N Forcadel

C Imbert

R Monneau

Homogenization of fully overdamped Frenkel-Kontorova models

Keywords: AMS Classification: 35B27, 35F20, 45K05, 47G20, 49L25, 35B10 particle systems, periodic homogenization, Frenkel-Kontorova models, Hamilton-Jacobi equations, hull function, cumulative distribution function, Slepčev formulation

In this paper, we consider the fully overdamped Frenkel-Kontorova model. This is an infinite system of coupled first order ODEs. Each ODE represents the microscopic evolution of one particle interacting with its neighbors and submitted to a fixed periodic potential. After a proper rescaling, a macroscopic model describing the evolution of densities of particles is obtained. We get this homogenization result for a general class of Frenkel-Kontorova models. The proof is based on the construction of suitable hull functions in the framework of viscosity solutions.

Introduction

In the present paper we are interested in systems of ODEs describing the motion of particles in interactions with their neighbors and submitted to a periodic potential. An important special case is the classical Frenkel-Kontorova (FK) model in its fully overdamped version. This physical model is a very simple and very important one. For a good overview on the Frenkel-Kontorova model, we refer the reader to the recent book [START_REF] Braun | Kivshar The Frenkel-Kontorova Model, Concepts, Methods and Applications[END_REF] of Braun and Kivshar and the article [START_REF] Floria | Dissipative dynamics of the Frenkel-Kontorova model[END_REF] of Floria and Mazo. We want to study the limit of the system of ODEs as the number of particles per length unit goes to infinity. As we shall see, this can be understood as an homogenization procedure.

The classical fully overdamped Frenkel-Kontorova model

The classical Frenkel-Kontorova model describes a chain of classical particles evolving in a one dimensional space, coupled with their neighbors and subjected to a periodic potential. If τ denotes time and U i (τ ) denotes the position of the particle i ∈ Z, one of the simplest FK models is given by the following dynamics

m d 2 U i dτ 2 + γ dU i dτ = U i+1 -2U i + U i-1 + sin (2πU i ) + L
where m denotes the mass of the particle, γ a friction coefficient, L is a constant driving force which can make the whole "train of particles" move and the term sin (2πU i ) describes the force created by a periodic 1 potential whose period is assumed to be 1. Notice that in the previous equation, we set to one physical constants in front of the elastic and the exterior forces. If we assume that m ≪ γ = 1, we can neglect the acceleration term and obtain for i ∈ Z

(1.1)

dU i dτ = U i+1 -2U i + U i-1 + sin (2πU i ) + L for τ > 0 .
This is the reason why we say that the dynamics of this model is fully overdamped. It can describe the friction between two materials. Indeed, this model was originally introduced in Kontorova, Frenkel [START_REF] Kontorova | On the theory of plastic deformation and doubling[END_REF] to describe the plasticity at a microscopic level. Such a model is sketched on figure 1. We would like next to give the flavour of the results we obtain in this paper. In order to do so, let us assume that at initial time, particles satisfy

i-1 i i+1 i+2 periodic potential
U i (0) = ε -1 u 0 (iε)
for some ε > 0 and some Lipschitz continuous function u 0 (x) which satisfies the following assumption: (A0) Initial gradient bounded from above and below 0 < 1/K 0 ≤ (u 0 ) x ≤ K 0 on R for some fixed K 0 > 0.

Such an assumption can be interpreted by saying that at initial time, the number of particles per length unit lies in (K -1 0 ε -1 , K 0 ε -1 ). It is then natural to ask what is the macroscopic behaviour of the solution U of (1.1) as ε goes to zero, i.e. as the number of particles per length unit goes to infinity. To this end we define the following function that describes the rescaled positions of the particles

(1.2) u ε (t, x) = εU ⌊ε -1 x⌋ (ε -1 t)
where ⌊•⌋ denotes the floor integer part. One of our main results states that the limiting dynamics as ε goes to 0 of (1.1) is determined by a first order Hamilton-Jacobi equation of the form

(1.3) u 0 t = F (u 0 x ) for (t, x) ∈ (0, +∞) × R, u 0 (0, x) = u 0 (x) for x ∈ R
where F is a continuous function to be determined. More precisely, we have the following homogenization result:

Theorem 1.1. (Homogenization of the FK model) For all L ∈ R, there exists a continuous function F : R → R such that, under assumption (A0), the function u ε converges locally uniformly towards the unique viscosity solution u 0 of (1.3).

Generalized Frenkel-Kontorova models

In order to present our main results in full generality, we first describe the generalizations of the classical FK model we deal with.

An important remark about (1.1) is that such an ODE system can be embedded into a single PDE. In order to see this, let us first give the following definition. For a given integer m ∈ N \ {0} and for a function v : R → R, we define [v] m (y) = (v(y -m), v(y -m + 1), ..., v(y + m)) .

With this notation in hand, we claim that solving (1.1) for the family of initial condition u 0,α (•) = u 0 (• + α), α ∈ [0, 1), is equivalent to solve

∂ τ U = F ([U ] 1
) for (τ, x) ∈ (0, +∞) × R submitted to the initial condition U (0, x) = u 0 (x) and where (1.4)

F (τ, V -1 , V 0 , V 1 ) = V -1 -2V 0 + V 1 + sin (2πV 0 ) + L .
We can then consider generalized FK models with interactions with the m-th nearest neighbors. Precisely, we look for solutions u(τ, y) to the following "finite difference-like" PDE (1.5) u τ = F (τ, [u(τ, •)] m ) for (τ, y) ∈ (0, +∞) × R .

In the present paper, we work with viscosity solutions, and even with possibly discontinuous ones (see Definition 2.1). Let us now make precise the assumptions we make on the function F : R × R 2m+1 → R that maps (τ, V ) to F (τ, V ).

(A1) Regularity F is continuous , F is Lispchitz continuous in V uniformly in τ .

(A2) Monotonicity F (τ, V -m , ..., V m ) is non-decreasing in V i for i = 0 . (A3) Periodicity F (τ, V -m + 1, ..., V m + 1) = F (τ, V -m , ..., V m ) , F (τ + 1, V ) = F (τ, V ) .
Remarks 1.2.

1. When F does not depend on τ , we simply denote by F (V ).

2. We see that these assumptions are in particular satisfied for the classical FK model (1.1) (see Equation (1.4)).

We next rescale the generalized FK model as we did for the classical one. Precisely, we now consider the following problem satisfied by u ε (t, x)

(1.6)        u ε t = F t ε , u ε (t, •) ε ε m for (t, x) ∈ (0, +∞) × R, u ε (0, x) = u 0 (x) for x ∈ R
where for some function v(x) we set

[v] ε m (x) = (v(x -mε), ..., v(x + mε)) .
We then have the following homogenization result Theorem 1.3. (Homogenization of generalized FK models) Under assumptions (A0),(A1),(A2),(A3), there exists a continuous function F : R → R such that the solution u ε to (1.6) converges locally uniformly towards the unique viscosity solution u 0 of (1.3).

We will explain in the next subsection how the so-called effective Hamiltonian F is determined. We will see that it has to do with the existence of so-called hull functions (see Theorem 1.5 below). But before giving further details, let us make several comments about this general homogenization result.

We would like first to shed light on an important fact. Assumption (A2) concerning the monotonicity of F is fundamental in our analysis. Indeed, This condition ensures that a comparison principle holds true for the solutions of (1.5) and this allow us to perform the homogenization limit in the framework of viscosity solutions. With this respect, Theorem 1.3 makes part of a huge literature concerning homogenization of Hamilton-Jacobi equations whose pionnering paper is the one of Lions, Papanicolaou, Varadhan [START_REF] Lions | Homogeneization of Hamilton-Jacobi equations[END_REF].

The homogenization of model with interactions with an infinite number of particles (i.e. the case m = +∞) was studied in Forcadel, Imbert, Monneau [START_REF] Forcadel | Homogenization of the dislocation dynamics and of some systems of particles with two-body interactions[END_REF] for a model describing dislocation dynamics.

Concerning the homogenization of equations with periodic terms in u/ε (which is the case of the models considered in the present paper), only very few results exist. Let us mention the recent result of Imbert, Monneau [START_REF] Imbert | Homogenization of first order equations with (u/ε)-periodic hamiltonians. Part I: local equations[END_REF] and the one of Barles [START_REF] Barles | Some homogenization results for non-coercive Hamilton-Jacobi equations[END_REF]. We can also mention the work of Boccardo, Murat [START_REF] Boccardo | Remarques sur l'homogénéisation de certains problèmes quasi-linéaires[END_REF] about the homogenization of elliptic equations and the one of Bacaër [START_REF] Bacaër | Convergence of numerical methods and parameter dependence of min-plus eigenvalue problems, Frenkel-Kontorova models and homogenization of Hamilton-Jacobi equations[END_REF].

Hull functions

In order to study the solutions of (1.5), it is classical to introduce the so-called (dynamical) hull function, i.e. a function h(τ, z) such that u(τ, y) = h(τ, py + λτ ) is a solution of (1.5). We refer for instance to the pionnering work of Aubry [START_REF] Aubry | The twist map, the extended Frenkel-Kontorova model and the devil's staircase[END_REF][START_REF] Aubry | Exact models with a complete Devil's staircase[END_REF], and Aubry, Le Daeron [START_REF] Aubry | The discrete frenkel-Kontorova model and its extensions[END_REF] where they studied (among other things) this notion in details.

Definition 1.4. (Hull function)

Given F satisfying (A1),(A2) and (A3), a positive number p ∈ (0, +∞) and a real number λ ∈ R, a locally bounded function h

: R 2 → R is a hull function for (1.5) if it satisfies for all (τ, z) ∈ R 2 (1.7)            h τ + λh z = F (τ, h(τ, z -mp), ..., h(τ, z + mp)), h(τ + 1, z) = h(τ, z), h(τ, z + 1) = h(τ, z) + 1, h z (τ, z) ≥ 0, |h(τ, z + z ′ ) -h(τ, z) -z ′ | ≤ 1 for all z ′ ∈ R .
In the case where F is independent on τ , we require that the hull function h is also independent on τ and we denote it by h(z).

Given p > 0, the following theorem explains how the effective Hamiltonian F (p) is determined by an existence/non-existence result of hull functions as λ ∈ R varies.

Theorem 1.5. (Effective Hamiltonian and hull function)

Given F satisfying (A1),(A2),(A3) and p ∈ (0, +∞), there exists a unique real λ for which there exists a hull function h (depending on p) satisfying (1.7). Moreover the real number λ, seen as a function F of p, is continuous on (0, +∞).

Qualitative properties of the effective Hamiltonian

In this subsection, we list the important qualitative properties of the effective Hamiltonian defined thanks to Theorem 1.5. Keeping in mind the first FK model we described (1.1), we are in particular interested in the behaviour of F when it is computed by replacing F with F + L. We will give several results about the function F seen as a function of L. Let us state a precise result.

Theorem 1.6. (Qualitative properties of F for general F ) Consider a non-linearity F satisfying (A1),(A2),(A3). Given p > 0 and L ∈ R, let F (L, p) denote the effective Hamiltonian defined thanks to Theorem 1.5 where F is replaced with F + L.

Then F : R 2 → R is continuous and we have the following properties:

a1. (Bound) There exists a constant C > 0 such that for all (L, p) ∈ R × (0, +∞)

|F (L, p) -L| ≤ C(1 + p) . a2. (Monotonicity in L) F (L, p) is non-decreasing in L . a3. (Antisymmetry in V ) If for all (τ, V ) ∈ R × R 2m+1 , F (τ, -V ) = -F (τ, V ), then F (0, p) = 0 for any p > 0 . a4. (Periodicity in p) Assume that for all (τ, V ) ∈ R × R 2m+1 (1.8) F (τ, V -m -m, ..., V m + m) = F (τ, V -m , ..., V m ) , then F (L, p + 1) = F (L, p) . a5. (Continuous hull function / no plateau of L → F (L, p))
Assume that for some (L 0 , p) ∈ R × (0, +∞), there exists a continuous hull function h(τ, z). Then for all

L = L 0 F (L, p) = F (L 0 , p) .
We next say more about Property (a5) about the characterization of plateaux of the function F seen as a function of L. Let us first consider the following example for m = 1 (1.9)

F = F (τ, V -1 , V 0 , V 1 ) = α (V 1 -2V 0 + V -1 ) + β sin (2πV 0 ) + γ cos (2πτ )
which satisfies in particular condition (1.8). In particular for this model, the full picture is 1-periodic in p.

For some suitable constants α, β, γ > 0, numerical simulations (see Braun [START_REF] Braun | Kivshar The Frenkel-Kontorova Model, Concepts, Methods and Applications[END_REF] page 334 and the references cited therein), seem to show that the map L → F (L, p) may have many plateaux as illustrated on figure 2. See also Hu, Qin, Zheng [START_REF] Hu | Rotation number of the overdamped Frenkel-Kontorova model with ac-driving[END_REF] and chapter 11 (homeomorphism of the circle) of the book [START_REF] Katok | Introduction to the Modern Theory of Dynamical Systems[END_REF] of Katok and Hasselblatt, for an interesting attempt of explanation of this behaviour. From (a5), we deduce in particular that the hull function is not continuous in space at points corresponding to these plateaux.

Here are further results related to this issue in the case where F does not depend on time τ .

Theorem 1.7. (Further plateau properties when F does not depend on τ ) Under the assumptions of Theorem 1.6, assume moreover that F does not depend on τ . Then we have the following properties:

b1. (No plateau in L if F = 0) There exists a constant C > 0 such that for all (L, p) ∈ R × (0, +∞) ∂F ∂L (L, p) ≥ |F (L, p)| |L| + C(1 + p) .
b2. (0-plateau property) Assume that the map v → F (v, ..., v) is not constant and that F satisfies property (1.8). Then there exists L 0 ∈ R and δ > 0 such that In the case where F does not depend on the time τ , we see from (b1) that the map L → F (L, p) has at most a single plateau at the level F = 0, as illustrated on figure 3.

F (L, p) = 0 for all (L, p) s.t. L ∈ (L 0 -δ, L 0 + δ), p ∈ N \ {0} .
Let us now turn to some comments on the 0-plateau. For model (1.9) with α = 1, γ = 0, more is known when p is a Diophantine number, i.e. satisfies for some κ, ν > 0 ∀a ∈ Z, ∀b ∈ Z \ {0} , |bp -a| ≥ κ|b| -ν . These Diophantine numbers (when considered for any κ, ν > 0) have full measure. It is proven in De La Llave [START_REF] De | KAM Theory for equilibrium states in 1-D statistical mechanics models[END_REF], that if p is Diophantine, then there exists an analytic hull function for L = 0 with F (0, p) = 0 as soon as |β| is smaller than a constant depending on κ, ν. In particular, we deduce from (a5), that the map L → F (L, p) has no plateau at all in this case.

As it is well-known (see [START_REF] Aubry | The twist map, the extended Frenkel-Kontorova model and the devil's staircase[END_REF]), this result is only valid for β small enough, because the hull function has to satisfy (see (1.7) and recall that λ = F (0, p) = 0)

(h(z + p) -h(z)) -(h(z) -h(z -p)) = -β sin (2πh(z)) .
Moreover h satisfies h(z + 1) = h(z) + 1 and is non-decreasing, which for instance for p ∈ (0, 1], implies that

|β sin (2πh(z)) | ≤ 2.
Therefore h can not take the value 1/4 for |β| > 2, and then h has to be discontinuous for |β| > 2. This is the well-known breaking of analyticity. See also [START_REF] Aubry | Exact models with a complete Devil's staircase[END_REF][START_REF] Kao | Exact Solution of Frenkel-Kontorova Models with a Complete Devil's Staircase in Higher Dimensions[END_REF] for some explicit computations of the hull functions for particular potentials.

Even for |β| = 0 arbitrarily small, (b2) shows that the map L → F (L, p) has a 0-plateau for integers p > 0. From (a5), this implies in particular the breaking of continuity for the hull function corresponding to such p and L.

As an example, in model (1.9) with α = 1, γ = 0, for any β > 0 and L = β > 0, p = 1, the following function h(z) = -1 4 + ⌊z⌋ is a discontinuous hull function with F (L, p) = 0. This shows that for the same model, the 0-plateau property of the map L → F (L, p) can be very sensitive to the values of p (and of its irrationality).

Organization of the article

In Section 2, we recall the notion of viscosity solutions. In Section 3, we prove the homogenization results, namely Theorems 1.1 and 1.3. In Section 4, we prove the ergodicity of the problem; precisely, we prove Theorem 1.5 on the hull function. In Section 5, we build Lipschitz sub-and super-hull function, using an approximate Hamiltonian. In Section 6, we prove Theorems 1.6 and 1.7 on the qualitative properties of the effective Hamiltonian. Finally in the appendix A, we propose a discussion on the relation between the hull function for our problem and the correctors for a "dual" approach of the problem: the so-called Slepčev formulation.

Viscosity solutions

This section is devoted to the definition of viscosity solutions for equations such as (1.5), (1.6) and (1.7). In order to construct hull functions when proving Theorem 1.5, we will also need to consider a perturbation of (1.7) with linear plus bounded initial data. For all these reasons, we define a viscosity solution for a generic equation whose Hamiltonian G satisfies proper assumptions.

Before making precise assumptions, definitions and fundamental results we will need later (such as stability, comparison principle, existence), we refer the reader to Barles [5] and the user's guide of Crandall, Ishii, Lions [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF] for an introduction to viscosity solutions.

Main assumptions and definitions

Consider for 0 < T ≤ +∞ the following Cauchy problem (2.10) u τ = G(τ, [u(τ, •)] m , inf y ′ ∈R (u(τ, y ′ ) -py ′ ) + py -u(τ, y), u y ) for (τ, y) ∈ (0, T ) × R u(0, y) = u 0 (y) for y ∈ R for a general non-linearity G. The most important example we have in mind is the following

G(τ, V, a, q) = F (τ, V ) + δ(a 0 + a)q
for some constants η, δ ≥ 0, a 0 , a, q ∈ R and where F appears in (1.5),(1.6), (1.7).

We make the following assumptions on G.

(A1') Regularity    G is continuous , For all R > 0, G(τ, V, a, q) is Lispchitz continuous in (V, a) uniformly in (τ, q) ∈ R × [-R, R] . (A2') Monotonicity G(τ, V -m , ..., V m , a, q) is non-decreasing in a and V i for i = 0 . (A3') Periodicity For all (τ, V, a, q) ∈ R × R 2m+1 × R × R G(τ, V -m + 1, V -m+1 + 1, ..., V m + 1, a, q) = G(τ, V -m , V -m+1 , ..., V m , a, q) G(1 + τ, V, a, q) = G(τ, V, a, q)
In view of (2.10), it is clear that, if G effectively depends on the variable a, solutions must be such that the infimum of u(τ, y) -p • y is finite for all time τ . We will even only consider solutions u satisfying for some C(T ) > 0: for all τ ∈ [0, T ) and all y, y ′ ∈ R

(2.11) |u(τ, y + y ′ ) -u(τ, y) -py ′ | ≤ C .
When T = +∞, we may assume that (2.11) holds true for all time T 0 > 0 for a family of constants C 0 > 0.

Since we have to solve a Cauchy problem, we have to assume that the initial datum satisfies the assumption (A0') (Initial condition)

u 0 satisfies (A0); it also satisfies (2.11) if G depends on a.
Finally, we recall the definition of the upper and lower semi-continuous envelopes, u * and u * , of a locally bounded function u u * (τ, y) = lim sup

(t,x)→(τ,y) u(t, x) and u * (τ, y) = lim inf (t,x)→(τ,y) u(t, x) .
We can now define viscosity solutions for (2.10).

Definition 2.1. (Viscosity solutions)

Let u 0 : R → R be a continuous function and u : R + × R → R be a locally bounded function such that (2.11) holds true if G depends on a.

-The function u is a subsolution (resp. a supersolution) of (2.10) on an open set Ω ⊂ (0, T ) × R if u is upper semi-continuous (resp. lower semi-continuous) and for all (τ, y) ∈ Ω and all test function φ ∈ C 1 (Ω) such that u -φ attains a strict local maximum (resp. a local minimum) at the point (τ, y), then we have

(2.12) φ τ (τ, y) ≤ G(τ, [u(τ, •)] m , inf y ′ ∈R (u(τ, y ′ ) -py ′ ) + py -u(τ, u), φ y (τ, y)) (resp. ≥) .
-The function u (resp. v) is said to be a subsolution (resp. supersolution) on [0, T ) × R, if u is a subsolution (resp. v is a supersolution) on Ω = (0, T ) × R and if moreover it satisfies for all y ∈ R u(0, y) ≤ u 0 (y) (resp. ≥) .

-A function u is said a viscosity solution of (2.10) if u * is a subsolution and u * is a supersolution.

Remark 2.2. A locally bounded function u is also (classically) called a subsolution (resp. supersolution) if its upper semi-continuous envelope (resp. lower semicontinuous envelope) is a subsolution in the sense of the previous definition.

The first main property of this notion of solution is its stability when passing to the limit. More precisely, a family of subsolutions (u ε ) ε>0 that is uniformly locally bounded from above is stable when passing to the so-called relaxed upper semi-limit u defined as follows

u(τ, y) = lim sup ε * u ε (τ, y) = lim sup (t,x)→(τ,y), ε→0 u ε (t, x).
Such a relaxed upper semi-limit is well-defined as soon as the family of functions u ε is uniformly locally bounded from above. Remark that u is upper semicontinuous and if u ε does not depend on ε (u ε = u for all ε > 0), we recover the upper semi-continuous envelope of the function u. In the same way, we can define the relaxed lower semi-limit of a family of lower semicontinuous functions that are uniformly locally bounded from below. The main discontinuous stability result for viscosity solutions is stated as follows.

Proposition 2.3. (Stability of viscosity solutions)

Assume (A1'), (A2') and T < +∞. Assume that (u ε ) ε is a sequence of subsolutions (resp. supersolutions) of Eq. (2.10) on (0, T ) × R satisfying (2.11) with the same constant C > 0. Then the relaxed upper semi-limit u is a subsolution (resp. u is a supersolution) of (2.10) on (0, T ) × R.

We will also use stability of subsolutions by passing to the supremum. Let us be more specific.

Proposition 2.4. (Stability of viscosity solutions (II)) Assume (A1'), (A2') and T < +∞. Assume that (u α ) α∈A is a family of subsolutions (resp. supersolutions) of Eq. (2.10) on (0, T ) × R satisfying (2.11) with the same constant C > 0. Then sup α∈A u α is a subsolution (resp. u is a supersolution) of (2.10) on (0, T ) × R.

We skip the proofs of both propositions since they are straightforward adaptations of classical ones (see for instance [START_REF] Barles | Solutions de viscosité des équations de Hamilton-Jacobi[END_REF]).

Comparison principles and existence

This subsection is devoted to state comparison principles that are used throughout the paper and to get the main existence results for the PDEs at stake.

We first state two comparison principles for the generic Hamilton-Jacobi equation (2.10). One is stated on the whole space while the second one is stated on bounded sets.

Proposition 2.5. (Comparison principle)

Assume (A0'), (A1') and (A2'). Assume that u and v are respectively a subsolution and a supersolution of

(2.10) on [0, T ) × R. Then we have u ≤ v on [0, T ) × R.
For a given point (τ 0 , y 0 ) ∈ (0, T ) × R and for all r, R > 0, let us set

Q r,R = (τ 0 -r, τ 0 + r) × (y 0 -R, y 0 + R) .
Proposition 2.6. (Comparison principle on bounded sets) Assume (A1') and (A2') and that G(τ, V, a, q) does not depend on the variable a. Assume that u is a subsolution (resp. v a supersolution) of (2.10) on the open set Q r,R ⊂ (0, T ) × R. Assume also that

u ≤ v on Q r,R+m \ Q r,R . Then u ≤ v on Q r,R .
Remarks 2.7.

-Here we need to increase the domain with a distance m, because the equation is nonlocal in space (recall that each particle has interactions with its m nearest neighbors on the left and on the right).

-We could ask to have only

u ≤ v on Q r,R+m \ Q r,R ∩ {τ < τ 0 + r}.
We now turn to the construction of solution. We recall the celebrated Perron's method.

Proposition 2.8. (Existence by Perron's method) Assume (A1') and (A2'). Assume that u is a subsolution (resp. v is a supersolution) of (2.10) on (0,

T ) × R such that u ≤ v on (0, T ) × R .
Let C be the set of all supersolutions ṽ of (2.10) on (0, T ) × R satisfying (2.11) with C corresponding to u and v and such that ṽ ≥ u . Let w(τ, y) = inf {ṽ(τ, y) such that ṽ ∈ C} .

Then w is a (discontinuous) solution of (2.10) on (0, T ) × R satisfying u ≤ w ≤ v and (2.11).

We skip the proofs of Propositions 2.5, 2.6 and 2.8 since they are completely classical.

The important corollary of the proposition is the following well-posedness result for (2.10).

Corollary 2.9. (Existence and uniqueness for the Cauchy problem) Assume (A0'), (A1'), (A2') and (A3'). Then there exists a unique solution u of (2.10) on [0, +∞) × R. Moreover u is continuous.

Remark 2.10. As we will see in the proof, the solution u we construct satisfies (2.11) with a constant C that depends on the one corresponding to the barriers we will construct. Hence, the constant C in (2.11) for u depends on u 0

Proof of Corollary 2.9.

In order to apply Proposition 2.8, we need to construct barriers. In view of assumptions (A1') and (A3'), the constant G 0 defined by

(2.13) G 0 = sup τ ∈R, |q|≤K0
|G(τ, 0, 0, q)| is finite. Moreover, using (A1'), let us introduce the constants

K 1 and K 2 such that for all τ, a, b ∈ R, V, W ∈ R 2m+1 , q ∈ (-K 0 , K 0 ), (2.14) |G(τ, V, a, q) -G(τ, W, b, q)| ≤ K 1 |V -W | ∞ + K 2 |a -b| with |W | ∞ = sup k=-m,...,m |W k |.
Then we have the following lemma whose proof is postoned.

Lemma 2.11. (Existence of barriers) Assume (A0'), (A1'), (A2') and (A3'). There exists a constant C > 0 such that

u + (τ, y) = u 0 (y) + Cτ and u -(τ, y) = u 0 (y) -Cτ
are respectively supersolution and subsolution of (2.10) on [0, T ) × R for any T > 0.

Moreover, we can choose

(2.15) C = K 2 C 1 + C 0 (m, K 0 , K 1 , G 0 )
where K 2 , K 1 and G 0 are given respectively in (2.14) and (2.13). Here C 1 is given in (A0').

From Lemma 2.11 and Proposition 2.8, we get the existence of a function u which is a solution of (2.10) on (0, +∞) × R and satisfies u -≤ u ≤ u + . Therefore the initial condition is satisfied. Moreover u * (0, •) = u * (0, •) and from the comparison principle (Proposition 2.5), we get that u * ≤ u * for all time which implies that u is continuous. Finally, still from Proposition 2.5, we deduce the uniqueness of the solution of (2.10) on [0, +∞) × R.

We now turn to the proof of the Lemma.

Proof of Lemma 2.11. We set u ± (τ, y) = u 0 (y) ± Cτ for some C to be fixed later. We have

|G(τ, [u ± (τ, •)] m (y)), inf y ′ ∈R u ± (τ, y ′ ) -py ′ + py -u ± (τ, y), u ± y (τ, y))| = |G(τ, [u ± (τ, •) -⌊u ± (τ, y)⌋] m (y), inf y ′ ∈R (u 0 (y ′ ) -py ′ ) + py -u 0 (y), (u 0 ) y (y))| ≤ K 2 C 1 + K 1 + |G(τ, [u ± (τ, •) -u ± (τ, y)] m (y), 0, (u 0 ) y (y))| ≤ K 2 C 1 + K 1 + G 0 + K 1 mK 0 =: K 2 C 1 + C 0
where we have used the periodicity assumption (A3') for the second line, assumption (A0') for the third line, and for the last line, we have used

|u ± (τ, y ′ ) -u ± (τ, y)| ≤ K 0 |y ′ -y|.
When G(τ, V, a, q) is independent on a, we can simply choose K 2 = 0. This ends the proof of the Lemma.

Convergence

This section is devoted to the proof of the main homogenization result (Theorem 1.3). The proof relies on the existence of hull functions (Theorem 1.5) and qualitative properties of the effective Hamiltonian (Theorem 1.6). As a matter of fact, we will use the existence of Lipschitz sub-and super-hull functions for (see Proposition 5.3). All these results are proved in the next sections.

We start with some preliminary results. The following result is a straightforward corollary of Lemma 2.11 by a change of variables: Lemma 3.1. (Barriers uniform in ε) Assume (A0),(A1),(A2) and (A3). Then there is a constant C > 0, such that for all ε > 0, the solution u ε to (1.6) satisfies for all t > 0 and x ∈ R

|u ε (t, x) -u 0 (x)| ≤ Ct .
We have Lemma 3.2. (ε-bounds on the gradient) Assume (A0),(A1),(A2) and (A3). Then the solution u ε of (1.6) satisfies for all t > 0, x ∈ R, z > 0

(3.16) ε z εK 0 ≤ u ε (t, x + z) -u ε (t, x) ≤ ε zK 0 ε for all (t, x) ∈ [0, +∞) × R Remark 3.3.
In particular we find that the solution u(t, x) is non-decreasing in x.

Proof of Lemma 3.2.

We prove the bound from below (the proof is similar for the bound from above). We first remark that (A0) implies that the initial condition satisfies (3.17)

u 0 (x + z) ≥ u 0 (x) + z/K 0 ≥ u 0 (x) + kε with k = z εK 0
From (A3), we know that for ε = 1, the equation is invariant by addition of integer to the solutions. After the rescaling, equation (1.6) is invariant by addition of constants kε with k an integer. For this reason the solution with initial data u 0 + kε is u ε + kε. Similarly the equation is invariant by translations. Therefore the solution with initial data u 0 (x + z) is u ε (t, x + z). Finally, from (3.17) and the comparison principle (Proposition 2.5), we get u ε (t, x + z) ≥ u ε (t, x) + kε which proves the bound from below. This ends the proof of the lemma.

We now turn to the proof of Theorem 1.3.

Proof of Theorem 1.3. Let u (resp. u) denote the relaxed upper (resp. lower) semi-limit associated with the family of functions (u ε ) ε>0 . These functions are well defined thanks to Lemma 3.1. We also get from this lemma and Lemma 3.2 that both functions w = u, u satisfy for all t > 0, x,

x ′ ∈ R |w(t, x) -u 0 (x)| ≤ Ct , K -1 0 |x -x ′ | ≤ w(t, x) -w(t, x ′ ) ≤ K 0 |x -x ′ | . (3.18)
We are going to prove that u is a subsolution of (1.3) on R + × R. Similarly, we can prove that u is a supersolution of the same equation. Therefore, from the comparison principle for (1.3), we get that u 0 ≤ u ≤ u ≤ u 0 . And then u = u = u 0 , which shows the expected convergence of the full sequence u ε towards u 0 . We now prove in several steps that u is a subsolution of (1.3) on (0, +∞) × R. We classically argue by contradiction by assuming that u is not a subsolution on (0, +∞) × R. Then there exists (t, x) ∈ (0, +∞) × R and a test function φ ∈ C 1 such that

(3.19)        u(t, x) = φ(t, x) u ≤ φ on Q r,2r (t, x), with r > 0 u ≤ φ -2η on Q r,2r (t, x) \ Q r,r (t, x), with η > 0 φ t (t, x) = F (φ x (t, x)) + θ,
with θ > 0 where we recall that Q r,R (t, x) denotes for r, R > 0

Q r,R (t, x) = (t -r, t + r) × (x -R, x + R) .
Let p denote φ x (t, x). From (3.18), we get

(3.20) 0 < 1/K 0 ≤ p ≤ K 0 .
Combining Theorems 1.5 and 1.6 (in particular a.1 and a.2), we get the existence of a hull function h associated with p such that

λ = F (p) + θ 2 = F (L, p) with L > 0 .
Indeed, we know from these results that the effective Hamiltonian is non-decreasing in L, continuous and goes to ±∞ as L → ±∞.

We now apply the perturbed test function method introduced by Evans [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF] in terms here of hull functions instead of correctors. Precisely, let us consider the following twisted perturbed test function

φ ε (t, x) = εh t ε , φ(t, x) ε .
Here the test function is twisted similarly as in [START_REF] Imbert | Homogenization of first order equations with (u/ε)-periodic hamiltonians. Part I: local equations[END_REF]. In order to get a contradiction, we first assume that h is smooth and is continuous in z uniformly in τ ∈ R. In view of the third line of (1.7), we see that this implies that h is uniformly continuous in z (uniformly in τ ∈ R). For simplicity, and since we will construct approximate hull functions with such a regularity, we just assume that h is Lipschitz continuous in z (uniformly in τ ∈ R). We will next see how to treat the general case.

Case 1: h is smooth and Lipschitz continuous in z

Step 1.1: φ ε is a supersolution of (1.6) on a neightbourhood of (t, x) When h is smooth enough (i.e. C 1 here), it is sufficient to check directly the supersolution property of φ ε for (t, x) ∈ Q r,r (t, x). We have, with τ = t/ε and z = φ(t, x)/ε,

(3.21) φ ε t (t, x) -F τ, φ ε (t, •) ε ε m (x) = h τ (τ, z) + φ t (t, x)h z (τ, z) -F τ, h τ, φ(t, •) ε ε m (x) = (φ t (t, x) -λ) h z (τ, z) + L + F (τ, [h(τ, •)] p m (z)) -F τ, h τ, φ(t, •) ε ε m (x) ≥ (φ t (t, x) -λ) h z (τ, z) + L -L F [h(τ, •)] p m (z) -h τ, φ(t, •) ε ε m ∞
where we have used that Equation (1.7) is satisfied by h to get the third line and (A1) to get the fourth one; here, L F denotes the Lipschitz constant of F with respect to V for the norm | • | ∞ on R 2m+1 . Let us next estimate, for j ∈ {-m, . . . , m} and ε such that mε ≤ r,

h(τ, z + jp) -h τ, φ(t, x + jε) ε = h(τ, z + jp) -h(τ, z + jp + o r (1))
where o r (1) only depends on the modulus of continuity of φ x on Q r,r (t, x). Hence, if h is Lipschitz continuous with respect to z uniformly in τ , we conclude that we can choose ε small enough so that

(3.22) L -L F [h(τ, •)] p m (z) -h τ, φ(t, •) ε ε m ∞ ≥ 0 .
Combining (3.21) and (3.22), we obtain

φ ε t (t, x) -F τ, φ ε (t, x) ε ε m (x) ≥ (φ t -λ) h z (τ, z) = θ 2 + φ t (t, x) -φ t (t, x) h z (τ, z) = θ 2 + o r (1) h z (τ, z) ≥ 0 .
We used the non-negativity of h z , the fact that θ > 0 and again the fact that φ is C 1 , to get the result on Q r,r (t, x) for r > 0 small enough. Therefore, when h is smooth and Lipschitz continuous on z uniformly in τ , φ ε is a viscosity supersolution of (1.6) on Q r,r (t, x).

Step 1.2: getting the contradiction By construction, we have φ ε → φ as ε → 0, and therefore from (3.19), we get for ε small enough

u ε ≤ φ ε -η ≤ φ ε -εk ε on Q r,2r (t, x) \ Q r,r (t, x)
with the integer

k ε = ⌊η/ε⌋ .
Therefore, for mε ≤ r, we can apply the comparison principle on bounded sets (Proposition 2.6) to get

(3.23) u ε ≤ φ ε -εk ε on Q r,r (t, x) .
Passing to the limit as ε goes to zero, we get

u ≤ φ -η on Q r,r (t, x)
which gives a contradiction with u(t, x) = φ(t, x) in (3.19). Therefore u is a subsolution of (1.3) on (0, +∞)×R and this ends the proof of the theorem.

Case 2: general case for h

In the general case, we can not check by a direct computation that φ ε is a supersolution on Q r,r (t, x). The difficulty is due to the fact that h(τ, z) may not be Lipschitz continuous in the variable z. This kind of difficulties was overcomed in [START_REF] Imbert | Homogenization of first order equations with (u/ε)-periodic hamiltonians. Part I: local equations[END_REF] by using Lipschitz super-hull functions, i.e. functions satisfying (1.7) with ≥ instead of = in the first line. Indeed, it is clear from the previous computations that it is enough to conclude. In [START_REF] Imbert | Homogenization of first order equations with (u/ε)-periodic hamiltonians. Part I: local equations[END_REF], such regular super-hull functions (as a matter of fact, regular super-correctors) were build as exact solutions of an approximate Hamilton-Jacobi equation. Moreover this Lipschitz hull function is a supersolution for the exact hamiltonian with a slightly bigger λ.

Here we conclude using a similar result, namely Proposition 5.3. Notice that the fact that h is smooth is not a restriction, the previous argument being completely valid in the viscosity sense since p satisfies (3.20). See [START_REF] Imbert | Homogenization of first order equations with (u/ε)-periodic hamiltonians. Part I: local equations[END_REF] for further details. This ends the proof of the theorem.

We continue with the proof of Theorem 1.1.

Proof of Theorem 1.1.

Remark that the initial condition satisfies

u 0 (y) -ε⌈K 0 ⌉ ≤ u ε (0, y) ≤ u 0 (y)
Therefore the comparison with the solution u ε of (1.6) gives

u ε -ε⌈K 0 ⌉ ≤ u ε ≤ u ε on [0, +∞) × R .
Using the convergence of u ε to u 0 given in Theorem 1.3, we deduce that u ε → u 0 . This ends the proof of the theorem.

Ergodicity and construction of hull functions

In this section, we prove Theorem 1.5 that defines the effective Hamiltonian F and states the existence of hull functions.

As we shall see, for given real numbers (L, p), the constant F (L, p) is (classically) defined as the "time slope" (in a sense to be made precise, see Proposition 4.1) of the solution of an initial Cauchy problem. This is the reason why the Hamiltonian is said to be ergodic.

Since approximate Lipschitz continuous hull functions must be constructed (see the proof of convergence in the preceding section), we work with the general (approximate) Hamiltonian G considered in Section 2. Hence, the Cauchy problem we work with is (2.10).

Ergodicity

In this subsection, we successively prove two propositions. The first one (Proposition 4.1) asserts that ergodicity holds true for G as soon as we are able to control space oscillations of the solution u of (2.10). The next proposition (Proposition 4.2) asserts that we are indeed able to control space oscillations and that the solution u satisfies additional important properties.

Let us first start with Proposition 4.1. (Time oscillations controlled by space oscillations) Assume (A0'), (A1'), (A2') and (A3'), and let u be a solution of (2.10) on R + × R. Assume that there exists constants p > 0 and an integer C 1 ≥ 1 such that we have the following control on the space oscillations: for all τ > 0, y, y ′ ∈ R,

(4.24) |u(τ, y + y ′ ) -u(τ, y) -py ′ | ≤ C 1
Then there exists λ ∈ R such that for all τ > 0, y ∈ R

(4.25) |u(τ, y) -u(0, 0) -py -λτ | ≤ C 2 with C 2 = 8C 1 + 2M
where

(4.26) M = sup {|G(τ, V -m , ..., V m , ±C 1 , p)| : τ > 0, V 0 ∈ R, V k = kp ± C 1 + V 0 for k = 0} .
Moreover we have

(4.27) |λ| ≤ M .
Proof of Proposition 4.1. The proof follows line by line the one given in [START_REF] Imbert | Homogenization of first order equations with (u/ε)-periodic hamiltonians. Part I: local equations[END_REF] in a different context. For the reader's convenience, we write all the details below.

In order to control time oscillations, let us introduce the following two continuous functions defined for

T > 0 λ + (T ) = sup τ ≥0 u(τ + T, 0) -u(τ, 0) T and λ -(T ) = inf τ ≥0 u(τ + T, 0) -u(τ, 0) T which satisfy -∞ ≤ λ -(T ) ≤ λ + (T ) ≤ +∞.
Step 1: Estimate on the time derivative of the space oscillations Let us consider

(4.28) m(τ ) = sup y∈R (u(τ, y) -py) = u(τ, y(τ )) -py(τ )
if the supremum is reached at some y(τ ) ∈ R. Then we have in the viscosity sense

m τ ≤ G(τ, [u(τ, •)] m (y(τ )), C 1 , p) .
If the supremum in (4.28) is reached at infinity, we get the same result, up to replace u with u ∞ (τ, y) = lim sup * (u(τ, y + c n )) n≥1 for some suitable sequence (c n ) n≥1 going to infinity. Using moreover that (4.24) implies |u(τ, y(τ

) + k) -u(τ, y(τ )) -kp| ≤ C 1 ,
we deduce that Finally from (4.29),(4.30) and (4.24), we deduce that λ ± (T ) are finite.

m τ ≤ G(τ, V -m , ..., V m , C 1 , p) with V 0 = u(τ, y(τ )) , V k = kp + C 1 + V 0 for k = 0 .
Step 2: Estimate on λ + -λ - By definition of λ ± (T ), for all δ > 0, there exists t ± ≥ 0 such that

λ ± (T ) - u(t ± + T, 0) -u(t ± , 0) T ≤ δ .
Let us pick l ∈ Z such that 0 ≤ a := t -+ l -t + < 1 and let us set ũ(τ, y) = u(τ -l, y) .

Case 1: T ≥ 1 Then we have t + ≤ t -+ l < t + + T ≤ t -+ l + T .

Let us define k ∈ Z such that 2C 1 < ũ(t -+l, 0)+k -u(t + +a, 0) ≤ 3C 1 . Then from (4.24) and the invariance of the equation by addition of integers (see assumption (A3)), we deduce that for all y ∈ R, we have

(4.31) 0 < ũ(t -+ l, y) + k -u(t + + a, y) ≤ 5C 1
Therefore from the comparison principle (Proposition 2.5), we deduce with

T ′ = T -a 0 ≤ ũ(t -+ l + T ′ , y) + k -u(t + + a + T ′ , y) ≤ 5C 1
and then from (4.31), we get

(4.32) -5C 1 ≤ ũ(t -+ l + T ′ , y) -ũ(t -+ l, y) -(u(t + + a + T ′ , y) -u(t + + a, y)) ≤ 5C 1 .
Let us consider m(τ ) = sup From (4.29), we deduce that

m(t -+ l + T ) ≤ m(t -+ l + T ′ ) + M a ≤ C 1 + m(t -+ l + T ′ ) + M a which implies that ũ(t -+ l + T, y) -py ≤ ũ(t -+ l + T ′ , y) -py + M a + C 1 i.e. ũ(t -+ l + T, y) ≤ ũ(t -+ l + T ′ , y) + M a + C 1 .
Similarly, using (4.30), we get

ũ(t -+ l + T, y) ≥ ũ(t -+ l + T ′ , y) -M a -C 1
and even (4.33)

|u(t + + a, y) -u(t + , y)| ≤ M a + C 1 .
Together with (4.32), we get

-7C 1 -2M a ≤ ũ(t -+ l + T, y) -ũ(t -+ l, y) -(u(t + + T, y) -u(t + , y)) ≤ 7C 1 + 2M a
which implies, for y = 0,

|λ + (T ) -λ -(T )| ≤ 2δ + 7C 1 + 2M a T .
Because δ > 0 is arbitrary small and a ∈ [0, 1), we deduce that

(4.34) |λ + (T ) -λ -(T )| ≤ 7C 1 + 2M T
Case 2: T < 1 Using (4.33) with a = T , we deduce that

|u(t + + T, y) -u(t + , y)| ≤ C 1 + M T .
Similarly, we have

|ũ(t -+ l + T, y) -ũ(t -+ l, y)| ≤ C 1 + M T . Therefore |λ + (T ) -λ -(T )| ≤ 2δ + 2C 1 + 2M T T .
Again, because δ > 0 is arbitrary small, we deduce in particular that (4.34) is still true for T ∈ [0, 1).

Step 3: (λ ± (T )) T is a Cauchy sequence Let us consider T 1 , T 2 > 0 such that T 2 /T 1 = P/Q with P, Q ∈ N \ {0}. Remark that the following inequality holds true

λ + (P T 1 ) = sup τ ≥0 i=1,...,P u(τ + iT 1 , 0) -u(τ + (i -1)T 1 , 0) P T 1 ≤ i=1,...,P λ + (T 1 ) P = λ + (T 1 ) .
Similarly, we get λ -(QT 2 ) ≥ λ -(T 2 ). Then we have

λ + (T 1 ) ≥ λ + (P T 1 ) = λ + (QT 2 ) ≥ λ -(QT 2 ) ≥ λ -(T 2 ) ≥ λ + (T 2 ) - 7C 1 + 2M T 2 .
By symmetry, we deduce that

(4.35) |λ + (T 2 ) -λ + (T 1 )| ≤ max 7C 1 + 2M T 1 , 7C 1 + 2M T 2
and similarly

(4.36) |λ -(T 2 ) -λ -(T 1 )| ≤ max 7C 1 + 2M T 1 , 7C 1 + 2M T 2 .
Since the functions T → λ ± (T ) are continuous, inequalities (4.35)-(4.36) remain valid in the case T 2 /T 1 ∈ (0, +∞).

Step Moreover we have for all τ ≥ 0, y, y ′ ∈ R,

u(τ, y + 1/p) = u(τ, y) + 1 u y (τ, y) ≥ 0 |u(τ, y + y ′ ) -u(τ, y) -py ′ | ≤ 1 . (4.39) Proof of Proposition 4.2.
We perform the proof in three steps.

Step 1: u(τ, y) is non-decreasing in y First, remark that the equation satisfied by u is invariant by translations in y and for all b ≥ 0, we have

u 0 (y + b) ≥ u 0 (y) .
Therefore, from the comparison principle, we get u(τ, y + b) ≥ u(τ, y) which shows that the solution u(τ, y) is non-decreasing in y.

Step 2: control of the space-oscillations We have u 0 (y + 1/p) = u 0 (y) + 1 .

Therefore from the comparison principle and from the integer periodicity (A3') of G, we get that u(τ, y + 1/p) = u(τ, y) + 1 .

Because u(τ, y) is non-decreasing in y, we deduce that for all b ∈ [0, 1/p]

0 ≤ u(τ, b) -u(τ, 0) ≤ 1
Let now y ∈ R, that we write py = k + a with k ∈ Z and a ∈ [0, 1). Then we have

u(τ, y) -u(τ, 0) = k + u(τ, a/p) -u(τ, 0) which implies, for b ∈ [0, 1/p), u(τ, y) -u(τ, 0) -py = -a + u(τ, b) -u(τ, 0)
and then |u(τ, y) -u(τ, 0) -py| ≤ 1 .

Finally, we deduce (4.39) by using the invariance by translations in y of the problem.

Step 3: control of the time-oscillations We can now apply Proposition 4.1 to control the time-oscillations by the space-oscillations. We get the existence of some λ ∈ R such that

|u(τ, y) -u(0, 0) -py -λτ | ≤ 8 + 2M = C 3 .
This ends the proof of the proposition.

Construction of hull functions for general Hamiltonians

In this subsection, we construct hull functions for the genreal Hamiltonian G. As we shall see, this is straightforward after we constructed time-space periodic solutions of (4.40) below; see Proposition 4.3 and Corollary 4.4 below. We conclude this subsection by proving that the time slope we constructed in Proposition 4.2 is unique and that the map p → λ is continuous.

Given p > 0, we consider the equation in R × R

       u ∞ (τ, y + 1/p) = u ∞ (τ, y) + 1 u ∞ (τ + 1, y) = u ∞ (τ, y + λ/p) (u ∞ ) y (τ, y) ≥ 0 |u ∞ (τ, y + y ′ ) -u ∞ (τ, y) -py ′ | ≤ 1 .
Eventually, when G is independent on τ , we can choose u ∞ independent on τ .

By considering h(τ, z) = u ∞ (τ, (z -λτ )/p), we immediately get the following corollary Corollary 4.4. (Existence of hull functions) Assume (A1'), (A2') and (A3'). There exists a hull function h for (2.10) satisfying

|h(τ, z) -z| ≤ 2⌈2M + 8⌉ = 2⌈C 3 ⌉
where M is given by (4.26) with C 1 = 1.

Remark 4.5. The definition of hull function for (2.10) is very similar to Definition 1.4. The only difference is the equation satisfied by h which is replaced here by

h τ + λh z = G(τ, h(τ, z -mp), . . . , h(τ, z + mp), inf z ′ (h(τ, z ′ ) -z ′ ) + z -h(τ, z), ph z ).
Proof of Proposition 4.3.

The proof is performed in three steps. In the first one, we construct sub-and supersolutions of (4.40) in R × R with good translation invariance properties (see the first two lines of (4.42)). We next apply Perron's method in order to get a (discontinuous) solution satisfying the same properties. Finally, in step 3, we prove that if G does not depend on τ , then we can construct such a solution such that it does not depend on τ either.

Step 1: global sub-and supersolution By Proposition 4.2, we know that the solution u of (2.10) with initial data u 0 (y) = py satisfies on [0, +∞)×R (4.43)

   u y ≥ 0, |u(τ, y) -py -λτ | ≤ 2M + 8 = C 3 , |u(τ, y + y ′ ) -u(τ, y) -py ′ | ≤ 1 .
We first construct a subsolution and a supersolution of (4.40) for τ ∈ R (and not only τ ≥ 0) that also satisfy the first two lines of (4.42), i.e. satisfy for all k, l ∈ Z, To do so, we consider the sequence, for n ∈ N,

u n (τ, y) = u(τ + n, y) -λn and consider u = lim sup n→+∞ * u n u = lim inf n→+∞ * u n .
Now a way to construct semi-solutions satisfying (4.44) is to consider

(4.45) u ∞ (τ, y) = sup k,l∈Z (u(τ + k, y -kλ/p + l/p) -l) (4.46) u ∞ (τ, y) = inf k,l∈Z (u(τ + k, y -kλ/p + l/p) -l)
Notice that u ∞ and u ∞ satisfy moreover (4.43) on R × R. Therefore we have in particular

u ∞ ≤ u ∞ + 2⌈C 3 ⌉ .
Step 2: existence by Perron's method Applying Perron's method we see that the lowest supersolution u ∞ above u ∞ is a solution of (4.43) on R × R and satisfies

u ∞ ≤ u ∞ ≤ u ∞ + 2⌈C 3 ⌉ .
We next prove that u ∞ satisfies (4.42). Moreover let us consider

(4.47) ũ∞ (τ, y) = inf k,l∈Z (u ∞ (τ + k, y -kλ/p + l/p) -l)
By construction ũ∞ is a supersolution and is again above the subsolution u ∞ . Therefore from the definition of u ∞ , we deduce that ũ∞ = u ∞ which implies that u ∞ satisfies (4.44), i.e the first two equalities of (4.42).

Similarly, we can consider û∞ (τ,

y) = inf b∈[0,+∞) u ∞ (τ, y + b)
which is again a supersolution above the subsolution u ∞ . Therefore û∞ = u ∞ which implies that u ∞ is non-decreasing, i.e. the third line of (4.42) is satisfied. Finally, the function u ∞ -⌈C 3 ⌉ still satisfies (4.42) but also (4.41).

Step 3: Further properties when G is independent on τ When G does not depend on τ , we can apply Steps 1 and 2 with k ∈ Z in (4.45), (4.46) and (4.47) replaced with k ∈ R. This implies that the hull function h does not depend on τ . This ends the proof of the proposition.

Proposition 4.6. (Definition and continuity of the effective Hamiltonian) Given p > 0, and under the assumptions (A1'), (A2') and (A3'),

-there exists a unique λ such that there exists a function u ∞ ∈ L ∞ loc (R × R) solution of (4.40) on R × R and satisfying

(4.48) |h(0, z) -z| ≤ 1 ; -if λ is seen as a function G of p (λ = G(p)), then this function G : (0, +∞) → R is continuous.
Proof of Proposition 4.6.

Step 1: Uniqueness of λ Given some p ∈ (0, +∞), assume that there exist λ 1 , λ 2 ∈ R with their corresponding hull functions h 1 , h 2 . Then define for i = 1, 2

u i (τ, y) = h i (τ, λ i τ + py)
which are both solutions of equation (2.10) on [0, +∞) × R. Using the fact that h i (τ, z + 1) = h i (τ, z) + 1 and the monotonicity of the hull functions in the variable z, we see that for each h i (up to a substraction of an integer and a translation of h i in the variable z) we can assume that (4.48) holds true. Then we have

u 1 (0, y) ≤ u 2 (0, y) + 2
which implies (from the comparison principle) for all (τ, y)

× [0, +∞) × R u 1 (τ, y) ≤ u 2 (τ, y) + 2 .
Using the fact that h i (τ + 1, z) = h i (τ, z), we deduce that for τ = k ∈ N and y = 0 we have

h 1 (0, λ 1 k) ≤ h 2 (0, λ 2 k) + 2
which implies by (4.48)

λ 1 k ≤ λ 2 k + 4 .
Because this is true for any k ∈ N, we deduce that

λ 1 ≤ λ 2 .
The reverse inequality is obtained exchanging h 1 and h 2 . We finally deduce that λ 1 = λ 2 , which proves the uniqueness of the real λ, that we call G(p).

Step 2: Continuity of the map p → G(p) Let us consider a sequence (p n ) n such that p n → p > 0. Let λ n = G(p n ) and h n be the corresponding hull functions. From Corollary 4.4, we can choose these hull functions such that

|h n (τ, z) -z| ≤ 2⌈2M (p n ) + 8⌉
and we have

|λ n | ≤ M (p n )
where we recall that M (p) is defined in (4.26). We deduce in particular that there exists a constant

C 4 > 0 such that |h n (τ, z) -z| ≤ C 4 and |λ n | ≤ C 4 .
Let us consider a limit λ ∞ of (λ n ) n , and let us define Again we see that v is in C 2+α,1+α . In particular v is a viscosity solution of (5.53).

h = lim sup n→+∞ * h n . This function h is such that u(τ, y) = h(τ, λ ∞ τ + py)
Step 1: bound from below on the gradient Let us now define m(τ ) = inf y∈R v(τ, y) .

Then we have in the viscosity sense:

m τ ≥ K 1 min(0, m) -K 1 m -δ(m -p)m m(0) = p > 0
where we have used the monotonicity assumption (A2) to get the term K 1 min(0, m). The fact that 0 is subsolution implies that v ≥ m ≥ 0 .

Step 2: bound from above on the gradient Similarly we define m(τ ) = sup y∈R v(τ, y) .

Then we have in the viscosity sense

m τ ≤ K 1 m -δ(m -p)m m(0) = p > 0
where we have used Step 1 to ensure that |v| ≤ m. The fact that p + K 1 /δ is a supersolution implies that

v ≤ m ≤ p + K 1 /δ .
Case B: η = 0 and F general We simply consider a C 1 approximation F η of F and call u η the solution of (5.52) with F replaced with F η , for η > 0. From Case A, we have

(5.54) 0 ≤ (u η ) y ≤ p + K 1 /δ + o η (1)
Then we call u = lim sup

η→0 * u η u = lim inf η→0 * u η
Then u and u are respectively sub and supersolutions of (5.52) with η = 0. Therefore

u ≤ u ≤ u
But by construction we have u ≤ u. Therefore u = u = u and passing to the limit in (5.54), we see that u satisfies (5.51). This ends the proof of the proposition.

Then we have Proposition 5.2. (Existence of Lipschitz approximate hull functions) Assume (A1), (A2) and (A3). Given p > 0, δ > 0 and a 0 ∈ R, then there exists a Lipschitz hull function h(τ, z) satisfying

(5.55)    0 ≤ h z ≤ 1 + K 1 /(pδ) h(τ, z + 1) = h(τ, z) + 1 h(τ + 1, z) = h(τ, z)
and there exists λ ∈ R such that (5.56)

h τ + λh z = F (τ, [h(τ, •)] p m ) + δp a 0 + inf z ′ ∈R (h(τ, z ′ ) -z ′ ) + z -h(τ, z)) h z and (5.57) |h(τ, z ′ ) -z ′ + z -h(τ, z)| ≤ 1 .
Moreover there exists a constant M 0 > 0, only depending on F and p > 0, such that For any δ > 0, let h ± δ be the Lipschitz hull function obtained in Proposition 5.2 for a 0 = ±1, and λ ± δ the corresponding value of the effective Hamiltonian. Then we have

(
(h + δ ) τ + λ + δ (h + δ ) z ≥ F (τ, [h + δ (τ, •)] p m ) and λ ≤ λ + δ → λ as δ → 0 (h - δ ) τ + λ - δ (h - δ ) z ≤ F (τ, [h - δ (τ, •)] p m
) and λ ≥ λ - δ → λ as δ → 0 where λ = F (p).

Proof of Proposition 5.3. Inequalities ±λ ± δ ≥ ±λ follow from the comparison principle. In view of the bounds (5.58) and (5.59) on λ ± δ and h ± δ we have (in particular they are uniform as δ goes to zero), it is clear that the convergence λ ± δ → λ holds true as δ → 0. It suffices to adapt Step 2 of the proof of Proposition 4.6.

Qualitative properties of the effective Hamiltonian

Proof of Theorem 1.6: a1,a2,a3,a4. We recall that we have hull functions h solutions of

h τ + λh z = L + F (τ, [h(τ, •)] p m ) with λ = F (L, p).
The continuity of the map (L, p) → F (L, p) is easily proved as in step 2 of the proof of Proposition 4.6. a1. Bound This is a straightforward adaptation of step 1 of the proof of Proposition 4.1.

a2. Monotonicity in L

The monotonicity of the map L → F (L, p) follows from the comparison principle on u(τ, y) = h(τ, λτ + py) where h is the hull function and λ = F (L, p). a3. Antisymmetry in V We just remark that if a hull function h solves

h τ + λh z = F (τ, [h(τ, •)] p m ) then h(τ, z) = -h(τ, -z) satisfies hτ -λ hz = -F (τ, -h(τ, z + mp), ..., -h(τ, z -mp)) = F (τ, [ h(τ, •)] p m )
By the uniqueness of λ, we deduce that λ = -λ and then F (0, p) = λ = 0. a4. Periodicity in p It is sufficient to remark that, given p > 0, if h is a hull function for λ = F (L, p), then h is also a hull function for p + 1 with the same λ.

Remark 6.1. (Potential with zero mean value)

In Theorem 1.6 point 3, if we do not assume the antisymmetry in V but that F has the following form:

F (V ) = -g ′ 0 (V 0 ) + i=1,...,m -g ′ i (V 0 -V -i ) + g ′ i (V 0 -V i )
where g 0 ∈ W 2,∞ (R; R) is a 1-periodic function and g i ∈ W 2,∞ (R; R) for i = 1, ..., m are convex functions, then it is expected as in [START_REF] Forcadel | Homogenization of the dislocation dynamics and of some systems of particles with two-body interactions[END_REF]Theorem 2.6] that F (0, p) = 0, but it is not proved here.

Before to prove the point (a5) of Theorem 1.6, let us prove the following easier result, which also shows that the Lipschitz constant of the hull function is related to the inverse of the bound from below of the gradient in L of the effective hamiltonian Proposition 6.2. (Lipschitz hull function / bound from below on ∂F ∂L ) Given (L 0 , p) ∈ R × (0, +∞), asssume that there exists a corresponding hull function h which satisfies for some K 3 ≥ 1 0 ≤ h(τ, z + a) -h(τ, z) ≤ K 3 a for any (a, z) ∈ [0, +∞) × R .

Then we have for all

L ∈ R |F (L + L 0 , p) -F (L 0 , p)| ≥ |L| K 3 .
Proof of Proposition 6.2.

Up to redefine F , we can assume that L 0 = 0. Then we have with λ = F (0, p):

h τ + λh z = F (τ, [h(τ, •)] p m (z)) This implies h τ + λh z ≤ L + F (τ, [h(τ, •)] p m (z)) with L = ( λ -λ)K 3 . From the comparison principle, we deduce that λ ≤ F (L, p), i.e. L/K 3 ≤ F (L, p) -F (0, p)
which gives the result for positive L. We get similarly the corresponding inequality for negative L. This ends the proof of the proposition.

Proof of Theorem 1.6: a5. Continuous hull function/no plateau of L → F (L, p) Up to redefine F , we can assume that L 0 = 0. We assume that h is continuous with the following spacemodulus of continuity ω: for all τ ≥ 0, z ′ ≥ 0, z ∈ R,

(6.60) 0 ≤ h(τ, z + z ′ ) -h(τ, z) ≤ ω(z ′ )
and solves, for λ = F (0, p), (6.61)

h τ + λh z = F (τ, [h(τ, •)] p m (z)) .
Then we define for α > 0 the sup-convolution (in space only)

h α (τ, z) = sup y∈R h(τ, y) - |z -y| 2 2α .
We (classically) show that h α is a Lipschitz continuous subsolution of equation ( 6.61) perturbed by some error term.

Step 1: the basic viscosity inequality satisfied by h α More precisely, let ϕ ∈ C 1 (R 2 ) such that h α ≤ ϕ with equality at (τ 0 , z 0 ) and let y 0 ∈ R be such that (6.62)

h(τ 0 , z 0 ) ≤ h α (τ 0 , z 0 ) = h(τ 0 , y 0 ) - |z 0 -y 0 | 2 2α .
Then we have h(τ, y) ≤ ϕ(τ, z 0 ) + |z 0 -y| 2 2α =: ϕ(τ, y) with equality at (τ 0 , y 0 ) .

This implies ϕ τ + λ ϕ y ≤ F (τ 0 , [h(τ 0 , •)] p m (y 0 )) at (τ 0 , y 0 ) . By definition of h α , we have

h α (τ 0 , z 0 + kp) ≥ h(τ 0 , y 0 + kp) - |z 0 -y 0 | 2 2α .
We deduce that

ϕ τ (τ 0 , z 0 ) + λ y 0 -z 0 α ≤ F τ 0 , |z 0 -y 0 | 2 2α + h α (τ 0 , •) p m (z 0 )
where we have used (6.62) and the monotonicity assumption (A2) on F . We classically have ϕ z (τ 0 , z 0 ) = (y 0 -z 0 )/α ≥ 0 (recall that h is non-decreasing). This gives the basic viscosity inequality satisfied by h α (6.63)

ϕ τ (τ 0 , z 0 ) + λϕ z (τ 0 , z 0 ) ≤ F τ 0 , |z 0 -y 0 | 2 2α + h α (τ 0 , •) p m (z 0 ) .
Step 2: getting a bound from below on the effective Hamiltonian Using the Lipschitz constant K 1 > 0 defined in (5.50), we get, from (6.63),

ϕ τ (τ 0 , z 0 ) + λϕ z (τ 0 , z 0 ) ≤ K 1 |z 0 -y 0 | 2 2α + F (τ 0 , [h α (τ 0 , •)] p m (z 0 )) . This implies ϕ τ (τ 0 , z 0 ) + λϕ z (τ 0 , z 0 ) ≤ L + F (τ 0 , [h α (τ 0 , •)] p m (z 0 ))
for any ( λ, L) such that (6.64)

L ≥ K 1 |z 0 -y 0 | 2 2α + ( λ -λ) y 0 -z 0 α .
Now using (5.57) and (6.62), we get

|z 0 -y 0 | 2 2α ≤ h(τ 0 , y 0 ) -h(τ 0 , z 0 ) ≤ ω(y 0 -z 0 ) ≤ 1 + |y 0 -z 0 | which implies |y 0 -z 0 | ≤ 4 √ α for α ≤ 2.
Consider now L > 0 and λ such that (6.64) holds true and

λ ≥ λ α := λ + √ α 4 L -K 1 ω(4 √ α) .
We then have, in the viscosity sense, for all (τ, z) ∈ R 2 ,

h α τ + λ α h α z ≤ L + F (τ, [h α (τ, •)] p m (z)) .
Therefore, for any L > 0, we have

F (L, p) ≥ λ α > λ = F (0, p)
for α small enough.

Step 3: the bound from above on the effective Hamiltonian Proceeding by inf-convolution, we get similarly the expected result for negative L. This ends the proof of the theorem.

Remark 6.3. We can also get explicit estimates to bound |F (L, p) -F (0, p)| from below, using the modulus of continuity ω(•).

Proof of Theorem 1.7. b1. No plateau in L if F = 0 Consider L 2 > L 1 and the corresponding hull functions h i (z) independent on time and satisfying

λ i (h i ) z = L i + F ([h i (•)] p m (z)), i = 1, 2
for the corresponding λ i = F (L i , p). We assume that λ 1 > 0 and we already know that λ 2 ≥ λ 1 > 0. Let us define

F 0 = sup V0∈R |F (V 0 , ..., V 0 )| .
Remark now that (5.57) implies

|h i (z + kp) -h i (z) -kp| ≤ 1 and then |F ([h i (•)] p m (z))| ≤ F 0 + K 1 (mp + 1) Therefore 0 ≤ (h 1 ) z ≤ λ -1 1 (|L 1 | + F 0 + K 1 (mp + 1)) . Hence (λ 1 + δ(L 2 -L 1 ))(h 1 ) z ≤ L 2 + F ([h 1 (•)] p m (z)) for δ ≤ λ 1 (|L 1 | + F 0 + K 1 (mp + 1)) -1 . This implies that λ 2 ≥ λ 1 + δ(L 2 -L 1 ), i.e. λ 2 -λ 1 L 2 -L 1 ≥ λ 1 (|L 1 | + F 0 + K 1 (mp + 1)) -1
This implies the result for F > 0. We get a similar result for F < 0. b2. 0-plateau property Because V 0 → F (V 0 , ..., V 0 ) is assumed not constant, we see that there exists

L 0 ∈ R such that inf V0∈R F (V 0 , ..., V 0 ) < -L 0 < sup V0∈R F (V 0 , ..., V 0 )
Up to redefine F , we can assume that L 0 = 0 to simplify. Recall also that for (L, p), the (possibly discontinuous) hull function h satisfies λh z = L + F ([h(•)] p m (z)) Now for p ∈ N \ {0}, and using property (1.8), we deduce that

λh z = L + F (h(z), ..., h(z)) .
Consider λ = 0. Assume for instance that λ > 0. Then h is Lipschitz continuous. Moreover, h is nondecreasing. Then using a test function φ which touches h at z in a region where F (h(z), ..., h(z)) < 0, we get a contradiction for |L| small enough. This shows that λ ≤ 0. Similarly, we show that λ ≥ 0. Therefore F (L, p) = λ = 0 for L small enough. This ends the proof of the theorem.

We have moreover the following result Proposition 6.4. (Uniqueness of the continuous hull functions) Assume (A1), (A2) and (A3). Assume also that there exist δ 0 > 0 and k 0 ∈ {-m, ..., m} \ {0} such that (6.65) ∂F ∂V k0 (τ, V ) ≥ δ 0 > 0 for any V = (V -m , ..., V m ) ∈ R 2m+1

and we consider hull functions for some fixed irrational p > 0.

If there exists a continuous hull function h(τ, z), then every hull function is continuous and is equal to h, up to a fixed translation in z. In that case, the hull function is moreover stricly monotone in z, i.e. satisfies h(τ, z ′ ) > h(τ, z) if z ′ > z Remark 6.5. We do not know if Proposition 6.4 is still true without assuming the continuity of the hull function, but only assuming that p is irrational.

Remark 6.6. The classical FK model (1.4) gives an example of non-uniqueness of hull functions which can be discontinuous for F (p) = 0. Indeed for f = 0 and p = 1, the following functions (for any a ∈ (0, 1))

h 1 (z) = ⌊z⌋ and h 2 (z) = ⌊z⌋ if 0 ≤ z -⌊z⌋ < a, 1 2
+ ⌊z⌋ if a ≤ z -⌊z⌋ < 1 are two admissible discontinuous hull functions.

Proof of Proposition 6.4. i) Uniqueness of the hull function Assume that h 1 and h 2 are two hull functions, with h 2 continuous. We can slide h 2 (τ, z + a) above h * 1 (τ, z) for a large enough. Then we decrease a until some a * to get a contact between h 2 (τ, z + a * ) and h * 1 (τ, z) at some point (τ 0 , z 0 ). Up to redefine h 2 , we can assume that a * = 0. for some z(τ ) ∈ R. Recall that we have in the viscosity sense

(h i ) τ + λ(h i ) z = F (τ, [h i (τ, •)] p m (z))
Then up to a dedoubling of variable in time and in space, we can indentify the space derivatives at (τ, z(τ )) of h 2 and h 1 which implies (this is a routine exercice to justify this in the viscosity framework): In particular, from the fact that b(τ 0 ) = 0, we deduce that b(τ ) = 0 for τ ≤ τ 0 .

Moreover, we deduce that the function g(τ, z) = h 2 (τ, z) -h * 1 (τ, z) satifies g(τ, z(τ ) + k 0 p) = 0 for τ ≤ τ 0

Step 2: conclusion We can now reapply step 1 iteratively to z(τ ) + k 0 pl for l = 1, 2, .... We deduce that for all l ∈ N and for τ ≤ τ 0 , g(τ, z(τ ) + k 0 pl) = 0 .

Because p is irrational, we deduce that h * 1 is equal to the continuous function h 2 on a set which is dense in (-∞, τ 0 ] × R. Therefore h * 1 is continuous on (-∞, τ 0 ] × R. But recall that u 1 (τ, y) = h 1 (τ, λτ + py) solves

(u 1 ) τ = F (τ, [u 1 (τ, •) m (z)] .
Because the right hand side is bounded, this implies that u 1 is Lipschitz in time. On the other hand, we have u 1 (τ, y) is non-decreasing in y, so u 1 = u * 1 only if u 1 has a jump in space at some point (τ 1 , y 1 ). This would imply that u * 1 has also a jump at the same point. This is impossible, because u * 1 is continuous as a consequence of the continuity of h * 1 . Therefore u 1 and h 1 are continuous. Hence h 1 = h * 1 = h 2 on (-∞, τ 0 ] × R and then on R × R, using the periodicity in time of the hull functions. ii) Strict monotonicity of the hull function We simply apply i) with h 1 (τ, z) = h(τ, z) and h 2 (τ, z) = h(τ, z + a 0 ) ≥ h 1 (τ, z) for some a 0 > 0. Assume by contradiction, the existence of a contact point between h 1 and h 2 . Point i) implies that h 1 = h 2 , i.e. h(τ, z + a 0 ) = h(τ, z). This implies that h(τ, z + ka 0 ) = h(τ, z) for any k ∈ Z which is impossible. Therefore, we have h(τ, z + a 0 ) > h(τ, z) for a 0 > 0 which ends the proof of the proposition.

A Appendix: the hull function versus Slepčev formulation

In this Section we present a kind of "dual formulation" of the equations, called the Slepčev formulation and satisfied by the inverse in space of the functions. This presentation is done formally, but can be made rigorous.

(1.68), it should be possible to show that u ε converges to u 0 whose the inverse is a solution of (1.67). The case of infinite gradient should be treated by an approximation argument by comparison with functions with large, but finite, gradient.

Similarly the case p → 0 could be treated following the lines of Imbert, Monneau [START_REF] Imbert | Homogenization of first order equations with (u/ε)-periodic hamiltonians. Part I: local equations[END_REF] in the hull function approach. This could also be treated in the "Slepčev formulation" dealing with solutions with initial data in BU C loc , rather than Lipschitz initial data. where for any a > 0 we set

M a k [v](Y ) = [-a,a] dZ E k (v(Y + Z) -v(Y ))
Then we see that if u(τ, y) solves (1.5), then its inverse (in space) ρ(τ, Y ) solves the following non-local and non linear equation

Figure 1 :

 1 Figure 1: Sketch of the chain or particles for the classical FK model

LFFigure 2 :FFigure 3 :

 23 Figure 2: Sketch of F as a function of L when F depends on τ

Finally, from the definition ( 4 .

 4 26) of M , we get (4.29) m τ ≤ M . Similarly we have (4.30) m τ ≥ -M with m(τ ) = inf y∈R (u(τ, y) -py) .

  τ, y) -py) .

  4: Conclusion Therefore inequalities (4.35)-(4.36) and (4.34) imply the existence of the following limits lim T →+∞ λ + (T ) = lim T →+∞ λ -(T ) = λ and we deduce that(4.37) |λ ± (T ) -λ| ≤ 7C 1 + 2M T .Combining (4.37) with (4.24), we get withT = τ |u(τ, y) -u(0, 0) -py -λτ | ≤ 8C 1 + 2M .Finally, we deduce easily from (4.29)-(4.30) that |λ| ≤ M . This ends the proof of the proposition. Proposition 4.2. (Ergodicity) Assume (A0'), (A1'), (A2') and (A3'), and let u be a solution of (2.10) on [0, +∞) × R with initial data u 0 (y) = py with p > 0. Then there exists λ ∈ R such that |λ| ≤ M where M is defined in (4.26) with C 1 = 1 and for all (τ, y) ∈ [0, +∞) × R, (4.38) |u(τ, y) -py -λτ | ≤ C 3 = 2M + 8 .

( 4 .

 4 40) u τ = G(τ, [u(τ, •)] m , inf y ′ ∈R (u(τ, y ′ ) -py ′ ) + py -u(τ, y), u y ) . Then we have the following result Proposition 4.3. (Existence of time-space periodic solutions of (4.40)) Assume (A1'), (A2') and (A3') and consider p > 0. Then there exists a function u ∞ solving (4.40) on R × R and a real number λ ∈ R satisfying for all τ, y ∈ R, |u ∞ (τ, y) -py -λτ | ≤ 2⌈2M + 8⌉ (4.41) |λ| ≤ M with M defined by (4.26) with C 1 = 1. Moreover u ∞ satisfies (4.42)

( 4 .

 4 44) U (τ + k, y) = U (τ, y + λ k p ) and U (τ, y + l p ) = U (τ, y) + l .

is a subsolution of ( 4 .

 4 40) on R × R. On the other hand, if h denotes the hull function associated with p and λ = G(p), then u(τ, y) = h(τ, λτ + py) is a solution of (4.40) on R × R. Finally, as in Step 1, we conclude that λ ∞ ≤ λ . Similarly, considering h = lim inf n→+∞ * h n we can show that λ ∞ ≥ λ . Therefore λ ∞ = λ and this proves that G(p n ) → G(p); the continuity of the map p → G(p) follows and this ends the proof of the proposition. Proof of Theorem 1.5. Just apply Proposition 4.6 with G = F .

Step 1 :

 1 strong maximum principle at the contact point Let us consider b(τ ) = inf z∈R (h 2 (τ, z) -h * 1 (τ, z)) = h 2 (τ, z(τ )) -h * 1 (τ, z(τ ))

d

  dτ b(τ ) ≥ F (τ, [h 2 (τ, •)] p m (z(τ ))) -F (τ, [h * 1 (τ, •)] p m (z(τ ))) ≥ δ 0 (h 2 (τ, z(τ ) + k 0 p) -h * 1 (τ, z(τ ) + k 0 p)) ≥ δ 0 b(τ ) .

1. 2

 2 The generalized FK modelWe define for anyk ∈ Z \ {0} E k (x) = H(x) + H(x -k) -1 and E 0 = 0 and the operator for v(Y ) M k [v](Y ) = lim a→+∞ M a k [v](Y )

( 1 .

 1 70)ρ τ = -|ρ Y |F (τ, Y -M -m [ρ(τ, •)](Y ), ..., Y -M m [ρ(τ, •)](Y ))This equation is still monotone in ρ and could be treated directly with a suitable "Slepčev formulation".AknowledgementsThis work was supported by the ACI "Dislocations" (2003-2007) and the ANR MICA (2006-2009).

1

  CERMICS, Paris Est-ENPC, 6 & 8 avenue Blaise Pascal, Cité Descartes, Champs sur Marne, 77455 Marne la Vallée Cedex 2, France. 2 Projet Commands, CMAP-INRIA Futurs, Ecole Polytechnique, 91128 Palaiseau and ENSTA, UMA, 32 Bd Victor, 75739 Paris Cedex 15, France. 3 CEREMADE, UMR CNRS 7534, université Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris Cedex 16, France

  Moreover, when F does not depend on τ , we can choose the hull function h such that it does not depend on τ either.Proof of Proposition 5.2. This is a simple corollary of Proposition 4.2, Proposition 4.3 and Proposition 5.1; this leads to an improvement of the statement of Corollary 4.4. This proves in particular the bound on h z . Lipschitz continuity in time of h follows from the PDE satisfied by h. Indeed, it permits to get a uniform bound on h τ . This ends the proof of the proposition.

	5.58)	|λ| ≤ M 0 + δ(|a 0 | + 1)p
	and for all (τ, z) ∈ R × R,
	(5.59)	|h(τ, z) -z| ≤ M 0 + 4δ(|a 0 | + 1)p .
	We finally have
	Proposition 5.3. (Sub-and super-Lipschitz hull functions)

Construction of Lipschitz approximate hull functions

When proving the convergence Theorem 1.3, we explained that, on one hand, it is necessary in order to apply Evans' perturbed test function method, to deal with hull functions h(τ, z) that are uniformly continuous in z (uniformly in τ ).; on the other hand, given some p > 0, we also know some Hamiltonian F , with effective hamiltonian F (p), such that every corresponding hull function h is necessarily discontinuous in z (see the end of the introduction). that a hull function h solves, with λ = F (p),

We overcome this difficulty as in [START_REF] Imbert | Homogenization of first order equations with (u/ε)-periodic hamiltonians. Part I: local equations[END_REF]. As a matter of fact, the argument is simplified here: approximate Hamiltonians are defined in a simpler way.

Let us be more specific now. We show in this section that we can build approximate Hamiltonian G δ with corresponding effective Hamiltonian λ δ = G δ (p), and corresponding hull functions h δ , such that

We will show that it is enough to choose

with a 0 ∈ R (in fact, we will consider a 0 = ±1). Using (A1), we know that there exists a constant

We have the following regularity result Proposition 5.1. (Bound on the gradient) Assume (A1), (A2), (A3) and p > 0. Then the solution u of (2.10) with G = G δ defined by (5.49) and u 0 (y) = py satisfies

Proof of Proposition 5.1. For all η ≥ 0, we consider the more general equation

For η > 0, it is possible to show by the classical fixed point method that there exists a unique solution u of (5.52) in C 2+α,1+α for any α ∈ (0, 1). Moreover u satisfies

Then, if we define v = u y we see by derivation with respect to y, that v solves

distribution of particles positions of particles

The classical FK model

Let us start with the solution U i (τ ) of (1.1). Then we can define the "cumulative distribution of particles"

where H is the Heavyside function defined by

Here ρ(τ, •) is nothing else than the inverse (in space) of the function

Then we can check that the discontinuous function ρ solves the following equation

where the non-local operator M is defined for v(Y ) by

where for any a > 0 we set

is independent on a for any a sufficiently large (depending on v and Y ). Equation (1.66) has to be understood in the sense of Slepčev viscosity solutions as in Forcadel, Imbert, Monneau [START_REF] Forcadel | Homogenization of the dislocation dynamics and of some systems of particles with two-body interactions[END_REF].

More generally, if a continuous function ρ solves equation (1.66) and satisfies for some δ > 0

Another approach to the homogenization of system (1.1) consists in doing the homogenization of equation (1.66) following the lines of [START_REF] Forcadel | Homogenization of the dislocation dynamics and of some systems of particles with two-body interactions[END_REF]. Consider the rescaled function

where ρ ε (t, •) appears to be the inverse (in space) of u ε (t, •) defined in (1.2). Under suitable assumptions, it is possible to show that ρ ε converges to ρ 0 which solves the following equation:

(1.67)

Here ρ 0 (t, •) is the inverse (in space) of the function u 0 (t, •) which solves (1.3). Taking the derivatives of the identity:

a simple computation shows that (1.68)

Moreover the quantity θ = ρ 0 X can be interpreted as the density of particles and satisfies the following conservation law (the derivative of (1.67)):

The cell equation corresponding to equation (1.66) is found setting ρ(τ, Y ) = µτ + qY + v(Y ). We see that the corrector v satisfies

with µ = H(q) and v is 1-periodic. Therefore, if we set

and if w satisfies for some δ > 0: 0 < δ ≤ w Y ≤ 1/δ then we see (from (1.7)) that the hull function h is nothing else than the inverse of w, i.e.

(1.69) h(w(Y )) = Y and -µ/q = λ with p = 1/q which again is exactly the relation (1.68).

If both w and h are monotone, then a discontinuity of the hull function corresponds to a zero gradient of w and a discontinuity of w corresponds to a zero gradient of h. But in general, we do not know how to exclude the possibility for h and for w to be non-monotone and then (1.69) could be no longer true.

Let us also remark that, while the case p → +∞ seems difficult to deal with in the "hull function approach", this corresponds to a density q = 1/p going to zero with a corresponding effective Hamiltonian H(0) = 0 (because F is bounded). Therefore this case is well-posed for the formulation in ρ ε and could be proven naturally using directly the "Slepčev formulation". Another proof should be possible working in the "hull function approach" with initial data in BU C loc with gradient bounded from below. Using the relation