Recall that we can take N > N 0 , with N 0 the term of initialization in the Nash-Moser scheme sufficiently large such that K 1 < N 2 0 and E N 2 is never an empty set.

Under assumption 5.1.2, the set E N 2 has the finite cardinality

card(E N 2 ) ≤ CN d+1 r +1 .
In fact, we estimate

card(E N 2 ) ≤ � |l|≤N � j r ≤C|l| c 4 j d ≤ � |l|≤N � j≤C|l| 1/r c 4 j d ≤ � |l|≤N C|l| d+1 r ≤ CN d+1 r +1
.

Let E N 1 , E N 2 be the subspaces of H s, p corresponding respectively to the indices sets E N 1 and E N 2 and let Π E N 1 , Π E N 2 be the orthogonal projectors respectively on E N 1 and E N 2 .

Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension

We write L (N ) (v(ε, λ)) = D (N ) + R (N ) , with D the diagonal part which in the Hilbert basis {e ılt ϕ j,k } l,j,k of L 2 (T, H 0 C ) we write as

D (N ) := (1 -ελ) 2 ω2 ∂ tt + A = diag |l|≤N,j,k {-(1 -ελ) 2 ω2 l 2 + ω 2 j,k }, (5.121)
and

R (N ) := -εΠ ⊥ D u f (ε, h ε (λ) + v(ε, λ, h ε (λ))) | E ⊥(N ) s, p .
(5.122)

Using that �h ε (λ)� s1, p ≤ C(s 1 , p)R and �v(ε, λ, h ε (λ))� s1, p → 0 as ε → 0, we estimate for ε small enough

�R (N ) � 0,0 = ε�Π ⊥ D u f (ε, h ε (λ) + v(ε, λ, h ε (λ)))� (op) 0,0 ≤ ε�Π ⊥ D u f (ε, h ε (λ) + v(ε, λ, h ε (λ)))� L ∞ (T,L(H 0 )) ≤ εC(�h ε (λ)� s1, p, �v(ε, λ, h ε (λ))� s1, p) ≤ Cε (5.123)
where we have used also that ∀ t ∈ T, by (5.10)

�D u f (ε, h ε (λ)(t) + v(ε, λ, h ε (λ))(t))� L(H 0 ) ≤ C(�h ε (λ)(t)� p, �v(ε, λ, h ε (λ))(t)� p) ≤ C(�h ε (λ)� s1, p, �v(ε, λ, h ε (λ))� s1, p)
which implies

�D u f (ε, h ε (λ) + v(ε, λ, h ε (λ))� L ∞ (T,L(H 0 )) := sup t∈T �D u f (ε, h ε (λ)(t) + v(ε, λ, h ε (λ))(t))� L(H 0 )
≤ C(�h ε (λ)� s1, p, �v(ε, λ, h ε (λ))� s1, p).

(5.124)

More precisely, we write

L (N ) (v(ε, λ)) = � B (N ) (ε, λ) R (N ) 1 (ε, λ) R (N ) 2 (ε, λ) C (N ) (ε, λ) � (5.125)
where

B (N ) (ε, λ) := Π E N 1 L (N ) (v(ε, λ)) | E N 1 , R (N ) 1 (ε, λ) := Π E N 1 L (N ) (v(ε, λ)) | E N 2 = Π E N 1 (R (N ) ) | E N 2 C (N ) (ε, λ) := Π E N 2 L (N ) (v(ε, λ)) | E N 2 , R (N ) 2 (ε, λ) := Π E N 2 L (N ) (v(ε, λ)) | E N 1 = Π E N 2 (R (N ) ) | E N 1 .
We have that B (N ) (ε, λ) and C (N ) (ε, λ) are self-adjoint and R

(N ) 2 (ε, λ) = (R (N ) 1 (ε, λ)) T .

infinite dimension

In order to prove that

A 1 ε := � N ≥N0 � λ ∈ Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ) : ∃ (L (N ) (v(ε, λ))) -1 and �(L (N ) (v(ε, λ))) -1 � 0,0 ≤ 1 2
N µ0 γ � is a large set, we will apply the following Lemma which is a particular case (q = 1) of a more general result proved in Appendix B, see Lemma B.2.3.

Lemma 5.5.1. Let L(ε, λ) be a self-adjoint operator defined on a Hilbert space of the form

L(ε, λ) = � B(ε, λ) R 1 (ε, λ) R 2 (ε, λ) C(ε, λ) � with C(ε, λ) defined on a M -finite dimensional space.
Assume that B(ε, λ) is invertible and that there exist positive constants

c 0 , C 1 , C 2 , C 3 , C 4 such that (i) �R i (ε, λ)� 0,0 ≤ εC 1 , (ii)
d dλ C(ε, λ) ≥ εc 0 I, (iii) �B -1 (ε, λ)� 0,0 ≤ 2 C2 , (iv) � d dλ R i (ε, λ)� 0,0 ≤ εC 3 , (v)
� d dλ (B -1 )(ε, λ)� 0,0 ≤ εC 4 . Then for ε small enough, there exists a constant K > 0 such that for all δ > 0 small enough meas

� λ ∈ Λ :� ∃ (L (N ) (ε, λ)) -1 or �(L(ε, λ)) -1 � 0,0 ≥ 1 δ � ≤ K δM εc 0 .
We show that we have all the hypotheses required and so the measure estimate for A 1 ε will follow easily for ε small enough. We write in the following B(λ) and C(λ) instead of B (N ) (ε, λ) and C (N ) (ε, λ) and � • � = � • � 0,0 (both for the operator norm in L(L 2 (T, H 0 )) and for the norm of functions in L 2 (T, H 0 )) to simplify the notations. e ilt w l , we have

�-∂ tt w, w� L 2 (T,H 0 ) = � |l|≤N l 2 ≥K1 l 2 �w l � 2 0 ≥ K 1 � |l|≤N l 2 ≥K1 �w l � 2 0 ≥ K 1 �w� 2 (5.127)
where we recall that we use the notation

� • � = � • � 0,0 .
Using the C ∞ -regularity of f and the C 1 -regularity of h and v with respect to λ, with �h ε (λ)� s1, p ≤ C(s 1 , p)R, �∂ λ h ε (λ)� s1, p ≤ C(s 1 , p)L (that we have by

�∂ λ h ε (λ)� 0,0 ≤ L) and using �v(ε, λ)� s1, p, �∂ (λ,h) v(ε, λ)� s1, p → 0 as ε → 0, we can prove � � � d dλ b(ε, λ) � � � L ∞ (T,L(H 0 )) ≤ C. (5.128)
for a constant C = C(�h ε (λ)� s1, p, �v(ε, λ)� s1, p, �h ε (λ)� s1, p, �∂ (λ,h) v(ε, λ)� s1, p).

Estimate (5.128) implies for some appropriate constant K > 0

� d dλ b(ε, λ)w, w � L 2 (T,H 0 ) ≤ � � � d dλ b(ε, λ) � � � 2 L ∞ (T,L(H 0 )) �w� 2 ≤ K�w� 2 .
(5.129) By (5.127) and (5.129), taking K 1 large enough with respect to K, K 1 > K,

we have hypothesis (ii).

For proving (iii), we show first that

(l, j, k) ∈ E N 1 ⇒ | -(1 -ελ) 2 ω2 l 2 + ω 2 j,k | ≥ C ω2 �l� 2 (5.130)
for some appropriate small constant C > 0, where �l� := max{1, |l|}.

By definition of E N 1 , we have to distinguish two different cases for (l, j, k).

• First case: |l| ≤ N and l 2 < K 1 .

By (5.32), for ε small enough, we estimate

| -(1 -ελ) 2 ω2 l 2 + ω 2 j,k | ω2 �l� 2 ≥ ω-2 � Cγ �l� 2 + �l� τ0+2 -εc � ≥ ω-2 � Cγ K 1 + K τ 0 +2 2 1 -εc � ≥ C
Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension taking ε small enough and some C > 0.

• Second case: |l| < N and ω j,k > K 2 ω|l|.

We have

| -(1 -ελ) 2 ω2 l 2 + ω 2 j,k | > -(1 -ελ) 2 ω2 l 2 + K 2 2 ω2 l 2 > ω2 �l� 2
if we take K 2 large enough.

Then, we have proved (5.130) with a constant C properly chosen.

By (5.130), we deduce that B(λ) is invertible and that estimate (iii) holds for an appropriate choice of C 2 .

In fact, we write

B(λ) = D B (λ) + R B (λ) (5.131) with D B (λ) = diag (l,j,k)∈E N 1 {-(1 -ελ) 2 ω2 l 2 + ω 2 j,k } (5.132)
and by (5.123)

�R B (λ)� ≤ εC R . (5.133) Since D -1 B (λ) = diag (l,j,k)∈E N 1 {(-(1 -ελ) 2 ω2 l 2 + ω 2 j,k ) -1 }, (5.134)
by (5.130), there exists a constant C 2 > 0 such that for ε small enough

�D -1 B � ≤ 1 C 2 .
(5.135) By Lemma B.1.1, we obtain (iii): if ε is small enough such that

�D -1 B (λ)R B (λ)� ≤ �D -1 B (λ)��R B (λ)� ≤ εC R C 2 < 1 2 , (5.136)
then B(λ) is invertible and

�B -1 (λ)� ≤ 2 C 2 .
(5.137)

In order to prove (iv), it is enough to show that

� � � � d dλ � L (N ) (v(ε, λ)) -D (N ) � � � � � < εC 3 .
(5.138) Using (5.128), we have for some appropriate constant C 3 large enough

� � � � d dλ � L (N ) -D (N ) � w � � � � ≤ � � � � ε d dλ � b(ε, λ) � w � � � � ≤ ε � � � d dλ b(ε, λ) � � � L ∞ (T,L(H 0 )) �w� ≤ εC 3 �w�.
Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension

Finally, we want to prove (v).

Applying Lemma B.1.2, we write

d dλ (B -1 )(λ) = -B -1 (λ) � d dλ B(λ) � B -1 (λ) (5.139)
where by (5.131)

d dλ B(λ) = d dλ D B (λ) + d dλ R B (λ) (5.140)
and by (5.138)

� � � d dλ R B (λ) � � � ≤ εC 3 .
(5.141)

Write on the one hand

B -1 (λ) = (D B (λ) + R B (λ)) -1 = (D B (λ)(I + D -1 B (λ)R B (λ))) -1 = (I + D -1 B (λ)R B (λ)) -1 D -1 B (λ) (5.142)
and on the other hand

B -1 (λ) = (D B (λ) + R B (λ)) -1 = ((I + R B (λ)D -1 B (λ))D -1 B (λ)) = D -1 B (λ)(I + R B (λ)D -1 B (λ)) -1 , (5.143)
using (5.139) and (5.140), we obtain

� � � d dλ (B -1 )(λ) � � � = � � �B -1 (λ) � d dλ B(λ) � B -1 (λ) � � � = �(I + D -1 B (λ)R B (λ)) -1 D -1 B (λ) � d dλ D B (λ) + d dλ R B (λ) � D -1 B (λ)(I + R B (λ)D -1 B (λ)) -1 � ≤ �(I + D -1 B (λ)R B (λ)) -1 �× �� � �D -1 B (λ) d dλ D B (λ)D -1 B (λ) � � � + � � �D -1 B (λ) d dλ R B (λ)D -1 B (λ) � � � � × �(I + R B (λ)D -1 B (λ)) -1 �.
We have to estimate

� � �D -1 B (λ) d dλ D B (λ)D -1 B (λ) � � �, � � �D -1 B (λ) d dλ R B (λ)D -1 B (λ) � � �.
By (5.132),

d dλ D B (λ) = diag (l,j,k)∈E N 1 � d dλ � -(1 -ελ) 2 ω2 l 2 + ω 2 j,k �� = diag (l,j,k)∈E N 1 � 2ε(1 -ελ)ω 2 l 2 �
Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension and by (5.134), we have

D -1 B (λ)D � B (λ)D -1 B (λ) = diag (l,j,k)∈E N 1 � 2ε(1 -ελ)ω 2 l 2 (-(1 -ελ) 2 ω2 l 2 + ω 2 j,k ) 2 � .
By (5.130) and (5.141)

� � �D -1 B (λ) d dλ D B (λ)D -1 B (λ) � � � ≤ ε C 2 and � � �D -1 B (λ) d dλ R B (λ)D -1 B (λ) � � � ≤ ε C 2 . Now, applying Lemma B.1.1, we obtain �(I + D -1 B (λ)R B (λ)) -1 � ≤ 2
and

�(I + R B (λ)D -1 B (λ)) -1 � ≤ 2 if ε is such that εC R C < 1 2
. Hence, we conclude the proof of (v) with

� � � d dλ (B -1 )(λ) � � � ≤ 4ε C 2 + 4ε C 2 ≤ εC 4 (5.144) for a constant C 4 ≥ 8/C 2 .
By Lemmas 5.5.1 and 5.5.2, we can deduce meseaure estimates for the set

A 1 ε . In fact, recall A 1 ε := � N ≥N0 � λ ∈ Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ) : ∃ (L (N ) (v(ε, λ))) -1 and �(L (N ) (v(ε, λ))) -1 � 0,0 ≤ 1 2 N µ0 γ � then (A 1 ε ) c := � N ≥N0 � λ ∈ Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ) : �(L (N ) (v(ε, λ))) -1 � 0,0 > 1 2 N µ0 γ � .
Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension

We estimate

meas((A 1 ε ) c ) ≤ � N ≥N0 meas � λ ∈ Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ) : �(L (N ) (v(ε, λ))) -1 � 0,0 > 1 2 N µ0 γ � ≤ � N ≥N0 K εc 0 2γ N µ0 CN d+1 r +1 = � N ≥N0 K � γ εc 0 N d+1 r +1-µ0 ≤ K � γ εc 0 N d+1 r +2-µ0 0 (5.145)
which is small if we take N 0 = N 0 (ε) large enough and µ 0 > d+1 r + 2 such that

K � γ εc 0 N d+1 r +2-µ0 0 << 1.
(5.146)

Note that ∀ ε ∈ [0, ε), ε ≤ ε 1 ,
γ 4 ε q < 1 2
if we take q large enough.

Measure estimates for the set A 2 ε

We want to prove here that we have good properties of separation of the singular sites as required in the hypothesis (5.60) of Proposition 5.3.4 with an appropriate choice of two constants 0 < δ < 1, 0 < β < 1 for a set of parameters λ of positive measure.

We want to prove positive measure estimates for the set

A 2 ε := � N ≥N0 � λ ∈ Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ) : S N = ∪ r Γ N,
� = � λ ∈ Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ) : S N = ∪ r Γ N,r such that ∀ r, diam(Γ N,r ) ≤ N δ and ∀ r � = r � , dist(Γ N,r , Γ N,r � ) ≥ N β � .
Let us introduce the following set:

∀ N ∈ N ΛS N := � λ ∈ Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ) : S N = ∪ r Γ N,r such that ∀ r, diam(Γ N,r ) ≤ N δ 2 and ∀ r � = r � , dist(Γ N,r , Γ N,r � ) ≥ N β � . It is clear that ∀ N ∈ N, ΛS N ⊂ Λ S N .
Moreover, we have the following result.

Lemma 5.5.3.

If λ ∈ ΛS N , then λ ∈ Λ S K , ∀ K ∈ [N/2, N ] ΛS N ⊂ Λ S K ∀ K ∈ � N 2 , N � . (5.148) Proof. If λ ∈ ΛS N , then S N = ∪ r Γ N,r with ∀ r, diam(Γ N,r ) ≤ N δ 2 , ∀ r � = r � , dist(Γ N,r , Γ N,r � ) ≥ N β . Fix K ∈ [N/2, N ]
. By definition of S K , we write

S K = S N ∩ [-K, K] = (∪ r Γ N,r ) ∩ [-K, K] = ∪ r (Γ N,r ∩ [-K, K]) = ∪ r Γ K,r
where we have defined Γ K,r := (Γ N,r ∩ [-K, K]).

Using δ ∈ (0, 1) and

K ∈ [N/2, N ], we have ∀ r diam(Γ K,r ) ≤ diam(Γ N,r ) ≤ N δ 2 ≤ � N 2 � δ ≤ K δ and ∀ r � = r � dist(Γ K,r , Γ K,r � ) ≥ dist(Γ N,r , Γ N,r � ) ≥ N β ≥ K β .
Hence, by definition of Λ S K , we have that

ΛSN ⊂ Λ SK ∀ K ∈ � N 2 , N � .
Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension Using Lemma 5.5.3, we have the following inclusions:

∀ N k := 2 k N 0 , k ≥ 0, ΛS N 0 ⊂ Λ S K , ∀ K ∈ � N 0 2 , N 0 � ΛS N 1 ⊂ Λ S K , ∀ K ∈ � N 0 , 2N 0 � . . . ΛS N k ⊂ Λ S K , ∀ K ∈ � 2 k-1 N 0 , 2 k N 0 � .
(5.149) By (5.147) and (5.149), we write For m = N η k 2 with η ∈ (0, 1β) and δ = η + β, by all previous considerations and using inclusion (5.150), we have

A 2 ε = � N ≥N0 Λ S N ⊃ � N k =2 k N0 k≥0 ΛS N k . ( 5
|l 1 -l| ≤ N β , |l i+1 -l i | ≤ N β , |l p -l � | ≤ N β .
� N k =2 k N0 k≥0 G N η k /2,N k ⊂ � N k =2 k N0 k≥0 ΛS N k ⊂ A 2 ε .
Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension

As a result,

(A 2 ε ) c ⊂ � N k =2 k N0 k≥0 (G N η k /2,N k ) c ⊂ � N k =2 k N0 k≥0 B N η k /2,N k .
(5.151)

We want to find measure estimates of the set B N η k /2,N k for an appropriate choice of η.

Fix l ∈ Z, let us consider the set Ω l := {λ ∈ Λ ∩ (λ 0δ 0 , λ 0 + δ 0 ) : l is a singular site for L}.

Define the function

✷ Ω l : Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ) → R λ � →    1 if λ ∈ Ω l 0 if λ � ∈ Ω l , it is clear that F (λ) := ✷ Ω -N (λ) + ✷ Ω -(N -1) (λ) + • • • + ✷ Ω N (λ)
represents the number of singular sites for L in the interval [-N, N ].

Note that, by definition, if l ∈ Z is a singular site

∃ (j, k) ∈ N × N / | -(1 -ελ) 2 ω2 l 2 + ω 2 j,k | < 1
then also -l is singular too. This implies that we can write

F (λ) := ✷ Ω0 (λ) + 2(✷ Ω1 (λ) + • • • + ✷ Ω N (λ)).
By previous definitions, we write where �l� := max{1, |l|}.

B m,N = {λ ∈ Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ) : F (λ) ≥ m}, � Λ∩(λ0-δ0,λ0+δ0) F dλ ≥ � B m
In order to simplify the proof of (5.153), we will use Assumption 5.1.2 written for σ = 1:

r > 1 2 , |δ j | ≤ c 3 |p j | .
Recall

Ω l : = {λ ∈ Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ) : l is a singular site for L} = {λ ∈ Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ) : ∃ (j, k) ∈ N × N / | -(1 -ελ) 2 ω2 l 2 + ω 2 j,k | < 1}, let us define Ω � l := {λ ∈ Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ) : ∃ j ∈ N / | -(1 -ελ) 2 ω2 l 2 + p 2 j | < c(c 3 )}
for an appropriate constant c(c 3 ), where p j nd c 3 are introduced in assumption 5.1.2.

We have

Ω l ⊂ Ω � l .
(5.154)

For ω = (1ελ)ω, this is a consequence of the following result.

Lemma 5.5.4. Fix j ∈ N large enough. Under assumption 5.1.2 written for σ = 1, there exists an appropriate constant c(c 3 ) such that

|ω 2 l 2 -p 2 j | > c(c 3 ) ⇒ |ω 2 l 2 -ω 2 j,k | > 1 (5.155)
for all k such that ω j,k ∈ I j = [p jδ j , p j + δ j ].

Proof. By assumption 5.1.2, |δ j p j | < c 3 and for j ∈ N large enough, |δ j | < c 3 .

For ω j,k ∈ I j = [p jδ j , p j + δ j ], we estimate

|ω 2 j,k -p 2 j | = |ω j,k -p j ||ω j,k + p j | ≤ 2|δ j |(|p j | + |δ j |) ≤ 2|δ j p j | + 2δ 2 j ≤ 2c 3 + 2c 2 3 . infinite dimension
Hence, we have

|ω 2 l 2 -ω 2 j,k | = |ω 2 l 2 -p 2 j + p 2 j -ω 2 j,k | ≥ |ω 2 l 2 -p 2 j | -|p 2 j -ω 2 j,k | > c(c 3 ) -(2c 3 + 2c 2
3 ).

Choosing c(c 3 ) := 2c 3 + 2c 2 3 + 1 we obtain the Lemma.

Write

Ω � l = ∪ ∞ j=1 Ω � l,j
with

Ω � l,j := {λ ∈ Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ) : | -(1 -ελ) 2 ω2 l 2 + p 2 j | < c(c 3 )}, we have meas(Ω � l ) ≤ ∞ � j=1 meas(Ω � l,j ). (5.156) Take λ ∈ Ω � l,j , for C = C(Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 )) | -(1 -ελ) 2 ω2 l 2 + p 2 j | = | -(1 -ελ)ω|l| + p j |((1 -ελ)ω|l| + p j ) < c(c 3 ) =⇒ | -(1 -ελ)ω|l| + p j | < c(c 3 ) ((1 -ελ)ω|l| + p j ) < C �l� =⇒ � � �ελ -1 + p j ω|l| � � � < C �l� 2 =⇒ λ ∈ � 1 ε � 1 - p j ω|l| - C �l� 2 � , 1 ε � 1 - p j ω|l| + C �l� 2
�� .

( For l ∈ Z given, since {p j } → ∞, there is only a finite number of p j which may verify (5.157): the set

S l := � j : dist(p j , ω|l|(1 -ελ)) < C |l| � ⊂ � j : p j ∈ � ω|l|(1 -εb) - C |l| , ω|l|(1 -εa) + C |l| �� ⊂ � j : p j ∈ � 0, ω|l|(1 -εa) + C |l| �� has a finite cardinality.
By Assumption (5.6) written for σ = 1,

p j ≥ c 1 j r -c 2 , r > 1 2 .
Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension Hence, if j ∈ S l we have

c 1 j r -c 2 ≤ ω|l|(1 -εa) + C |l| =⇒ j r ≤ c -1 1 � ω|l|(1 -εa) + C |l| + c 2 � =⇒ j r ≤ C(ω|l| + 1) =⇒ j ≤ C�l� 1 r
and so card(S l ) ≤ C�l�

meas(Ω l ) ≤ meas(Ω � l ) ≤ ∞ � j=1 meas(Ω � l,j ) ≤ � j∈S l meas(Ω � l,j ) ≤ � j∈S l 2C ε�l� 2 ≤ 2C ε�l� 2 �l� 1 r ≤ C ε�l� 2-1 r ≤ C ε �l� -ν (5.160)
for ν := 2 -1 r > 0 since 1 r < 2. By (5.151), (5.152) and (5.160), taking 1ν < η < 1, we estimate

meas((A 2 ε ) c ) ≤ � N k =2 k N0 k≥0 meas(B N η k /2,N k ) ≤ � N k =2 k N0 k≥0 � N k l=-N k meas(Ω l ) N η k 2 ≤ � N k =2 k N0 k≥0 C ε N 1-ν k N η k ≤ � k≥0 C ε 2 k(1-ν-η) N 1-ν-η 0 ≤ C ε N (1-ν-η) 0
(5.161) Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension which is small if we take N 0 = N 0 (ε) large enough such that γ 4 ε q < 1 2 if we take q large enough.

N 1-ν-η 0 ε << 1. (5.162) Note that ∀ ε ∈ [0, ε), ε < ε 1 ,
We have concluded the proof of the main Theorem 5.1.6. Suppose Assumption 5.1.2 and the existence of (0, 0

) � = (λ 0 , h 0 ) ∈ Λ × E 0 such that    D h Ψ(0, λ 0 , h 0 ) = 0, dim(KerD 2 h Ψ(0, λ 0 , h 0 )) = 1
, for all fixed ε ∈ [0, ε) small enough, Theorem 5.3.8 and Theorem 5.4.9 give us the existence of a Cantor family of functions Taking N 0 = N 0 (ε) large enough with respect to ε such that N 0 (ε) → ∞ as ε → 0, estimates (5.145) and (5.161) imply that

u ε ∈ C 1 (Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ), H s1+2, p) u ε (λ) = h ε (λ) + v(ε, λ, h ε (λ)) with �u ε (λ) -h0 (λ)� s1, p → 0 as ε → 0 for some h0 : Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ) → E 0 , h0 (λ 0 ) = h 0 which for parameters λ ∈ A ε , u ε (λ)
A ε = A 1 ε ∩ A 2 ε has asymptotically full Lebesgue measure in Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ), i.e. meas{(Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 )) \ A ε } → 0 as ε → 0. Moreover, u ε ∈ C 1 (Λ ∩ (λ 0 -δ 0 , λ 0 + δ 0 ), H s, p) for all s ≥ s 1 .
Chapter 6. Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifolds i.e. integrating by parts

�∇g� 2 L 2 (M,C) + �V g, g� L 2 (M,C) ≥ β 0 �g� 2 L 2 (M,C) , ∀ g ∈ L 2 (M, C).
• the non-linear quasi-periodic forcing term f is in

C ∞ (T ν × M × R, R).
Let us introduce a function F :

T ν × M × R → R such that ∂ u F(ϕ, x, u) = f (ϕ, x, u), ϕ ∈ T ν . (6.4) • the frequency vector ω ∈ R ν is colinear to a fixed vector ω ∈ R ν , |ω| ≤ 1, namely ω = λω, λ ∈ � 1 2 , 3 2
� (6.5)
with ω which verifies the diophantine condition: there exist τ 0 > 1 and

γ 0 ∈ (0, 1) such that |ω • l| ≥ γ 0 |l| τ0 , ∀ l ∈ Z ν \ {0} (6.6)
where |l| := max{|l 1 |, . . . , |l ν |}.

For ε = 0, u = 0 is a solution of (6.1). If ε � = 0 and f (ωt, x, 0) � = 0 then u = 0 is no more a solution.

We look for quasi-periodic in time solutions of frequencies ω of (6.1), namely functions u(ωt) with u(•) defined on T ν .

Since ω verifies the diophantine condition (which implies that in particular its components are rationally independent, i.e. ω • l � = 0, ∀ l ∈ Z ν \ {0}), the colinear frequency ω is diophantine too and so {ωt} t∈T is dense in T ν .

This implies that looking for quasi-periodic in time solutions of frequencies ω of (6.1) means looking for (2π) ν -periodic solutions of

(ω • ∂ ϕ ) 2 u -∆u + V (x)u = εf (ϕ, x, u) ϕ ∈ T ν (6.7) that is, for ω = λω λ 2 (ω • ∂ ϕ ) 2 u -∆u + V (x)u = εf (ϕ, x, u) ϕ ∈ T ν . (6.8)
Fix p > d/2 + 2, the solutions will be looked for in the Sobolev spaces

H s := H s (T ν , H p(M, R)) = � u(ϕ, x) = � l∈Z ν e ıl•ϕ u l (x); u l ∈ H p(M, C), u -l = ūl and �u� 2 s := � l∈Z ν �l� 2s �u l � 2 H p (M) < ∞ � Chapter 6.
Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifolds where s ≥ s 0 > ν/2 and �l� := max{1, |l|}.

Note that for all l ∈ Z ν , u -l = ūl ∈ H p(M, C) implies u(ϕ, x) ∈ R.

H s is a space of (2π) ν -periodic in time functions with values in the Sobolev space H p(M, C), with p > d/2 + 2.

As said in the section A.1 of Appendix A, for p ≥ d/2, the Sobolev space

H p(M, C) ⊂ C 0 (M, C
) is a Banach algebra with respect to multiplication of functions. Thanks to this property, for s ≥ s 0 > ν/2, H s is a Banach algebra too (see e.g. [START_REF] Berti | Nonlinear Oscillations in Hamiltonian PDEs[END_REF]) and there is the continuous embedding

H s (T ν , H p(M, R)) �→ C 0 (T ν , C 0 (M, R)) � C 0 (T ν × M, R) (6.9)
where C 0 (T ν × M, R) is endowed with the sup-norm

�u� L ∞ (T ν ,L ∞ (M)) := sup ϕ∈T ν �u(ϕ)� L ∞ (M) . ( 6
, ∀ u 1 , u 2 ∈ H s �u 1 u 2 � s ≤ C 1 (s 0 )�u 1 � s0 �u 2 � s + C 2 (s 0 , s)�u 1 � s �u 2 � s0 . (6.11)
The proof of (6.11) is given for example in Appendix of [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF] (see also [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF]).

With respect to the standard Moser-Nirenberg interpolation estimate in Sobolev spaces (see e.g. [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF]), in property (6.11) one of the constants is independent of s.

We recall that the linear operator -∆+V (x) is invertible and (-∆+V (x)) -1 is regularizing of order 2 in the spatial variable: more precisely

�(-∆ + V (x)) -1 u� s,p � +2 ≤ C�u� s,p � , (6.12)
∀ u ∈ H s,p � := H s (T ν , H p � (M, R)).

Note that when p � = p, we shall more simply denote � The spectrum spect(P ) of the unbounded, self-adjoint operator

• � s,p � = � • � s, p = � • � s the norm in H s .
P := � -∆ + V (x)
densely defined in L 2 (M), is discrete, real and every eigenvalue of P has a finite multiplicity. Lemma 2.1.1 in the chapter 2, due to Colin de Verdière and taken from [START_REF] Bambusi | Almost global existence for Hamiltonian semilinear Klein-Gordon equations with small Cauchy data on Zoll manifolds[END_REF], describes the asymptotic distribution of the eigenvalues of P when M is a d-dimensional Zoll manifold: they are confined in disjoint compact intervals I j spect(P ) ⊂ � j≥1 I j

I j := � 2π T j + α - c 0 j β , 2π T j + α + c 0 j β � , j ≥ 2
for some constants α ∈ R, c 0 > 0, β ∈ (0, 1), with cardinality card(spect(P

) ∩ I j ) ≤ C 0 j d-1 , C 0 > 0 counted with multiplicity. We call ω j,k , 1 ≤ k ≤ d j , d j ≤ C 0 j d-1
, the eigenvalues of P in each I j counted with multiplicity. There is an orthonormal Hilbert basis of L 2 (M, R) composed of corresponding eigenvectors ϕ j,k of each eigenvalues ω j,k .

Since the manifold M has no boundary, the Sobolev norms in H p(M, C) defined in (A.4) of Appendix A, can be written as

�g� 2 H p (M,C) = � � � � � � � � � j≥1 1≤k≤dj g j,k ϕ j,k � � � � � � � � 2 H p (M,C) := � j≥1 1≤k≤dj ω 2 p j,k |g j,k | 2
for all g ∈ H p(M, C), see [START_REF] Berti | An abstract Nash-Moser theorem with parameters and applications to PDEs[END_REF].

There exist γ 1 ∈ (0, 1), τ 1 > 1 such that for α of Lemma 2.1.1 which appears in the definition of intervals I j , we define the set

Λ := Λ 1 ∩ Λ 2 (6.14)
where

Λ 1 := � λ ∈ � 1 2 , 3 2
� : � � �λ(ω • l) - 2π T p � � � ≥ γ 1 |l| τ1 , ∀ l ∈ Z ν \ {0}, ∀ p ∈ Z � and Λ 2 := � λ ∈ � 1 2 , 3 2
� : � � �λ(ω • l) - 2π T p -2α � � � ≥ γ 1 |l| τ1 , ∀ l ∈ Z ν \ {0}, ∀ p ∈ Z � .
Recall that we take τ 1 > 1 large enough in order to have Λ � = ∅. In particular, we take τ 1 large enough with respect to τ 0 of condition (6.6)

� Λ c 1 ∩ � 1 2 , 3 2
�� = O(γ 1 ), meas � Λ c 2 ∩ � 1 2 , 3 2
�� = O(γ 1 )
which imply (6.15) by

Λ := Λ 1 ∩ Λ 2 .
We look for quasi-periodic solutions of (6.1) of frequencies ω = λω and λ ∈ Λ.

Note that in the search of periodic (ν = 1) and quasi-periodic (ν > 1) solutions for some non linear PDEs like (6.1) in which the forced term f depends on time, the frequency ω = λω is not connected to the eingeinvalues ω j,k of the linear operator which appears in the PDE (linear operator P = � -∆ + V (x) in our case). It can be considered as an external parameter of the problem.

The main result of this chapter is: Theorem 6.1.3. Let M be a Zoll manifold and let us consider equation (6.7)

with V (x) satisfying (6.2) and f (ϕ, x, u) ∈ C ∞ (T ν × M × R). There exist an index s 1 > ν/2 and ε 0 = ε 0 (s 1 ) small enough such that for all ε ∈ [0, ε 0 ), there exist a map u ε ∈ C 1 (Λ, H s1 ) with �u ε (λ)� s1 → 0 as ε → 0 and a Cantor-like set A ε ⊂ Λ with asymptotically full Lebesgue measure, i.e.

meas(Λ

\ A ε ) → 0 as ε → 0, (6.16)
such that, for all λ ∈ A ε , u ε (λ) is a solution of (6.7) with ω = λω.

Moreover, u ε ∈ C 1 (Λ, H s ) for all s ≥ s 1 .
Similar results for quasi-periodic solutions of NLW equation are obtained by Berti, Corsi and Procesi [START_REF] Berti | An abstract Nash-Moser theorem and quasi-periodic solutions for NLW and NLS on compact Lie groups and homogeneous manifolds[END_REF] on Lie groups via Nash-Moser and by Grébert, Paturel [START_REF] Grébert | KAM for the Klein-Gordon equation on S d[END_REF] for the Klein-Gordon equation on S d via KAM method. Moreover, in a work in preparation, Berti and Bolle [START_REF] Berti | Quasi-periodic solutions for autonomous NLW on T d with a multiplicative potential[END_REF] prove existence of quasi-periodic solutions for NLW with multiplicative potential on T d in the autonomous case. potential on Zoll manifolds

Invertibility of linearized operators

We will solve Equation (6.7) using a Nash-Moser scheme. In every step of the scheme, we have to invert approximately linearized operators

L(ε, λ, u) := λ 2 (ω • ∂ ϕ ) 2 -∆ + V (x) -ε∂ u f (ϕ, x, u)
where u ∈ (∩ s≥0 H s ) and the notion of operator for the term

∂ u f (ϕ, x, u) is explained in Remark 6.2.1.
This section is dedicated to the study of invertibility of these linearized operators.

Before explaining what we will prove, we introduce some useful definitions.

Define the Hilbert space

L 2 (T ν , L 2 (M, R)) = � u(ϕ) = � l∈Z ν e ıl•ϕ u l , u l ∈ L 2 (M, C) | u -l = ūl and �u� 2 0,0 := � l∈Z ν �u l � 2 L 2 (M) < ∞ � , we have L 2 (T ν , L 2 (M, R)) � L 2 (T ν × M, R)
in accordance with the Parseval's identity for which � • � 0,0 is equal to the

standard norm on L 2 (T ν , L 2 (M, R)) � l∈Z ν �u l � 2 L 2 (M) = 1 (2π) ν � T ν �u(ϕ)� 2 L 2 (M) dϕ = 1 (2π) ν � T ν ×M |u(ϕ, x)| 2 dϕ dx.
See section A.2 in Appendix A for details on the definitions of the Sobolev spaces.

We shall denote �•� 0,0 the L 2 -operator norm for a linear and continuous operator from L 2 (T ν , L 2 (M, R)) into itself:

�L� 0,0 := sup u� =0 �Lu� 0,0 �u� 0,0 . (6.17)
As said in Remark 6.1.1, we use the notation � • � 0,0 both for the operator norm (6.17) of a linear continuous operator L from L 2 (T ν , L 2 (M, R)) into itself and for the norm of a function in L 2 (T ν , L 2 (M, R)). In every estimate, we have to understand from the context what norm we consider.

We insert here a useful remark in order to clarify some notations that we will use in the sequel.

Chapter 6. Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifolds Remark 6.2.1. Take the non-linear function f ∈ C ∞ (T ν × M × R, R), we consider the non-linear operator

f : C 0 (T ν , C 0 (M, R)) → C 0 (T ν , C 0 (M, R)) ⊂ L 2 (T ν , L 2 (M, R)) u � → f (u)
where f (u) is defined by:

(f (u))(ϕ, x) := f (ϕ, x, u(ϕ, x)).
In this setting, we can consider the � • � 0,0 -norm of the function f (u) and, if

f (u) ∈ H s,p ⊂ L 2 (T ν , L 2 (M, R)), also its � • � s,p -norm .
By an abuse of notation, we will write f (ϕ, x, u) for refering to the function

f (u) ∈ L 2 (T ν , L 2 (M, R)) giving sense to �f (ϕ, x, u)� 0,0 and �f (ϕ, x, u)� s,p . For all u ∈ C 0 (T ν , C 0 (M, R)), we take ∂ u f (u) ∈ C 0 (T ν , C 0 (M, R)) defined as (∂ u f (u))(ϕ, x) := ∂ u f (ϕ, x, u(ϕ, x))
and we consider the linear operator L ∂uf (u) of "multiplication" by ∂ u f (u).

Since ∂ u f (u) ∈ C 0 (T ν , C 0 (M, R)), the operator L ∂uf (u) : L 2 (T ν , L 2 (M, R)) → L 2 (T ν , L 2 (M, R)) v � → ∂ u f (u)v
is continuous and by definition of operator norm (6.17), it is clear that

�L ∂uf (u) � 0,0 ≤ �∂ u f (u)� L ∞ (T ν ×M) . (6.18)
Moreover, for all s > ν/2, p > d/2 we have the embedding (6.9) and (recall the definition of operator norm (6.13)) we obtain

�L ∂uf (u) � s,p ≤ C(s, p)�∂ u f (u)� s,p . (6.19)
Estimate (6. [START_REF] Brezis | Free vibrations for a nonlinear wave equation and a Theorem of P. Rabinowitz[END_REF]) is a consequence of the algebra structure of the Sobolev spaces

H s,p which implies ∀ v ∈ H s,p �L ∂uf (u) v� s,p = �∂ u f (u)v� s,p ≤ C(s, p)�∂ u f (u)� s,p �v� s,p .
By an abuse of notation, we will write ∂ u f (ϕ, x, u) for denoting both the function

∂ u f (u) ∈ C 0 (T ν , C 0 (M, R))
and the linear operator L ∂uf (u) of "multiplication" by ∂ u f (u).

In order to distinguish the operator norm from the norm of the function, only for the term ∂ u f (ϕ, x, u) we will write potential on Zoll manifolds

• �∂ u f (ϕ, x, u)� (op) 0,0
for refering to the operator norm (6.17)

• �∂ u f (ϕ, x, u)� 0,0 for refering to the norm of the function in L 2 (T ν , L 2 (M, R)).

We will do the same for the operator norm �∂ u f (ϕ, x, u)� (op) s,p and the norm of the function �∂ u f (ϕ, x, u)� s,p .

With these notations, estimates (6.18) and (6.19) are written as

�∂ u f (ϕ, x, u)� (op) 0,0 ≤ �∂ u f (ϕ, x, u)� L ∞ (T ν ×M) (6.20)
and for s > ν/2, p > d/2

�∂ u f (ϕ, x, u)� (op) s,p ≤ C(s, p)�∂ u f (ϕ, x, u)� s,p . (6.21)
We conclude this remark with the following tame estimates (see [START_REF] Berti | Cantor families of periodic solutions of wave equations with C k nonlinearities[END_REF] for references)

�∂ u f (ϕ, x, u)� s,p ≤ C(s, p, �u� s0,p )(1 + �u� s,p ) ∀ s ≥ s 0 > ν/2, ∀ p > d/2 (6.22)
which imply, using (6.20), (6.21) and the embedding (6.9) that

�∂ u f (ϕ, x, u)� (op) 0,0 ≤ �∂ u f (ϕ, x, u)� L ∞ (T ν ×M) ≤ C(s, p)�∂ u f (ϕ, x, u)� s,p ≤
A : D(A) ⊂ L 2 (T ν , L 2 (M, C)) → L 2 (T ν , L 2 (M, C)) defined by: ∀ u 1 , u 2 ∈ D(A) ⊂ L 2 (T ν , L 2 (M, R)) A(u 1 + ıu 2 ) := Au 1 + ıAu 2 .
Let ξ be a real parameter. We define the ξ-parametrized family of operators

L(ε, λ, u, ξ) : D(L) ⊂ L 2 (T ν , L 2 (M, C)) → L 2 (T ν , L 2 (M, C)) L(ε, λ, u, ξ) := λ 2 (ω • ∂ ϕ + ıξ) 2 -∆ + V (x) -ε∂ u f (ϕ, x, u) (6.25)
where ∂ u f (ϕ, x, u) is seen as an operator of "multiplication" with respect to Remark 6.2.1.

Chapter 6. Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifolds

The operator -∆ + V (x) is self-adjoint in L 2 (M). Moreover, the fact that ∀ (ϕ, x) we have f (ϕ, x, u(ϕ, x)) ∈ R and (6.4)

f (ϕ, x, u) = ∂ u F(ϕ, x, u) imply that ∂ u f (ϕ, x, u) is self-adjoint in L 2 (T ν , L 2 (M, C)).
Hence, the operators L(ε, λ, u, ξ) are self-adjoint with respect to the scalar

product in L 2 (T ν , L 2 (M, C)).
Of course, for ξ = 0

L(ε, λ, u, 0) = L(ε, λ, u).
Recall that ϕ j,k are the eigenvectors of the operator

� -∆ + V (x) in L 2 (M, C).
In the complex Hilbert basis {e ıϕ•l ϕ j,k } l,j,k of L 2 (T ν , L 2 (M, C)), we write the operator L(ε, λ, u, ξ) as the matrix

L(ε, λ, u, ξ) = diag (l,j,k) {-λ 2 (ω • l + ξ) 2 + ω 2 j,k } -ε∂ u f (ϕ, x, u). (6.26)
In the following, we will often identify an operator with his associated matrix in the Fourier basis.

Let us define the "matrix coefficients" L n m of the linear (and unbounded) operator

L(ε, λ, u, ξ): ∀ g ∈ D(-∆ + V (x)) L n m (g) := 1 (2π) ν � T ν e -ım•ϕ L(ε, λ, u, ξ)(e ın•ϕ g) dϕ.
(6.27)

Note that (6.27) recalls the definition of "matrix coefficients" of a linear and continuous operator (see chapter 3).

Let us introduce the subspaces E

(N ) 0 ⊂ L 2 (T ν , L 2 (M, C)) E (N ) 0 := � u(ϕ, x) = � l∈Z ν |l|≤N e ıl•ϕ u l (x), u l ∈ L 2 (M, C), u -l (x) = ūl (x) � and consider the corresponding L 2 -orthogonal projection Π (N ) 0 . Of course, E (N ) 0
⊂ H s,0 for all s ≥ 0 (in this sense it does not depend on s).

We define the linear, self-adjoint and unbounded operators

L N (ε, λ, u, ξ) := Π (N ) 0 L(ε, λ, u, ξ)| E (N ) 0
which in the complex Hilbert basis {e ıϕ•l ϕ j,k } l,j,k , we write as

L N (ε, λ, u, ξ) = diag (|l|≤N,j,k) {-λ 2 (ω • l + ξ) 2 + ω 2 j,k } -εΠ (N ) 0 ∂ u f (ϕ, x, u)| E (N ) 0 . Fix now l 0 ∈ Z ν .
Before introducing the next definition, we note that the Fourier coefficients in Chapter 6. Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifolds time (L l0 ) n m of the operator L l0 (ε, λ, u, ξ) := e -ıl0•ϕ L(ε, λ, u, ξ)e ıl0•ϕ (defined properly as in (6.27)) verify

∀ g ∈ D(-∆ + V (x)) (L l0 ) n m (g) := 1 (2π) ν � T ν e -ım•ϕ L l0 (ε, λ, u, ξ)(e ın•ϕ g) dϕ = 1 (2π) ν � T ν e -ım•ϕ e -ıl0•ϕ L(ε, λ, u, ξ)(e ıl0•ϕ e ın•ϕ g) dϕ = 1 (2π) ν � T ν e -ı(m+l0)•ϕ L(ε, λ, u, ξ)(e ı(n+l0)•ϕ g) dϕ = L n+l0 m+l0 (g).
This allows us to say that L l0 (ε, λ, u, ξ) is associated to a matrix "centered at l 0 ".

Moreover, since ε∂ u f (ϕ, x, u) is an operator of "multiplication" (see Remark 6.2.1), for any g ∈ L 2 (M, C) we have

e -ıϕ•l0 ε∂ u f (ϕ, x, u)e ıϕ•l0 g = e -ıϕ•l0 e ıϕ•l0 ε∂ u f (ϕ, x, u)h = ε∂ u f (ϕ, x, u)g which implies that (L l0 ) n m = L n+l0 m+l0 = � λ 2 (ω • ∂ ϕ + ıξ) 2 -∆ + V (x) � (n+l0) (m+l0) - � ε∂ u f (ϕ, x, u) � n m . (6.28)
We can now define the N -submatrices centered at l 0

L N,l0 (ε, λ, u, ξ) := Π (N ) 0 (L l0 (ε, λ, u, ξ)) | E (N ) 0 = Π (N ) 0 (e -ıl0•ϕ L(ε, λ, u, ξ)e ıl0•ϕ ) | E (N ) 0
that, using (6.28), we write in the basis {e ıϕ•l ϕ j,k } l,j,k as

L N,l0 (ε, λ, u, ξ) = diag (|l|≤N,j,k) {-λ 2 (ω • (l + l 0 ) + ξ) 2 + ω 2 j,k } -εΠ (N ) 0 ∂ u f (ϕ, x, u)| E (N ) 0 = diag (|l|≤N,j,k) {-λ 2 (ω • l + ω • l 0 + ξ) 2 + ω 2 j,k } -εΠ (N ) 0 ∂ u f (ϕ, x, u)| E (N ) 0 = L N (ε, λ, u, ξ + ω • l 0 ). Since λ � = 0, λ ∈ [1/2, 3/2] L N (ε, λ, u, ξ) is invertible ⇔ LN (ε, λ, u, ξ) is invertible where LN (ε, λ, u, ξ) := L N (ε, λ, u, ξ) λ 2 = diag (|l|≤N,j,k) � -(ω • l + ξ) 2 + ω 2 j,k λ 2 � - ε λ 2 Π (N ) 0 ∂ u f (ϕ, x, u)| E (N ) 0 .
Chapter 6. Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifolds More generally, we define

L(ε, λ, u, ξ) := L(ε, λ, u, ξ) λ 2 .
Recall for n ∈ Z ν , the definition of seminorms (see chapter 3)

[L(n)] := max l-m=n �L m l � L(L 2 (M)) (6.29)
and for s ∈ R the definition of norms .30) In all this section, we consider the unknown function u depending on λ in a C 1 way. Moreover, we introduce an index s 1 ≥ s 0 > ν/2 and we assume that

|L| 2 s := � n∈Z ν �n� 2s [L(n)] 2 . ( 6
�u� s1 + �∂ λ u� s1 < 1. (6.31)
The index s 1 will be the lower index in the Nash-Moser scheme.

Remark 6.2.2. The bound �u� s1 ≤ 1 implies the following estimates:

�∂ u f (ϕ, x, u)� (op) 0,0 ≤ c (6.32) |∂ u f (ϕ, x, u)| s1 ≤ c (6.33)
for some constants c = c(s 1 , p) > 0.

Estimate (6.32) is obtained from (6.23).

Estimate (6.33) follows directly by

|∂ u f (ϕ, x, u)| s1 = � n∈Z ν �n� 2s1 [∂ u f (ϕ, x, u)(n)] 2 = � n∈Z ν �n� 2s1 � max l-m=n �(∂ u f (ϕ, x, u)) m l � L(L 2 (M)) ) � 2 = � n∈Z ν �n� 2s1 � max l-m=n �(∂ u f (ϕ, x, u)) l-m � L(L 2 (M)) � 2 = � n∈Z ν �n� 2s1 �(∂ u f (ϕ, x, u)) n � 2 L(L 2 (M)) ≤ � n∈Z ν �n� 2s1 �(∂ u f (ϕ, x, u)) n � 2 L ∞ (M) ≤ � n∈Z ν �n� 2s1 �(∂ u f (ϕ, x, u)) n � 2 H p ≤ �∂ u f (ϕ, x, u)� s1
and by the tame estimates (6.22) from which we conclude

|∂ u f (ϕ, x, u)| s1 ≤ �∂ u f (ϕ, x, u)� s1 ≤ C(s 1 , p, �u� s0 )(1 + �u� s1 ) ≤ C(s 1 , p).

potential on Zoll manifolds

Let us introduce the following definitions. Definition 6.2.3. Fix τ ≥ 0, the linear operator L(ε, λ, u, ξ) is said to be "N -L 2 -regular" if and only if LN (ε, λ, u, ξ) is invertible and

� L-1 N (ε, λ, u, ξ)� 0,0 ≤ N τ .
An operator which is not N -L 2 -regular is said "N -L 2 -singular".

Recall s 1 ≥ s 0 > ν/2 is the index of condition (6.31). Definition 6.2.4. Fix τ � > 2τ +1 and δ ∈ [0, 1), the linear operator L(ε, λ, u, ξ) is said to be "N-regular" if and only if LN (ε, λ, u, ξ) is invertible and

| L-1 N (ε, λ, u, ξ)| s ≤ N τ � +δs ∀ s ∈ [s 0 , s 1 ].
An operator which is not N-regular is said "N-singular". Definition 6.2.5. Fix θ ∈ R + . An index l 0 ∈ Z ν is said to be a "regular site"

for the operator L(ε, λ, u, ξ) if and only if

� � � -(ω • l 0 + ξ) 2 + ω 2 j,k λ 2 � � � ≥ θ ∀ j ∈ N \ {0}, k ∈ [1, d j ].
An index which is not regular is said a "singular site". Definition 6.2.6. An index l 0 ∈ Z ν is said to be a "N-regular site" for the operator L(ε, λ, u, ξ) if and only if LN,l0 (ε, λ, u, ξ) is a "N-regular operator", i.e. LN,l0 (ε, λ, u, ξ) is invertible and

| L-1 N,l0 (ε, λ, u, ξ)| s ≤ N τ � +δs ∀ s ∈ [s 0 , s 1 ] ⇐⇒ | L-1 N (ε, λ, u, ξ + ω • l 0 )| s ≤ N τ � +δs ∀ s ∈ [s 0 , s 1 ].
An index which is not N-regular is said a "N-singular site". Definition 6.2.7. An index l 0 ∈ Z ν is said to be a "N-good site" for the operator L(ε, λ, u, ξ) if and only if l 0 is a regular site for L(ε, λ, u, ξ) or

∀ l � ∈ Z ν s.t. |l � -l 0 | ≤ N, l � is a N-regular site for L(ε, λ, u, ξ).
An index which is not N-good is said a "N-bad site":

l 0 is a N-bad site ⇐⇒    l 0 is a singular site, ∃ l � , |l � -l 0 | ≤ N, s.t. l � is a N-singular site.

potential on Zoll manifolds

The parameters θ ∈ R + , τ > 0 and s 1 ≥ s 0 will be chosen large enough in order to prove our results (see Remark 6.2.17).

The main result of this section is Proposition 6.2.18: a "good property" of separation of N -bad sites for L(ε, λ, u, ξ) allows us to prove that

L(ε, λ, u, ξ) N � -L 2 -regular ⇒ L(ε, λ, u, ξ) N � -regular
for N � = N χ > N , for some χ properly chosen.

In order to prove it, we need several useful results about:

• a lower bound of the distance between two different singular sites for the operator L(ε, λ, u, ξ) (Lemma 6.2.8 for ξ = 0 and Lemma 6.2.9 for ξ � = 0)

• given some ρ > 0, measure estimates for the set of parameter ξ ∈ • a bound of the number of N -singular sites l 0 for L(ε, λ, u, ξ) such that

[-N ρ , N ρ ] for which L(ε, λ, u, ξ) is N -L 2 -
(ξ + ω • l 0 ) ∈ [-N ρ , N ρ ],
for some ρ ≥ 1 (Lemma 6.2.15).

Finally, using the Multiscale Theorem 6.2.16, we will prove Proposition 6.2.18.

We start from the following: Lemma 6.2.8. Let us consider β ∈ (0, 1) of Lemma 2.1.1, γ 1 ∈ (0, 1) and τ 1 > 1 which appear in definition (6.14) of Λ.

If l 1 � = l 2 are two different singular sites for L(ε, λ, u), then

|l 1 -l 2 | ≥ C(γ 1 ) max{|ω • l 1 |, |ω • l 2 |} 1 η
where η := τ1 β .

Proof. By definition 6.2.5 of singular sites for L(ε, λ, u), there exist (j 1 , k 1 ) and

(j 2 , k 2 ) such that � � �(ω • l 1 ) 2 - ω 2 j1,k1 λ 2 � � � ≤ θ, � � �(ω • l 2 ) 2 - ω 2 j2,k2 λ 2 ) � � � ≤ θ. (6.34)
The first step is to prove

|l 1 -l 2 | ≥ (Cγ 1 ) 1 τ 1 min{j 1 , j 2 } β τ 1 . (6.35)
Chapter 6. Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifolds

If j 1 = 1 and j 2 ≥ 2, then min{j 1 , j 2 } = 1 and using |l 1l 2 | > 1 (because l 1 � = l 2 ), we have directly (6.35) for an appropriate constant C > 0.

The case j 1 ≥ 2 and j 1 = 1 is the same.

Suppose now j 1 , j 2 ≥ 2.

For λ ∈ [1/2, 3/2], by (6.34) we have

|(ω • l 1 ) 2 -ω 2 j1,k1 | ≤ λ 2 θ ≤ 3θ, |(ω • l 2 ) 2 -ω 2 j2,k2 )| ≤ λ 2 θ ≤ 3θ. Since |(ω • l 1 ) 2 -ω 2 j1,k1 | ≤ 3θ ⇐⇒ |ω • l 1 + ω j1,k1 ||ω • l 1 -ω j1,k1 | ≤ 3θ,
using Lemma 2.1.1 from which |ω ji,ki | ∼ j i , we can obtain two different estimates:

• if ω • l 1 > 0, then |ω • l 1 + ω j1,k1 | > |ω j1,k1 | > 0 and |ω • l i -ω ji,ki | ≤ 3θ |ω • l 1 + ω j1,k1 | ≤ 3θ |ω ji,ki | ≤ Cθ j i • if ω • l 1 < 0, then |ω • l 1 -ω j1,k1 | > |ω j1,k1 | > 0 and |ω • l i + ω ji,ki | ≤ 3θ |ω • l 1 -ω j1,k1 | ≤ 3θ |ω ji,ki | ≤ Cθ j i .
Proceeding similarly for the couple of indices (j 2 , k 2 ), we have for i = 1, 2, that .36) We have to distinguish different cases.

|ω • l i -ω ji,ki | ≤ Cθ j i or |ω • l i + ω ji,ki | ≤ Cθ j i . ( 6
• Case |ω • l 1 -ω j1,k1 | ≤ Cθ j1 and |ω • l 2 -ω j2,k2 | ≤ Cθ j2 Write |ω • (l 1 -l 2 ) -(ω j1,k1 -ω j2,k2 )| = |ω • l 1 -ω j1,k1 -(ω • l 2 -ω j2,k2 )| ≤ Cθ j 1 + Cθ j 2 ≤ Cθ min{j 1 , j 2 } . Using Lemma 2.1.1 from which ω j1,k1 -ω j2,k2 = 2π T (j 1 -j 2 ) + O(j -β 1 + j -β 2 ), we have � � �ω • (l 1 -l 2 ) - 2π T (j 1 -j 2 ) � � � ≤ Cθ min{j 1 , j 2 } + C min{j 1 , j 2 } β ≤ C min{j 1 , j 2 } β (6.37)
since β ∈ (0, 1) and θ is a fixed real positive number. Chapter 6. Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifolds

• Case |ω • l 1 + ω j1,k1 | ≤ Cθ j1 and |ω • l 2 + ω j2,k2 | ≤ Cθ j2
Proceeding similarly to the previous case, we estimate

|ω • (l 1 -l 2 ) -(ω j2,k2 -ω j1,k1 )| ≤ Cθ min{j 1 , j 2 } and ω j2,k2 -ω j1,k1 = 2π T (j 2 -j 1 ) + O(j -β 2 + j -β 1 ) implies � � �ω • (l 1 -l 2 ) - 2π T (j 2 -j 1 ) � � � ≤ C min{j 1 , j 2 } β . (6.38) • Case |ω • l 1 -ω j1,k1 | ≤ Cθ j1 and |ω • l 2 + ω j2,k2 | ≤ Cθ j2 Write |ω • (l 1 -l 2 ) -(ω j1,k1 + ω j2,k2 )| = |ω • l 1 -ω j1,k1 -(ω • l 2 + ω j2,k2 )| ≤ Cθ j 1 + Cθ j 2 ≤ Cθ min{j 1 , j 2 } , using Lemma 2.1.1 from which ω j1,k1 + ω j2,k2 = 2π T (j 1 + j 2 ) + 2α + O(j -β 1 + j -β 2 )
we obtain

� � �ω • (l 1 -l 2 ) - 2π T (j 1 + j 2 ) -2α � � � ≤ Cθ min{j 1 , j 2 } + C min{j 1 , j 2 } β ≤ C min{j 1 , j 2 } β . (6.39) • Case |ω • l 1 + ω j1,k1 | ≤ Cθ j1 and |ω • l 2 -ω j2,k2 | ≤ Cθ j2
Proceeding similarly to the previous case, we obtain

� � �ω • (l 2 -l 1 ) - 2π T (j 2 + j 1 ) -2α � � � ≤ C min{j 1 , j 2 } β . (6.40)
For λ ∈ Λ := Λ 1 ∩ Λ 2 with Λ defined in (6.14), we have

� � �ω • l - 2π T p � � � ≥ γ 1 |l| τ1 , ∀ l ∈ Z ν \ {0}, ∀ p ∈ Z and � � �ω • l - 2π T p -2α � � � ≥ γ 1 |l| τ1 , ∀ l ∈ Z ν \ {0}, ∀ p ∈ Z
and by estimates (6.37), (6.38), (6.39), (6.40) we obtain in every case

γ 1 |l 1 -l 2 | τ1 ≤ C min{j 1 , j 2 } β potential on Zoll manifolds which implies estimate (6.35) also for j 1 , j 2 ≥ 2 |l 1 -l 2 | ≥ (Cγ 1 ) 1 τ 1 min{j 1 , j 2 } β τ 1 .
Hence, using (6.34) and Lemma 2.1.

1 from which |ω • l i | ∼ |ω ji,ki | ∼ j i , we obtain |l 1 -l 2 | ≥ (Cγ 1 ) 1 τ 1 min{|ω • l 1 |, |ω • l 2 |} β τ 1 .
We suppose now that |ω • l 1 | ≤ |ω • l 2 | just to fix the ideas. We can have two different cases:

• Case |ω • l 1 | ≥ 1 2 |ω • l 2 | which implies |l 1 -l 2 | ≥ (Cγ 1 ) 1 τ 1 min{|ω • l 1 |, |ω • l 2 |} β τ 1 ≥ (Cγ 1 ) 1 τ 1 |ω • l 1 | β τ 1 ≥ (Cγ 1 ) 1 τ 1 � 1 2 |ω • l 2 | � β τ 1 ≥ C(γ 1 )|ω • l 2 | β τ 1 • Case |ω • l 1 | < 1 2 |ω • l 2 | from which we have |l 1 -l 2 | ≥ 1 |ω| |ω • l 1 -ω • l 2 | > 1 2|ω| |ω • l 2 | > 1 3 |ω • l 2 | ≥ C|ω • l 2 | β τ 1 since |ω| = λ|ω| ≤ 3/2 for assumption (6.5), |ω • l 2 | ≥ C > 0 and β τ1 < 1 (β ∈ (0, 1), τ 1 > 1).
Hence, we have proved that

|l 1 -l 2 | ≥ C(γ 1 ) max{|ω • l 1 |, |ω • l 2 |} β τ 1 .
We conclude, using ω = λω and λ ∈ [1/2, 3/2], that

|l 1 -l 2 | ≥ C(γ 1 ) max{|ω • l 1 |, |ω • l 2 |} 1 η .
More generally, with a similar proof, we have the following: Chapter 6. Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifolds It is convenient to have a bound of the measures of these sets. We will have it using that the operator L is self-adjoint in L 2 (T, L 2 (M)) (for this reason we introduce the L 2 -operator norm � • � 0,0 in the definitions of these sets) and using an argument of variation of his eigenvalues. Lemma 6.2.11. For any ρ ∈ R, we have the following Lebesgue measure estimate

meas{M 0 c,n (ε) ∩ (Λ × [-N ρ n , N ρ n ])} ≤ Kc(N ν-τ +d+ρ n + N ρ(d+1)+ν-τ n ) (6.42)
where d is the dimension of the Zoll maifold M and K is a positive constant.

Proof. We will prove estimate (6.42) applying Lemma B.2.3 in Appendix B to the self-adjoint operators in L 2 (T ν , L 2 (M, C)) Indices N n disappear to simplify the notations.

LNn (ε, λ, u, ξ) := diag (|l|≤Nn,j,k) � -(ω • l + ξ) 2 + ω 2 j,k λ 2 � - ε λ 2 Π (Nn) 0 ∂ u f (ϕ, x, u)| E (Nn) .
The matrices B(λ) and C(λ) are self-adjoint and R2 (λ) = ( R1 (λ)) T . Moreover, by (6.44), C(λ) is finite dimensional.

We show now that we have all the hypotheses (with q = 0) which allow us to apply Lemma B.2.3 and Remark B.2.4 to LNn (ε, λ, u, ξ).

• (i) ∃ C > 0 such that � Ri (λ)� 0,0 ≤ εC Hypothesis (i) comes directly from (6.47).

• For all w ∈ E (Nn) 0

, using that ω j,k → ∞, λ ∈ [1/2, 3/2], we have

� diag (j,k) � - 2ω 2 j,k λ 3 � w, w � L 2
≤ -C�w� 2 0,0 , (6.49) Moreover, using embedding (6.9), tame estimates (6.22) and (6. By (6.49), (6.50) and (6.51), taking ε = ε(s 1 ) small enough, we obtain (ii).

• (iii) ∃ C > 0 such that � B-1 (λ)� 0,0 ≤ 2

C

In order to prove hypothesis (iii), we show firstly that (l, j, k) � -1 � .

∈ E n 1 (ξ) ⇒ � � �-(ω •l +ξ) 2 + ω 2 j,k λ 2 � � � ≥ max � 1 2 (|ω •l|+|ξ|+1) 2 , ω 2
By (6.52) and (6.53)

� � � - 2ω 2 j,k λ 3 � -(ω • l + ξ) 2 + ω 2 j,k λ 2 � -2 � � � ≤ 2ω 2 j,k λ 3 � ω 2 j,k 6
� -2 ≤ C ω 2
≤ sup |l|≤Nn � � � � -(ω • l + ξ � ) 2 + (ω • l + ξ) 2 � I L 2 (M) � � � L 2 (M) ≤ sup |l|≤Nn � � � � -2(ω • l)ξ � + 2(ω • l)ξ -(ξ � ) 2 + ξ 2 � I L 2 (M) � � � L 2 (M) ≤ sup |l|≤Nn � � � � -2(ω • l)(ξ � -ξ) -((ξ � ) 2 -ξ 2 ) � I L 2 (M) � � � L 2 (M) ≤ � � � � 2N n N -ρ-τ n + 2N -τ n � I L 2 (M) � � � L 2 (M)
≤ 2N -τ n (6.66)

where I L 2 (M) is the identity map in L 2 (M).

Since ξ ∈ M 0 n (ε, λ) ∩ [-N ρ n , N ρ n ], we have � L-1 Nn (ε, λ, u, ξ)� 0,0 ≥ N τ n which implies that ∃ v � = 0 : � LNn (ε, λ, u, ξ)v� 0,0 ≤ N -τ n �v� 0,0 . Moreover, we do not specify the required conditions on parameters (see Remark 6.2.17 for details on their choice).

Fix δ ∈ (0, 1). Theorem 6.2.16 (The Multiscale Theorem). Assume δ ∈ (0, 1), a constant C 1 > 2 and appropriate conditions on

• χ large enough and such that χδ > C 1

• τ � large enough with respect to τ

• s 1 large enough with respect to τ, χ, C 1 .

For any given Υ > 0, there exist θ = θ(Υ, s 1 ) large enough (appearing in Definition 6.2.5) and N 0 (Υ, θ, s 1 ) ∈ N such that: for all N ≥ N 0 , if the linear operator LN � = LN � (ε, λ, u, ξ), where N � = N χ , satisfies (H1) � L-1

N � � 0,0 ≤ (N � ) τ (H2) | LN � -diag( LN � )| s1 ≤ Υ (H3)
there is a partition of N -bad sites for LN � , B N = ∪ α Ω α , with

diam(Ω α ) ≤ N C1 , dist(Ω α , Ω α � ) ≥ N 2 , ∀ α � = α � , then LN � is N � -regular.
Moreover, the following estimate holds

∀ s ≥ s 1 , | L-1 N � | s ≤ C(s)(N � ) τ � � (N � ) δs + | LN � -diag( LN � )| s � . ( 6
.72) Remark 6.2.17. Some comments on the choice of parameters in the Multiscale Theorem:

Chapter 6. Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifolds

• the exponent τ will be taken large enough in order to have (H1) for a large enough set of parameters lambda and the exponent τ � must be large enough with respect to τ

• the exponent χ, which measures the scale N � = N χ , must be large with respect to the size of the bad blocks Ω α , i.e. with respect to C 1 . The intuitive meaning is that, for χ large enough, the "resonant effects" due to the bad sites are tamed at the new larger scale

• the constant θ which defines the regular sites (see Definition 6.2.5) must be large enough with respect to Υ, i.e. with respect to the off-diagonal part RN � := LN �diag( LN � ) (in our application, | RN � | s1 will be bounded by a small constant)

• the Sobolev index s 1 must be large enough with respect to χ and τ : if the decay is sufficiently strong (i.e. | LN �diag( LN � )| s1 is sufficiently small), then the "interaction" between different blocks of bad sites is weak enough.

Take now ρ = 2η + 2, with η = τ1 β of Lemma 6.2.8. ]) ⊂ � Q q=1 J q , where J q are intervals with meas{J q } ≤ N Let start with n = 0. We have to prove:

• (i) 0 L(ε, λ, u, ξ) N 0 -L 2 -regular ⇒ L(ε, λ, u, ξ) N 0 -regular and ∀ s ≥ s 1 , | L-1 N0 | s ≤ C(s)N τ � 0 � N δs 0 + | LN0 -diag( LN0 )| s �
We will obtain (i) 0 imposing some conditions on N 0 , which will be the term of initialization in the Nash-Moser scheme that we have fixed in and consider every equivalence class as a block of our partition.

The blocks are separated with a distance more than N 2 n and in every block there are at most N (2η+2)(d+2)+3ν+3 n sites. Moreover, for all l 0 ∈ B n , in the block [l 0 ] there are the N 2 n -chains of bad sites, l 1 , . . . , l p for some p ∈ N, which contain l 0 and with a distance between two consecutive sites in the same chain less then N 2 n . Hence, we have

diam{[l 0 ]} ≤ N (2η+2)(d+2)+3ν+3 n N 2 n ≤ N (2η+2)(d+2)+3ν+5
n and hypothesis (H3) is verified with C 1 := (2η + 2)(d + 2) + 3ν + 5.

For δ ∈ (0, 1), in order to apply the Multiscale Theorem 6.2.16 we have to take χ large enough such that χδ > C 1 ≥ (2η + 2)(d + 2) + 3ν + 5, i.e.

χ >> (2η + 2)(d + 2) + 3ν + 5.

This condition is not in contradiction with condition (6.70) written in this case for ρ := 2η + 2,

χ < 2η + 2 + τ τ 1 ,
if we take τ large enough such that τ > χτ 1 -(2η + 2). (6.84)

The Multiscale Theorem 6.2.16 gives us that L(ε, λ, u, ξ) is N n+1 -regular and estimates (6.72) for LNn+1 .

• (ii) n+1 (M n+1 (ε, λ) ∩ [-N 2η+2 n+1 , N 2η+2 n+1 ]) ⊂ � Q q=1 J q with meas{J q } ≤ N -(2η+2)-τ n+1 and Q ≤ 8K � N (2η+2)(d+2)+ν+2 n+1
We have it directly by (i) n+1 which implies that

M n+1 (ε, λ) ⊂ M 0 n+1 (ε, λ)
and complexity estimates come from Corollary 6.2.14 for ρ = 2η + 2.

The Nash-Moser scheme

We want to solve the forced NLW equation (6.8) with the application of the Nash-Moser Theorem 4.1.1 to F (ε, λ, u) := λ 2 (ω • ∂ ϕ ) 2 u -∆u + V (x)uεf (ϕ, x, u) = 0 (6.85) potential on Zoll manifolds finding for all ε fixed and small enough, a Cantor family of quasi-periodic solutions u ε (λ) for a positive measure set of parameters λ ∈ Λ.

We have to prove that F in (6.85) verifies the assumptions required in Theorem 4.1.1.

In our case, the scales of Banach spaces in the Nash-Moser scheme are the following Sobolev spaces X s = H s (T ν ; H p-2 (M, R))

X � s = H s (T ν ; H p(M, R))
with p > d/2 + 2, where this condition on p is introduced in order to have the algebra structure in H p-2 (M, C).

The subspaces E In what follows, we generally assume Sobolev indices s, p ∈ N even if not specified.

For s 1 > ν/2 and for u satisfying (6.31), fix ε 0 = ε 0 (s 1 ) small enough. with

R N = L N (ε, λ � , u � ) -L N (ε, λ, u) = ((λ � ) 2 -λ 2 )(ω • ∂ ϕ ) 2 -ε(Π N p-2 ∂ u f (ε, x, u � ) -Π N p-2 ∂ u f (ε, x, u)).
Recall that λ, λ � ∈ [1/2, 3/2], using (6.92) we estimate ∀ w ∈ E

= ((λ � ) 2 -λ 2 )(ω • ∂ ϕ + iξ) 2 -ε(Π M p-2 ∂ u f (ε, x, u � ) -Π M p-2 ∂ u f (ε, x, u)),
we can prove in the same way as in (6.94) that �((λ � ) 2λ 2 )(ω • ∂ ϕ + iξ) 2 w� 0,0 ≤ CM -µ+4η+3 �w� 0,0 .

Using (6.95), we obtain �R M (ξ)� 0,0 ≤ CM -µ+4η+3 .

Taking N 0 large enough and

µ > 4η + 3 + τ ⇐⇒ τ � > τ -2δs 1 + p -4η -1 we have ∀ N 0 ≤ M ≤ N �L -1 M (ε, λ, u, ξ)R M (ξ)� 0,0 ≤ �L -1 M (ε, λ, u, ξ)� 0,0 �R N (ξ)� 0,0 ≤ CM τ M -µ+4η+3 ≤ CN τ -µ+4η+3 0 ≤ 1 2 .
Note that we have �L -1 M (ε, λ, u, ξ)� 0,0 < M τ c because ξ ∈ � M 0 2c,M (ε, λ)(u) � c .

Hence Lemma B.1.1 gives that �L -1 M (ε, λ � , u � , ξ)� 0,0 ≤ 2�L -1 M (ε, λ, u, ξ)� 0,0 and proceeding as done in (6.104), we obtain that ∀ N 0 ≤ M ≤ N � L-1 M (ε, λ � , u � , ξ)� 0,0 ≤ 8� L-1 M (ε, λ, u, ξ)� 0,0 .

Hence, for all ξ ∈ � M 0 2c,M (ε, λ)(u) � c , we have Measure estimates for M 0 c/4,M (ε, λ � )(u � ) follow from measure estimates for M 0 2c,M (ε, λ)(u) that we have because (λ, u) ∈ J N .

� L-1 M (ε, λ � , u, � ξ)� 0,0 ≤ 8� L-1 M (ε, λ, u, ξ)� 0,0 ≤ 8 M τ 2c ≤ 4M τ c . ( 6
With these definitions, the � • � X -norm verifies the C-scalar property:

�λu� X = |λ|�u� X , ∀ λ ∈ C. (A.20)
Take u ∈ H s (T, X) with Fourier coefficients u l ∈ X.

For

Similarly, we can prove (A.19). composition

Proof. We prove this result in three steps.

• L is invertible ⇐⇒ T := C -R 2 B -1 R 1 is invertible Taking g = (g 1 , g 2 ) ∈ X, we look for h = (h 1 , h 2 ) ∈ X such that L(ε, λ)h = g. For

L(ε, λ) = � B(ε, λ) R 1 (ε, λ) R 2 (ε, λ) C(ε, λ) � , that means    Bh 1 + R 1 h 2 = g 1 R 2 h 1 + Ch 2 = g 2 .
(B.11)

Since B(ε, λ) is invertible, the first equation is equivalent to h 1 = B -1 g 1 -

B -1 R 1 h 2 .
Putting it in the second equation, we obtain

R 2 B -1 g 1 -R 2 B -1 R 1 h 2 + Ch 2 = g 2 ⇔ (C -R 2 B -1 R 1 )h 2 = g 2 -R 2 B -1 g 1 .
To find h 2 and solve the system (B.11), we have to invert the operator

T := C -R 2 B -1 R 1 .
Hence L(ε, λ) is invertible if and only if T (ε, λ) is invertible.

• �L -1 � ≤ C(�T -1 � + 1) By (B.11), we have h = L -1 (ε, λ)g if and only if

   h 1 = B -1 g 1 -B -1 R 1 T -1 (g 2 -R 2 B -1 g 1 ) h 2 = T -1 (g 2 -R 2 B -1 g 1 ).
Using (i) and (iii), we estimate

�h 1 � X = �B -1 g 1 -B -1 R 1 T -1 (g 2 -R 2 B -1 g 1 )� X ≤ 2 C 2 �g 1 � X + 2εC 1 C 2 �T -1 � � �g 2 � X + 2εC 1 C 2 �g 1 � X � ≤ 2 C 2 �g� X + C�T -1 ��g� X and �h 2 � X = �T -1 (g 2 -R 2 B -1 g 1 )� X ≤ �T -1 � � �g 2 � X + 2εC 1 C 2 �g 1 � X � ≤ C�T -1 ��g� X composition
we performe the decomposition

H s (T, X) = E N ⊕ E ⊥ N
where E ⊥ N is the orthogonal subspace of E N with respect to the scalar product in L 2 (T, X). Proof. We prove this lemma by induction on s.

For s = 1, we can fix α 1 = 1 and α i = 0 for all i ≥ 2 without loss of generality.

Using the definition of � • � L ∞ (T,X) -norm and (A.18), we have directly

I = � � T � �h � 1 (t)� X �h 2 (t)� X . . . �h k (t)� X � 2 dt � 1 2 ≤ �h 2 � L ∞ (T,X) . . . �h k � L ∞ (T,X) � � T �h � 1 (t)� 2 X � 1 2 ≤ C � �h 1 � 1,X k � j=1 j� =i �h j � s0,X
� .

We suppose now that the result is true for all p ≤ s and we prove it for s + 1.

If α j = 0 for all j ∈ A ⊂ [1, k], then we write

I = � � T � � j∈A �h j (t)� X k � i=1 i� ∈A �h (αi) i (t)� X � 2 dt � 1 2 ≤ � j∈A �h j � L ∞ (T,X) � � T � k � i=1 i� ∈A �h (αi) i (t)� X � 2 dt � 1 2 ≤ C � j∈A �h j � s0,X � � T � k � i=1 i� ∈A �h (αi) i (t)� X � 2 dt � 1 2 .
Hence, we can reduce to the case such that α i � = 0 for all i = 1, . . . , k and � k i=1 α i = s + 1. Without loss of generality, we can fix α 1 = min i {α i }.

Let us introduce s �

0 := min{1, s 0 }. For s 0 > 1/2, s � 0 > 1/2 too and the continuous embedding (A.17) holds.

We estimate

I = � � T � �h (α1) 1 (t)� X �h (α2) 2 (t)� X . . . �h (α k ) k (t)� X � 2 dt � 1 2 ≤ �h (α1) 1 � L ∞ (T,X) � � T � �h (α2) 2 (t)� X . . . �h (α k ) k (t)� X � 2 � 1 2 ≤ C�h (α1) 1 � s � 0 ,X � � T � �h (α2) 2
(t)� X . . . �h

(α k ) k (t)� X � 2 � 1 2 composition
and, using the inductive hypothesis on p = � k i=2 α i = s + 1α 1 ≤ s, we obtain

I ≤ C�h (α1) 1 � s � 0 ,X � � T � �h (α2) 2
(t)� X . . . �h

(α k ) k (t)� X � 2 � 1 2 ≤ C�h (α1) 1 � s � 0 ,X C(k, s) � k � i=2 �h i � s+1-α1,X k � j=2 j� =i �h j � s � 0 ,X � ≤ C(k, s) � k � i=2 �h 1 � s � 0 +α1,X �h i � s+1-α1,X k � j=2 j� =i �h j � s � 0 ,X � . (B.26)
Since s � 0 := min{1, s 0 }, we have

s � 0 < s � 0 + α 1 < s + 1, s � 0 < s + 1 -α 1 < s + 1
and we can use Lemma B.3.2 obtaining for all i = 2, . . . , k �h 1 � s � 0 +α1,X �h i � s+1-α1,X ≤ C(�h 1 � s � 0 ,X �h i � s+1,X + �h 1 � s+1,X �h i � s � 0 ,X ).

Using � • � s � 0 ,X ≤ � • � s0,X , we conclude by (B.26) that

I ≤ C(k, s) � k � i=2 (�h 1 � s � 0 ,X �h i � s+1,X + �h 1 � s+1,X �h i � s � 0 ,X ) k � j=2 j� =i �h j � s � 0 ,X � ≤ C(k, s) � k � i=1 �h i � s+1,X k � j=1 j� =i �h j � s � 0 ,X � ≤ C(k, s) � k � i=1 �h i � s+1,X k � j=1 j� =i �h j � s0,X � .
Finally, estimates (B.25) are a direct consequence of (B.24): (t), . . . , h

� � T � �u (α1) (t)� X �u (α2) (t)� X . . . �u (α k ) (t)� X � 2 dt � 1 2 ≤ C(k, s)�u� s,X �u� k-
(α k ) k (t)] � � 2 Y dt � 1 2 ≤ � � T C(k, �u(t)� X ) � �h (α1) 1 (t)� X . . . �h (α k ) k (t)� X � 2 dt � 1 2 ≤ C(k, s, �u� s0,X ) � k � i=1 �h i � s,X k � j=1 j� =i �h j � s0,X � .
As a consequence, estimates (B.27) imply for

h 1 = • • • = h k = u
�G(u, u (α1) , . . . , u (α k ) )� 0,Y ≤ C(k, s, �u� s0,X )�u� s,X .

  a solution of the range equation only for parameters (ε, λ, h) in an appropriate set A ∞ . Moreover in Theorem 5.4.9, under appropriate non-degenerate conditions on the problem (see Definition 5.4.4), we deduce by the Implicit Function Theorem the existence of solutions h ε = h ε (λ) of the bifurcation equation which are smooth with respect to λ.

Chapter 1 .

 1 IntroductionChapter 6: Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifolds

. 10 ) 2 .

 102 Chapter Periodic solutions of the autonomous NLW equation with potential on Zoll manifoldsWe consider the case where S is finite card(S) < +∞.

Lemma 2 . 1 . 3 .

 213 If a function u ∈ H s1, p is solution of equation (2.5), then T a u ∈ H s1, p is solution too.

2 .

 2 Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds with the corresponding L 2 -orthogonal projectors Π

Remark 2 . 4 . 3 .

 243 Let us consider (ε, ω, h) ∈ A ∞ and v = v(ε, ω, h)solution of the (P )-equation associated to parameters (ε, ω, h). Then h solves the (Q)-equation (2.70) ⇐⇒ DΨ ε (ω, h) = 0.(2.72)

.111) Chapter 2 .

 2 Periodic solutions of the autonomous NLW equation with potential on Zoll manifoldsOf course, as done for ε � = 0 in the previous section, we can prove by Mountain Pass theorem that the equation D Ψ0 (h) = 0 (and so the (Q 0 )-equation) has a non-trivial solution, under hyphotesis (2.93).

  +β sin(a),-α sin(a)+β cos(a)) .

  .121), by definition (2.123) we deduce ψ0 (α, β) = ψ0 (ρ a (α, β)) i.e. ψ0 is a radial function. Introduce ψ0 : R → R r � → ψ0 (r) := ψ0 (r, 0), (2.124) we have ψ0 (α, β) = ψ0 ( � α 2 + β 2 ). (2.125) (ii) By definition (2.123) of ψ0 (α, β) and (2.125), we have

  matrix A d (w) is close to the spectrum of the A d (D) which by hypothesis has all simple eigenvalues. By previous Lemma, in order to have a function w(•) ∈ C ∞ (M) such that verifies hypothesis of Lemma 2.5.13, it is enough to prove the existence of (d -1) different points x i and of a vector (λ 1 , λ 2 , . . . , λ d-1 ) ∈ R d-1 \ {0} for which the Dirac meseaure D = � d-1 i=1 λ i δ xi verifies that A d (D) has only simple eigenvalues. By iteration we can prove the following result. Lemma 2.5.16. For all 2

Chapter 2 .

 2 Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds we can apply Lemma 2.5.14 to the operator A + εB since the Π F B |F has only simple eigenvalues by inductive hypothesis and we obtain that for ε > 0 small enough, the operator A + εB has only simple eigenvalues.

Chapter 2 .

 2 Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds Moreover, the eigenvaules ω j → ∞ as j → ∞ (see Lemma 2.1.1) which may verify (2.131) for some m ∈ [0, 1] are only in finite number n = O(|l| d ).

  See section A.2 in Appendix A for details about the definitions of the space L 2 (T, H) and of the � • � 0 -norm. We consider the Banach space L(L 2 (T, H)) := {L : L 2 (T, H) → L 2 (T, H), L linear and continuous}, with operator norm �L� 0 := sup 0� =u∈L 2 (T,H) �Lu� 0 �u� 0 . (3.2) Note that we use the notation � • � 0 both for the operator norm (3.2) of a linear operator L ∈ L(L 2 (T, H)) and for the norm of a function in L 2 (T, H). In every estimate, we have to understand from the context what norm we consider.

Remark 3 . 1 . 3 .

 313 If L = L P , then |L P | s = |||P ||| s,L(H) .

. 7 )

 7 See Remark 3.1.4 for some comments on the constant C of the � • � s -norm.See also section A.2 in Appendix A for details about the definitions of the Sobolev space and of the � • � s -norm. Chapter 3. An abstract analysis for some linear invertible operators For s = 0, H 0 (T, H) = L 2 (T, H).

Remark 3 . 1 . 4 .

 314 As in the definition (3.5) of the | • | s -norm for the linear and continuous operators, the role of the constant C > 0 (independent of s) in the definition of the Sobolev norm � • � s -norm is minor. It is chosen large enough so that the continuous embedding (3.8) implies the following inequality for a

Lemma 3 . 1 . 14 .

 3114 Let L ∈ M B C , with B, C finite sets. If s 0 > 1 2 , then �L� 0 ≤ |L| s0 .

. 18 )

 18 By the identification H B ≡ M {0} B and (3.16), we write

  of separation of the singular sites in(3.24) withdiam(Γ r ) ≤ N δ ∀ r.Hence (U � ) k l = 0 if |k -l| > N δ and we can apply Lemma 3.1.16 to U � to obtain

  .46) By (3.38), (3.45) and Lemma 3.1.11, for s 1 satisfying (3.25) we can take ε 0 small enough and independent of N such that

  we want to estimate �h� s ≤ �h R � s + �h S � s . (3.48) Chapter 3. An abstract analysis for some linear invertible operators Note that, ∀ s ≥ s 0 |L R S | s ≤ |T | s . (3.49) Since h S = U -1 (w S -L R S (L R ) -1 w R ), estimates (3.43) and (3.44) imply

Lemma 4 . 2 . 1 .

 421 (Contraction, first step). There is ε 1 (γ) ∈ (0, ε 0 ] sufficiently small (the one of Theorem 4.1.1) such that ∀ ε ∈ [0, ε 1 ) and for any N 0 large enough depending on ε, γ, s 1 and satisfying (4.8), G 0 is a contraction in the complete metric space

Chapter 4 .

 4 An abstract Nash-Moser Theorem function ψ 0 : [0, ε 1 ) × Λ → [0, 1] takes, for all ε ∈ [0, ε 1 ), the values 1 on N (Λ ε,0 , γN -σ/2 0

42 )

 42 and by (P 1 ) n , v n (0, λ) = 0, we deduce from (F 1) and the uniqueness of the fixed point, thatλ ∈ N (Λ 0,n+1 , 2γN -σ/2 n+1 ) =⇒ gn+1 (0, λ) = 0. (4.43) Lemma 4.3.2. (Estimate in high norm) For all ε ∈ [0, ε 1 ) and for all λ ∈ N (Λ ε,n+1 , 2γN -σ/2 n+1 ), we have

Chapter 4 .

 4 An abstract Nash-Moser Theorem We already have by (4.40) and gn+1 ≤ ρ n+1 ≤ N -2σ-1 n+1 |r n | s1 + |R n (g n+1 )| s1 ≤ C(s 1 ) (4.45) from which we have B n (|r n | s1 + |R n (g n+1 )| s1 ) ≤ C(s 1 )B n . (4.46) Now, we write by (4.38), (4.46) and previous estimates

Chapter 4 .

 4 An abstract Nash-Moser Theorem By (4.44), χ 0 δ < 1, definition of s in (4.2) and using properly �g n+1 � s1 ≤ ρ n+1 ≤ N -2σ-1 n+1 or(4.41)

2 ( 4

 24 .70) taking σ > µ 0 -1 + ν and N 0 large enough. By Lemma B.1.1, estimate (4.70) implies

Remark 5 . 1 . 1 .

 511 We use the notation � • � s,p both for the operator norm (5.5) of a linear continuous operator L ∈ L(H s,p ) and for the Sobolev norm of a function in H s,p . In every estimate, we have to understand from the context what norm we consider. See also Remark 5.1.3 for more informations about the notations used for some operators.

1 .

 1 M is a Zoll manifold, we have studied this equation in chapter 2 and Lemma 2.1.1 assures that the eigenvalues of (-∆ + V (x)) verify Assumption 5.1.2 with r = Remark 5.1.3. Applying Proposition B.3.3 to the spaces X = H p = Y and to the function

. 18 )

 18 According to Remark 3.1.2, we have that the matrix coefficients of the operator D u f (ε, u) ∈ L(H s (T, H p )) and the Fourier coefficients of the function D u f (ε, u) : T → L(H p ) verify the following equality:

. 30 )

 30 Introduce the following frequency-amplitude relation between ω, ω and ε, justified by Remark 5.1.4: there exists λ ∈ Λ := [a, b] ⊂ R such that ω = (1ελ)ω.(5.31)

p⊥ 0 ,Remark 5 . 1 . 4 .

 0514 2π-periodic in time functions too, such that h ε (λ) → h0 (λ) as ε → 0, for some h0 ∈ E 0 and �v ε (λ)� s1, p → 0 as ε → 0. In the decomposition (5.33), solutions h ∈ E 0 of the linear equation (5.28) have frequency ω. The non-linearity εf (ε, u), of order ε, is expected to modify the frequency ω of solutions u of equation (5.1) by a quantity of order ε. The frequency-amplitude relation (5.31) follows this expectations.

Lemma 5 . 1 . 5 .

 515 If a function u ∈ H s1, p is solution of equation (5.1), then T a u ∈ H s1, p is solution too.

  a, b] of R and h ∈ B R (0) in a open ball with centre in the origin and radius R in the finite dimensional linear space E 0(λ, h) ∈ (Λ × B R (0), | • | + � • � 0,0 ). Since E 0 is finite dimensional, all norm in E 0 are equivalent and for h ∈ E 0 �h� 0,0 ≤ R =⇒ �h� s,p ≤ c(s, p)R, ∀ s ≥ 0, 0 ≤ p ≤ p. (5.44) Note that in this chapter we use the Nash-Moser scheme 4.1.1 which is more general than the Nash Moser scheme 2.3.1 used in chapter 2: we can consider two different scales of Banach spaces, in relation to the loss of regularity of the operator A and we have not to introduce an operator which regularize the function F in order to apply the Nash-Moser scheme (that was the role of the operator Q :

.65) Remark 5 . 3 . 5 .

 535 By definition (5.59) of | • | s -norm and definition (5.20) of ||| • ||| s,L(H 0 ) -norm, using the equality (5.19), we write

Chapter 5 .Remark 5 . 4 . 1 .

 5541 Periodic solutions of some non-linear autonomous equations in infinite dimension Let us consider h ∈ E 0 . For any v ∈ E s1, p⊥ 0 and for ε � = 0, we have that h solves the (Q)-equation (5.88) ⇐⇒ Dh Φ(ε, λ, h + v) = 0 (5.89)

Chapter 5 .Remark 5 . 4 . 2 .

 5542 Periodic solutions of some non-linear autonomous equations in infinite dimension Let us consider (ε, λ, h) ∈ A ∞ and v(ε, λ, h) solution of the (P )-equation asociated to parameters (ε, λ, h). Then for ε � = 0 h solves the (Q)-equation (5.91) ⇐⇒ DΨ(ε, λ, h) = 0

Remark 5 . 4 . 3 .

 543 The functional Ψ is invariant under the action of the time translations (T a ) a∈T

D 2 ΨE 0 =

 20 (0, λ 0 , h 0 )[H, ḣ0 ] Z h0 ⊕ span( ḣ0 ), using (5.106) and (5.107), we deduce that

Chapter 5 .

 5 Periodic solutions of some non-linear autonomous equations in infinite dimension By (5.101) and Remark 5.4.5, the condition (5.99) is the usual non-degeneratecondition for D Ψ(0, •, •)    D Ψ(0, λ 0 , h 0 ) = 0, KerD 2 Ψ(0, λ 0 , h 0 ) = {0}.(5.108) Under the hypothesis of the existence of (λ 0 , h 0 ) which verifies (5.108), we will solve for ε � = 0( Dh Φ)(ε, λ, h + v(ε, λ, h)) = 0and so the bifurcation equation (5.91), using the following version of the Implicit function Theorem for a 1-parameter family of functions and Lemma 5.4.8. Theorem 5.4.6 (IFT for family of smooth functions). Let us consider a parameter ε ∈ [0, ε 1 ) and a family of smooth functions {F ε } ε∈[0,ε1) ,

  110)where the convergence in the space C 0 (U, Z) is the convergence with respect tothe � • � L ∞ (U,Z) -norm defined ∀ G ∈ C 0 (U, Z) as �G� L ∞ (U,Z) := sup (x,y)∈U �G(x, y)� Z .(5.111)

  defined on (a, b) × B 1 (0) and with values in L(Z h0 , R), verifies all hypotheses of Implicit Function Theorem 5.4.6.Proof. For all ε ∈ [0, ε 1 ), hypotheses (5.109), (5.110), (5.112) and (5.113) follow immediately if we consider the verctor which represents F ε (λ, H) ∈ L(Z h0 , R) in L 2 (T, H 0 ) by Riesz Theorem.

Remark 5 . 4 . 8 .

 548 For all ε ∈ (0, ε), if (ε, λ, h ε (λ)) ∈ A ∞ , the function h ε (λ) ∈ B R (0) verifies also( Dh Φ)(ε, λ, h ε (λ) + v(ε, λ, h ε (λ))) =0 and so, by Remark 2.4.1, h ε (λ) is a non-trivial smooth solution of the bifurcation equation (5.91). Proof. Recall Z h0 := � span( ḣ0 ) � ⊥ with respect to the scalar product in L 2 := L 2 (T, H 0 ), equation (5.115) means that

  Using the time translation invariance (5.94) Ψ(ε, λ, T a h) = Ψ(ε, λ, h), derivating with respect to a and calculating for a = 0, we haveDΨ(ε, λ, h) ḣ = 0 (5.117) Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite dimension which implies, by definition (5.93), DΨ(ε, λ, h) ḣ = 0. (5.118) By definition (5.92) of Ψ, Ψ(ε, λ, h) := Φ(ε, λ, h + v(ε, λ, h)), and by definitions (5.90) and (5.93), we have DΨ(ε, λ, h) ḣ = DΦ(ε, λ, h + v(ε, λ, h))( ḣ + Dv(ε, λ, h) ḣ). (5.119) Write (5.118) and (5.119) in h ε (λ), we obtain

Assumption 5 . 1 . 2 [ 2 j,k . 5 . 5 . 1 1 ε

 51225511 Distribution of eigenvalues of operator A].There exist some positive constants σ ∈ (0, 1], c 1 , c 2 , c 3 , c 4 , d ≥ 1 and I j := [p jδ j , p j + δ j ] withp j ≥ c 1 j rc 2 , |δ j | ≤ c 3 |p j | σ and cardinality card(spect( √ A) ∩ I j ) ≤ c 4 j dwhere the eigenvalues of √ A are counted with multiplicity.Hypothesis card{σ(√ A) ∩ I j } ≤ c 4 j d says that in each interval I j there are at most c 4 j d eigenvalues counted with multiplicity.In what follows, we callω j,k , 1 ≤ k ≤ d j , d j ≤ c 4 j d ,the eigenvalues of √ A in each I j counted with multiplicity and ϕ j,k the eigenfunction of A associated to each ω Measure estimates for the set A Let us recall the operator defined on E

Lemma 5 . 5 . 2 .

 552 The linear and self-adjoint operator L (N ) (v(ε, λ)) written as(5.125) verifies hypotheses (i) -(v) of Lemma 5.5.1 with C(λ) defined on the subspace E N 2 of finite dimension M ≤ CN d+1 r +1 .Proof. Hypothesis (i) is satisfied by(5.123).Define b(ε, λ) := D u f (ε, h ε (λ) + v(ε, λ, h ε (λ)).Using (5.120), we compute∀ w ∈ E N 2 d dλ L (N ) (v(ε, λ))[w] = -2ε(1ελ)ω 2 w tt -εΠ ⊥ d dλ (b(ε, λ))w.(5.126) Chapter 5. Periodic solutions of some non-linear autonomous equations in infinite

there is not a contradiction between ( 5 (

 5 on the term of initialitation in the Nash-Moser scheme N 0

r satisfies ( 5 S

 5 N := � l ∈ [-N, N ], l singular site for L(v(ε, λ)) with constant 1 � infinite dimension and Λ S N := � λ ∈ Λ ∩ (λ 0δ 0 , λ 0 + δ 0 ) : S N = ∪ r Γ N,r satisfies (5.60)

  .150) Fix now the interval [-N, N ], let l and l � be singular sites in [-N, N ]. We define the following equivalence relation: l ≡ l � if and only if there exists p ∈ N and l 1 , l 2 , . . . , l p singulars sites such that

  By definition, the equivalence classes Γ N,1 , Γ N,2 , . . . Γ N,r , . . . verifydist(Γ N,r , Γ N,r � ) ≥ N β for r � = r �and if we assume that the number of singular sites in every equivalence classΓ N,r is less than N η 2 for some η, then ∀ r diam(Γ N,r ) ≤ N η 2 N β = N δ2with δ := η + β. The choice of β will be done in order to have η + β < 1. Define G m,N := {λ ∈ Λ ∩ (λ 0δ 0 , λ 0 + δ 0 ) : there are at most m singular sites in every Γ N,r in the inteval [-N, N ]}, B m,N := {λ ∈ Λ∩(λ 0 -δ 0 , λ 0 +δ 0 ) : there are at least m singular sites in [-N, N ]}, we obviously have (B m,N ) c ⊂ G m,N .

Chapter 5 .

 5 Periodic solutions of some non-linear autonomous equations in infinite dimension thenmeas(B m,N ) ≤ � N i=-N meas(Ω i ) m .(5.152)We shall find an estimate for meas(Ω l ) of typemeas(Ω l ) ≤ c �l� -ν ε ,for some ν > 0 (5.153)

  Λ := [a, b].

  154), (5.156), (5.158) and (5.159), we obtain an estimate like (5.153):

there is not a contradiction between ( 5 (

 5 on the term of initialitation in the Nash-Moser scheme N 0

  are solutions of the equation (5.1) with frequencies ω = ω(λ) close to ω ω = (1λε)ω.

(6. 43 ) 1 �⊂ L 2 (:Chapter 6 . 2 (:�

 431262 Fix ξ ∈ [-N ρ n , N ρ n ]. For an appropriate constant K large enough, we consider the setsE n 1 (ξ) := {(l, j, k) : |l| ≤ N n ; j > K(|ω • l| + |ξ| + 1), 1 ≤ k ≤ d j }, E n 2 (ξ) = {(l, j, k) : |l| ≤ N n ; j ≤ K(|ω • l| + |ξ| + 1), 1 ≤ k ≤ d j }where the choice 1 ≤ k ≤ d j ≤ C 0 j d-1 is motivated by Lemma 2.1.1.By definition,card(E n 2 (ξ)) ≤ C � N ν n × (N n + |ξ|) × (N n + |ξ|) d-≤ CN ν n (N n + |ξ|) d . + |ξ|) d ≤ CN ν n (N n + |ξ|) d . (6.44) Let E 1 (ξ), E 2 (ξ) be the subspaces of E (Nn) 0 T ν , L 2 (M, C)) corresponding respectively to the indices sets E n 1 (ξ) and E n 2 u(ϕ, x) = � (l,j,k)∈E n 1 (ξ)e ıl•ϕ ϕ j,k (x)u l,j,k , u l,j,k ∈ C   Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifoldsE u(ϕ, x) = � (l,j,k)∈E n 2 (ξ) e ıl•ϕ ϕ j,k (x)u l,j,k , u l,j,k ∈ C   and let Π E1 , Π E2 be the L 2 -orthogonal projectors respectively on E 1 (ξ) andE 2 (ξ).We write LNn (ε, λ, u, ξ) = DNn + RNn , with DNn the diagonal partDNn := diag (|l|≤Nn,j,k) � -(ω • l + ξ) RNn � 0,0 ≤ ε λ 2 �∂ u f (ϕ, x, u)� (op) 0,0 ≤ εc. (6.47)More precisely, according to the decompositionE ) := Π E1 LNn (ε, λ, u, ξ) | E 1 (ξ) , R1(λ) := Π E1 LNn (ε, λ, u, ξ) | E 2 (ξ) = Π E1 ( RNn ) | E 2 (ξ) C(λ) := Π E2 LNn (ε, λ, u, ξ) | E 2 (ξ) , R2 (λ) := Π E2 LNn (ε, λ, u, ξ) | E 1 (ξ) = Π E2 ( RNn ) | E 1 (ξ) .

  ii) ∃ C > 0 such that d dλ C(λ) ≤ -CI, where I is the identity map inL 2 = L 2 (T ν , L 2 (M, C)) � L 2 (T ν × M) (here we are considering Remark B.2.4) We will prove that in this case d dλ LNn (ε, λ, u, ξ) verifies an estimate like Chapter 6. Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifolds (ii) and not only d dλ C(λ). Derivating (6.43), we have d dλ LNn (ε, λ, u, ξ) = diag (j,k) ) ∂ u f (ε, x, u) � .

∂ 2 ≤≤

 2 u f (ϕ, x, u)w, w� L εC�∂ u f (ϕ, x, u)� L ∞(T×M) �w� 2 0,0 ≤ εC�∂ u f (ϕ, x, u)� s1 �w� 2 0,0 ≤ εC(s 1 , p, �u� s0 )(1 + �u� s1 )�w� 2 εC(s 1 , p, �u� s1 , �∂ λ u� s1 )�w� 2

6 . 1 �� 2 ,λ 3 ∂.Chapter 6 . 1 B 1 �� -(ω • l + ξ) 2 + ω 2 j,k λ 2

 612361122 Hence, B(λ) is invertible. In fact, write B(λ) = DB (λ) + RB (λ)(6.54) with DB (λ) = diag (l,j,k)∈E n -(ω • l + ξ) RB (λ)� 0,0 ≤ εC R , (6.56) using (6.52), we estimate� D-1 B (λ)� 0,0 ≤ 2. (6.57) By Lemma B.1.1, taking ε such that � D-1 B (λ) RB (λ)� 0,0 ≤ � D-1 B (λ)� 0,0 � RB (λ)� 0,0 ≤ 2εC R < 1 we have that B(λ) is invertible because B-1 (λ) = ( DB (λ) + RB (λ)) -1 = ( DB (λ)(I + D-1 B (λ) RB (λ))) -1 = (I + D-1 B (λ) RB (λ)) -1 D-1 B (λ) (6.58)and in norm we have � B-1 (λ)� 0,0 ≤ 4. (6.59) Chapter 6. Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifolds• (iv) ∃ C > 0 such that � d dλ Ri (λ)� 0,0 ≤ C Proceeding as in the proof of (ii), using (6.50) and (6.51) we obtain u f (ϕ, x, u)constant C.• (v) ∃ C > 0 such that � d dλ B-1 (λ)� 0,0 ≤ C Applying Lemma B.1.2 to B-1 (λ), we have d dλ ( B-1 )(λ) = -B-1 (λ) d dλ B(λ) B-1 (λ)where by (6.54) 58), write on the one handB-1 (λ) = (I + D-1 B (λ) RB (λ)) -1 D-1 B (λ)and on the other handB-1 (λ) = ( DB (λ) + RB (λ)) -1 = ((I + RB (λ) D-1 B (λ)) DB (λ)) -1 = D-1 B (λ)(I + RB (λ) D-1 B (λ)) -1, by (6.62), we obtain + RB (λ) D-1 B (λ)) -1 Quasi-periodic solutions of the forced NLW equation with potential on (λ) = diag (l,j,k)∈E n

1 B 2 .

 12 (λ)R B (λ)) -1 � 0,0 ≤ 2 and�(I + R B (λ) D-1 B (λ)) -1 � 0,0 ≤ 2 if ε verifies �R B (λ) D-1 B (λ)� 0,0 ≤ ε2C R <1Hence, we conclude the proof of (v) with the hypotheses of Lemma B.2.3 are satisfied: we obtain that for allξ ∈ [-N ρ n , N ρ n ], if c N τ n is small enough (i.e. N 0 large enough) meas � λ ∈ Λ :� ∃ L-1 Nn (ε, λ, u, ξ) or � L-1 Nn (ε, λ, u, ξ)� 0,0 ≥ N τ n c � ≤ KcN ν-τ n (N n + |ξ|) d . � (ω • ∂ ϕ + ıξ � )

Chapter 6 .

 6 Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifolds and it is impossible because we take l 1 and l 2 with modulus less than N n+1 .Hence, for ρ ≥ 1 andλ ∈ Λ \ Λ ρ n , card{l 0 ∈ Z ν , |l 0 | ≤ N n+1 : (ω • l 0 + ξ) ∈ M 0 n (ε, λ) ∩ [-N ρ n , N ρ n ]} ≤ Qand by Corollary 6.2.14,Q ≤ 8K � N ρ(d+2)+ν+2 n .In order to prove the main result of this section, we need the following Multiscale Theorem. We shall give the proof of the theorem in Appendix C: it is a consequence of the Multiscale Theorem C.1.11, which is an adapted version to our functional setting of the Multiscale result 4.1 of[START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF].

Proposition 6 . 2 . 18 .

 6218 Let λ ∈ Λ \ (∪ k≤n Λ 2η+2 k ). Then, we have (i) L(ε, λ, u, ξ) N n -L 2regular ⇒ L(ε, λ, u, ξ) N nregular and ∀ s ≥ s 1 , | L-1 Nn | s ≤ C(s)N τ � n � N δs n + | LNndiag( LNn )| s � (ii) (M n (ε, λ) ∩ [-N 2η+2 n , N 2η+2 n

-

  (2η+2)-τ n and Q ≤ 8K � N (2η+2)(d+2)+ν+2 n .Proof. We prove the Proposition by induction on n.

( 6 . 1 - 2 j,k λ 2 ≤

 6122 [START_REF] Rabinowitz | Free vibrations for a semi-linear wave equation[END_REF].Write as usual LN0 (ε, λ, u, ξ) = DN0 -RN0 withDN0 := diag (|l|≤N0,j,k) � -(ω • l + ξ) (ε, λ, u, ξ) + RN0 .Chapter 6. Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifoldsSince L(ε, λ, u, ξ) is N 0 -L 2regular, we compute � L-1 N0 RN0 � 0,0 ≤ � L-1 N0 � 0,0 � RN0 � 0,0 ≤ N τ 0 � RN0 � 0(6.76) imposing on N 0 the condition N 0 ≤ C ε 1/τ , N 0 → ∞ for ε → 0 (6.77)for an appropriate constant C > 0 and noting that � RN0 � 0,0 ≤ cε by (6.32).Then DN0 is invertible: by (6.75) we can writeD-1 N0 = ( LN0 (ε, λ, u, ξ) + RN0 ) -1 = ( LN0 (ε, λ, u, ξ)(I + L-1 N0 (ε, λ, u, ξ) RN0 )) -1 = (I + L-1 N0 (ε, λ, u, ξ) RN0 ) -1 L-1 N0 (ε, λ, u, ξ)and by Lemma B.1.1 we obtain� D-1 N0 � 0,0 ≤ �(I + L-1 N0 (ε, λ, u, ξ) RN0 ) -1 � 0,0 � L-1 N0 (ε, λ, u, ξ)� 0,0 ≤ 2N τ 0 .(6.78) Since DN0 is diagonal, (6.73) and (6.78) imply that∀ l ∈ Z ν , |l| ≤ N 0 , ∀ j ∈ N, ∀ k ∈ [1, d j ], (ω • l + ξ) 2 + ω 2N τ 0 ,and as a consequence| D-1 N0 | s ≤ 2N τ 0 , ∀ s ≥ 0.In the same way, we writeLN0 (ε, λ, u, ξ) = DN0 -RN0 = DN0 (I -D-1 N0 RN0 ),and we have| D-1 N0 RN0 | s0 ≤ | D-1 N0 | s0 | RN0 | s0 ≤ 2N τ 0 | RN0 | s0 ≤ 1/2potential on Zoll manifoldsWe can writeB n = � l0∈Bn [l 0 ]

  u ∈ H s (T ν , H p � (M, R)) : u(ϕ, x) = � |l|≤N e il•ϕ u l (x), u l ∈ H p � (M, C)   with the corresponding L 2 -orthogonal projectors (in variable ϕ ) Π this chapter we use the Nash-Moser scheme 4.1.1 which is more general than the Nash Moser scheme 2.3.1 used in chapter 1: we consider two different scales of Banach spaces in relation to the loss of regularity of the operator -∆ + V (x) and we have not to introduce an operator which regularize the function F in order to apply the Nash-Moser scheme (that was the role of the operator Q := (-∆ + V (x) + I) -1 defined in (2.31)).

Lemma 6 . 3 . 1 . 2 F

 6312 If f ∈ C ∞ (T ν × M × R), the map F defined in (6.85) satisfies (4.1) with ν = : [0, ε 0 ) × Λ × H s1+2 (T ν , H p(M, R)) → H s1 (T ν , H p-2 (M, R))and (F 1) holds. Moreover F ∈ C 2 and the tame properties (F 2) -(F 4) hold (and as a consequence (F 5) -(F 7) too) for all s ∈ [s 1 , +∞). Proof. The proof is based on standard properties of the composition operators in Sobolev spaces that we prove in section B.3 of Appendix B. Chapter 6. Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifolds we write L N (ε, λ � , u � ) = L N (ε, λ, u) + R N (6.93)

2 �≤ 1 - 1 M 1 M 2 � λ � λ � 2 � L- 1 M

 2111221 ((λ � ) 2λ 2 )(ω • ∂ ϕ ) 2 w� 0,0 ≤ |(λ � ) 2λ 2 |N 2 �w� 0,0 ≤ |λ � -λ|(λ � + λ)N 2 �w� 0,0≤ CN -µ+1 �w� 0,0(6.94) and by C ∞ -regularity of f , for �u� s1 ≤ 1�ε(Π N p-2 ∂ u f (ε, x, u � ) -Π N p-2 ∂ u f (ε, x, u))w� 0,0 ≤ Cε�w� 0,0 �u � -u� L ∞ (T ν ,H p (M)) ≤ Cε�w� 0,0 �u � -u� s1 ≤ CN -(µ+1) �w� 0,0(6.95)where we have used the immersionH s1 (T ν , H p(M)) ⊂ C 0 (T ν , H p(M)) for s 1 > ν/2 and H p(M) ⊂ C 0 (M) for p > d/2.Hence by (6.94) and (6.95),�R N � 0,0 ≤ CN -µ+1 . (6.96)This implies that L N (ε, λ � , u � ) is invertible. In fact, writeL N (ε, λ � , u � ) = L N (ε, λ, u) + R N = L N (ε, λ, u)(I + L -1 N (ε, λ, u)R N ), by (6.96), we can estimate for (λ, u) ∈ J N �L -1 N (ε, λ, u)R N � 0,0 ≤ �L -1 N (ε, λ, u)� 0,0 �R N � 0,0 ≤ CN τ N -µ+1 ≤ CN τ -µ+1taking N 0 large enough and τµ + 1 < 0, i.e.τ < µ -1 < τ � + 2δs 1 + p + 1.(6.98) Chapter 6. Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifolds Note that condition (6.98) is not in contradiction with (6.84) if we take τ � large enough such thatτ � > τ -2δs 1p -1.(6.99) By (6.97) we can writeL -1 N (ε, λ � , u � ) = (L N (ε, λ, u)(I + L -1 N (ε, λ, u)R N )) -1 = (I + L -1 N (ε, λ, u)R N ) -1 L -1 N (ε, λ, u)and Lemma B.1.1 gives that�L -1 N (ε, λ � , u � )� 0,0 ≤ 2�L -1 N (ε, λ, u)� 0,0 ≤ N τ 2 . (6.100)There remains to prove that∀ N 0 ≤ M ≤ N meas{M 0 c/4,M (ε, λ � )(u � ) ∩ [-M (η+1) , M (η+1) ]} ≤ K 1 M (η+1)(d+1)+ν-τ +2 . (6.101)Recall the definition ofM 0 c,M (ε, λ)(u) := � ξ ∈ R :� ∃ L-1 M (ε, λ, u, ξ) or � L-1 M (ε, λ, u, ξ)� 0,0 ≥ M τ c � ,we will obtain (6.101) proving that for (λ � , u � ) which verify (6.92)� M 0 2c,M (ε, λ)(u) � c ⊂ � M 0 c/4,M (ε, λ � )(u � ) M (ε, λ � )(u � ) ⊂ M 0 2c,M (ε, λ)(u).From (6.100), for N 0 ≤ M ≤ N and N 0 large enough such that CN τ -µ+10 (ε, λ � , u � )� 0,0 ≤ 2�L -1 M (ε, λ, u)� 0,0 . (6.102) Moreover, if N 0 is large enough such that 2N -(µ+1) 0 ≤ 1/2, then |λ � -λ| ≤ 1/2. Since λ, λ � ∈ [1/2, 3/2], this implies that (ε, λ � , u � )� 0,0 ≤ (ε, λ, u)� 0,0 ≤ 8� L-1 M (ε, λ, u)� 0,0 . (6.104) More generally, write for all ξ ∈ � M 0 2c,M (ε, λ)(u) � c L M (ε, λ � , u � , ξ) = L M (ε, λ, u, ξ) + R M (ξ) Chapter 6. Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifolds with R M (ξ) = L M (ε, λ � , u � , ξ) -L M (ε, λ, u, ξ)

  M (ε, λ � )(u � ) ⊂ M 0 2c,M (ε, λ)(u).

  t∈T �e ılt u l � X =(A.20) sup t∈T |e ılt |�u l � X = �u l � X . By (A.21), � l∈Z �e ılt u l � L ∞ (T,X) = � l∈Z �u l � X ≤ C�u� s,X < ∞. (A.22)By a classical result, consequence of the completeness of the Banach space(C 0 (T, X), � • � L ∞ (T,X) ), the fact that � l∈Z �e ılt u l � L ∞ (T,X) < +∞ implies that u =� l∈Z e ılt u l is well defined in C 0 (T, X). Finally, we can write from (A.[START_REF] Craig | Newton's method and periodic solutions of nonlinear wave equations[END_REF] �u� L ∞ (T,X) �e ılt u l � L ∞ (T,X) ≤ C�u� s,X .

�

  Let Π N and Π ⊥ N be the orthogonal projectors respectively on E N and E ⊥ N for which ∀ s ≥ 0, ∀ d ≥ 0�Π N u� s+d ≤ N d �u� s , �Π ⊥ N u� s ≤ (N + 1) -d �u� s+d . (B.15) Write u ∈ H s (T, X) as u = Π N u+Π ⊥ N u, using (B.15), we estimate for s 1 ∈ [s 0 , s]�u� s1 ≤ �Π N u� s1 + �Π ⊥ N u� s1 ≤ N s1-s0 �u� s0 + (N + 1) -(s-s1) �u� s . (B.16) Estimate (B.16) holds ∀ N ∈ N.Define for all x ∈ R + , [x] the integer part of x, we have[x] ≤ x < [x] + 1.Hence we can write by (B.16), ∀ x ∈ R + �u� s1 ≤ [x] s1-s0 �u� s0 + ([x] + 1) -(s-s1) �u� s ≤ x s1-s0 �u� s0 + x -(s-s1) �u� s . (B.17)The function g(x) := x s1-s0 �u� s0 + x -(s-s1) �u� s has a minimum inx min = � (ss 1 ) (s 1s 0 ) �u� s �u� s0 � 1 s-s 0 .Write (B.17) for x = x min , we obtain�u� s1 ≤ x s1-s0 min �u� s0 + x -(s-s1) min �u� s ≤ C(s 0 , s) �� �u� s �u� s0 � s 1 -s 0 s-s 0 �u� s0 + � �u� s �u� s0 � -(s-s 1 )and usings 1 = (1θ)s 0 + θs which implies 1 -s 1s 0 ss 0 = ss 1 ss 0 = 1θ -(ss 1 ) ss 0 + 1 = s 1s 0 ss 0 = θ, we conclude �u� s1 ≤ C(s 0 , s)(�u� θ s �u� 1-θ s0 + �u� θ s �u� 1-θ s0 ) ≤ C(s 0 , s)�u� 1-θ s0 �u� θ s . composition As a consequence, for u ∈ C ∞ (T, X) � � T � �u (α1) (t)� X �u (α2) (t)� X . . . �u (α k ) (t)� X � 2 dt� 1 2 ≤ C(k, s, �u� s0,X )�u� s,X (B.25) with C(k, s, �u� s0,X ) = C(k, s)�u� k-1 s0,X .

27 )

 27 ) k)� 0,Y ≤ C(k, s, �u� s0,X )In particular, forh 1 = • • • = h k = u �G(u, u (α1) , . . . , u (α k ) )� 0,Y ≤ C(k, s, �u� s0,X )�u� s,X .(B.28)Moreover, for k = 0, the map G(u) ∈ C ∞ (T, Y ) defined by G(u)(t) := P (u(t)), ∀ t ∈ T verifies �G(u)� 0,Y ≤ C(s, �u� s0,X ) (B.29)and�G(u)� s,Y ≤ C(s, �u� s0,X )(1 + �u� s,X). (B.30) Proof. By hypothesis of Proposition B.3.3, P ∈ C ∞ (X, Y ) and for all 1 ≤ k ∈ N and functions u, h 1 , . . . , h k ∈ C ∞ (T, X), the map G(u, h 1 , . . . , h k )(t) := D k P (u(t))[h 1 (t), . . . , h k (t)] is well defined and G(u, h 1 , . . . , h k ) ∈ C ∞ (T, Y ). For all 1/2 < s 0 ≤ s ∈ N and for α 1 , . . . , α k ∈ N such that � k i=1 α i ≤ s, by hypothesis (B.21), estimate (A.18) and (B.24), we have �G(u, h k P (u(t))[h

  ). We use the notation � • � s,p both for the operator norm (2.28) of an operator L from H s,p into itself and for the Sobolev norm (2.27) of a function in H s,p . In every estimate, we have to understand from the context what norm we consider.

	Remark 2.2.1.

  and only if h verifies the bifurcation equation.

	Chapter 2. Periodic solutions of the autonomous NLW equation with potential
	on Zoll manifolds
	0 Remark 2.4.2. For any h ∈ E 0 , a function v ∈ E s1,⊥ equation if and only if it is a critical point of the restricted functional v ∈ solves the range E s1,⊥ 0 → Φ ε (ω, h + v) ∈ R, i.e. if and only if
	Moreover, we have also the following result which is the corresponding of
	Remark 2.4.1 for the (P )-equation.

  Periodic solutions of the autonomous NLW equation with potential on Zoll manifoldsBy Lemma B.1.1, we obtain (iii). In fact for ε small enough, we have

	Chapter 2.
					.150)
	by (2.145), we estimate	�D -1 B � ≤	1 C 2	.	(2.151)

  Periodic solutions of the autonomous NLW equation with potential on Zoll manifolds 2.6.2 Measure estimate for the set A 2

  4.1) with ν = 2 and (F 1) holds. Moreover, the tame properties (F 2) -(F 4) (and as a consequence (F 5) -(F 7) too) hold for all s ∈ [s 1 , +∞].Proof. Using the continuity of the operator A : H s, p → H s, p-2 , of the operator ∂ tt : H s+2, p → H s, p and the C ∞ -regularity of the map f : H s (T, H p) → H s (T, H p) (see Remark 5.1.3), the function defined in(5.42)

  is solution of the range equation and we can prove (as done in Remark 2.4.2) that DΦ(ε, λ, h ε

  .10) See section A.2 of Appendix A for details on the definition of the Sobolev space H s (T ν , H p (M, R)), p ∈ N, and for the proof of the imbedding (2.7).

Moreover, ∀ s ≥ s 0

  Moreover, given a linear continuous operator L from H s,p � into itself, �L� s,p � denotes the associated operator norm (�L� s , if p � = p): We use the notation � • � s,p � both for the operator norm (6.13) of a linear and continuous operator L from H s,p � into itself and for the Sobolev norm of a function in H s,p � . In every estimate, we have to understand from the context what norm we consider.See also Remark 6.2.1 for having more information about the notation used for the operator norm of an "operator of multiplication" by a function of H s,p � .

	�L� s,p � := sup u� =0 Remark 6.1.1. Chapter 6. Quasi-periodic solutions of the forced NLW equation with �Lu� s,p � �u� s,p � . potential on Zoll manifolds	(6.13)

  singular (Lemma 6.2.11 and Lemma 6.2.12, proved using the Fubini-Tonelli formula and Lemma B.2.3

of Appendix) and complexity estimates of this set (Lemma 6.2.13 and Corollary 6.2.14 with ρ ≥ 1)

  Chapter 6. Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifolds Since λ ∈ [1/2, 3/2], this implies that ≥ -(|ω • l| + |ξ|) 2 + 4(|ω • l| + |ξ| + 1) 2 λ 2 ≥ -(|ω • l| + |ξ|) 2 + 16(|ω • l| + |ξ| + 1) 2 9 ≥ -(|ω • l| + |ξ| + 1) 2 + 16(|ω • l| + |ξ| + 1) 2 9

	� � � -(ω • l + ξ) 2 +	ω 2 j,k λ 2	� � � ≥ -(ω • l + ξ) 2 +	ω 2 j,k λ 2
			≥	1 2	(|ω • l| + |ξ| + 1) 2
	and
	� � � -(ω • l + ξ) 2 +	ω 2 j,k λ 2	� � � ≥ -(ω • l + ξ) 2 +	ω 2 j,k λ 2
						≥ -	ω 2 j,k 4	+	ω 2 j,k λ 2
						≥ -	ω 2 j,k
						� . (6.52) j,k 6
	By Lemma 2.1.1, we have for some constant K large enough
	ω j,k ≥	2π T	j + α -	c 0 j β
		≥	2π T	K(|ω • l| + |ξ| + 1) + α -	c 0 j β	(6.53)

≥ 2(|ω • l| + |ξ| + 1) + α ≥ 2(|ω • l| + |ξ| + 1).

  t ∈ T fixed, we estimate by Cauchy-Schwarz inequality� l∈Z �u l � X = � l∈Z �l� s �l� -s �u l � X l∈Z �l� -2s converges, which is the case if 2s > 1, i.e. s > 1/2. Moreover, for t ∈ T fixed �e ılt u l � L ∞ (T,X) = sup

		≤	�	� l∈Z	�l� 2s �u l � 2 X	� 1/2 �	� l∈Z	�l� -2s	� 1/2	(A.21)
		≤ C�u� s,X
	provided if	�

  1 s0,X . Lemma B.3.5. Assume hypotheses of Proposition B.3.3.For all 1 ≤ k ∈ N and functions u, h 1 , . . . , h k ∈ C ∞ (T, X), consider the map G(u, h 1 , . . . , h k ) ∈ C ∞ (T, Y ) G(u, h 1 , . . . , h k ) : T → (Y, � • � Y )For all 1/2 < s 0 ≤ s ∈ N and α 1 , . . . , α k ∈ N such that � k i=1 α i ≤ s, the following estimate hold:

	�G(u, h

defined ∀ t ∈ T by G(u, h 1 , . . . , h k )(t) := D k P (u(t))[h 1 (t), . . . , h k (t)]. composition

< 1.

Chapter 6. Quasi-periodic solutions of the forced NLW equation with

Chapter 6

Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifolds 6.1 Main result: functional setting and general idea

We consider the non-linear wave equation with a quasi-periodic non-linear forcing term u tt -∆u + V (x)u = εf (ωt, x, u),

x ∈ M, t ∈ R (6.1)

where the frequency ω is a ν-dimensional vector in R ν , ν > 1, and M is a ddimensional connected, compact, Riemannian C ∞ -manifold without boundary, of Zoll type, namely the geodesic flow on the unit tangent bundle is periodic of minimal period T > 0.

We assume that:

• the potential V (x) satisfies

which implies the property

where I is the identity map in L 2 (M, C) (we refer to section A.1 of Appendix A for definitions of Sobolev spaces H p (M, R) ⊂ L 2 (M, R), p ∈ N, and their complexifications H p (M, C) ⊂ L 2 (M, C)).

We recall that (6.3) means �-∆g, g� L 2 (M,C) + �V g, g� L 2 (M,C) ≥ β 0 �g, g� L 2 (M,C) , ∀ g ∈ L 2 (M, C) Chapter 6. Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifolds Lemma 6.2.9. Let us consider β ∈ (0, 1) of Lemma 2.1.1, γ 1 ∈ (0, 1) and τ 1 > 1 which appear in definition (6.14) of Λ.

If l 1 � = l 2 are two different singular sites for L(ε, λ, u, ξ), then

where η := τ1 β .

Recall u = u(λ), for M ∈ N, let us define the sets

Fix now a natural integer 0 < N 0 ∈ N (that will be chosen properly in the following for the initial step of the Nash-Moser scheme) and define

, for some χ > 1 (6.41) where [ • ] denotes the integer part of a real number.

We remark that this scale N n is not the same scale of the Nash-Moser scheme which will be defined for some χ 0 > 0, χ 0 << χ.

In the following, if M = N n for some n, we will write M 0 n (ε, λ) instead of M 0 Nn (ε, λ) (and the same for all others sets) in order to simplifying the notations. By definition 6.2.6 we have: Remark 6.2.10. A site l 0 is N n -singular for L(ε, λ, u, ξ) if and only if (ξ + ω • l 0 ) ∈ M n (ε, λ).

In particular, a site l 0 is N n -singular for L(ε, λ, u) if and only if ω•l 0 ∈ M n (ε, λ).

For any constant c � = 0 in a appropriate bounded interval (that we will choose), we define also

Of course, for c = 1

Chapter 6. Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifolds Call

meas{M ξ } dξ, we conclude

Lemma 6.2.12. For all ρ ≥ 1, there exists Λ ρ n with meas{Λ ρ n } < N -2 n such that for all λ ∈ Λ \ Λ ρ n , we have the following estimate

n for an appropriate constant K � > 0.

Proof. For ρ ≥ 1, Lemma 6.2.11 implies

for an appropriate positive constant K � .

For ξ ∈ [-N ρ n , N ρ n ], by Fubini-Tonelli formula, we write

Define for K � > 0 of (6.65)

we have

Estimate (6.65) implies that

Chapter 6. Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifolds from which

We conclude observing that, by definition of Λ ρ n , for all λ ∈ Λ \ Λ ρ n ,

The following result is proved with same arguments used in Lemma 6.3 of [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF].

Lemma 6.2. [START_REF] Berti | An abstract Nash-Moser theorem and quasi-periodic solutions for NLW and NLS on compact Lie groups and homogeneous manifolds[END_REF]. For all ρ ≥ 1, we have that

where J q are intervals with meas{J q } ≤ N -ρ-τ n and

Proof. Take ξ ∈ M 0 n (ε, λ) ∩ [-N ρ n , N ρ n ] and consider the bounded interval

We have for all

� LNn (ε, λ, u, ξ � ) -LNn (ε, λ, u, ξ)� 0,0 ≤ � � �Π

(N ) 0

Chapter 6. Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifolds From (6.66) and (6.67), we have that for all parameters ξ � ∈ [ξ -N -ρ-τ n , ξ +

∃ v � = 0 : � LNn (ε, λ, u, ξ � )v� 0,0 ≤ 4N -τ n �v� 0,0 which implies � L-1 Nn (ε, λ, u, ξ � )� 0,0 ≥ 4N τ n . (6.68)

Recall the definition of M 0 4,n (ε, λ)

and so

Since each connected component of

as a union of disjoint intervals J m with length meas{J m } ≥ 2N -ρ-τ n and we have

We decompose each J m as a union of nonoverlapping intervals J q of length between N -ρ-τ n /2 and N -ρ-τ n . Then, by (6.69), we get a new covering

and since the intervals do not overlap,

} and this proves the lemma. Chapter 6. Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifolds From Lemmas 6.2.12 and 6.2.13, we obtain: Corollary 6.2.14. For all ρ ≥ 1, there exists Λ ρ n with meas{Λ ρ n } ≤ N -2 n such that for all λ ∈ Λ \ Λ ρ n , we have

In the following Lemma, we show that we have a bound of the number of

Lemma 6.2.15. Fix ξ ∈ R and the growth exponent χ of the scale N n defined in (6.41),

As a consequence, for all ρ ≥ 1 and

Proof. In the reductio ad absurdum, we suppose that there exist l 1 � = l 2 ,

and using (6.71)

Chapter 6. Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifolds because as before

is obtained by (6.33).

Hence, by Lemma 3.1.17 (and Remark 3.1.18), we obtain estimates (6.72)

Moreover, using (6.79), (6.80) implies

By definition 6.2.4, we have that L(ε, λ, u, ξ) is a N 0 -regular operator.

We have it directly by (i) 0 which implies that

and the complexity estimates come from Corollary 6.2.14 for ρ = 2η + 2.

We suppose now to have (i) n and (ii) n and we prove the inductive step.

We will prove it applying the Multiscale Theorem 6.2.16 to LNn+1 (ε, λ, u, ξ).

The heart of the proof is the partition in blocks of the N n -bad sites.

We have to check that LNn+1 (ε, λ, u, ξ) verifies all the hypotheses of the Multiscale Theorem 6.2.16.

The estimation (H1) about the L 2 -norm is done directly: L(ε, λ, u, ξ) is

We have (H2) by estimates

Chapter 6. Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifolds obtained by (6.33).

We prove now (H3). We need a good separation of the N n -bad sites for the operator LNn+1 (ε, λ, u, ξ).

Recall the definition of N n -bad sites for the operator L(ε, λ, u, ξ):

By these definitions, it is clear that the N n -bad sites for the operator LNn+1 (ε, λ, u, ξ) are also N n -bad sites for the operator L(ε, λ, u, ξ).

We consider a singular site l such that (ξ

]. It is well separated from any other singular site: take a singular site l 1 � = l (for which we suppose nothing about (ξ + ω • l 1 )), using Lemma 6.2.9, we have

Let us consider now the N n -singular sites l � such that (ξ + ω

By Remark 6.2.10, they are the sites for which (ξ + ω

By inductive hypothesis (i) n , we have

and then using Lemma 6.2.15, for λ

With these considerations we can prove a good separation in blocks of the N n -bad sites for the operator L(ε, λ, u, ξ).

Recall that u depends on λ, u = u(λ) and define

We write

Chapter 6. Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifolds where

taking N 0 (ε) large enough. Hence, recall the definition of N n -bad sites, we can write

where

. By (6.82), we obtain

(6.83)

Introduce now the following equivalence relation in B n : l ≡ l � if and only if there exists a N 2 n -chain in B n which connects l to l � , namely if and only if there exist p ∈ N and l 1 , l 2 , . . . , l p N n -bad sites such that

By definition of the equivalence relation, for l 0 ∈ B 1 n , the equivalence classe [l 0 ] is reduced to the singleton {l 0 } (there are not N n -bad sites different from l 0 which have a distance from l 0 less than N 2 n ). In the same way, if l 0 ∈ B 2 n , in the equivalence classe [l 0 ] there are only elements of B 2 n (if there was some site

n and it is impossible because we are supposing l 0 ∈ B 2 n ): by (6.83)

Hence, we have that for all See also see [START_REF] Berti | Cantor families of periodic solutions of wave equations with C k nonlinearities[END_REF] for references.

We take as approximating solution in the initial step the function ũ = 0 and hypotheses ( Ṽ 1) -( Ṽ 4) are easily verified for q = 1.

There remains to define the sets of parameters J

µ,K , the linearized operators

are invertible and the following estimates hold for a positive measure set of

Moreover, we have to prove Hypothesis (L1) of the Nash-Moser scheme.

Recall the definition of L(ε, λ, u)

and that since λ

with similar estimates for the norms of their inverses.

We start from the following proposition.

then for ε small enough we have the estimates (6.87) for LN (ε, λ, u) with a constant K(s): Since LN (ε, λ, u) verifies condition (6.88), using Lemma 3.1.12 (and Remark 3.1.18), we have

These estimates imply (6.89) proceeding as in Lemma 5.3.6.

Fix ε ∈ [0, ε 0 ). Fix constants c ≥ 1 and

where K � is the constant of Lemma 6.2.12.

Let us define for �u� s1 ≤ 1,

� Note that we write M 0 2c,M (ε, λ)(u) in order to remark that this set depends on u (we will have u = u(λ) only after the application of the Nash-Moser Theorem).

By definition, since

and so

Recall the definition of Λ 2η+2 M introduced in Lemma 6.2.12,

and we have that estimate (6.89) on � L-1 N (ε, λ, u)� s implies estimate (6.87) on �L -1 N (ε, λ, u)� s for all s ≥ s 1 , with the same µ := τ � + 2δs 1 + p + 2 and a constant

Hence, we have the inclusion

with µ := τ � + 2δs 1 + p + 2 as required in the Nash-Moser scheme.

We can now prove the last hypothesis for applying the Nash-Moser Theorem:

we will obtain hypothesis (L1) of the Nash-Moser scheme as a consequence of the following lemma.

Lemma 6.3.3. Let us consider µ := τ � + 2δs 1 + p + 2. Take (λ, u) ∈ J N and

Chapter 6. Quasi-periodic solutions of the forced NLW equation with potential on Zoll manifolds Using Lemma 6.3.3, we can prove the validity of the hypothesis (L1):

proceeding as done in the proof of inclusion (6.91) (and using K 1 ≤ c 4 K � ), we obtain that

� for an appropriate constant K � (s) > 0.

We have so proved that F (ε, λ, u) defined in (6.85) verifies all the hypotheses of the Nash-Moser Theorem 4.1.1.

Hence, there exists ε 0 > 0 small enough such that for all ε ∈ [0, ε 0 ), there exists a sequence of functions

, where the scale in the Nash-Moser scheme is defined as

Moreover, ūε (λ) are in H s for all s ≥ s 1 .

There remains to prove that the set A ∞ has a "large" measure.

Define

By property (P 2) n of the sequence {u k } in the Nash-Moser scheme, we have

and it implies

Hence, proceeding as in the last part of Lemma 6.3.3, we can prove that

and this implies, with Proposition 4.5.1, the inclusion

we can prove proceeding as in Lemma 6.2.12 that

for some positive constant C.

We write

By (6.107) and by Lemma 6.2.11 used with ξ = 0, ρ = 0 and c = 18, we estimate

which is small if we take N 0 = N 0 (ε) large enough and τ > ν + 1 + d such that

Hence the set A ε is large: by inclusion (6.106), A ∞ is large too.

We conclude the proof of the main Theorem 6.1.3 noting that, since N 0 = N (ε) verifies (6.77)

imply that A ε has asymptotically full Lebesgue measure, i.e.

meas(Λ

Appendix A

Sobolev spaces

A.1 Sobolev spaces on compact manifolds

Then there exists a finite atlas

where for all i ∈ I, σ i is a homeomorphism with the following property: for two charts (U i , σ i ), (U j , σ j ), i, j ∈ I, with U i ∩ U j not empty, the transition map τ i,j

Let us consider {ψ i } i∈J a C ∞ -partition of unity for M subordinate to the atlas

Let us consider a function g : M → R and let us introduce for all i ∈ I the map

defined as

For all i ∈ I, we have that supp{ρ i,g } is compact and supp{ρ i,g } ⊂ σ i (U i ).

We can now introduce the Sobolev space defined on a d-dimensional compact manifold M: for all p ∈ N, we define

We can define a scalar product on H p (M, R) by

and the associated Sobolev norm

Of course this norm depends on the atlas A, but a different choice of the atlas would give an equivalent norm.

Moreover, (H

We write

For p > d/2, the Sobolev space H p (M, R) is a Banach algebra with respect to multiplication of functions.

Let us consider also (C 0 (M, R), � • � L ∞ (M,R) ) the space of all continuous functions defined on M into R, endowed with the sup-norm

For p > d/2, there is a continuous embedding

equipped with an inner product g.

Let us consider the Sobolev spaces H p (M, R), p ∈ N, defined on (M, g) by (A.2).

For p = 0, we can endow L 2 (M, R) := H 0 (M, R) with the natural scalar

where �•, •� g is the inner scalar product induced by g and dx is the natural measure on M associated to g.

Finally, for all p ≥ 0, we define the complexification of Sobolev spaces as

Similarly, we define

A.2 Sobolev spaces on T with values in Hilbert spaces

Let (X, � • � X ) be a real (or complex) separable Hilbert space with inner

Let us consider a Hilbert basis {e n } n∈Z ⊂ X of X.

For T = R/2πZ, let us consider a function u : T → X. We define ũn :

Let us define the space

where

For u ∈ C ∞ (T, X) and α ∈ N, we denote by u (α) its α-th derivative with respect to t ∈ T.

For a positive integer s ∈ N + and u ∈ C ∞ (T, X), we introduce the definition of the Sobolev norm

and the definition of the Sobolev space H s (T, X) as the closure of C ∞ (T, X)

Finally, for u ∈ H s (T, X) and l ∈ Z, we define its Fourier coefficients u l ∈ X as

It can be proved that � |l|≤N u l e ılt converges in norm � • � s,X to the function u ∈ H s (T, X) and we can write

One can prove Parseval's identity

which implies by (A.7) that

More generally, one can prove

where �l� := max{1, |l|}.

We can now introduce the definition of the Sobolev norm also for all real indices s ∈ R + . We define

which for s = 0 it is equal to the L 2 -norm defined in (A.6) by (A.13) and for integers s ∈ N + it is equal to the definition (A.8) by property (A.14).

Hence, we can extend the definition (A.9)

By (A.11) and (A.15), we can write

For all 0 ≤ s � ≤ s, we have

Finally, let us introduce C 0 (T, X) the set of all continuous function from T to

We have the following important result.

Remark A.2.1. For all s > 1/2, there is a continuous embedding

Hence, we have

More general, for all s > 1/2 + k, there is the embedding

where C k (T, X) is the set of all functions u which are C k in the time variable

Proof. If X is a real Hilbert space, we consider the complexification of X

with scalar product defined by: ∀

In particular, X � is indowed with the following norm (that we still denote by

Take two operators L 1 and L 2 in L(E), we have

In the following we will write simply � • � for the operator norm instead of

Moreover, every time we estimate �L�, the operator L is supposed to be linear and continuous, even if not specified.

Lemma B.1.1. Let A and B be two operators in L(E) such that A is invertible and

Then, the operator (A + B) is invertible and

Proof. The fact that A + B is invertible is standard: it is enough to write

and to notice that I + A -1 B is invertible using

and the completeness of the Banach space L(E).

We compute directly

where we have used the Neumann series formula to write (I + A -1 B) -1 under property (B.1).

Hence, by (B.1) and (B.2) we conclude Then I is differentiable and

Moreover, if we consider the map b : ε ∈ (0, 1) → B(ε) ∈ O(E) and the composition

Proof. The Lemma follows from direct computation: by (B.2), taking �H� small enough such that �B -1 H� ≤ �B -1 ��H� < 1,

with �R(H)� ≤ C�H� 2 and (B.3) is proved.

B.2 Measure estimates

Let (X, � • � X ) be a Hilbert space and A, B be two linear operators defined on X. We will write A ≥ B for meaning

where �•, •� X is the scalar product in X.

then, for all δ > 0,

Recall that, since A(λ) is a selfadjoint operator, the operator norm of A(λ)

where µ 1 (λ), ... , µ M (λ) are the eigevalues of A(λ).

See Lemma 6.1 of [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF] (and also [START_REF] Kato | Perturbation Theory for Linear Operators[END_REF]) for the references of the Lemma B.

Let us consider now an infinite dimensional Hilbert space X which we decompose in

Let us define Π X1 and Π X2 the orthogonal projectors respectively on X 1 and

Let L be an operator defined on X. We write L as

where

By definition, R 2 = R ⊥ 1 and C is an operator defined on the finite dimesional Hilbert space X 2 .

Lemma B.2.3 (for Infinite dimensional operators). Let L(ε, λ) be a selfadjoint operator defined on a Hilbert space X = X 1 ⊕ X 2 , with X 2 of finite dimension M . According to the decomposition (B.9), we write L(ε, λ) as

Assume that B(ε, λ) is invertible and that there exist q ≥ 0 and constants

Then there exist ε 0 , δ 0 small enough and a constant K > 0 such that for all ε ∈ (0, ε 0 ) and δ ∈ (0, δ 0 )

for some constant C > 0.

Then

for an appropriate constant C > 0.

Moreover, take K > C and δ 0 small enough such that ∀ δ ∈ (0, δ 0 )

we have the inclusion

• Measure estimates

We write

By (B.6) and hypothesis (i) -(v), we take ε 0 small enough such that for all ε ∈ (0, ε 0 ) we have

By inclusion (B.12), we have S c L ⊂ S c T and (B.13) implies meas

3 is true also if we replace condition (ii) with

B.3 Operators of composition

The aim of this section is to prove the useful result about operators of composition given in the Proposition B.3.3.

We use the functional setting introduced in the section A.2 of Appendix A: we

When we consider only one Hilbert space X, we will denote by � • � s (instead of � • � s,X ) the norm in H s (T, X) in order to simplify the notations.

We insert here two lemmas which we will use in the proof of Proposition B.3.3.

For references of the following result see e.g. [START_REF] Moser | A rapidly convergent iteration method and non-linear partial differential equations[END_REF] (chapter 1).

Proof. Define for all N ∈ N, E N ⊂ H s (T, X) the subspace

The following result is taken from [START_REF] Berti | Cantor families of periodic solutions of wave equations with C k nonlinearities[END_REF] (see Lemma 2.2).

Lemma B.3.2. For all u ∈ H s (T, X) and r ≥ 0 such that Write

for some θ, θ � ∈ [0, 1]. We have that

and then θ + θ � = 1.

Using (B.14), we have

where we have used Young's inequality

In the same way, we prove (B.20).

We can now insert the main result of this section.

For simplicity, we consider the Sobolev spaces H s with integer indices s ∈ N.

Appendix B. Useful results about invertible operators and operators of composition

Consider the function P : u � → P (u) defined on the Sobolev space (H s (T, X), � • � s,X ), s ∈ N, by

For all 1/2 < s 0 ≤ s ∈ N + , the operator P maps (H s (T, X), � • � s,X ) into

More precisely, P ∈ C ∞ (H s (T, X), H s (T, Y )) with

and the following norm-estimates hold:

We divide the proof of Proposition B.3.3 in several lemmas.

We start giving some results for functions u, h 1 , . . . , h k ∈ C ∞ (T, X).

Recall that, take a function u ∈ C ∞ (T, X) and α ∈ N, we denote by u (α) its α-th derivative with respect to t ∈ T.

we have

Moreover, for k = 0, we define G(u)(t) := P (u(t)). By C ∞ -regularity of P beetwen the spaces X and Y , we have that G(u) ∈ C ∞ (T, Y ) and

There remains to estimate �G(u)� s,Y .

Define v(t) = G(u)(t) = P (u(t)), for s 0 ≤ s ∈ N, we write the Sobolev norm as in (A.8)

where, every l-th derivative v (l) (t) is a sum of terms of the form For all 1/2 < s 0 ≤ s ∈ N, for all 1 ≤ k ∈ N and for all functions u, h 1 , . . . , h k ∈ C ∞ (T, X), we have

Proof. Using the property (B.21) of P , the definition of � • � L ∞ (T,X) -norm and the estimate (A.18), we have directly

Proceeding as in Lemma B.3.5, we define

and for all 1/2 < s 0 ≤ s ∈ N, we write the Sobolev norm as

where every l-th derivative w (l) (t) is a sum of terms of the form

(t), . . . h

with q ∈ [0, l] and � q i=1 α i + � k j=1 β j = l. By estimates (B.27) written for �w (l) � 0,Y and (B.31), we conclude For all 1/2 < s 0 ≤ s ∈ N, for all 1 ≤ k ∈ N and for all functions u, ū, h 1 , h1 , . . . , h k , hk ∈ C ∞ (T, X), we have

(B.32) composition

Proof. Recall the definition of

it is enough to write

where

We have

Hence, (B.32) follows using Lemma B.3.6 to estimate Let us consider 1/2 < s 0 ≤ s ∈ N.

If we take u, h 1 , . . . , h k ∈ H s (T, X), then G(u, h 1 , . . . , h k ) ∈ H s (T, X).

Proof. For 1/2 < s 0 ≤ s ∈ N, we have the continuous embedding (A.17)

, by definition of G and regularity of

By density of C ∞ (T, X) in H s (T, X), there exist sequences {u n } n , {h i,n } n ∈ C ∞ (T, X) which converge in � • � s,X -norm respectively to u and h i for all i = 1, . . . , k. In particular, using the embedding (A.17),

converge in � • � L ∞ -norm respectively to u and h i for all i = 1, . . . , k and

We have to prove that G(u n , h 1,n , . . . , h k,n ) converges also in H s (T, X) in order

It is enough to prove that {G(u n , h 1,n , . . . , h k,n )} n is a Cauchy sequence in H s (T, X). The completeness of the Hilbert space H s (T, X) gives us the convergence.

Using estimates (B.32), we have

By these estimates, using that {u n } n , {h i,n } n ∈ C ∞ (T, X) are Cauchy sequences (because they are convergent), we obtain that {G(u n , h

Then P maps H s (T, X) into H s (T, Y ) for all 1/2 < s 0 ≤ s ∈ N.

We prove now the C ∞ -regularity of P . composition

We prove that P is in C k (H s (T, X), H s (T, Y )) for all k ∈ N by induction on k.

We start to prove that P ∈ C 0 (H s (T, X), H s (T, Y )).

Write

we have

which tends to 0 as �h� s,X → 0. Hence P is continuous.

We suppose now that P ∈ C k (H s (T, X), H s (T, Y )) with

and we prove that P ∈ C k+1 (H s (T, X), H s (T, Y )).

Using the regularity of P , we write

that means, by inductive hypothesis (B.33),

By Lemma B.3.6 and Remark B.3.9 with notations

for i = 1, . . . k, we have that

Appendix B. Useful results about invertible operators and operators of composition

and then is linear with respect to h.

In the same way, for

Lemma B.3.6 and Remark B.3.9 imply

and so

Hence, by definition of differentiability, the map u � → D k P (u)[h 1 , . . . , h k ] is differentiable with

Moreover the continuity of u � → D k+1 P (u)[h 1 , . . . , h k , h] can be proved with similar estimates.

We conclude by Lemma B.3.6 and Remark B.3.9 for which we estimate

Appendix C

The Multiscale Theorem C.1 Multiscale analysis for L(ε, λ, u, ξ)

The aim of this Appendix is to prove the Multiscale Theorem 6.2.16 of chapter 6. We will deduce it as a consequence of the Multiscale result 4.1 of [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF] that we adapt here to a new setting, obtaining the Multiscale Theorem C.1.11.

The main differences between this Appendix and the multiscale analysis in [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF],

concern the functional spaces and the introduction of new classes of regular and good sites.

We use the framework of chaper 6: we consider the functional Hilbert space

and �u� 2 0,0 :=

where the Fourier coefficients u l ∈ L 2 (M, C) are defined by

We have

The functional Hilbert space used in [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF]

) when the unkown function u of the non-linear Appendix C. The Multiscale Theorem Schrödinger equation is written as the vector u = (u + , u -) with each u + ,

Note that here we have a Fourier decomposition in one variable (the time ϕ ∈ T ν ) instead of a Fourier decomposition in two variables in [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF] (the time ϕ ∈ T ν and the space x ∈ T d ).

Let us consider a linear operator

We consider the C-linear complexification of L, that we still denote by L,

defined by

We associate to each linear and continuous operator

The coefficients L m l are defined by: for all h ∈ L 2 (M, C)

In the setting of [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF], the coefficients (L m l ) m,l of the linear operators considered, with m, l ∈ (Z ν+d × {0, 1}), are real numbers (see Definition 3.1 of [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF]). The Let B, C be subsets of Z ν . We introduce the subspace

In the following, we will write

, with L a linear operator (possibly unbounded).

For a linear and continuous operator L : L 2 B → L 2 C , its associated matrix is in the set of matrices of linear and continuous coefficients (L m l ) m∈B, l∈C

For a linear and continuous L ∈ M B C , we recall the definitions of semi-norm

Given F, F � ⊂ E ⊂ Z ν , we define the diameter of a set as

where |l| := max{|l 1 |, |l 2 |, . . . , |l ν |} and the "distance" beetwen F and F � as

Moreover, for L ∈ M E E , we define the "restricted" operators

where Π F is the L 2 -orthogonal projector onto L 2 F .

For N ∈ N being fixed, we introduce the following definitions of N -regular operators and N -good sites taken from [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF] which are different with respect to the ones in the chapter 6 (because we want firstly to adapt the Multiscale result of [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF] to our functional setting in order to prove, as a consequence, the Multiscale Theorem used in chapter 6).

Definition C.1.1. For an appropriate choice of τ � > 0, δ ∈ (0, 1/2) and

, is said to be a "N-regular operator" if and only if L is invertible and

Otherwise L is said a "N-singular operator".

Definition C.1.2 (Regular sites in Z ν ). Fix θ ∈ R + , an index l ∈ Z ν is said to be a "regular site" for the operator L if and only if L l l is invertible and

Otherwise the index l is said a "singular site".

Definition C.1.3 (N -Regular sites in Z ν ). An index l ∈ Z ν is said to be a "N-regular site" for the operator L ∈ M E E if and only if there is a subset

F is a N-regular operator. Otherwise the site l is said to be "N-singular".

Definition C.1.4 (N -good sites in Z ν ). An index l ∈ Z ν is said to be a "N-good site" for the operator L ∈ M E E if and only if it is a regular site or a N-regular site for L.

Otherwise the site l is said to be "N-bad".

Let R ⊂ E ⊂ Z ν (resp. S ⊂ E ⊂ Z ν ) be the set of all regular sites (resp. singular sites) for an operator

The following remark explains the reason why the Definition C.1.2 of regular (and singular) sites is not strong enough in order to prove the Multiscale Theorem in our setting.

Remark C.1.5. By Definition C.1.2, take a singular site l for an operator

We do not have informations about the L 2 -operator norm of L l l or about | • | snorm of L.

In the proof of Multiscale Theorem, we need a bound on the

For this reason, we introduce a new definition of singular site that allows us to have this bound.

Note that in the setting of [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF], the sites are in Z ν+d × {0, 1} and a site l is said "singular" with respect to the definition 4.2 of [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF] if and only if |L l l | ≤ θ, where L l l is a real number. It is clear in this setting that |diag(L S )| s ≤ θ for all s ≥ 0.

We will prove the Multiscale Theorem for the self-adjoint linear operators introduced in the chapter 6

where the notion of operator for the term ∂ u f (ϕ, x, u) is explained in Remark 6.2.1.

Recall that {ϕ j,k } j,k are the eigenfunctions of the linear operator -∆ + V (x) in L 2 (M, C) with corresponding eigenvalues ω 2 j,k :

(-∆ + V (x))ϕ j,k = ω 2 j,k ϕ j,k .

In the complex Hilbert basis {e ıϕ•l ϕ j,k } l,j,k of L 2 (T ν , L 2 (M, C)), we write L(ε, λ, u, ξ) as

We define the corresponding matrix coefficients of the linear and unbouded operator L(ε, λ, u, ξ) as

Remark C.1.6. If an index l ∈ Z ν is a regular sites for the operator L(ε, λ, u, ξ)

See Lemma 2.1.1 for the condition k ∈ [0, d j ].

Proof. By Definition C.1.2, take l a regular site for L(ε, λ, u, ξ), we have that Ll l is invertible and

Then Dl l is invertible: by (C.3) and (C.4), we can write

and by Lemma B.1.1 we obtain

For all index l ∈ Z ν , we define the following subspace of L 2 (M, C)

Moreover, by Lemma 2.1.1, ω j,k → +∞ as j → ∞ and it implies that E l (M) is a finite subspace of L 2 (M, C).

Performing the decomposition

where F l (M) := (E l (M)) ⊥ with respect to the scalar product in L 2 (M, C)

By previous considerations, if l is a regular site for the operator L(ε, λ, u, ξ), then u - l (x) = 0 and u l (x) = u + l (x). Let Π E l , Π F l be the L 2 (M, C)-orthogonal projectors respectly on E l (M) and

We write every linear coefficient Lm l of the operator L(ε, λ, u, ξ) as the block matrix

where

Let Of course, since E l (M) = {0} when l is a regular site for L(ε, λ, u, ξ), then sites l = (l, -1) are not defined for all l regular sites of L(ε, λ, u, ξ) and Lm l = ( Lm l ) + + , for l, m regular sites.

Let P : Z ν × {-1, +1} → Z ν be the projector on Z ν defined by: for all l = (l, ±1) ∈ Z ν × {-1, +1}

For all E ⊂ Z ν , we define the subset E � ⊂ Z ν × {-1, +1} as

Call Z := (Z ν ) � , using the decomposition (C.7), we write u ∈ L 2 (T ν , L 2 (M, C))

where u l (x) ∈ L 2 (M, C) is defined by

Given E ⊂ Z ν × {-1, +1}, we define the diameter of E as

and the "distance" beetwen two subsets F and F � of Z ν × {-1, +1} as dist(F, F � ) := inf l∈F,l � ∈F � |P(l) -P(l � )| = dist(P(F), P(F � )).

In the following, L may denote an operator from L 2 (T ν , L 2 (M, C)) into itself as well as an element of M E E , i.e. a matrix of coefficients ( Lm l ) l,m∈E . Moreover, for F � , F � ⊂ E we define the "restricted" operators LF as the matrix of coefficients ( Lm l ) l∈F , LF as the matrix of coefficients ( Lm l ) m∈F , LF � F as the matrix of coefficients ( Lm l ) l∈F,m∈F � .

We can now define a new class of regular sites for the operator L(ε, λ, u).

Definition C.1.7 (Regular sites in Z ν × {-1, 1}). Every index l = (l, +1) ∈ Z ν × {+1}, with l ∈ Z ν , is said to be a "regular site" for L(ε, λ, u, ξ).

Otherwise, every index l = (l, -1), with l ∈ Z ν such that E l (M) � = {0}, is said to be a "singular site" for L(ε, λ, u, ξ).

Remark C.1.8. Assume that l = (l, -1)

and by (C.6), we have for ε small enough

This property is used in the proof of Lemma C.1.13.

Definition C.1.9 (N -Regular sites in Z ν ×{-1, 1}). An index l ∈ Z ν ×{-1, 1} is said to be a "N -regular site" for the operator

Otherwise the site l is said to be "N -singular".

Definition C.1.10 (N -good sites in Z ν × {-1, 1}). An index l ∈ Z ν × {-1, 1} is said to be a "N -good site" for the operator L(ε, λ, u, ξ) ∈ M E E if and only if it is a regular site or a N -regular site for L(ε, λ, u, ξ).

Otherwise the site l is said to be "N -bad".

We can now state the following theorem, adapted version to our setting of the Multiscale result 4.1 of [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF].

and, setting q := τ � + ν + s 0 ,

For any given Υ > 0, there exist θ = θ(Υ, s 1 ) > 0 large enough and N 0 (Υ, θ, s 1 ) ∈ N such that: for all N ≥ N 0 and

there is a partition of N -bad sites for L(ε, λ, u, ξ),

Moreover, the following estimate holds

with a constant C(s) independent of N .

The Multiscale result says, roughly, the following. If L has a sufficient "off-diagonal decay" (assumption (H2) with s 1 large enough) and if the N -bad sites are sufficiently separated (assumption (H3)), then the L 2 -bound (H1) for L-1 implies that the "large" matrix L of size N � = N χ (with χ large enough) is N � -regular and satisfies estimates (C.12).

The proof of Multiscale Theorem C.1.11 is almost the same as the one of Multiscale Proposition 4.1 in [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF]. The difference between them is mainly in the functional setting and in the definition of bad sites. Moreover, in [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF], the result holds for all index S ≥ s 1 and ∀ N ≥ N 0 = N 0 (Υ, θ, S). The dependence of N 0 on S implies that the estimates for the � • � s -norm of the inverse L-1 are given for all s ∈ [s 1 , S] and with a constant C = 1 4 independent on s:

We give here the Multiscale result for all N ≥ N 0 (Υ, θ, s 1 ) and estimate (C.12) for all s ≥ s 1 with a constant C(s) which depends on s.

We will deduce it from estimate (C.13) proved in [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF] (see Lemma C.1.15).

We reproduce for convenience the steps of the proof of the Multiscale result divided in several lemmas (see [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF]). In each lemma we shall assume satisfied all the hypotheses of Theorem C.1.11.

We set

We insert now three lemmas used to prove the Multiscale Theorem C.1.11.

Lemma C.1.12 (Semi-reduction on the N -good sites). Let θ -1 Υ be small enough. There exist

for some c := c(s 1 ) > 0 and, for all s ≥ s 0 ,

Proof. The proof is similar as the one provided in Lemma 4.1 of [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF]. It is based on "resolvent identity" arguments as in [START_REF] Bourgain | Recent progress on quasi-periodic lattice Schrödinger operators and Hamiltonian PDE's[END_REF]. We often use properties of the | • | s -norm introduced in the chapter 3.

Lemma C.1.13 (Reduction on the N -bad sites). We have

where

and, for all s ≥ s 0 , in [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF] is required. In fact, for l singular site, using Remarck C.1.8, we obtain for all s ≥ s 0

where R is the set of regular sites in E.

Estimate C.14 is exactly the same as estimate (4.32) in [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF] and the rest of the proof proceeds as in Lemma 4.2 of [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF].

Lemma C.1.14 (Left inverse with decay). The matrix

Proof. The proof is the same as the one of Lemma 4.3 in [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF].

Proceeding as in the last part of the proof of the Multiscale Proposition 4.1 in [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF], we obtain for all index S ≥ s 1 the existence of a N 0 = N 0 (Υ, θ, S) such that ∀ N ≥ N 0 the following estimates hold:

We conclude the proof of Theorem C.1.11 with the following lemma.

Lemma C.1.15. Let N 0 = N 0 (Υ, θ, s 1 ) be the integer provided by Proposition 4.1 in [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF] with respect to the lower index s 1 . Then ∀ N ≥ N 0 = N 0 (Υ, θ, s 1 ), estimates (C.12) hold for all s ≥ s 1 .

Proof. Take s ≥ s 1 . By Proposition 4.1 in [START_REF] Berti | Quasi-periodic solutions with Sobolev regularity of NLS on T d with a multiplicative potential[END_REF], estimates (C.12) hold for all N ≥ N 0 (Υ, θ, s) with a constant C = 1 4 . We have to prove that estimates (C.12) hold also for N 0 (Υ, θ, s 1 ) ≤ N < N 0 (Υ, θ, s).

Let us consider N 0 (Υ, θ, s 1 ) ≤ N < N 0 (Υ, θ, s). C.2 Proof of Multiscale Theorem 6.2.16

In this section we prove the Multiscale Theorem 6. which are defined in chapter 6.

We have to verify that hypothesis (H3) of Theorem 6. In this setting LN � (ε, λ, u, ξ) is seen as a matrix LN � ∈ M E E with

and LN,l � (ε, λ, u, ξ) is senn as a matrix LN,l � ∈ M E � E � with

In order to do it, it is enough to prove the following two Lemmas. We will build l � = (l � 1 , l � 2 , . . . , l � ν ) ∈ Z ν choosing each of its components l � i with respect to the distance of each l i from the boundary of [-N � , N � ]. For all i = 1, . . . , ν, we take l � i as follows: Proof. It is useful to recall the Definition 6.2.7 of chapter 6.

An index l ∈ [-N � , N � ] ν is said to be a "N-good site" for the operator LN � (ε, λ, u, ξ)

with respect to Definition 6.2.7 if and only if l verifies For all index l of the form l = (l, +1) ∈ [-N � , N � ] ν × {+1} there is nothing to prove: by Definition C.1.7, l = (l, +1) is a regular site (and so a N -good site)

for LN � (ε, λ, u, ξ).

Let us consider indices of the form l = (l, -1) ∈ [-N � , N � ] ν × {-1}.

We have to prove that these l = (l, -1) are N -good sites for LN � (ε, λ, u, ξ).

Since their second component is -1, they are not regular for LN � (ε, λ, u, ξ).

We have to prove that they are N -regular for LN � (ε, λ, u, ξ): we have to find

and ( LN � ) F F is a N -regular operator. By previous considerations, indices (l, -1) are not defined for l which verify (C.17 Moreover, ( LN � ) F F = LN,l � is a N -regular operator. The Lemma is so proved. By Lemma C.2.2, we have that if an index l = (l, -1) ∈ [-N � , N � ] ν × {-1} is a N -bad site for LN � (ε, λ, u, ξ) with respect to Definition C.1.10 then the index l is a N -"bad site" for LN � (ε, λ, u, ξ) with respect to Definition 6.2.7.

Hence, if we have the good separation of the N -"bad site" for LN � (ε, λ, u, ξ) described by hypothesis (H3) of Multiscale Theorem 6.2.16 then we have also the good separation of the N -bad site for LN � (ε, λ, u, ξ) described by hypothesis (H3) of Multiscale Theorem C.1.11.

We can see the Multiscale result 6.2.16 as a consequence of the Multiscale Theorem C.1.11.

We conclude this Appendix noting that in the Mutiscale Theorem 6.2.16 of chapter 6, we take δ in the larger interval (0, 1) instead of (0, 1/2) taken in