Cyril Imbert

Régis Monneau

Homogenization of first order equations with (u/ε)-periodic Hamiltonians. Part I: local equations

Keywords: periodic homogenization, Hamilton-Jacobi equations, correctors, perturbed test function, coercivity. Mathematics Subject Classification: 35B10, 35B27, 35F20, 49L25

In this paper, we present a result of homogenization of first order Hamilton-Jacobi equations with (u/ε)-periodic Hamiltonians. On the one hand, under a coercivity assumption on the Hamiltonian (and some natural regularity assumptions), we prove an ergodicity property of this equation and the existence of non periodic approximate correctors. On the other hand, the proof of the convergence of the solution, usually based on the introduction of a perturbed test function in the spirit of Evans' work, uses here a twisted perturbed test function for a higher dimensional problem.

Introduction

Setting of the problem. In this paper, we study the limit as ε → 0 of the viscosity solution of the following first order Hamilton-Jacobi equation with N ∈ N:

u ε t = H u ε ε , x ε , ∇u ε for (t, x) ∈ (0, +∞) × R N , u ε (0, x) = u 0 (x) for x ∈ R N (1)
with u 0 (•) ∈ W 1,∞ (R N ) (bounded and Lipschitz continuous functions on R N ), and where u ε t stands for ∂u ε ∂t and ∇u ε or ∇ x u ε for the gradient vector ∂u ε ∂x1 , ..., ∂u ε ∂x N . Our motivation comes from a problem of periodic homogenization for a model of dislocations dynamics [START_REF] Imbert | u/ε)periodic hamiltonians. Part II: application to dislocation dynamics[END_REF], where the level sets of the solution describe dislocations. We consider the following asumptions on the Hamiltonian H:

• (A1). Regularity: the function H : R × R N × R N → R is Lipschitz continuous and satisfies for a constant γ ≥ 0 and for almost every (v, y, p) ∈ R × R N × R N :

|∇ y H(v, y, p)| ≤ γ(1 + |p|), |∂ v H(v, y, p)| ≤ γ, |∇ p H(v, y, p)| ≤ γ;
• (A2). Periodicity: for any (v, y, p) ∈ R × R N × R N :

H(v + l, y + k, p) = H(v, y, p) for any l ∈ Z, k ∈ Z N ;

• (A3). Coercivity:

H(v, y, p) -→ +∞ as |p| -→ +∞ uniformly for (v, y) ∈ R × R N .

Under Assumptions (A1)-(A2), there exists a unique bounded continuous viscosity solution u ε of (1) -see Section 3 for references on viscosity solutions and below for a discussion about the Regularity Assumption (A1). The aim of this paper is to prove a homogenisation result, i.e. we want to prove that the limit u 0 of u ε as ε → 0 exists and is the solution of a homogenized equation of the form:

u 0 t = H 0 ∇u 0 for (t, x) ∈ (0, +∞) × R N , u 0 (0, x) = u 0 (x) for x ∈ R N (2)
where H 0 is a continuous function to be determined. There exists a unique bounded continuous viscosity solution for such an equation (see [START_REF] Barles | Solutions de viscosité des équations de Hamilton-Jacobi[END_REF] for instance). To get such a result, the coercivity assumption (A3) is natural, see for instance the seminal work of Lions, Papanicolaou and Varadhan [START_REF] Lions | Homogeneization of Hamilton-Jacobi equations[END_REF], and Subsection 2.2. Typically, our model Hamiltonian is H(v, y, p) = c(y)|p| + g(v) with c ≥ 1 and c, g Lipschitz continuous and periodic functions

The literature about homogenization of Hamilton-Jacobi equations developed a lot since [START_REF] Lions | Homogeneization of Hamilton-Jacobi equations[END_REF] and it is difficult to give an exhaustive list of references. See [START_REF] Lions | Homogeneization of Hamilton-Jacobi equations[END_REF][START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF][START_REF] Evans | Periodic homogenisation of certain fully nonlinear partial differential equations[END_REF][START_REF] Fathi | Sur la convergence du semi-groupe de Lax-Oleinik[END_REF][START_REF] Roquejoffre | Comportement asymptotique des solutions d'équations de Hamilton-Jacobi monodimensionnelles[END_REF][START_REF] Panagiotis | Stochastic homogenization of Hamilton-Jacobi equations and some applications[END_REF][START_REF] Ishii | Homogenization of the Cauchy problem for Hamilton-Jacobi equations[END_REF][START_REF] Barles | On the large time behavior of solutions of Hamilton-Jacobi equations[END_REF][START_REF] Barles | Some counterexamples on the asymptotic behavior of the solutions of Hamilton-Jacobi equations[END_REF][START_REF] Barles | Space-time periodic solutions and long-time behavior of solutions to quasilinear parabolic equations[END_REF][START_REF] Alvarez | Viscosity solutions methods for singular perturbations in deterministic and stochastic control[END_REF][START_REF] Alvarez | Singular perturbations of nonlinear degenerate parabolic PDEs: a general convergence result[END_REF][START_REF] Caffarelli | Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media[END_REF][START_REF] Lions | Homogenization of "viscous" Hamilton-Jacobi equations in stationary ergodic media[END_REF] and references therein. However, to our best knowledge, there are very few papers concerning periodic homogenization with equations depending (periodicly) on u/ε. Let us cite [START_REF] Piccinini | G-convergence for ordinary differential equations[END_REF][START_REF] Livio | Homogeneization problems for ordinary differential equations[END_REF] where Piccinini studied the homogenization of ordinary differential equations of the type u ε t = f u ε ε and also more general ODEs. In [START_REF] Boccardo | Remarques sur l'homogénéisation de certains problèmes quasi-linéaires[END_REF][START_REF] Boccardo | Homogeneisation de problemes quasi-lineaires[END_REF], Boccardo and Murat studied the homogenization of second order equations which inlude div A u ε ε • ∇u ε = f . Main results. Let us now describe the main results of this paper. They concern the ergodicity of the Hamiltonian and the convergence of the oscillating solution towards the homogenized one. We next give more details. In order to determine the limit of the solution u ε of (1), the first task is to determine the limit equation [START_REF] Alvarez | Singular perturbations of nonlinear degenerate parabolic PDEs: a general convergence result[END_REF]. The so-called effective Hamiltonian H 0 is uniquely defined by the long time behaviour of an evolution equation in (0, +∞) × R N . Let us be more specific. For any p ∈ R N , consider the continuous viscosity solution w = w(t, y) of w τ = H(p • y + w, y, p + ∇w) for (τ, y) ∈ (0, +∞) × R N , w(0, y) = 0 for y ∈ R N .

Then we have the following ergodicity of the Hamiltonian.

Theorem 1. (Ergodicity) Under Assumptions (A1)-(A2)-(A3), for any p ∈ R n , there exists a unique λ ∈ R such that the continuous viscosity solution w of (3) satisfies: w(τ,y) τ converges towards λ as τ → +∞, locally uniformly in y. The real number λ is denoted by H 0 (p) and this defines a continuous function on R n . Moreover we have the following coercivity of H 0 :

H 0 (p) -→ +∞ as |p| -→ +∞.
We can now state the main result of this article.

Theorem 2. (Convergence result) Under Assumptions (A1)-(A2)-(A3), the solution u ε of (1) converges towards the solution u 0 of (2) locally uniformly in (t, x), where H 0 is defined in Theorem 1.

In order to prove the convergence of u ε towards u 0 , we try to construct a so-called corrector, that is a solution of the cell problem which, in our case, has the following form:

λ + v τ = H(λτ + p • y + v, y, p + ∇ y v) for (τ, y) ∈ R × R N . (4)
It turns out that we are only able to construct sub and supercorrectors, i.e. sub and supersolution of the cell problem -see Theorem 3 in Appendix. However, it is enough to prove our convergence result and Theorem 1 appears as a corollary of this result. Let us also point out that, at the contrary of the classical case, i.e. Hamiltonians of the form H(y, p), the (sub and super) correctors here are not periodic w.r.t. y in general and may be discontinuous. Moreover true correctors are necessarily time-dependent in general.

A specific technical difficulty of our problem is to deal with the case λ = p = 0. For this reason, the "discontinuous" sub and super-correctors of Theorem 3 are not directly used in our proof of convergence. On the contrary, the proof of Theorem 2 is based on the existence of smoother correctors which are exact correctors for approximate Hamiltonians in higher dimension.

Our approach to construct these non-periodic correctors in a periodic framework is somehow related to several different works. Let us cite the work of Müller [START_REF] Müller | Homogenization of nonconvex integral functionals and cellular elastic materials[END_REF] where the periodic homogenization of non-convex functionals involves some non-periodic correctors. See Caffarelli [START_REF] Caffarelli | Some Liouville theorems for PDE problems in periodic media[END_REF] for the construction of planar-like non-periodic minimal surfaces in periodic media. See also Berestycki, Hamel [START_REF] Berestycki | Front propagation in periodic excitable media[END_REF], where pulsating fronts are studied in periodic media, these pulsating fronts being very similar to our corrector solutions. Finally, let us mention that the construction of correctors is related to the long time behaviour of solutions to Hamilton-Jacobi equations; such a problem has been studied by many authors, with seminal works of Fathi [START_REF] Fathi | Sur la convergence du semi-groupe de Lax-Oleinik[END_REF] and Roquejoffre [START_REF] Roquejoffre | Comportement asymptotique des solutions d'équations de Hamilton-Jacobi monodimensionnelles[END_REF] -see also Barles and Roquejoffre [8] and Barles and Souganidis [START_REF] Barles | On the large time behavior of solutions of Hamilton-Jacobi equations[END_REF] and references therein.

For general references on homogenization, see the book of Bensoussan, Lions and Papanicolaou [START_REF] Bensoussan | Asymptotic analysis for periodic structures[END_REF], and the pionneering work of Murat and Tartar [START_REF] Murat | H-convergence[END_REF].

Extensions. Let us first mention that we chose to present our results by assuming that the initial condition and the Hamiltonian are Lipschitz continuous. However, it is well-known that the "uniformly continuous" framework is a natural extension of the "Lipschitz" one and, on one hand, we could handle uniformly continuous initial condition and, on the other hand, (A1) can be adapted to deal with more general Hamiltonians. Let us make a further comment on assumption (A1). Usually, one only needs to bound from above ∂ v H; but in order to ensure a strong maximum principle when perturbing the equation by a nonlocal operator, we strength the usual assumption.

The reader can also think about more general Hamiltonians, depending for instance on slow variables. Let us consider two integers 0 ≤ m ≤ N and denote x = (x 1 , ..., x m ), x = (x 1 , ..., x N ). Let us consider a viscosity solution of

u ε t = H u ε ε , x ε , u ε , t, ∇u ε for (t, x) ∈ (0, +∞) × R N , u ε (0, x) = u 0 (x) for x ∈ R N (5)
where the Hamiltonian H = H(v, y, u, t, p) satisfies classical assumptions ensuring comparison principles. Assuming moreover some Coercive Condition analogous to (A3), namely,

• (A3'). Coercivity: For p = (p 1 , ..., p m ), p = (p m+1 , ..., p N ) and p = (p , p ) we have

H(v, y , u, t, p , p ) -→ +∞ as |p | -→ +∞ uniformly for (v, y , u, t, p ) ∈ R × R m × R × R × R N -m
and that (v, y) → H(v, y, u, t, p) is periodic, we consider the limit problem:

u 0 t = H 0 u 0 , t, ∇u 0 for (t, x) ∈ (0, +∞) × R N , u 0 (0, x) = u 0 (x) for x ∈ R N (6)
where H 0 (v, t, p) is a continuous function which is defined as the limit of w(τ,y) τ as τ -→ +∞ for the solution of

w τ = H(p • y + w, y , v, t, p + ∇ y w) for (τ, y) ∈ (0, +∞) × R N , w(0, y) = 0 for y ∈ R N .
The homogenized Hamiltonian H 0 satisfies the following coercivity condition

H 0 (v, t, p , p ) -→ +∞ as |p | -→ +∞ uniformly for (v, t, p ) ∈ R × R × R N -m .
Then a straightforward adaptation of our proofs permits to prove that for an initial condition u 0 ∈ W 1,∞ (R N ), the solution u ε of (5) converges towards the solution u 0 of (6) locally uniformly in (t, x). In a forthcoming work, we will study extensions of these results to some general ODE or non-local PDE problems.

After this work was completed, Barles [4] provided much simpler proofs of our results in a special case. Moreover, the extensive use of these ideas permited him to obtain homogenization results for noncoercive Hamiltonians. In particular, he introduced a different higher dimensional problem (instead of the original problem in dimension N ) by considering a geometric equation in dimension N + 1.

Organisation of the paper. The paper is organised as follows. In Section 2, we present the main difficulties encountered on the present problem of homogenization and the main new ideas we introduced to solve it. In Section 3, we state various comparison principles and gradient estimates for first order Hamilton-Jacobi equations perturbed by a nonlocal operator. In Section 4, we prove the convergence result (Theorem 2) by using the existence of approximate sub and supercorrectors (Proposition 7). In Section 5, we state a result on correctors for approximate Hamiltonians (Proposition 8) that encompasses Proposition 7 and the ergodicity theorem (Theorem 1). In Section 6, we prove the corner stone of our contruction, namely Proposition 8. Finally in Appendix, we state an independent result, Theorem 3, about the existence of bounded sub and supercorrectors; we also give a second proof of Theorem 1. Notations. B r (x) is the open ball of radius r centered at the point x. x is the integer such that x -x ∈ [0, 1) for any real x.

Strategies of proofs

The proofs of the main results are quite technical. To keep an easy understanding without technicalities, we propose in this section to indicate to the reader the main arguments used in the proofs, as simply as possible, but with formal arguments.

Main ideas for the Ansatz used in the proof of convergence

The goal of this subsection is to give some heuristic explanations of the difficulties arising in the homogenization of (1), and the main arguments that we have introduced in our proof of convergence.

First try: the usual Ansatz. Let us first start with a naive approach to the problem. The fisrt Ansatz that we can try is the following one (for (t, x) close to (0, 0)):

u ε (t, x) u 0 (t, x) + ε v t ε , x ε (7)
where v is a corrector to determine. If we plug this expression of u ε into (1), we find formally with (τ, y) = t ε , x ε :

u 0 t (t, x) + v τ (τ, y) H u 0 (t, x) ε + v(τ, y), y, ∇ x u 0 (t, x) + ∇ y v(τ, y) .
Using the Taylor expansion u 0 (t,x) ε u 0 (0,0) ε + u 0 t (0, 0)τ + ∇ x u 0 (0, 0) • y, the notation λ = u 0 t (0, 0), p = ∇ x u 0 (0, 0), and decoupling the slow variables (t, x) from the fast variables (τ, y), we get:

λ + v τ H u 0 (0, 0) ε + λτ + p • y + v, y, p + ∇ y v .
From this easy computation, we can learn two things. The first one is that a good candidate for the corrector v seems to be a solution of the equation where we put the term u 0 (0,0) ε to zero, i.e. Equation (4). The second thing, is that Ansatz [START_REF] Barles | Solutions de viscosité des équations de Hamilton-Jacobi[END_REF] is not the right Ansatz, because the final result depends on the term u 0 (0,0) ε which is not acceptable. This is why, we will have to consider a twisted corrector.

Second try: the twisted corrector. We use the previous notations, and let p denote (p , p N ) with p = (p 1 , ..., p N -1 ). We also denote x = (x , x N ). In the case where p N = 0, we introduce a new Ansatz with an x N -twisted corrector which takes into account the distorsion created by the term u ε ε in the Hamiltonian:

u ε (t, x , x N ) A ε (t, x , x N ) := u 0 (t, x , x N ) + ε v t ε , x ε , u 0 (t, x , x N ) -λt -p • x εp N ( 8
)
where a Taylor expansion immediately shows that

y N = u 0 (t,x ,x N )-λt-p •x εp N
is a good approximation of

x N ε . If we denote y = x1 ε , ..., x N -1
ε and y = (y , y N ), we get:

A ε (t, x , x N ) ε = λτ + p • y + p N • y N + v.
Plugging the Ansatz in Equation ( 1), we get with v N = ∂v ∂y N and u N = ∂u ∂x N :

λ + v τ (τ, y) + v N (τ, y) • u 0 t (t,x)-λ p N H λτ + p • y + v, y, p + ∇ y v + v N (τ, y) • ∇ x u 0 (t,x)-p p N , p N + v N (τ, y) • u 0 N (t,x) p N .
From this computation, we now learn two more things. First, for (t, x) close enough to (0, 0) and a smooth function u 0 , we see that we can replace

u 0 t (t,x)-λ p N and ∇ x u 0 (t,x)-p p N
by 0, and

u 0 N (t,x) p N
by 1, creating an error term which is bounded by ||v N || ∞ . But the difficulty is that the corrector is only bounded, and may be even not continuous which makes things difficult to control the term ||v N || ∞ = ∂v ∂y N ∞ . This difficulty will be overcomed using a truncated Hamiltonian H K whose corresponding correctors are Lipschitz continuous in all variables. Secondly, the previous computation only works for p N = 0. In the case p N = 0, we could still consider a x i -twisted corrector if p i = 0, or even a t-twisted corrector if λ = 0. But in the case where λ = p = 0, we still have a difficulty. This case, and by the way the general case, will be solved by imbbedding the problem in a higher dimensional problem where by construction p N +1 = 0.

Third try: our definitive choice of the Ansatz: a twisted corrector in higher dimension. We first consider a modification H K of the Hamiltonian H such that H K is bounded and converges locally uniformly on compact sets to H as K → +∞. Let us fix p N +1 ∈ R\ {0}, and consider correctors V K (τ, y, y N +1 ) of:

λ K + ∂V K ∂τ = H K λ K τ + p • y + p N +1 • y N +1 + V K , y, p + ∇ y V K , p N +1 + ∂V K ∂y N +1 in R × R N × R.
(9) From the boundedness of H K , we can find a corrector V K which is Lipschitz continuous in all variables. We now consider the solution U ε (t, x, x N +1 ) to the following equation:

U ε t = H U ε ε , x ε , ∇U ε for (t, x, x N +1 ) ∈ (0, +∞) × R N × R, U ε (0, x, x N +1 ) = U 0 (x, x N +1 ) := u 0 (x) + p N +1 • x N +1 for (x, x N +1 ) ∈ R N × R. (10)
In particular, we have u ε (t, x) = U ε (t, x, 0). We now consider the following Ansatz:

U ε (t, x, x N +1 ) U 0 (t, x, x N +1 ) + ε V K t ε , x ε , U 0 (t, x, x N +1 ) -λ K t -p • x ε p N +1 (11)
where U 0 (t, x, x N +1 ) = u 0 (t, x) + p N +1 • x N +1 . We can then easily check that the Ansatz ( 11) is a good Ansatz for which we can control the error terms in the equation, using in particular the fact that λ K is close enough to λ for K large enough, and V K is Lipschitz continuous with respect to y N +1 . Finally this construction works for any p N +1 = 0 and to simplify the presentation we take p N +1 = 1.

Main ideas for the construction of correctors

Boundedness of the corrector and the Coercivity Condition (A3). In the very first result of homogenization for Hamilton-Jacobi equations, namely in [START_REF] Lions | Homogeneization of Hamilton-Jacobi equations[END_REF], a coercivity condition is used to ensure the existence of correctors. Let us comment this assumption. An example of a simple Hamiltonian without coercivity condition is H(v, y, p) = sin (2πy) in dimension N = 1. For this Hamiltonian, the solution of (1) with zero initial condition is

u ε (t, x) = t sin 2πx ε .
Because the limit as t → +∞ of u ε (t,x) t = sin 2πx ε depends on x, this shows that this Hamiltonian is not ergodic in the sense of Theorem 1. There is no homogenization (with strong convergence) for this example, or more generally for some hyperbolic equations (see Tassa [START_REF] Tassa | Homogenization of two-dimensional linear flows with integral invariance[END_REF], Tartar [START_REF] Tartar | Nonlocal effects induced by homogenization[END_REF][START_REF] Tartar | Homogenization and hyperbolicity[END_REF]). We also remark a fundamental property of this solution: the space oscillation of the solution u ε is proportional to the time t, and then is unbounded in time.

On the contrary, let us now consider a Hamiltonian H(v, y, p) for which we can bound the space oscillation of the solution by a constant C > 0 for all time. In the very particular case where the solution is constant in space, the ergodicity of the equation is reduced to the study of the long-time behaviour of the solution to an ODE. In the general case, the bound on the space oscillation of the solution allows us to compare easily the solution at two different times, which is enough to prove the ergodicity of the Hamiltonian.

We now explain how the Coercivity Condition (A3) can be used here to bound the corrector on the whole space for all time. Even if, as a matter of fact, we are not able to prove the existence of such a corrector, let us assume that we have a bounded corrector v which is a solution of (4) i.e.

λ + v τ = H(λτ + p • y + v, y, p + ∇v) for (τ, y) ∈ R × R N which is T -periodic in time where T = 1/|λ| with λ = H 0 (p) = 0. Considering the maximum of v, we first remark that: λ ≤ sup (v,y)∈R×R N H(v, y, p). (12)
Let us now define

v(y) = inf τ ∈R v(τ, y) = v(τ (y), y) for some τ (y) ∈ [0, T ). Then v satisfies λ ≥ H(λτ (y) + p • y + v, y, p + ∇v).
Using (A3), this gives (at least formally) a bound on p + ∇v, i.e.

|∇v| ≤ C 1 (|p|).

This implies that the space oscillation of v is bounded at short distance. The space oscillation of v at large distance is also bounded if we choose a corrector v satisfying moreover:

|v(τ, y + k) -v(τ, y)| ≤ 1 ∀k ∈ Z N , ∀(τ, y) ∈ R × R N
which follows from the periodicity of the Hamiltonian (see Assumption (A2)).

Under certain conditions, we can formally show that the corrector v can be chosen such that w(τ, y) := λτ + v(τ, y) satisfies λw τ ≥ 0. Together with the time periodicity of v, we deduce that 0 ≤ w(τ + s, y)w(τ, y) ≤ 1 for 0 ≤ s ≤ T , in the case λ > 0. In general, we get that

|v(τ + s, y) -v(τ, y)| ≤ 1.
Up to substract an integer to v, this is enough to derive the following L ∞ bound on the corrector:

|v(τ, y)| ≤ C 2 (|p|).
Construction of the correctors. The basic idea to build correctors is to consider solutions w to (3), i.e.

w τ = H(p • y + w, y, p + ∇ y w) for (τ, y) ∈ (0, +∞) × R N , w(0, y) = 0 for y ∈ R N .
If we are able to bound first the space oscillation of w uniformly in time, we can show that w(τ,y) τ → λ as τ → +∞, uniformly in y ∈ R N , and then consider a limit w ∞ (τ, y) of w(τ + k, y) as k → +∞. Roughlty speaking, we then show that v(τ, y) = w ∞ (τ, y) -λτ (or at least a limit of w ∞ (τ, y) -λτ ) is 1/|λ|-periodic in time. This last property can be proven using a strong maximum principle on a perturbed problem.

The perturbed problem consists first in considering a truncated Hamiltonian H K in place of H and to adding a non-local term ε I(w) on the right hand side of the equation to ensure the strong maximum principle for ε > 0. At the end, we take the limit as ε -→ 0 and, if necessary, as K -→ +∞.

Comparison principles and gradient estimates for non-local equations

In this section, we state various comparison principles and obtain gradient estimates for viscosity solutions of Hamilton-Jacobi equations perturbed by a 0-order nonlocal operator, under appropriate assumptions on the Hamiltonian. For a definition of viscosity solutions and their properties, see in particular the User's Guide [START_REF] Michael | User's guide to viscosity solutions of second order partial differential equations[END_REF] for viscosity solutions, the book of Barles [START_REF] Barles | Solutions de viscosité des équations de Hamilton-Jacobi[END_REF], the book of Bardi and Capuzzo-Dolcetta [START_REF] Bardi | Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations[END_REF] or the book of Lions [START_REF] Lions | Generalized solutions of Hamilton-Jacobi equations[END_REF].

We first introduce the 0-order non local operator:

I(v)(x) = R N dz J(z) (v(x -z) -v(x))
where the function J satisfies

   J is continuous, J(-z) = J(z) > 0 for every z ∈ R N , R N dz J(z) = 1 and I i := R N dz |z| i J(z) < +∞, i = 1, 2. ( 13
)
These conditions are for instance satisfied for the choice of the following function:

J(z) = c e -|z| with c = R N dz e -|z| -1 . Let F : R × R N × R N -→ R
and w 0 : R N -→ R be continuous functions, the second one satisfying for some constant C > 0:

|w 0 (x)| ≤ C(1 + |x|) for every x ∈ R N .
For T > 0, we consider the equation for ε ≥ 0:

w t = ε I(w(t, •)) + F (w, x, ∇w) (14)
for (t, x) ∈ (0, T ) × R N , with the following initial condition:

w(0, x) = w 0 (x) for x ∈ R N . ( 15
)
We say that a function w : [0, T ) × R N → R has at most a linear growth if there exists a constant C > 0 such that:

|w(t, x)| ≤ C(1 + |x|) for every (t, x) ∈ [0, T ) × R N .
Given such a function w, the function w * designates its upper-semicontinuous envelope (i.e. the smallest u.s.c. function above w) and the function w * its lower-semicontinuous envelope.

Throughout the paper, we use the following convention (from the theory of discontinuous solutions developed by Ishii): we say that a locally bounded function w is a subsolution (resp. supersolution) of an equation if its usc envelope (resp. lsc envelope) is a subsolution (resp. supersolution).

Next, we successively give a comparison principle for unbounded viscosity solutions of ( 14), a comparison principle on bounded domains and a strong maximum principle. The first result is adapted from [START_REF] Michael | User's guide to viscosity solutions of second order partial differential equations[END_REF]. Let us mention that the nonlocal term can be easily handled because I i < +∞ for i = 1, 2. See the appendix for details. Proposition 1. (Comparison principle, ε ≥ 0) Under Assumption (A1) and assuming that u 0 is Lipschitz continuous, if u and v are respectively sub and supersolutions of ( 14)-( 15) on (0, T ) × R N , which have at most a linear growth, then

u ≤ v on [0, T ) × R N .
The following result is easier than the previous one since we assume boundedness of the domain; it can be easily derived from classical results -see [START_REF] Barles | Solutions de viscosité des équations de Hamilton-Jacobi[END_REF] for instance and [START_REF] Imbert | A non-local regularization of first order hamilton-jacobi equations[END_REF] for integro-PDE's. This is the reason why we skip the proof.

Proposition 2. (Comparison principle on bounded open sets

, ε = 0) Under Assumption (A1), let Q ⊂ (0, T ) × R N a
bounded open set and the two functions u : Q -→ R and v : Q -→ R be respectively sub-and supersolutions of (14

) on Q. If u * ≤ v * on ∂Q, then u * ≥ v * on Q.
We next state a strong maximum principle for [START_REF] Boccardo | Remarques sur l'homogénéisation de certains problèmes quasi-linéaires[END_REF] when ε > 0. The techniques used in the following proof are classical but not really the result. This is the reason why we provide details. Proposition 3. (Strong maximum principle for ε > 0) Under Assumption (A1) and assuming that u 0 is Lipschitz continuous, let us consider functions w and w which are respectively super and subsolution of ( 14)-( 15) on (0, T ) × R N , are at most of linear growth, and satisfy

w * ≤ w * on (0, T ) × R N . If w * (t, x) = w * (t, x) for some point (t, x) ∈ (0, T ) × R N and if ε > 0, then we have w * = w * on (0, t] × R N .
Proof. Let us first notice that we can assume that ε = 1. Let us next define, for any α ≥ 0:

w α (t, x) = w * (t, x) -w * (t, x) + α |x -x| 2
and m α (t) := inf

x∈R N w α (t, x).
Note that m α (t) ≥ 0 and m α (t) = 0. We claim that this function satisfies, in the viscosity sense,

m α (t) + γm α (t) ≥ -C 1 α (16)
for some constant C 1 to be determined. In order to justify such a result, we first prove that the supremum is realised. Because w and w are at most of linear growth, we know that for every α ∈ (0, 1), there exists

x α (t) ∈ R N such that m α (t) = w α (t, x α (t)) and |x α (t)| ≤ C
α for some constant C > 0. In order to prove that ( 16) is satisfied, we need to consider the lsc envelope of m α . In view of the bound on x α (t), it is clear that m α is lsc and therefore coincides with its lsc envelope. Next, we assert that m α is the relaxed upper limit of the family of functions {m α,ε } ε>0 :

m α = lim inf * (m α,ε ) with m α,ε (t) = inf s≥0,x,y∈R N w * (t, x) -w * (s, y) + (t -s) 2 2ε + |y -x| 2 2ε + α|x -x| 2 .
Using once again the fact that w * and w * have linear growth, we can assert that the supremum defining m α,ε is attained.

Let us next consider a test function φ(t) such that m α -φ attains a strict local minimum at time t 0 > 0. This implies that there exists ε n → 0 and t n → t 0 such that m α,εn -φ attains a local minimum at time t n and m α,εn (t n ) → m α (t 0 ) as n → +∞. As explained above, the supremum defining m α,εn (t n ) is attained and

m α,εn (t n ) = w * (t n , x n ) -w * (s n , y n ) + (t n -s n ) 2 2ε n + |y n -x n | 2 2ε n + α|x n -x| 2 .
Classical results of penalization assert that the following properties hold true:

         (tn-sn) 2 2εn
→ 0 and in particular s n > 0 for n large enough,

|yn-xn| 2 2εn → 0, x n → x 0 such that m α (t 0 ) = w * (t 0 , x 0 ) -w * (t 0 , x 0 ) + α|x 0 -x| 2 , w * (t n , x n ) -w * (s n , y n ) → w * (t 0 , x 0 ) -w * (t 0 , x 0 ). ( 17
)
In the following, x 0 = x α (t 0 ) in accordance with the notations introduced previously. Consequently,

w * (t, x) -w * (s, y) + (t -s) 2 2ε + |y -x| 2 2ε + α|x -x| 2 -φ(t) ≥ w * (t n , x n ) -w * (s n , y n ) + (t n -s n ) 2 2ε + |y n -x n | 2 2ε + α|x n -x| 2 -φ(t n )
for any (s, x, y) ∈ (0; +∞) × R 2N and t close to t n . Using the fact that w (resp. w) is a supersolution (resp. subsolution) of ( 14)-( 15), we conclude that:

s n -t n ε n + φ (t n ) ≥ I(w * (t n , •))(x n ) + F (w * (t n , x n ), x n , p n + 2α(x n -x)) s n -t n ε n ≤ I(w * (s n , •))(y n ) + F (w * (s n , y n ), y n , p n ) with p n = (y n -x n )/ε n .
Substracting both inequalities yields:

φ (t n ) ≥ I(w * (t n , •))(x n ) -I(w * (s n , •))(y n ) -2γα|x n -x| -γ m α,εn (t n ) + (t n -s n ) 2 2ε n + |x n -y n | 2 2ε n + α|x n -x| 2 -γ |x n -y n | + |x n -y n | 2 ε n ≥ dzJ(z)(w * (t n , x n -z) -w * (s n , y n -z)) -(w * (t n , x n ) -w * (s n , y n )) -γ m α,εn (t n ) + (t n -s n ) 2 2ε n + |x n -y n | 2 2ε n + α|x n -x| 2 -γ |x n -y n | + |x n -y n | 2 ε n
where we used (A1). Using Fatou's lemma and (17) yields as n → +∞:

φ (t 0 ) ≥ I(w * (t 0 , •))(x α (t 0 )) -I(w * (t 0 , •))(x α (t 0 )) -2γα|x α (t 0 ) -x| -γ m α (t 0 ) -α|x α (t 0 ) -x| 2 ≥ I(w α (t 0 , •))(x α (t 0 )) -αI(| • -x| 2 )(x α (t 0 )) -γm α (t 0 ) + α inf r≥0 (γr 2 -2γr).
Using the fact that J(-z) = J(z), we get

I(| • -x| 2 )(x) = constant = I 2 := R N dz J(z)|z| 2
and setting C 1 = -inf r≥0 (γr 2 -2γr) + I 2 , we deduce that m α satisfies, in the viscosity sense,

m α (t) + γm α (t) ≥ -αC 1 + I(w α (t, •))(x α (t)) with I(w α (t, •))(x α (t)) ≥ 0.
In particular, we have:

m α (t) + γm α (t) ≥ -αC 1 .
By integration (using m α (t) = 0), we get:

m α (t) ≤ α C 1 γ e γ(t-t) -1 for t ∈ (0, t]. (18)
Using the fact that:

m α (t) = w * (t, x α (t)) -w * (t, x α (t)) + α |x α (t) -x| 2 and w * -w * ≥ 0,
we deduce that:

|x α (t) -x| 2 ≤ C 1 γ e γ(t-t) -1 for t ∈ (0, t]. (19)
First, we remark that m α (t) -→ m 0 (t) as α -→ 0, and then there exists x 0 (t) satisfying [START_REF] Droniou | Fractal first order partial differential equations[END_REF], such that

m 0 (t) = w 0 (t, x 0 (t)) = w * (t, x 0 (t)) -w * (t, x 0 (t)) for t ∈ (0, t]
We also have m 0 = lim sup * m α and arguing as previously implies that m 0 satisfies:

m 0 (t) + γm 0 (t) ≥ I(w 0 (t, •))(x 0 (t)) for t ∈ (0, t).
From [START_REF] Michael | User's guide to viscosity solutions of second order partial differential equations[END_REF], we also deduce that m 0 (t) = 0 for every t ∈ (0, t). Therefore for any t 0 ∈ (0, t), we can take a test function tangent from below to m 0 at t 0 , and deduce that:

0 ≥ I(w 0 (t 0 , •))(x 0 (t 0 )) = R N
dz J(z) w 0 (t 0 , x 0 (t 0 ) -z) with w 0 (t 0 , x 0 (t 0 )) = 0 and w 0 (t 0 , •) ≥ 0.

Because J is continuous and satisfies J > 0, we deduce that w 0 (t 0 , x) = 0 for almost every x ∈ R N . Because w 0 is lower semi-continuous, we deduce that w 0 (t 0 , x) = 0 for every x ∈ R N . Now this result is true for every t 0 ∈ (0, t), and then still by lower semi-continuity (and using w 0 ≥ 0) w 0 (t, x) = 0 for every (t, x) ∈ (0, t] × R N i.e. w * (t, x) = w * (t, x) for every (t, x) ∈ (0, t] × R N . This ends the proof of Proposition 3.

We next state and prove existence of a solution of ( 14)- [START_REF] Caffarelli | Some Liouville theorems for PDE problems in periodic media[END_REF].

Proposition 4. (Existence and uniqueness of a solution, ε ≥ 0) If u 0 ∈ W 1,∞ and F satisfies (A1), there exists a unique viscosity solution w of ( 14)-( 15) on (0, T ) × R N satisfying |w(t, x) -u 0 (x)| ≤ Ct for some C > 0. Moreover this solution w is continuous.

Proof. The existence is classical via Perron's method if one constructs barriers. Let us consider

w ± (t, x) = ±Ct + u 0 (x) with C = sup (v,y,p)∈R 2N +1 ,|p|≤ u0 1,∞ F (v, y, p) ± ε u 0 1,∞ I 1 .
We consider E the set of all subsolutions u of ( 14) such that u ≤ w + on [0, T )) × R N . This set is nonempty since it contains w -. Define

w(t, x) = sup{u(t, x) : u ∈ E}
It is a subsolution. It is also classical to check that w is a supersolution -it relies on a "bump construction", described for instance in [START_REF] Michael | User's guide to viscosity solutions of second order partial differential equations[END_REF]. The uniqueness and the continuity follow from the comparison principle (Proposition 1).

When constructing the correctors, we will need some gradient estimates, but in an integral form. Let us give a precise definition. Definition 1. (Gradient estimate) For a function w 0 : R N -→ R, we say that

ξ • ∇w 0 ≤ M on R N if and only if w 0 (x + hξ) -w 0 (x) ≤ hM for all h ≥ 0, x ∈ R N .
We next state and prove two results concerning gradient estimates satisfied by the viscosity solution of ( 14)-( 15) under certain conditions on the Hamiltonian F . Proposition 5. (A priori bound on the gradient, ε ≥ 0) Under assumptions of Proposition 4, let us assume moreover that there exists ξ ∈ R N with |ξ| = 1, M ≥ 0 and a function F 0 such that

F (u, x, p) = F 0 (x -(ξ • x) ξ, p) for all (u, x, p) ∈ R × R N × R N if ξ • p ≥ M.
If ξ • ∇w 0 ≤ M on R N in the sense of Definition 1, then the solution w of ( 14)-( 15) on (0,

T ) × R N satisfies ξ • ∇w ≤ M on [0, T ) × R N .
Before proving this proposition, let us derive a straightforward corollary.

Corollary 1. Under the assumptions of Proposition 4, let us assume moreover that there exists a closed set Ω that is starshaped w.r.t. the origin and a function F 0 such that

F (u, x, p) = F 0 (p) for all (u, x, p) ∈ R × R N × R N if p / ∈ Ω.
If ∇w 0 ∈ Ω on R N then the solution w of ( 14)-( 15) on (0,

T ) × R N satisfies ∇w ∈ Ω on [0, T ) × R N .
Let us now turn to the proof of the proposition.

Proof of Proposition 5. The proof proceeds by reduction to a simpler case and by approximation. First, there is no restriction in assuming that ε = 1 and ξ = (1, 0, . . . , 0). Hence, if x = (x 1 , x ) and p = (p 1 , p ) with x , p ∈ R N -1 :

∂ 1 w 0 ≤ M, F (u, x, p) = F 0 (x , p) for all (u, x, p) ∈ R × R N × R N if p 1 ≥ M (20)
and we want to estimate from above ∂ 1 w. Next, we consider

w δ 0 ∈ C ∞ (R N ), F δ ∈ C ∞ (R × R N × R N
) and such that: w δ 0 → w 0 and F δ → F locally uniformly as δ → 0, F δ satisfies ( 20) and (A1) with M + δ and γ + δ respectively,

∂ 1 w 0 ≤ M + δ on R N .
Then if w δ is the viscosity solution of:

w δ t = I(w δ (t, •)) + F δ (w δ , x, ∇w δ ) + δ∆w δ
for (t, x) ∈ (0, T ) × R N , with the following initial condition:

w δ (0, x) = w δ 0 (x) for x ∈ R N , we can prove that w δ ∈ C ∞ ([0, T ) × R N
) by adapting the classical theory of parabolic equations [START_REF] Ladyženskaja | Linear and quasilinear equations of parabolic type[END_REF] -see [START_REF] Imbert | A non-local regularization of first order hamilton-jacobi equations[END_REF][START_REF] Droniou | Fractal first order partial differential equations[END_REF] for such an adaptation but with a different integral term. Let us next write down the equation satisfied by v(t, x) = ∂ 1 w δ (t, x):

∂ t v = I(v(t, •)) + ∂ w F δ (w δ , x, v, ∇ w δ )v + ∂ x1 F δ (w δ , x, v, ∇ w δ ) + i ∂ pi F δ (w δ , x, v, ∇ w δ )∂ i v + δ∆v
where ∇ w = (∂ 2 w, . . . , ∂ N w) ∈ R N -1 . Now remark that M + δ is a supersolution of such an equation and the comparison principle yields v ≤ M + δ which implies that:

w δ (x 1 + h, x ) ≤ w δ (x 1 , x ) + h(M + δ).
Passing to the limit as δ → 0 permits to achieve the proof. Proposition 6. (Monotonicity of the solution, ε ≥ 0) Under assumptions of Proposition 4, let us assume moreover that there exists ξ ∈ R N with |ξ| = 1, and a function F 1 such that:

F (u, x, p) = F 1 (u, x -(ξ • x) ξ, p) for all (u, x, p) ∈ R × R N × R N .
If ξ • ∇w 0 ≤ 0 on R N in the sense of Definition 1, then the solution w of ( 14)-( 15) on (0,

T ) × R N satisfies ξ • ∇w ≤ 0 on [0, T ) × R N .
Proof. The proof is very similar to the proof of Proposition 5. We do the same reduction and the same approximation and v satisfies the same equation but with ∂ x1 F ≡ 0. It is therefore clear that v = 0 is a supersolution and we conclude the same way.

The proof of the convergence

This section is dedicated to the proof of Theorem 2. Before presenting it, we first imbed the problem in a higher dimensional one. Precisely, we consider the solution U ε of

U ε t = H U ε ε , x ε , ∇ x U ε for (t, x, x N +1 ) ∈ (0, +∞) × R N × R, U ε (0, x, x N +1 ) = u 0 (x) + x N +1 for (x, x N +1 ) ∈ R N × R. (21)
There exists a unique viscosity solution of such an equation under assumptions (A1)-(A3). Then we have the following lemma.

Lemma 1. (Link between problems on R N and on

R N +1 ) We have u ε (t, x) = U ε (t, x, 0). Proof. u ε [x N +1
] the solution of (1) on R N with initial condition

u ε [x N +1 ](0, x) = u 0 (x) + x N +1 .
We now build the function:

V (t, x, x N +1 ) = u ε [x N +1 ](t, x).
Let us first justify that V is a continuous function with respect to x N +1 . To see this, consider the function u ε [x N +1 ](t, x) + e γt/ε δ where γ is given by (A1), and prove that it is above u ε [x N +1 + δ](t, x). It suffices to remark that it is a supersolution of (1) (by using Assumption (A1)) and to use the comparison principle (Proposition 1). Hence, V is upper semicontinuous. We prove analogously that V is lower semicontinuous and we conclude that V is continuous. We now check easily (using test functions) that V is a solution of [START_REF] Evans | Periodic homogenisation of certain fully nonlinear partial differential equations[END_REF]. By the comparison principle applied to [START_REF] Evans | Periodic homogenisation of certain fully nonlinear partial differential equations[END_REF], we deduce that V = U ε and then u ε (t, x) = U ε (t, x, 0). This ends the proof of the Lemma.

As underlined in the third try of Subsection 2.1, the proof of convergence will use Lipschitz continuous approximate sub and super-correctors on R×R N +1 . More precisely, we will use the following proposition.

∈ R, ∃β 0 ∈ R, such that λ(β 0 ) = λ 0 ( 22
)
For any fixed β ∈ R, there exist real numbers λ + K (β), λ - K (β), a constant C = C(p) > 0 (independent on K and β) and bounded super and sub-correctors

V + K , V - K depending on β, such that λ(β) = lim K→+∞ λ + K (β) = lim K→+∞ λ - K (β)
with λ + K (β) and λ - K (β) satisfying ( 22) and, for τ ∈ R, Y = (y, y N +1 ) ∈ R N × R and P = (p, 1) ∈ R N × R:

|V + K (τ, Y )| ≤ C, |V - K (τ, Y )| ≤ C λ + K + ∂V + K ∂τ ≥ β + H λ + K τ + P • Y + V + K , y, p + ∇ y V + K for (τ, Y ) ∈ R × R N +1 , (23)
λ - K + ∂V - K ∂τ ≤ β + H λ - K τ + P • Y + V - K , y, p + ∇ y V - K for (τ, Y ) ∈ R × R N +1 . ( 24
)
For any λ 0 ∈ R, there exist real numbers β ± 0 and

β ± K such that      λ ± K (β ± K ) = λ(β ± 0 ) = λ 0 β ± K -→ β ± 0 as K → +∞ ∂V ± K ∂y N +1 ≤ C K (25)
for the correctors V + K and V - K respectively associated to β + K and β - K , and for some constant

C K = C(K, p) > 0.
Proof of Theorem 2. Classically, we prove that U + = lim sup * U ε is a subsolution of

W t = H 0 (∇ x W ) for (t, x, x N +1 ) ∈ (0, +∞) × R N × R, W (0, x, x N +1 ) = u 0 (x) + x N +1 for (x, x N +1 ) ∈ R N × R. (26)
Analogously, we can prove that U -= lim inf * U ε is a supersolution of [START_REF] Ladyženskaja | Linear and quasilinear equations of parabolic type[END_REF]. By the barriers (uniform in ε) given in Proposition 4, we deduce that U + (0, x) = U -(0, x). The comparison principle for (2) (see Proposition 1) thus implies that U + ≤ U -. Since U -≤ U + always holds true, we conclude that the two functions coincide with U 0 , the unique continuous viscosity solution of ( 26). This last fact is equivalent to the local convergence of U ε towards U 0 . From the invariance by translations we also deduce that U 0 (t, x, x N +1 + a) -a is also a solution for any a ∈ R. This proves that U 0 (t, x, x N +1 ) = U 0 (t, x, 0) + x N +1 . Therefore U 0 (t, x, 0) is the solution u 0 of (2). By Lemma 1, this proves in particular the local convergence of u ε to u 0 .

Step 1: U + is a subsolution. Let us thus prove that U + is a subsolution of [START_REF] Ladyženskaja | Linear and quasilinear equations of parabolic type[END_REF]. First, by the comparison principle, we easily check that for any sequence k ε ∈ Z, we have

U ε (t, x, x N +1 + εk ε ) = εk ε + U ε (t, x, x N +1 )
For any a ∈ R, we then choose the sequence such that εk ε → a. Passing to the limit as ε → 0, we deduce by definition of U + that U + (t, x, x N +1 + a) = a + U + (t, x, x N +1 ) and then

U + (t, x, x N +1 ) = U + (t, x, 0) + x N +1 . ( 27
)
Applying Evans' technique of the perturbed test function [START_REF] Evans | The perturbed test function method for viscosity solutions of nonlinear PDE[END_REF][START_REF] Evans | Periodic homogenisation of certain fully nonlinear partial differential equations[END_REF], we argue by contradiction: we consider a test function φ ∈ C 2 ((0; +∞) × R N +1 ) such that U + -φ attains a strict zero local maximum at (t 0 , X 0 ) with t 0 > 0, X 0 ∈ R N +1 , and we suppose that there exists θ > 0 such that:

φ t (t 0 , X 0 ) = H(∇ x φ(t 0 , X 0 )) + θ.
In the following, we set p = ∇ x φ(t 0 , X 0 ) and λ 0 = φ t (t 0 , X 0 ). With the notations of Propositon 7, we see that we have

λ 0 = λ(0) + θ = λ(β 0 ) for β 0 > 0.
We deduce from Proposition 7 that there exists

β + K ∈ R such that λ 0 = λ + K (β + K ) and β + K ≥ β 0 /2 for K large enough.
We next construct a perturbed test function in the spirit of Evans' seminal work. Here, this is a x N +1 -twisted perturbed test function:

φ ε (t, x, x N +1 ) = φ(t, x, x N +1 ) + εV + K t ε , x ε , φ(t, x, x N +1 ) -λ 0 t -p • x ε
where V + K is a super-corrector given by Proposition 7 for β = β + K . From [START_REF] Lions | Generalized solutions of Hamilton-Jacobi equations[END_REF], we also know that

∂φ ∂x N +1 (t 0 , X 0 ) = 1.
Let us define, for r small enough: V r := (t 0 -r, t 0 + r) × B r (X 0 ) ⊂ (0, +∞) × R N +1 . We claim that the following lemma (whose proof is postponed) holds true.

Lemma 2. (φ ε is a supersolution on V r K ) There exist r K > 0 and ε 0 > 0 such that for all 0 < ε < ε 0 , the function φ ε is a supersolution of

φ ε t = H φ ε ε , x ε , ∇ x φ ε with (t, x, x N +1 ) ∈ (0; +∞) × R N +1 (28) on V r K .
Now, since (t 0 , X 0 ) is a strict local zero maximum of U + -φ, for 0 < r ≤ r K small enough, we have: U + -φ ≤ -2η on ∂V r for some η > 0. Then U ε -φ ≤ -η on ∂V r for ε small enough. From the bound on the corrector given in Proposition 7, we deduce that U ε ≤ φ ε -η + Cε on ∂V r . By definition, U ε is a solution of ( 21) and by Lemma 2, φ ε is a supersolution. Remark that the function

φ ε + ε -η+Cε
ε is still a supersolution. By the comparison principle (Proposition 2), we conclude that

U ε ≤ φ ε +ε -η+Cε ε on V r . Letting ε → 0, we get in particular at (t 0 , X 0 ): U + (t 0 , X 0 ) ≤ φ(t 0 , X 0 )-η which is a contradiction.
Step 2: U -is a supersolution. Let us thus prove that U -is a supersolution of [START_REF] Ladyženskaja | Linear and quasilinear equations of parabolic type[END_REF]. We proceed as in Step 1 with U + -φ attains a strict local zero minimum at (t 0 , X 0 ) with θ < 0 such that:

φ t (t 0 , X 0 ) = H(∇ x φ(t 0 , X 0 )) + θ and find β - K ∈ R such that λ 0 = λ - K (β - K ) = λ(β 0 ), which satisfies β - K ≤ β 0 /2 < 0 for K large enough. We define φ ε (t, x, x N +1 ) = φ(t, x, x N +1 ) + εV - K t ε , x ε , φ(t, x, x N +1 ) -λ 0 t -p • x ε
where V - K is a corrector given by Proposition 7 for β = β - K . We claim that the following lemma holds true (its proof is similar to the proof of Lemma 2): Lemma 3. (φ ε is a subsolution on V r K ) There exist r K > 0 and ε 0 > 0 such that for all 0 < ε < ε 0 , the function φ ε is a subsolution on V r K , i.e. satisfies

φ ε t ≤ H φ ε ε , x ε , ∇ x φ ε on V r K .
By using such a lemma, we can get a contradiction as in Step 1. The proof of Theorem 2 is now complete.

Proof of Lemma 2. We want to prove that φ ε is a supersolution of [START_REF] Lions | Homogeneization of Hamilton-Jacobi equations[END_REF]. Consider a test function ψ and a point (t, X) ∈ V r such that φ ε -ψ attains a local minimum at (t, X) with X = (x, x N +1 ):

φ ε (t, X) -ψ(t, X) ≤ φ ε (t, X) -ψ(t, X) i.e. V + K t ε , x ε , φ(t, X) -λ 0 t -p • x ε - 1 ε (ψ(t, X) -φ(t, X)) ≤ V + K t ε , x ε , φ(t, X) -λ 0 t -p • x ε - 1 ε (ψ(t, X) -φ(t, X)) (29)
Let us define F (t, X) = φ(t, X)-λ 0 t-p•x. We have ∂F ∂x N +1 (t 0 , X 0 ) = ∂φ ∂x N +1 (t 0 , X 0 ) = 1. Consequently, there exists r 0 > 0 such that the map

Id × F : V r0 -→ U ⊂ R × R N × R (t, x, x N +1 ) -→ (t, x, F (t, x, x N +1 ))
is a C 1 -diffeomorphism from V r0 onto its range U , and let us call G : U -→ R the map such that

Id × G : U -→ V r0 (t, x, ξ N +1 ) -→ (t, x, G(t, x, ξ N +1 ))
is the inverse of Id × F .

Let us consider the variables τ = t/ε, Y = (y, y N +1 ) with y = x/ε and y N +1 = F (t, X)/ε and define

Γ ε (τ, Y ) = 1 ε (ψ (ετ, εy, G(ετ, εy, εy N +1 )) -φ (ετ, εy, G(ετ, εy, εy N +1 ))) . Let τ = t ε , y = x ε , y N +1 = F (t,X) ε , Y = (y, y N +1
). Then (29) implies

V + K (τ , Y ) -Γ ε (τ , Y ) ≤ V + K (τ, Y ) -Γ ε (τ, Y ) for all (τ, Y ) in a neighborhood of (τ , Y )
i.e. V + K -Γ ε reaches a local minimum at (τ , Y ). Estimate [START_REF] Ishii | Homogenization of the Cauchy problem for Hamilton-Jacobi equations[END_REF] implies in particular that

∂Γ ε ∂y N +1 (τ , Y ) ≤ C K . (30)
Since V + K is a viscosity solution of ( 24), we conclude that (with P = (p, 1) ∈ R N +1 , λ 0 = λ + K (β + K ) and β + K ≥ β 0 /2 > 0):

λ 0 + Γ ε τ (τ , Y ) ≥ β 0 2 + H(λ 0 τ + P • Y + V + K (τ , Y ), y, p + ∇ y Γ ε (τ , Y )). ( 31
)
Using the fact that G(t, x, F (t, x, x N +1 )) = x N +1 , simple computations yield:

       λ 0 τ + P • Y + V + K (τ , Y ) = φ ε (t,X) ε λ 0 + Γ ε τ (τ , Y ) = ψ t (t, X) + λ 0 -φ t (t, X) 1 + ∂Γ ε ∂y N +1 (τ , Y ) p + ∇ y Γ ε (τ , Y ) = ∇ x ψ(t, X) + p -∇ x φ(t, X) 1 + ∂Γ ε ∂y N +1 (τ , Y ) . (32)
Therefore we get

ψ t (t, X) + λ 0 -φ t (t, X) 1 + ∂Γ ε ∂y N +1 (τ , Y ) ≥ β0 2 + H φ ε (t,X) ε , x ε , ∇ x ψ(t, X) + p -∇ x φ(t, X) 1 + ∂Γ ε ∂y N +1 (τ , Y )
Using the uniform continuity of H on R × R N × B 2C K (0), bounds [START_REF] Müller | Homogenization of nonconvex integral functionals and cellular elastic materials[END_REF], the C 1 regularity of φ and the fact that λ 0 = φ t (t 0 , X 0 ), p = ∇ x φ(t 0 , X 0 ), we deduce that there exists 0 < r K ≤ r 0 , such that (with

β 0 > 0): ψ t (t, X) ≥ β 0 4 + H φ ε (t, X) ε , x ε , ∇ x ψ(t, X) for (t, X) ∈ V r K .
This proves that φ ε is a supersolution on V r K and ends the proof of the Lemma 2.

The proof of Lemma 3 is similar and we skip it.

which satisfies ȟδ (K) > 0 for δK > 0 large enough. We then define for µ -≥ 1

Ȟδ K (v, y, P ) :=    Ȟδ (v, y, P ) if Ȟδ (v, y, P ) ≤ ȟδ (K) ȟδ (K) + µ -Ȟδ (v, y, P ) -ȟδ (K) if ȟδ (K) ≤ Ȟδ (v, y, P ) ≤ 2 ȟδ (K) (1 + µ -) ȟδ (K) if 2 ȟδ (K) ≤ Ȟδ (v, y, P ) (40)
and the compact set

Ωδ K = p N +1 ∈ R, |p N +1 | ≤ 2 ȟδ (K) -inf Ȟ δ with inf Ȟ = -2h(K). (41)
We can easily check that

Ωδ K ⊃ p N +1 ∈ R, ∃(v, y, p) ∈ R × R N × R N , H δ (v, y, p, p N +1 ) ≤ 2h δ (K) ⊃ B K (0).
Then we have in particular

Ȟδ K (•, •, P )    = Ȟδ (•, •, P ) for |p N +1 | ≤ K ≥ Ȟδ (•, •, P ) for p N +1 ∈ Ωδ K = -m δ,- K for p N +1 ∈ R\ Ωδ K (42) if we choose µ -≥ 1 such that µ -= µ δ,- K with (1 + µ δ,- K ) ȟδ (K) = -m δ,- K := 2 ȟδ (K) + 2h(K) -inf H ≥ 2 ȟδ (K). (43)
Let us define the compact set

Ω δ,- K = P = (p, p N +1 ) ∈ R N × R, |p| ≤ r K , |p N +1 | ≤ 2 ȟδ (K) + 2h(K) δ . (44)
We can easily check that Ω δ,- K ⊃ B K (0). Let us finally define the bounded and uniformly continuous Hamiltonian H δ,- K (v, y, P ) = -Ȟδ K (v, y, P )

which satisfies (using the definition (34) of H -δ )

H δ,- K (•, •, P )                = H -δ (•, •, P ) for |P | ≤ K ≤ H -δ (•, •, P ) for P ∈ Ω δ,- K = f δ K (p N +1 ) for P ∈ R N +1 \Ω δ,- K = m δ,- K for |p N +1 | ≥ 2 ȟδ (K)+2h(K) δ ∈ m δ,- K , 2h(K) for every P ∈ R N +1 (46)
where f δ K is a Lipschitz continuous function defined by

f δ K (p N +1 )      = -δ|p N +1 | + 2h(K) if |p N +1 | ≤ ȟδ (K)+2h(K) δ = -ȟδ (K) + µ -δ|p N +1 | -2h(K) -ȟδ (K) if ȟδ (K)+2h(K) δ ≤ |p N +1 | ≤ 2 ȟδ (K)+2h(K) δ = -(1 + µ -) ȟδ (K) if |p N +1 | ≥ 2 ȟδ (K)+2h(K)
δ Finally, we can easily check for later use that for K > 1 + |p| 2 large enough and for δK > 0 large enough, we have for every (v, y, q

) ∈ R × R N × R N : h(|p|) ± δ ≥ H δ,± K (v, y, q, 1) =⇒ H δ,± K (v, y, q, 1) = ±δ + H(v, y, q) , (47)
H δ,+ K (•, •, Q) ≥ H(•, •, q) for all Q = (q, q N +1 ) ∈ Ω δ,+ K , (48)
H δ,- K (•, •, Q) ≤ H(•, •, q) for all Q = (q, q N +1 ) ∈ Ω δ,- K . (49)
We now state the following fundamental result which will be used as the corner stone of our construction and will be proved in Section 6.

which still satisfy |λ ± | ≤ C, |V + ± | ≤ C, |V - ± | ≤ C
and passing to the limit in (52) we get, in the viscosity sense,

λ ± + ∂V ± + ∂τ ≤ H λ ± τ + P • Y + V ± + , y, p + ∇ y V ± + for (τ, Y ) ∈ R × R N +1 , λ ± + ∂V ± - ∂τ ≥ H λ ± τ + P • Y + V ± -, y, p + ∇ y V ± - for (τ, Y ) ∈ R × R N +1 . Let us define W ± α (τ, y) = λ ± τ + p • y + V ± α for α = ±. Let us choose k ∈ N such that k ≥ 2C.
Then the comparison principle (Proposition 1) applied to the supersolution W + -+ k and the subsolution W -

+ implies that W + -+ k ≥ W - + and then λ + ≥ λ -. Similarly, comparing W - -+ k with W + + we get λ -≥ λ + which proves that λ + = λ -=: λ. Let us consider the solution w of w τ = H(p • y + w, y, p + ∇w) for (τ, y) ∈ (0, +∞) × R N , w(0, y) = 0 for y ∈ R N (58)
and W solution of

W τ = H(P • Y + W, y, p + ∇ y W ) for (τ, Y ) ∈ (0, +∞) × R N +1 , W (0, Y ) = 0 for Y ∈ R N +1 .
By Lemma 1, we have w(τ, y) = W (τ, y, 0). The comparison principle implies

W + + -k ≤ W ≤ W + -+ k which proves that W (τ,y) τ -→ λ as τ -→ +∞ uniformly for Y ∈ R N +1
, and then w(τ,y) τ -→ λ as τ -→ +∞ uniformly for y ∈ R N . This ends the proof of Theorem 1.

Proof of Proposition 7. Simply apply Proposition 8 for K large enough, with δ = 1/ √ K, and set

V ± K = V 1 √ K ,± K
. Checking that β ± K -→ β ± 0 is very similar to the proof of Theorem 1.

Proof of ergodicity and construction of approximate correctors

In this section, we prove the existence of exact correctors for approximate Hamiltonians, namely Proposition 8.

We first introduce the 0-order non local operator:

I(v)(Y ) = R N +1 dZ J(Z) (v(Y -Z) -v(Y ))
where the function J satisfies [START_REF] Boccardo | Homogeneisation de problemes quasi-lineaires[END_REF].

Next, we consider the solution of the following equation L ∞ a priori bounds on the solution. Let us define R

∂W δ,± K ∂τ = ε I W δ,± K (τ, •) + β + H δ,± K P • Y + W δ,± K , y, P + ∇W δ,± K on (0, +∞) × R N +1 W δ,± K (0, Y ) = 0 for Y ∈ R N +1 (
δ,± K = inf R ≥ 0, Ω δ,± K ⊂ B R (0) and C δ,± K := 1 + √ N + 1 • |P | + R δ,± K . Then we have for all (τ, Y, Y ) ∈ R × R N +1 × R N +1 : |W δ,± K (τ, Y ) -W δ,± K (τ, Y )| ≤ C δ,± K . ( 60
)
Moreover there exist real numbers λ δ,± K (β, ε) such that the maps β -→ λ δ,± K (β, ε) are continuous, nondecreasing and with λ δ,± K = λ δ,± K (β, ε) and for any τ , τ ≥ 0, Y , Y ∈ R N +1 :

|W δ,± K (τ , Y ) -W δ,± K (τ, Y ) -λ δ,± K • (τ -τ ) | ≤ 7C δ,± K . (61)
A priori bounds on the derivatives of the solution. Moreover, W δ,+ K is Lipschitz continuous w.r.t. (τ, Y ) and we have the following a priori bounds:

P + ∇W δ,+ K (τ, Y ) ∈ Ω δ,+ K , and 0 ≤ 1 + ∂W δ,+ K ∂y N +1 (τ, Y ) ≤ 2h δ (K) -inf H δ , (62)
-ε C δ,+ K + β + inf H ≤ ∂W δ,+ K ∂τ (τ, Y ) ≤ ε C δ,+ K + β + M δ,+ K , (63)
-ε C δ,+ K + β + inf H ≤ λ δ,+ K (β, ε) ≤ ε C δ,+ K + β + M δ,+ K (64)
and

P + ∇W δ,- K (τ, Y ) ∈ Ω δ,- K , and 0 ≤ 1 + ∂W δ,- K ∂y N +1 (τ, Y ) ≤ 2 ȟδ (K) + 2h(K) δ , (65)
-ε C δ,- K + β + m δ,- K ≤ ∂W δ,- K ∂τ (τ, Y ) ≤ ε C δ,- K + β + 2h(K), (66)
-ε C δ,- K + β + m δ,- K ≤ λ δ,- K (β, ε) ≤ ε C δ,- K + β + 2h(K) ( 67
)
where m δ,- K and M δ,+ K are respectively defined by ( 43) and (39). Further properties of the solution. The correctors satisfy

W δ,± K (τ, y, y N +1 + 1) = W δ,± K (τ, y, y N +1 ), |W δ,± K (τ, y + k, y N +1 ) -W δ,± K (τ, y, y N +1 )| ≤ 1, for every k ∈ Z N . (68)
If p = P/Q with P ∈ Z N and Q ∈ N\ {0}, then

W δ,± K (τ, y + Qk, y N +1 ) = W δ,± K (τ, y, y N +1 ) for every k ∈ Z N . ( 69
)
Proof of Proposition 9. We perform the proof in several steps.

Step 1: Existence, uniqueness and a priori bounds on the gradient. By Proposition 4, we know that there exists a unique and continuous solution W δ,± K to (59). We now remark that U δ,±

K = P • Y + W δ,± K satisfies ∂U δ,± K ∂τ = ε I U δ,± K (τ, •) + β + H δ,± K (U, y, ∇U ) for (τ, Y ) ∈ (0, +∞) × R N +1 (70)
We know that

∇U δ,± K (0, Y ) = P ∈ Ω δ,± K for every Y ∈ R N +1
with Ω δ,± K starshaped with respect to the origin, and

H δ,+ K (•, •, Q) = constant if Q ∈ R N +1 \Ω δ,+ K . (71)
By Corollary 1 and the fact that the function r is nondecreasing, we deduce that

∇U δ,+ K (τ, Y ) ∈ Ω δ,+ K for every (τ, Y ) ∈ [0, +∞) × R N +1
Similarly we conclude that

∇U δ,- K (τ, Y ) ∈ Ω δ,- K for every (τ, Y ) ∈ [0, +∞) × R N +1
where (71) is replaced by

H δ,- K (•, •, Q) = f δ K (q N +1 ) if Q = (q, q N +1 ) ∈ R N +1 \Ω δ,- K constant if |q N +1 | ≥ 2 ȟδ (K)+2h(K)
δ and we use the fact that

Ω δ,- K = B r K (0) × q N +1 ∈ R, |q N +1 | ≤ 2 ȟδ (K) + 2h(K) δ
Finally, we remark that

0 ≤ ∂U δ,± K ∂y N +1 (0, Y ) = 1 for every Y ∈ R N +1
and then by Proposition 6, we deduce that

0 ≤ ∂U δ,± K ∂y N +1 (τ, Y ) for every (τ, Y ) ∈ [0, +∞) × R N +1
All these results on U δ,± K prove (62) and (65) for W δ,± K . It is now easy to get (63) and (66).

Step 2: Properties of the solution by integer translations. We first remark that U δ,± K (0, y, y N +1 + 1) -1 = U δ,± K (0, y, y N +1 ).

From the invariance of (70) by translation in y N +1 , and by integer addition to the solution, we deduce that for all τ ≥ 0 : U δ,± K (τ, y, y N +1 + 1) -1 = U δ,± K (τ, y, y N +1 ). This proves the first line of (68). Moreover, for a given k ∈ Z N , we set p • k = l + α, with l ∈ Z and α ∈ [0, 1). Then we have

l ≤ U δ,± K (0, y + k, y N +1 ) -U δ,± K (0, y, y N +1 ) ≤ l + 1
From the comparison principle, and the various invariances by integer translations of the equation, we deduce that for all τ ≥ 0 :

l ≤ U δ,± K (τ, y + k, y N +1 ) -U δ,± K (τ, y, y N +1 ) ≤ l + 1.
This proves the second line of (68).

Finally if p = P/Q with P ∈ Z N and Q ∈ N\ {0}, then

U δ,± K (0, y + Qk, y N +1 ) -P • k = U δ,± K (0, y, y N +1 )
and then U δ,± K (τ, y + Qk, y N +1 ) -P • k = U δ,± K (ψ, y, y N +1 ) This proves (69).

Step 3: Control of the oscillations in space and consequences. Let us consider Y, Y ∈ R N +1 , and let us write

Y -Y = L + γ with L ∈ Z N +1 , γ ∈ [0, 1) N +1 . Then |W δ,± K (τ, Y ) -W δ,± K (τ, Y )| ≤ |W δ,± K (τ, Y + L + γ) -W δ,± K (τ, Y + γ)| + |W δ,± K (τ, Y + γ) -W δ,± K (τ, Y )| ≤ 1 + √ N + 1 • |P | + R δ,± K =: C δ,± K
where in the last line, we used (68) for estimate by integer translations, and the estimates on the gradient (62)-( 65) to bound the variation of W δ,± K on the cube [0, 1) N +1 . As a consequence of estimates (60), estimate (63)-(66) are true in the viscosity sense (and then in the sense of Definition 1).

Step 4: Control of the oscillations in time. We first set w = W δ,± K to simplify the notation. In order to control the oscillations in time of w, we define two continuous functions by: 63),(66), we deduce that

λ + (T ) = sup τ ≥T w(τ + T, 0) -w(τ -T, 0) 2T and λ -(T ) = inf τ ≥T w(τ + T, 0) -w(τ -T, 0) 2T which satisfy λ -(T ) ≤ λ + (T ). From (
-ε C δ,+ K + β + inf H ≤ λ -(T ) ≤ λ + (T ) ≤ ε C δ,+ K + β + M δ,+ K (72)
if w = W δ,+ K , and

-ε C δ,- K + β + m δ,- K ≤ λ -(T ) ≤ λ + (T ) ≤ ε C δ,- K + β + 2h(K) (73)
if w = W δ,- K . By definition of λ ± (T ), for any δ > 0, there exists τ ± ≥ T such that:

λ ± (T ) - w(τ ± + T, 0) -w(τ ± -T, 0) 2T ≤ δ.
From (60), we see that w satisfies for τ ≥ T :

|w(τ -T, Y ) -w(τ -T, 0)| ≤ C 0 := C δ,± K ≥ 1. (74)
Let us define k ∈ Z such that 2C 0 < w(τ --T, 0) + k -w(τ + -T, 0) ≤ 3C 0 . Then from (74), we deduce that for every

Y ∈ R N 0 < w(τ --T, Y ) + k -w(τ + -T, Y ) ≤ 5C 0
From the comparison principle, we deduce that for every

Y ∈ R N 0 ≤ w(τ -+ T, Y ) + k -w(τ + + T, Y ) ≤ 5C 0 .
Therefore we deduce

-5C 0 ≤ (w(τ -+ T, Y ) -w(τ --T, Y )) -(w(τ + + T, Y ) -w(τ + -T, Y )) ≤ 5C 0 and then |λ + (T ) -λ -(T )| ≤ 2δ + 5C 0 2T and because δ > 0 is arbitrarily small we deduce that |λ + (T ) -λ -(T )| ≤ 5C 0 2T . (75)
Now let us consider T 1 > 0 and 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 ≥P T1 P i=1 w(τ + 2iT 1 -P T 1 , 0) -w(τ + 2(i -1)T 1 -P T 1 , 0) 2P T 1 ≤ P i=1 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 ) - 5C 0 2T 2 .
By symmetry we deduce that

|λ + (T 2 ) -λ + (T 1 )| ≤ max 5C 0 2T 2 , 5C 0 2T 1 (76)
and similarly

|λ -(T 2 ) -λ -(T 1 )| ≤ max 5C 0 2T 2 , 5C 0 2T 1 . (77)
Since λ ± are continuous, we can extend inequalities (76)-(77) to the case T 2 /T 1 / ∈ Q. Eventually, inequalities (76),( 77) and (75) imply the existence of the following limits lim

T →+∞ λ + (T ) = lim T →+∞ λ -(T ) = λ and we deduce that |λ ± (T ) -λ| ≤ 5C 0 2T . (78)
Combining ( 78) and (74), we conclude that for any τ, σ ≥ 0 and

Y, Z ∈ R N +1 , |w(τ, Y ) -w(σ, Z) -λ(τ -σ)| ≤ 7C 0 .
This proves (60) with w = W δ,± K , C 0 = C δ,± K , and λ := λ δ,± K . Moreover, by (72)-(73), we deduce (64)-(67). This ends the proof of Proposition 9.

Proof of Proposition 8. We perform the proof in several steps.

Step 1: Construction of a global solution. We first consider the sequence of functions for n ∈ N

w n (τ, Y ) = W δ,± K (τ + n, Y ) -k n with k n ∈ Z such that W δ,± K (n, 0) -k n ∈ [0, 1)
where W δ,± K is given by Proposition 9 for ε > 0. The first derivatives of w n with respect to space and time are then bounded independently on n, and we can extract a subsequence which converges to a function w ∞ which is defined on the whole space and for all time, and satifies

∂w ∞ ∂τ = ε I (w ∞ (τ, •)) + β + H δ,± K (P • Y + w ∞ , y, P + ∇w ∞ ) for (τ, Y ) ∈ R × R N +1 .
Moreover w ∞ is globally Lipschitz continuous in space and time, with a priori estimates given in Proposition 9.

Step 2: Construction of a "periodic in time" solution. We consider the vector e 0 = (λ, 0, ..., 0, 1) ∈ R×R N ×R and an arbitrary vector ν 0 = (ν 0 τ , 0, ..., 0, ν 0 N +1 ) ∈ R×R N ×R such that e 0 •ν 0 = λν 0 τ +ν 0 N +1 > 0, and define

u ∞ (τ, Y ) = w ∞ (τ, Y ) + P • Y is (σ 0 e 0 )-periodic.
In particular, when λ = 0, we can choose ν 0 = (sign(λ), 0, ..., 0, 0) ∈ R × R N +1 . This shows that v ∞,∞ (τ, Y ) is 1 |λ| -periodic in τ and 1-periodic in y N +1 (which was already known for W δ,± K in Proposition 9). In the general case, we simply consider ν 0 = (ν 0 τ , 0, ..., 0, ν 0 N +1 ) ∈ R × R N × R with ν 0 τ = 0 and e 0 • ν 0 > 0 which implies that v ∞,∞ (τ, Y ) is (σ 0 ν 0 )-periodic and 1-periodic in y N +1 .

Step 3: Improving the periodicity by the sliding method. First, for a reference on the sliding method, see in particular Berestycki, Nirenberg [START_REF] Berestycki | On the method of moving planes and the sliding method[END_REF]. For any vector ν = (ν τ , 0, ..., 0,

ν N +1 ) ∈ R × R N × R such that e 0 • ν > 0, we define ũσ (τ, Y ) = u ∞,∞ ((τ, Y ) -σν).
We set σ * = inf σ ≥ 0, ũσ ≤ ũ for σ ≥ σ .

The estimate (81) is still true for u ∞,∞ with ν 0 replaced by ν, and then we deduce that σ * < +∞. As in step 2, we get the existence of a sequence P n such that ũσ * (P n ) -u ∞,∞ (P n ) -→ 0, and can define a new sequence of functions (which are translations of u ∞,∞ ) such that the limit u ∞,∞,∞ has the same properties as u ∞,∞ and satisfies moreover

u ∞,∞,∞ (• -σ * ν) = u ∞,∞,∞ and u ∞,∞,∞ (• -σν) ≤ u ∞,∞,∞ for every σ ≥ σ * .
From estimate (82) applied to u ∞,∞,∞ , ν 0 replaced by ν, and σ 0 replaced by σ * , for integers k -→ +∞, we deduce that σ * = 0, and then u ∞,∞,∞ is nondecreasing in the direction ν.

We can now do this construction for a sequence of vectors

ν n = (ν n τ , 0, ..., 0, ν n N +1 ) ∈ R × R N × R, with e 0 • ν n > 0 and ν n -→ ±(e 0 ) ⊥ with (e 0 ) ⊥ = (-1, 0, ..., 0, λ) ∈ R × R N × R.
Up to extract at each step a subsequence of the translated functions, we get at the limit of this process, a function that we simply denote by u, such that u is nondecreasing in each direction ±(e 0 ) ⊥ . This means that u is independent on the direction (e 0 ) ⊥ , i.e. only depends on the coordinates (τ, Y ) -((τ,Y )•(e 0 ) ⊥ ) (e 0 ) ⊥ |(e 0 ) ⊥ | 2 , i.e. u(τ, y, y N +1 ) = u(0, y, y N +1 + λτ ).

(83)

In particular the function u(τ, Y ) -P • Y satisfies on R × R N +1 all the properties given in Proposition 9.

Finally we define the corrector as V δ,± K (τ, Y ) := u(τ, Y ) -λτ -P • Y.

Step 4: The limit ε -→ 0. We now take the limit ε -→ 0 and up to extraction of a subseqquence, we get a limit corrector still denoted by V δ,± K (τ, Y ) and a limit λ δ,± K =: λ δ,± K (β), such that w ± (τ, Y ) = V δ,± K (τ, Y ) + λ δ,± K τ satisfies on R × R N +1 all the properties given in Proposition 9 for ε = 0. Moreover, from (83), we deduce the following invariance of the corrector V δ,± K (τ, y, y N +1 ) = V δ,± K (0, y, y N +1 + λ δ,± K τ ) (84) and we have

λ δ,± K + ∂V δ,± K ∂τ = β + H δ,± K λ δ,± K τ + P • Y + V δ,± K , y, P + ∇V δ,± K for (τ, Y ) ∈ R × R N +1 . ( 85
)
Step 5: New estimates on λ δ,± K and consequences. Let us consider the supremum of V δ,± K . We have two cases: either the supremum is reached at a point P 0 or "it is reached at infinity". Case 1: sup R×R N +1 V δ,± K = V δ,± K (P 0 ). Let us set P 0 = (τ 0 , Y 0 ) with Y 0 = (y 0 , y 0 N +1 ). Looking at this point, we deduce that λ δ,± K ≤ β + H δ,± K (λ δ,± K τ 0 + P • Y 0 + V δ,± K (P 0 ), y 0 , P ).

which proves (53).

Step 7: Monotonicity and continuity of λ δ,± K (β). The proof of the monotonicity and the continuity of λ δ,± K (β) is similar to the proof of Theorem 1. This ends the proof of Proposition 8.

A Appendix: ergodicity (again) and construction of super and subcorrectors for the original Hamiltonian for some constant C independent of δ small enough and K large enough. Let us call λ = λ ± the limit of λ δ,± K (see the proof of Theorem 1 to check the equality λ = λ + = λ -) for a subsequence of (δ, K) -→ (0, +∞). If λ = 0, we know that, in this limit, λ δ,± K = 0, and we can define v δ,+ K,+ (τ, y) = inf where p = Ke µt ∇ζ(x -y). Notice that we can still write the viscosity inequalities if t = T or s = T . Substracting both inequalities yields:

Ke µt ζ(x -y) ≤ F (u(t, x), x, p + ∇β R (x)) -F (v(s, y), y, p)

+ dzJ(z)(u(t, x -z) -v(s, y -z) -(u(t, x) -v(s, y))).

We deduce from M K,ν ≥ 0 that u(t, x) ≥ v(s, y) and by construction, u(t, x -z) -v(s, y -z) -(u(t, x) -v(s, y)) ≤ β R (x -z) -β R (x).

Using these estimates and (A1) permits to obtain: Let us exhibit a contradiction if 0 < M = sup R N {u -v} ∈ (-∞; +∞]. In such a case, for α, ε, ν small enough, we have M α,ε,ν ≥ M > 0. The supremum is attained at (t, s, x, y) and the initial condition ensures that t > 0, s > 0. From M α,ε,ν ≥ 0 and (91), we deduce that:

K √ 2
α 2 |x| 2 ≤ K(1 + |x -y|) - |x -y| 2 2ε ≤ K + sup r≥0 Kr - 1 2ε r 2 = K + C ε .
We conclude that for a fixed ε > 0, αx → 0 as α → 0. From writing both viscosity inequalities and substracting them, it comes:

η ≤ γ|αx| + γ |x -y| + |x -y| 2 ε + α 2 I 2 .
Letting successively ν, α and ε go to 0, usual penalization results permit to obtain the contradiction: η ≤ 0 and we are done.

Aknowledgement. The authors thank O. Alvarez, P. Cardaliaguet, M. Kobylanski, C. Marchi, F.

Murat and E. Rouy for fruitful discussions.

Proposition 7 .

 7 (Lipschitz continuous in y N +1 approximate sub and super-correctors in dimension N + 1) Let p ∈ R N . For any β ∈ R, let λ(β) be the constant defined by Theorem 1 for the Hamiltonian β + H. Then λ(β) is nondecreasing in β, and ∀λ 0

  59) and prove the following result: Proposition 9. (A priori estimate for the problem with initial conditions and ε ≥ 0) Let p ∈ R N , P = (p, 1) ∈ R N × R. Let us consider the troncated Hamiltonians H δ,+ K and H δ,- K defined by (35) and (45) for K > 1 + |p| 2 large enough and for δK > 0 large enough, and H satisfying (A1)-(A2)-(A3). For any given β ∈ R, let us consider the solution W δ,± K to (59).

Theorem 3 ..

 3 (Existence of sub and super-correctors) Under Assumptions (A1)-(A2)-(A3), consider p ∈ R N and let λ = H 0 (p). Then there exists a bounded supersolution v + (resp. subsolution v -) of (4).Before to make the proof of Theorem 3, let us show a simple corrollary:A second proof of Theorem 1. Let us consider the solution w ofw τ = H(p • y + w, y, p + ∇w) for (τ, y) ∈ (0, +∞) × R N , w(0, y) = 0 for y ∈ R N . (90)With v ± given by Theorem 3 and an integer k ≥ C ≥ |v ± |, the comparison principle impliesv -+ λτ -k ≤ w ≤ v + + λτ + kwhich proves that w(τ,y) τ -→ λ as τ -→ +∞ uniformly for y ∈ R N . This ends the proof of Theorem 1Proof of Theorem 3. Case 1 : λ = 0 Let us apply Proposition 8 with β = 0. We have |λ δ,± K | ≤ C, |V δ,± K | ≤ C

))

  V δ,+ K (σ, y, y N +1 ), v δ,- K,-(τ, y) = sup ( y N +1 ∈R, τ =σ+ V δ,- K (σ, y, y N +1 )which are respectively super and subsolutions, i.e. satisfyλ K τ + p • y + v δ,+ K,+ , y, p + ∇v δ,+ K,+ , 1 + λ δ,τ + p • y + v δ,- K,-, y, p + ∇v δ,- K,-, 1 + λ δ,y) ∈ R × R N .In the limit (δ, K) -→ (0, +∞), we setv + = lim inf * v δ,+ K,+ , v -= lim sup * v δ,- K,-which still satisfy |v ± | ≤ C.

( 1 +

 1 |x -y|) ≤ Kζ(x -y) ≤ Ke µt ζ(x -y) ≤ γ(1 + |p|)|x -y| + γ|∇β R (x)| + I[β R ](x) ≤ 2γ|x -y| + Cγ + CI 1 ≤ C(1 + |x -y|).Choosing K large enough yields the desired contradiction.The second step consists in adapting the classical proof of the comparison principle, i.e. by considering the following penalized supremum:M α,ε,ν = sup t,s∈(0,T ],x,y∈R N u(t, x) -v(s, y) -(t -s)

Approximate cell problems

In this section, we explain that approximate correctors of Proposition 7 (used in the proof of convergence) are in fact exact correctors for approximate and bounded Hamiltonians. The construction of these approximate correctors also permits to prove ergodicity (Theorem 1).

First, we define two nondecreasing functions by using (A3):

Because the Hamiltonian H is continuous, we deduce that the function h is continuous and the function r is nondecreasing, upper semi-continuous and (therefore) continuous from the right. Moreover these functions satisfy in particular h(r(h)) ≥ h and r(h(r)) ≥ r.

Definition of H δ,+ K . For every δ ∈ R and P = (p, p N +1 ), we define:

For δ > 0 fixed for the sequel, we set h δ (K) = h(K) + δK. Using (A3), for every K > 0 large enough we have h δ (K) ≥ h(K) > 0 and we define for some µ + ≥ 1 to be chosen later:

Let us define

with inf H := inf (v,y,q)∈R×R N ×R N H(v, y, q). We can easily check that

Then the bounded and uniformly continuous Hamiltonian H δ,+ K satisfies in particular

Definition of H δ,- K . We first define Ȟ(v, y, p) := -min (2h(K), H(v, y, p)) .

Then we proceed similarly as in the definition of H δ,- K , with H replaced by Ȟ. We define for δ > 0 Ȟδ (v, y, P ) = δ |p N +1 | + Ȟ(v, y, p) ȟδ (K) = -inf H + δK Proposition 8. (Lipschitz continuous correctors for an approximate Hamiltonian in higher dimension) Let p ∈ R N and P = (p, 1) ∈ R N × R. Let us consider the troncated Hamiltonians H δ,+ K and H δ,- K defined by [START_REF] Panagiotis | Stochastic homogenization of Hamilton-Jacobi equations and some applications[END_REF] and (45) for K > 1 + |p| 2 large enough and for δK > 0 large enough, and H satisfying (A1)-(A2)-(A3). For any β ∈ R, there exist real numbers λ δ,+ K (β), λ δ,- K (β) and bounded approximate sub and supercorrectors V δ,+ K , V δ,- K depending on β, satisfying λ δ,± K (β) is nondecreasing in β, and ∀λ 0 ∈ R, ∃β δ,± K,0 ∈ R, such that λ δ,± K (β δ,± K,0 ) = λ 0 (50) and inf

(where the function h is defined by ( 33)) and

W 1,∞ a priori bounds on the correctors. We can construct bounded Lipschitz continuous correctors with:

where r and h are defined by [START_REF] Livio | Homogeneization problems for ordinary differential equations[END_REF] and

where m δ,- K and M δ,+ K are defined respectively by (43) and by (39). Further properties of the correctors. The correctors satisfy

We next deduce from this proposition Theorem 1 about the ergodicity of the problem and Proposition 7 about the existence of approximate correctors (the proposition we used in the proof of convergence).

Proof of Theorem 1. Let us apply Proposition 8 with β = 0. We have

for some constant C independent of δ small enough and

K = 0 and is independent on τ if λ δ,± K = 0 (by (56)) and satisfies (52). Let us call λ ± any limit of λ δ,± K for a subsequence of (δ, K) -→ (0, +∞) and

By construction we have u 0 = u ∞ + 1. We define

By the uniform Lipschitz continuity of u ∞ in space and time, we deduce that σ 0 > 0. Moreover, w ∞ satisfies (61), i.e.

and then

Then using the fact that P = (p, 1) ∈ R N × R, we get:

Therefore, for σ large enough, we deduce that u σ < u ∞ . We conclude that σ 0 < +∞.

From the definition of σ 0 , we deduce that there exists a sequence

Up to extraction of a subsequence, we can assume that u ∞,n converges to a function u ∞,∞ and Z n converges to Z ∞ ∈ [0, 1] N +1 such that:

where we used at the last line the invariance of (79) by translations in τ and y N +1 , and its invariance by addition of integers to the solution. From the strong maximum principle (only applied for ε > 0), we deduce that

and then we deduce (taking k -→ +∞) that σ 0 (e 0 • ν 0 ) = 1. As a consequence of our proof, we get that

Using the fact that

where h is defined by [START_REF] Livio | Homogeneization problems for ordinary differential equations[END_REF].

Case 2: There exists

In this case, up to consider the limit of translated functions, we come back to case 1 for this limit function which is still a solution of the PDE (85). This proves (86) in case 2.

Similarly, looking at the infimum of V δ,± K , we get:

Then ( 86) and ( 87) implies (51). By substraction of ( 51) to ( 63) and (66), we get respectively ( 54) and (55).

Step 6: New L ∞ bound on the corrector. Let us consider

The function v only depends on y (and not on τ ) because of (84). By contruction v is a supersolution of (85). Therefore we have

We deduce that (88) is true almost everywhere. From the fact that V δ,± K is 1-periodic in y N +1 and satisfies

where we used successively (84), (89) twice, (57) and (88). Finally, up to substract an integer to v(0, 0), we can assume that |v(0, 0)| ≤ 1 and then we get with τ = 0, Y = 0

Therefore, we get a bounded supersolution v + and a bounded subsolution v -. Case 2 : λ = 0 We proceed as in the second proof of Theorem 1, choosing some β > 0 such that λ(β) = lim λ δ,± K (β) > 0. For λ(β) > 0 arbitrarily small, this shows (by the comparison principle and the limit λ(β) → 0) that the solution w to the initial value problem is bounded from above. Similarly, for λ(β) < 0 arbitrarily close to zero, we get that the solution w is bounded from below. Finally w(τ, y) is bounded for all τ > 0 and y ∈ R N . We then define w n (τ, y) = w(n + τ, y) and as n → +∞, we define v + = lim inf * w n , v -= lim sup * w n which are respectively super and subsolutions on the wole space and time. This ends the proof of Theorem 3.

B Appendix: proof of Proposition 1

The following proof is classical but we provide it for the reader's convenience.

When ε = 0, the result is classical and when ε > 0, we do not restrict ourselves by assuming that ε = 1.

We proceed in two steps. First, we prove that there exists a constant K > 0 such that:

To obtain such a result, it suffices to obtain the following inequality:

where ζ(z) = 1 + |z| 2 . Since u and v are at most of linear growth, there exists L > 0 such that:

Let us consider the family of functions β R ∈ C 2 (R N ) parametrized by R ≥ 1 and introduced in [START_REF] Michael | User's guide to viscosity solutions of second order partial differential equations[END_REF]; we assume that they satisfy for some C > 0, which does not depend on R:

For any K > 0, consider the following penalized supremum:

x,y∈R N {u(t, x) -v(t, y) -Ke µt ζ(x -y) -β R (x)} with µ > 0 to be chosen later. It suffices to prove that M K ≤ 0 for some K large enough, not depending on R; indeed, by letting R → +∞ pointwise, we can conclude. In order to prove the existence of such a K, we argue by contradiction (as usual) and we suppose that M K > 0 for any K > 0. Hence, we dedouble the time variable: for any ν > 0, consider: