Homogenization of First Order Equations with u/ -Periodic Hamiltonians Part II: Application to Dislocations Dynamics

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1 Communications in Partial Differential Equations, 33: , 2008 Copyright Taylor & Francis Group, LLC ISSN print/ online DOI: / Homogenization of First Order Equations with u/-periodic Hamiltonians Part II: Application to Dislocations Dynamics CYRIL IMBERT 1, RÉGIS MONNEAU 2, AND ELISABETH ROUY 3 1 Polytech Montpellier & Institut de Mathématiques et de Modélisation, UMR CNRS 5149, Université Montpellier II, Montpellier, France 2 CERMICS, ENPC, Marne la Vallée, France 3 Ecole Centrale de Lyon, Ecully, France This paper is concerned with a result of homogenization of a non-local first order Hamilton Jacobi equation describing the dislocations dynamics. Our model for the interaction between dislocations involves both an integro-differential operator and a (local) Hamiltonian depending periodicly on u/. The first two authors studied in a previous work homogenization problems involving such local Hamiltonians. Two main ideas of this previous work are used: on the one hand, we prove an ergodicity property of this equation by constructing approximate correctors which are necessarily non periodic in space in general; on the other hand, the proof of the convergence of the solution uses here a twisted perturbed test function for a higher dimensional problem. The limit equation is a nonlinear diffusion equation involving a first order Lévy operator; the nonlinearity keeps memory of the short range interaction, while the Lévy operator keeps memory of long ones. The homogenized equation is a kind of effective plastic law for densities of dislocations moving in a single slip plane. eywords Dislocations dynamics; Hamilton Jacobi equations; Integrodifferential operators; Non-periodic approximate correctors; Periodic homogenization. Mathematics Subject Classification 35B10; 35B27; 35F20; 4505; 47G20; 49L Introduction Setting of the Problem. In this paper, we study a non-local Hamilton Jacobi equation describing the dislocations dynamics. We would like to say what happens Received July 1, 2006; Accepted February 1, 2007 Address correspondence to Cyril Imbert, Polytech Montpellier & Institut de Mathématiques et de Modélisation, Université Montpellier II, CC 051, Place E. Bataillon, UMR CNRS 5149, Montpellier cedex 5, France; imberts@univ-montp2.fr 479

2 480 Imbert et al. when 0 to the solution of: ( ( ) [ ]) ( ) x u t u = c + M t u u +h u in + N (1) u 0x= u 0 x on N where M is a 0 order non-local operator M defined by M Ux = Ux + dzjzux + z N where J C N is an even nonnegative function that satisfies N dzjz = 1 and: there exists R 0 > 0 and a function g>0 s.t. for z R 0 Jz = 1 ( ) z z g (2) N +1 z Throughout the paper, we assume: cx is Lipschitz continuous and 1-periodic; hu p is bounded Lipschitz continuous w.r.t. u p, is 1-periodic w.r.t. u and hu p/p is bounded; u 0 W 2 N. The aim is to prove a homogenization result i.e., to prove that the limit u 0 of u as 0 exists and is the (unique) solution of a homogenized equation of the form: { t u 0 = H 0 u 0 in + N u 0 0x= u 0 x on N with u 0 W 2 N. Our aim is two-folded: to determine the so-called effective Hamiltonian H and to prove the convergence of u towards u 0. Main Results. As usual in periodic homogenization, the limit (or effective) equation is determined by a cell problem, and more precisely by the long time behaviour of the solution of an evolution equation related to this cell problem. And as usual, this problem is determined by a formal computation, by an ansatz. In Imbert and Monneau (2008), we described the successive tries of ansatz we did to find the proper cell equation for our local homogenization problem and analogous calculi can be done here. It turns out that the proper evolution equation is, for any p N and L : w = HL y p y + w p + w w in + N where HL y p y + w p + w w = cy + L + Mw p + w (3) + hp y + w p + w w0y= 0 on N where M is the non-local operator M introduced above with = 1.

3 Homogenization of First Order Equations 481 Theorem 1 (Ergodicity). For any L and p N, there exists a unique such that the continuous viscosity solution of (3) satisfies: wy converges towards as +, locally uniformly in y. The real number is denoted by H 0 L p. A superscript 0 appears in the effective Hamiltonian. The reason is that we will have to study the ergodicity of a family of Hamiltonians in order to prove the convergence (H 0 L p = HL p 0 with the notations of Section 4 see below). We now can give the precise form of the effective equation: { t u 0 = H 0 I 1 u 0 t u 0 in + N (4) u 0 0x= u 0 x on N where I 1 is a Lévy operator of order 1 defined for any function U Cb 2N by: ( ) 1 z I 1 Ux = Ux + z Ux x Ux z z r z g dz N +1 z ( ) 1 z + Ux + z Ux z r z g dz N +1 z for any r>0. Because J is even and satisfies (2), M can be seen like a regularized version of I 1 : the singularity of the unbounded measure dz = g ( ) z z 1 dz is z N +1 removed. We will see in the proofs that this Lévy operator I 1 only keeps the memory of the long range behaviour of the operator M. On the contrary the effective Hamiltonian H will keep the memory of the short range behaviour of M. Qualitative properties of the effective Hamiltonian will be given in Forcadel et al. (submitted) and numerical simulation will be presented in Ghorbel et al. (to appear). The second main result of this paper is the following convergence result. Theorem 2 (Convergence). The bounded continuous viscosity solution u of (1) converges locally uniformly in t x towards the bounded viscosity solution u 0 of (4). In order to prove the convergence of u towards u 0, we try to construct a so-called corrector, that is a bounded solution of the cell problem which, in our case, has the following form: + v = cy + L + Mv p + v+h + p y + v p + v in + N Let us recall that, formally, v y = w y where w satisfies (3). In the proof of convergence, it turns out that we need to consider regular correctors but we are not able to construct them. However, we can construct regular suband supercorrectors (i.e., Lipschitz continuous sub and supersolution of the cell problem) and it is enough to conclude. Moreover, Theorem 1 is a by-product of this construction. Let us also point out that, at the contrary of the classical case, i.e., Hamiltonians of the form Hy p, the (sub and super) correctors here are not periodic w.r.t. y in general. Moreover (true) correctors are necessarily timedependent. Let us also mention that we construct sub- and supercorrectors as approximate correctors of approximate cell problems.

4 482 Imbert et al. A specific technical difficulty of our problem is to deal with the case = p = 0. In order to overcome it, we consider cell problems in a higher dimensional space. Another technical point leads us to solve a family of cell problems, indexed by a real number. Comments. This non-local Equation (1) is related to the local equation: ( u t u = F x ) u in 0 + N u 0x= u 0 x on N that the first two authors studied in Imbert and Monneau (2008) under the assumption that F is coercive in p = u. As pointed out in Barles (2006), this assumption appears in all the papers dealing with homogenization of first-order Hamilton Jacobi equations. First of all, Lions et al. (1986) completely solve the problem for F that are independent of u and coercive in p. The literature about homogenization of Hamilton Jacobi equations is highly developed and it is difficult to give an exhaustive list of references. As far as the first order equations are concerned, the interested reader is also referred to Evans (1989, 1992), Fathi (1998), Roquejoffre (1998), Barles and Souganidis (2000a,b), and references therein. Inspired in particular by some ideas from Imbert and Monneau (2008), Barles (2006) managed to get nice homogenization results for noncoercive Hamiltonians and gave, among other things, simpler proofs of the main results of Imbert and Monneau (2008) in a restricted case where it is possible to imbed the problem in dimension N + 1 for a geometric equation. In the present paper, the coercivity is somehow replaced with the nonlocal term. Indeed, coercivity together with a perturbation by a non-local operator (in order to get the strong maximum principle and apply the sliding method) are used in Imbert and Monneau (2008) in order to get estimates on space oscillations of the correctors. This is obtained here thanks to M [ ] u u (see pp. 509, 510 and Eq. (55)). Possible Extensions. In order to avoid unnecessary technicality, we chose to focus on a particular equation related to dislocation dynamics. With some trivial changes, it is for instance possible to treat the case of Frank Read source considering the homogenization of the following equation t u = ( ( x c ) [ u + M t ] ) ( ( u x u f + h ) u ) where curl f 0 represents a source term (physically in dimension N = 2). Using again the coercivity of the operator M, it should be possible to deal with the homogenization of t u = ( ( x c ) [ u + M t ] ) u u +h( u where F MCM is a mean curvature differential operator. ) + F MCM u D 2 u Organization of the Article. The paper is organized as follows. In Section 2.1, we give more details about the mathematical models yielding to the study of (1). In

5 Homogenization of First Order Equations 483 Section 3, we prove various comparison principles and existence results. Section 4 is dedicated to the proof of the convergence (Theorem 2) by assuming the existence of Lipschitz continuous sub- and supercorrectors (Proposition 5). Section 5 is the core of the paper; we construct approximate cell problems in the spirit of Imbert and Monneau (2008) and we construct the Lipschitz continuous supercorrectors as exact correctors of the approximate cell problems (Proposition 6). In Section 6, we prove the ergodicity of the problem (Theorem 1) and give some properties of the homogenized Hamiltonian such as its continuity (Proposition 4). Eventually, Section 2.2 is devoted to mechanical interpretation of the homogenization results obtained in this paper. Notation. The ball of radius r centered at x is classically denoted by B r x. When x is the origin, B r 0 is simply denoted B r. The cylinder t t + B r x is denoted Q r t x. The indicator function of a subset A B is denoted 1 A : it equals 1onA and0onb\a. x and x denote respectively the floor and ceil integer parts of a real number x. It is convenient to introduce the unbounded measure on N defined on N \0 by: dz = 1 ( ) z z g dz N +1 z and such that 0 = 0. For the reader s convenience, we recall here the three integro-differential operators appearing in this work: M Ux = Ux + z UxJzdz N MUx = Ux + z UxJzdz N ( ) 1 z I 1 Ux = Ux + z Ux x Ux z z r z g dz N +1 z ( ) 1 z + Ux + z Ux z r z g dz N +1 z To finish with, G G > 0 are positive numbers such that hu p G, hu p G p. 2. Physical Application to Dislocations Dynamics 2.1. Motivation Dislocations are line defects in crystals. Their typical length is of the order of 10 6 m and their thickness of the order of 10 9 m. When the material is submitted to shear stress, these lines can move in the crystallographic planes and their complicated dynamics is one of the main explanation of the plastic behaviour of metals. In the last decade the physics of dislocations have enjoyed a new boom, essentially based on new possibilities of simulations on ever more powerful computers. In the same time new tentatives have been done to build physical

6 484 Imbert et al. theories describing the collective behaviour of disocations. In particular models for the dynamics of dislocations densities have been proposed (see for instance Groma and Balogh, 1999; Groma et al., 2003; ratochvil, 1996; röner, 1958; Sethna, 2004 etc.). In the present paper we are interested in describing the effective dynamics for a collection of dislocations lines with the same Burgers s vector and all contained in a single slip plane x 3 = 0 of coordinates x = x 1 x 2, and moving in a periodic medium (see Fig. 1). At the end of this derivation, we will see that the dynamics of dislocations is described by Equation (1) in dimension N = 2. Several obstacles to the motion of dislocations lines can exist in real life: precipitates, inclusions, other pinned dislocations or other moving dislocations, etc. We will describe all these obstacles by a given and periodic field c 1 x (5) that we assume for simplicity not to depend on time. Another natural force exists: this is the Peach oehler force acting on a dislocation j. This force is the sum of the interactions with the other dislocations k for k j, and of the self-force created by the dislocation j itself (see Fig. 2 for a schematic representation of the interactions; to be closer to reality we should draw a spring between each pair of dislocations). The level set approach for describing dislocation dynamics consists in considering a function v such that the dislocation k is basically described by the level set v = k. Let us first assume that v is smooth. As explained in Alvarez et al. (2006), the Peach oehler force at the point x created by a dislocation j is well-described by the expression c 0 1 v>j where 1 v>j is the characteristic function of the set v>jwhich is equal to 1 or 0. In a general setting, the kernel c 0 can change sign. In the special case where Figure 1. Dislocations in a slip plane.

7 Homogenization of First Order Equations 485 Figure 2. Schematic representation of a 1D model for dislocations related with springs. the dislocations have the same Burgers vector and move in the same slip plane, a monotone formulation (see Alvarez et al., 2004, 2006; Da Lio et al., 2006) is physically acceptable. Indeed, the kernel can be chosen as c 0 = J 0 where J is nonnegative. The negative part of the kernel is somehow concentrated at the origin as a Dirac mass. Moreover, we assume that J satisfies 2 J = 1 and the symmetry J z = Jz. The kernel J can be computed from physical quantities (like the elastic coefficients of the crystal, the Burgers vector of the dislocation line, the slip plane of the dislocation, the Peierls Nabarro parameter, etc.). Here 0 is a Dirac mass where we set formally 1 if vx > j v>j x = if vx = j 2 0 if vx < j We remark in particular that the Peach oehler force is discontinuous on the dislocation line in this modeling. Let us now assume that for integers N 1 N 2 0, we have N 1 1/2 <v<n 2 + 1/2. Then the Peach oehler force at the point x on the dislocation j (i.e., vx = j) created by dislocations for k = N 1 N 2 is given by the sum ( (J N ) 2 0 k= N 1 1 v>k ) x = ( ( J v vx 1 )) x 2 where is the ceil integer part. In the general case, this Peach oehler force can be rewritten on the dislocations lines as v + J Ev (6)

8 486 Imbert et al. where E is a kind of integer part (up to an additive constant 1/2) defined by Ev = k + 1/2 if v k k + 1 with k Defining the normal velocity to dislocations lines as the sum of the periodic field (5) and the Peach oehler force (6), we see that the dislocations line v = j for integer j, is formally a solution of the following level set equation: v t = c 1 v + J Evv (7) In this paper we do not study the homogenization of Equation (7) whose mathematical framework is not convenient because the integer part E is discontinuous. In Forcadel et al. (submitted), a slightly different equation is derived and a formulation à la Slepcev is used to solve it despite the discontinuity of E. First, we replace the discontinuous integer part E in (7), by an approximation which is a smooth function E which satisfies and E +1 = E < E 1/ E j = j and E j + 1/2 = j + 1/2 for j (8) and we see that u = E v satisfies with u t = c 1 u + J uu+hu u (9) hu u = u E 1 uu (10) Equations (9) (10) is still not well-posed in the framework of viscosity solutions because, like for Burgers equation, it can create discontinuous solutions in finite time. We want to avoid this kind of mathematical difficulty which is artificial in the present modeling. We keep in mind that in order to describe well the dynamics, our model has to satisfy the following two properties: the function u has to be close to a half integer k + 1/2 on a region between two dislocations lines u = k and u = k + 1, and to have a small transition layer around any dislocation line; the function h has to be close to expression (10) outside the transition layer, i.e., for small gradients of u, and the contribution of h to the right hand side of (9) has to be neglectible inside the transition layer, i.e., for large gradient of u. This is the reason why the second thing we do in order to satisfy these assumptions and to get a good mathematical model, is to change the expression of the function h to get a bounded function with bounded derivatives. The boundedness of the derivative of h with respect to u insures mathematically that the maximum principle is applicable (see Theorem 3). We set, for instance, for >0 large enough: hu p = ( 1 e p/)( u E 1 u) (11)

9 Homogenization of First Order Equations 487 Conclusion. Finally the model that we study in this paper is Equation (9), with the function h given by (11) or even for more general h which are smooth enough, bounded with bounded derivatives and 1-periodic in u. To simplify the presentation, we will also assume that hu p/p is bounded, an assumption which is essentially technical. Remark 1. We can remark that u E 1 u is a bistable nonlinearity on the interval 1/2 1/2 like in Hamel et al. (2005). Remark 2. We can also have in mind a Phase Field derivation of the model studied in the present paper. In a Phase Field approach, the dislocations are already represented by the transitions of a continuous function u. The associated energy is typically like 1 Eu = u 2 + Wu + c 1 u where 1 2 u is a fractional derivative of u, and W is a 1-periodic potential. The dynamics of u is given by i.e., u t = E uu u t = c 1 + Luu+hu u with L = u is a nonlocal operator and hu u = W uu In the model that we study in the present paper, the non-local term Lu is approximated by u + J u, and to prevent some shocks like in Burgers equation, the term hu u is approximated by a function globally Lipschitz in u u. In Forcadel et al. (submitted), a sharp interface model is studied thanks to a Slepcev formulation Mechanical Interpretation of the Homogenization Let us briefly explain the meaning of the homogenization result. The homogenized Equation (4) can be interpreted as a (generalized) plastic law, i.e., a relationship between the plastic strain velocity and the stress. In the homogenized Equation (4), u 0 is a plastic strain, t u 0 is the plastic strain velocity, u 0 is the dislocations density and I 1 u 0 is the internal stress created by the density of dislocations contained in a slip plane. From a mechanical point of view, a shear stress is created in the three dimensional space by the density of dislocations. The trace of the shear stress on

10 488 Imbert et al. Table 1 Crystal elasto-visco-plasticity Homogenized model Resolved plastic strain x 1 x 2 0 x 3 u 0 x 1 x 2 0 x 3 Nye tensor of = b 0 x 3 = b u 0 0 x 3 dislocations densities Exterior applied stress ext Microscopic resolved shear stress c 1 c Resolved exterior ext e 0 c applied stress Displacement Strain v = v 1 v 2 v 3 e= ev e 0 0 x 3 with ev = 1 v + t v 2 and e 0 = 1 e 2 3 b + b e 3 Total elastic energy E= 1 ee+ 3 2 ext e E= ( ) u0 I 1 u 0 c u 0 Macroscopic stress = e+ ext Resolved macroscopic = e 0 = I 1 u 0 + c shear stress u Plastic law = f 0 = H 0 ( I1 u 0 + c t t u 0) d Energy decay E = d f 0 E = H 0 u 0 0 dt 2 dt 2 where H 0 has been computed for the velocity c 1 c with zero mean value. The fact that H 0 u 0 0 is a consequence of the monotonicity of H 0 in its first argument and from (28) when h 0. the slip plane is precisely given by I 1 u 0 (see Alvarez et al. (2006) for similar computations for a single dislocation line). Another way to justify the fact that the operator I 1 is a kind of half Laplacian is to remark that the physical model permits to see it as the Dirichlet Neumann operator associated with the elasticity equation in a half space, that is more or less the Laplace equation. Let us be more precise now. Let e 1 e 2 e 3 denote an orthonormal basis and x 1 x 2 x 3 the corresponding coordinates. We consider dislocations lines contained in the plane x 3 = 0, with a Burgers vector b that is also contained in this plane. From a mechanical point of view (see for instance Lardner, 1974), we have a table of equivalence between our homogenized model and classical models in mechanics for elasto-visco-plasticity of crystals where is the tensor of constant elastic coefficients (Table 1). Identifying the function g. We recall that I 1 u 0 x = dz gz/z 2 z N +1 u0 x + z u 0 x z u 0 x1 B1 z where the function g is related to the behaviour of the function J at large scales: Jz = gz/z/z N +1 for z >R 0

11 Homogenization of First Order Equations 489 with N = 2 in our case. Moreover the function gz = 1 gz/z is given by (see z Alvarez et al., 2006; Da Lio et al., 2006) where gz z z = D 2 Gz z z z = z 2 z 1 G 1 2 = 1 B d 3 with B = B/ and B = t e 0 1 e 0 e 0 e 0 In the special case of isotropic elasticity, we have and for b =be 1,weget ijkl = ij kl + ik jl + il jk Gz = b2 4z z2 2 + b2 z2 1 gz = 4z 3 z z2 22 with = 1 where = 1 1/2 is the Poisson ratio Two Classical Plastic Laws. Let us now recall two classical plastic laws usually used in mechanics. The Orowan law (see Sedlacek, 2003, p. 3739) is f = with the density of mobile dislocations. The Norton law with threshold (see François et al., 1995) is (for C m > 0) f = C sign c + m for not too large (in fact, here m>1 because this kind of law already contains the Frank-read process of creation of dislocations). We can compare these laws with our homogenized law: f = H 0 u 0 where u 0 is a kind of generalized dislocations density. Conclusion. The main features of the homogenization process is that H 0 p p for large enough but for small, we recover a threshold phenomenon which is due to the microstructure. The consequence is that at large scales the stress is computed as in mechanics, but the only effect of the microstructure is to introduce some nonlinearities in the plastic law with some threshold effects. Finally in this framework the dislocations density (u 0 ) is simply a derivative of the plastic strain u 0.

12 490 Imbert et al. 3. Results About Viscosity Solutions for Non-Local Equations In this paper, we have to deal with Hamilton Jacobi equations involving integral operators. The classical notion of viscosity solution can be adapted. It is clear how to do it for (1) and as far as (4) is concerned, the reader is referred to Sayah (1991) for instance. In this section, we state (and prove if necessary) comparison principles and existence results for such equations that will be used later in the proofs. Let us first recall the definition of relaxed lower semi-continuous (lsc for short) and upper semi-continuous (usc for short) limits of a family of functions u which is locally bounded uniformly w.r.t. : lim sup u t x = lim sup u s y and lim inf u t x = lim inf 0s ty x 0s ty x u s y If the family contains only one element, we recognize the usc envelope and the lsc envelope of a locally bounded function u: u t x = lim sup us y and u t x = lim inf us y s ty x s ty x Let us recall that we define viscosity solutions of (1) similarly as in Imbert and Monneau (2008), where for instance a usc function u is said to be a viscosity subsolution if the corresponding viscosity inequality is satisfied by the test function, keeping the function u in the evaluation of the regular nonlocal term M [ ] u t Viscosity Solutions for 4 We first recall the definition of a viscosity solution for (4). The difficulty is that the measure is singular at the origin and the function must be at least C 11 with respect to x in order that I 1 ut x makes sense. We refer the reader to Sayah (1991) for instance. Let us introduce first the following definition of viscosity solution associated to a cut-off radius r: Definition 1 (r-viscosity Solution). Let us fix r>0. A usc (resp. lsc) function u + N is a r-viscosity subsolution (resp. supersolution) of (4) if for any t x t > 0, any 0t, and any test function C 2 + N such that u attains a maximum (resp. minimum) at the point t x on Q r t x, then we have t t x H 0 L r x t x 0 (resp. 0 where L r = B r t x + z t x t x zdz + N \B r ut x + z ut x dz. A continuous function is a r-viscosity solution of (4) if it is a r-viscosity suband a supersolution of the equation. It is classical that the maximum can be supposed to be global (i.e., attained on + N and not only on the cylinder) and this will be used several times. We also have the following property (see Sayah, 1991 for instance):

13 Homogenization of First Order Equations 491 Proposition 1 (Equivalence of the Definitions). Let r>0 and r > 0. A continuous (resp. usc, lsc) function u is a r-viscosity solution (resp. subsolution, supersolution) of (4) if and only if it is a r -viscosity solution (resp. subsolution, supersolution) of (4). Because of this proposition, if we do not need to emphasize the positive r we use, we simply call them viscosity solutions (resp. subsolutions, supersolutions). In order to prove a comparison result for (4), we need to be able to consider semiconcave test functions for semiconvex subsolutions. Proposition 2. Consider a bounded semiconvex subsolution u of (4) and a continuous function, semiconcave w.r.t. t x. Then at a local strict maximum point t x of u, we can ensure that I 1 ut x is finite when ut x 0, and then t t x H 0 I 1 ut x ut x Remark 3. Recall that a semiconvex function (in our case, the function u) is differentiable at a maximum point. Proof. First, we notice that u and are differentiable at t x because of their regularity. If ut x = t x = 0, there is nothing to prove. Hence, we now assume that ut x 0. Let us consider a cylinder Q r t x where u attains a strict maximum and let us approximate M = sup sy Qr txus y s y = ut x t x by M = sup sy Qr txus y s y = ut x t x where is the supconvolution of and : t x = { sup s y 1 y N s 2 x y2 1 } t s2 2 We know that is C 11 w.r.t. t x (see for instance Lasry and Lions, 1986) and t x t x as 0. Moreover, we can control independently of the constant of semiconcavity of (this is used later in the proof). Now, is a proper test function for u and we therefore have: t t x H 0 r + L r t x (12) where r = t x + z t x t x zdz B L r = ut x + z ut x dz + ut x + z ut x dz B r \B N \B r = ut x + z ut x p zdz B r \B + ut x + z ut x dz N \B r

14 492 Imbert et al. where p = ut x = t x (we used the fact that is symmetric with respect to the origin). If s y is a point such that: t x = s y 1 x 2 y 2 1 s 2 t 2, we know that: t t x = t s y and s y t x =o In particular, because is semiconcave and differentiable at t x t x t x = ut x. Using the semiconvexity of u, the definition of M and the semiconcavity of, we conclude that there exists C>0 (independent of and ) such that: Cz 2 ut x + z ut x p z t x + z t x p z Cz 2 The dominated convergence theorem asserts that r + L r has a limit as 0 and 0 successively, and this limit equals Lr 00 = I 1 ut x. Hence (12) yields the desired result Comparison Principles In this section, we state comparison principles we will need later. The first theorem is a comparison principle for (1). Let us mention that, as usual in the viscosity solution theory, the fact that the derivative with respect to u of the function hu p is bounded uniformly in the gradient p is important. We would also like to shed light on the fact that, because the integral term is not singular, classical techniques can be used to prove the following comparison principle. Theorem 3. Consider a bounded usc subsolution u 1 and a bounded lsc supersolution u 2 of (1). Ifu 1 0x u 0 x u 2 0x with u 0 W 1, then u 1 u 2 on 0 + N. Remark 4. We will use later a comparison principle for a higher dimensional problem, precisely for (20), in the class of functions ft x x N +1 such that for all t x x N +1 0T N : ft x x N +1 CT1 +x N +1 Such a comparison principle can also be proved. Remark 5. We will use later comparison principles for two other equations very similar to (1) (see (50) and (65)). The reader can check that these results can be easily adapted from classical techniques. Remark 6. Let us finally point out that when passing from (50) to (51), the nonlocal operator is modified and the solution U is a function that is the sum of a linear function and of a bounded one (and the linear part is a priori known). In such a context and with such a modification of the nonlocal operator, a comparison principle also holds true.

15 Homogenization of First Order Equations 493 The third comparison principle is stated on a bounded domain Q. Because we deal with a nonlocal equation, we need to be able to compare the sub- and the supersolution everywhere outside Q. Theorem 4 (Comparison Principle on Bounded Domains for (1)). Consider a bounded usc function uq and a bounded lsc function vq that are respectively a sub- and supersolution of (1) on a bounded open domain Q + N.Ifu v outside Q, then u v on Q. The main result of this section is a comparison result for (4). We will use Proposition 2 to prove it. The technical difficulty comes from the singularity of the integral term I 1. Theorem 5. Consider a bounded usc subsolution u and a bounded lsc supersolution v of (4). If u0x u 0 x v0x with u 0 W 1, then u v on 0 + N. Remark 7. Remark 4 is also valid when passing from (4) to (32). Proof. Consider M = sup t 0Tx N ut x vt x, suppose that M>0 and let us exhibit a contradiction. First, we approximate the previous supremum by M = sup t 0Tx N u t x v t x where u (resp. v ) is the sup-convolution (resp. infconvolution) of u (resp. v) in the space variable x. It is well-known that u (resp. v ) is a semi-convex subsolution (resp. semi-concave supersolution) of (4); in particular, this means that u t is convex (resp. v t is concave) and we have, at a point t x of differentiability of u and v, u t x + z u t x u t x z 1 2 z2 for all z N (13) v t x + z v t x v st x z 1 2 z2 for all z N (14) Now consider M = sup { } ts 0Tx u t x v N s x x t s2 2 T t where is a C 2 function with bounded first and second derivatives, such that there exists C such that x C1 + x and x + as x. The supremum is attained at t s x. Choosing the parameters small enough, we can ensure that M M/2 > 0. Classical results about penalization (see Crandall et al., 1992 for instance) show that t s2 0as 0. Suppose that there are sequences n 0, n 0, n 0, m 0 such that t = 0ors = 0. Then by letting first n tend to +, we obtain: 0 <M M m supu m 0x v m 0 x x N supu 0 m x u 0 m x 0 x N as m +

16 494 Imbert et al. which is a contradiction. Therefore, we can assume that t s>0, and with B = B 1 0 we have ) ( t s + T t 2 u t x D 1+ u t x ) ( t s v s x D 1 v t x u t x x = v s x = p u t x + z u t x u t x z1 B z v s x + z v s x p z1 B z x + z x x z1 B z (15) Apply Proposition 2 and get the following viscosity inequalities: t s + ( T t H 0 u t x + z u t x 2 ) u t x z1zdz p + x t s ( H 0 v s x + z v s x p z1zdz ) I 1 x p + x Subtracting these two inequalities yield H 0 ( v s x + z v s x v s x z1zdz p ) T 2 H 0 I I 1 x p + x H 0 I p (16) with I = v s x + z v s x v s x z1zdz = I 1 + I 2 and I 1 = v s x + z v s x v s x zdz z 1 I 2 = v s x + z v s xdz z 1 Let us estimate these two integrals by using (13) (15): ( ) 1 C I B 1 dz z N +1 dz B 1 z C N +1 I 2 2v dz N \B 1 B 1 x + z x x zdz I 1

17 Homogenization of First Order Equations 495 2v We now use the classical estimate about inf-convolutions: p. We therefore see that the right hand side of (16) tends to zero as 0 for fixed. We then get the desired contradiction: 0. T Existence Results Theorem 6. Consider u 0 W 2 N and c W 1 N. For >0, there exists a (unique) bounded continuous viscosity solution u of (1) satisfying u 0x= u 0 x. Moreover, u is locally bounded in t x, uniformly in and: lim sup u 0x= lim inf u 0x= u 0 x Proof. From the classical theory of viscosity solutions, we know that it suffices to construct barriers in order to apply Perron s method. To do so, we prove that there exists a constant C>0 (independent of ) such that u + t x = u 0 x + Ct (resp. u t x = u 0 x Ct) is a supersolution (resp. subsolution) of (1). It is therefore possible to construct a bounded viscosity solution of (1) by adapting (Imbert, 2005, Theorem 3). It suffices to adapt (Imbert, 2005, Theorem 3). It also implies that lim sup u 0x= lim inf u 0x= u 0 x. Let us now determine C. First, we prove that for any C>0, [ ] M u0 + Ct C 1 where C 1 depends only on R 0 Nu 0 2 (in particular it does not depend on C or ). To see this, we write, [ ] M u0 + Ct ( u0 x + z = Jz B R0 ( u0 x + z + Jz B 1/ \B R0 u 0x ( u0 x + z + Jz N \B 1/ ) dz (17) u ) 0x dz (18) u 0x ) dz (19) and we estimate each term. The first one, (17), is treated as follows: ( u0 x + z Jz u ) 0x dz B R0 u 0 Jzzdz R 0 u 0 B R0 Next we estimate (18), by using (2): ( u0 x + z Jz u ) 0x dz R 0 z 1/ = u 0 x + y u 0 xdy R 0 y 1

18 496 Imbert et al. = u 0 x + y u 0 x u 0 x ydy R 0 y 1 D 2 y 2 u 0 B 1 2 dy Eventually, we estimate (19): ( u0 x + z Jz u ) 0x dz z 1/ y 1u = 0 x + y u 0 x dy 2u 0 N \B 1 dy Then define C 1 = R 0 u 0 +D 2 y u 0 2 dy + 2u B N \B 1 dy. Plugging u 0 + Ct into (1), we see that it is a supersolution if C c + C 1 u 0 + G. It also ensures that u 0 Ct is a subsolution of (1). Proposition 3. The homogenized equation (4) has a unique bounded continuous viscosity solution u 0. Proof. Adapting the argument of Imbert (2005) (already adapted from the classical argument), we can construct a solution by Perron s method if we construct good barriers. But since u 0 is W 2, the two functions u ± t x = u 0 x ± Ct are respectively a super and a subsolution for C sup { H 0 } L p L D N u 0 2 p u 0 with D N only depending on the dimension. Moreover we have u + 0x= u 0x= u 0 x. 4. The Proof of Convergence This section is devoted to the proof of Theorem 2. As announced in Introduction, the proof is essentially the same as the one for local equations in Imbert and Monneau (2008): imbed the problem in a higher dimensional one, argue by contradiction, consider a twisted perturbed test function and combine the relaxed semi-limit technique with the existence of Lipschitz continuous approximate correctors to exhibit a contradiction. The additional difficulty in comparison with Imbert and Monneau (2008) lies in the way to handle the non-local term. As announced above, let us consider a higher dimensional problem: [ ( ) [ ]] ( ) x U t U = c + M t x N +1 U x U +h xu in + N +1 U 0X= u 0 x + x N +1 on N +1 (20) where X = x x N +1. Let us exhibit the link between the problems in N N +1. and in

19 Homogenization of First Order Equations 497 Lemma 1. If u and U denote respectively the solutions of (1) and (20), then we have U t x x N +1 u xn t x +1 (21) ) a a U (t x x N +1 + = U t x x N +1 + (22) This lemma is a straightforward consequence of a comparison principle for (20) and of invariance by -integer translations w.r.t. x N +1. We previously explained that the existence of regular sub- and supercorrectors of a family of cell problems is necessary. Precisely, we consider for P = p 1 N +1 : + V = HL y + P Y + V P + V V + in 0 + N +1 where H = cy + L + MV y N +1 p + y V (23) + h + P Y + V p + y V V0Y= 0 on N +1 where Y = y y N +1. We also need to make more precise how the real number given by Theorem 1 for the Hamiltonian H + depends on its variables. The two following properties are by-products of the construction of approximate correctors we will do in the next section. Proposition 4 (Properties of the Effective Hamiltonian). Let p N and L. For any, let HL p be the constant defined by Theorem 1 for the Hamiltonian H +. Then H is continuous w.r.t. L p and nondecreasing w.r.t. L and and HL p ± as L ± if p 0 (24) HL p ± as ± (25) In particular, if denotes HL p, then { is nondecreasing and continuous w.r.t. 0 0 such that 0 = 0 (26) For = 0, we have H 0 L p = HL p 0, and Moreover if 01 N c = 0 and h 0, then H 0 L 0 = 0 for any L (27) H 0 0p= 0 for any p (28) Proposition 5 (Existence of Approximate Correctors). For any fixed p N, and >0 large enough, there exist real numbers, +, a constant C>0

20 498 Imbert et al. (independent on, and p) and bounded super and subcorrectors V +, V (depending on ) i.e., a super and a subsolution of (23) respectively with = ±, such that = lim + + = lim + (29) with + and satisfying (26) and for any Y + N +1 : V ± Y C (30) For any 0, there exists 0 ± ± such that ± ± = ± 0 = 0 Y V ± D ± ± 0 as + for the supercorrector V + and the subcorrector V respectively associated to + and, and for some constant D = D p > 0. Proof of Theorem 2. Classically, we prove that U + = lim sup U is a subsolution of t W = H 0 I 1 Wt x N +1 x W in 0 + N +1 (32) W0X= u 0 x + x N +1 on N +1 We will need the following lemma. Lemma 2. (31) If u 0 and U 0 denote respectively the solutions of (4) and (32), then we have: U 0 t x x N +1 = u 0 t x + x N +1 (33) U 0 t x x N +1 + a = U 0 t x x N +1 + a (34) Like Lemma 1, it is a consequence of invariance by integer translations and of a comparison principle for the higher dimensional problem (see Remark 7). Using barriers analogous to the ones of Theorem 6, we know that the family of functions U >0 is locally bounded so that U + is everywhere finite. Analogously, we can prove that U = lim inf U is a supersolution of (32). Using a second time the barriers, that are uniform with respect to, we deduce that U + 0X= U 0X= u 0 x + x N +1. The comparison principle for (32) 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 (32). This last fact is equivalent to the local convergence of U towards U 0. By Lemmas 1 and 2, this proves in particular the local convergence of u towards u 0. Arguing by contradiction, we prove that U + is a 2-viscosity subsolution of (32). We consider a test function such that U + attains a strict global zero maximum at t 0 X 0 with X 0 = x 0 xn 0 +1 and t 0 > 0 and we suppose that there exists >0 such that: t t 0 X 0 = HL 0 x t 0 X 0 +

21 where L 0 = + x 2 Homogenization of First Order Equations 499 t 0 x 0 + x x 0 N +1 t 0x 0 x 0 N +1 xt 0 x 0 x 0 N +1 xdx x 2 U + t 0 x 0 + x x 0 N +1 U + t 0 x 0 x 0 N +1 dx (35) First, let us consider >0 such that: HL 0 x t 0 X 0 + = HL 0 + x t 0 x Secondly, let us consider + 0 > 0 and + such that + 0 = + + = t t 0 X 0 and choose large enough so that /2 > 0. Then let V + be the supersolution of (23) given by Proposition 5 for = + and for L = L 0 +. In the following, = +, + p = x t 0 X 0 and V = V +. For r< 1, we consider the function Ft X = t X p x t and we 2 define a x N +1 -twisted perturbed test function by: ( t t X + V t X = x ) Ft X + k on t 0 /2 2t 0 B 1 X 0 U t X outside where k will be chosen later. We are going to prove that is a supersolution of (20) on r = Q rr t 0 X 0 for some r = r < 1/2 properly chosen, satisfying in particular r t 0 /2 2t 0 B 1 X 0. Let us first focus on the boundary conditions. For small enough (i.e., 0 < 0 r<r), since U + attains a strict maximum at t 0 X 0, we can ensure that: ( t U t X t X + V x ) Ft X r on t 0 /3 3t 0 B 2 X 0 \ r (37) for some r = o r 1 >0. Hence, choosing k = r, we ensure that U outside r. Let us next study the equation. From (22), we deduce that U + t x x N +1 + a = U + t x x N +1 + a Hence, we first derive that: x N +1 t 0 X 0 = 1 (36) Consider a test function such that attains a global zero minimum at t X r. Then for all t x + N +1 : ( t V x ) F t X 1 ( t t X V x ) Ft X 1 t X We have the map F t x N +1 0 X 0 = x N +1 t 0 X 0 = 1. Consequently, there exists r 0 > 0 such that Id F r0 r0 N

22 500 Imbert et al. t x x N +1 t x Ft x x N +1 is a C 1 -diffeomorphism from r0 onto its range r0. Let G r0 be the map such that Id G r0 r0 t x N +1 t x Gt x N +1 is the inverse of Id F. Let us consider the variables = t/ Y = y y N +1 with y = x/ and y N +1 = Ft X/ and define Y = 1 y G y y N +1 y G y y N +1 Let = t y = x y N +1 = F tx Y = y y N +1. Then: V Y Y V Y Y for all Y r0 From Proposition 5, we know that V is D -Lipschitz continuous with respect to Y. This implies that: Y Y D (38) Simple computations yield with P = p 1 N +1 : + Y = t t X yn +1 Y t t 0 X 0 t t X p + y Y = x t X yn +1 Y x t 0 X 0 x t X + P Y + V Y = t X k Using (39) and (38), Equation (23) yields: [ ( x t t X + o r 1 c ( t X + h ) + T nl ] x t X + o r 1 (39) ) x t X + o r (40) with the nonlocal term T nl = L MV y N +1 y We then have the following result whose proof is postponed: Lemma 3. The quantity M [ t x N +1 ] x is bounded independently of and we have [ ] M t x N +1 x L MV ȳ N +1 ȳ = T nl (41)

23 Homogenization of First Order Equations 501 Hence, from (40), we obtain with /2 > 0 [ ( ) [ ] ] x t t X + o r 1 c + M t x N +1 x x t X + o r 1 ( ) t X + h x t X + o r Therefore, there exists r > 0 such that for any r such that 0 < 0 r < r r is a supersolution of (20) on r. By Theorem 4, we conclude that U on r, i.e., we get: U t X t X + V+ k on r and we obtain the desired contradiction by passing to the upper limit at t 0 X 0 using the fact that U + t 0 X 0 = t 0 X 0 :0 r. Proof of Lemma 3. We proceed in two steps. Step 1. Bound on M uniformly in. We write: [ ] M t x N +1 x ( t x + z xn +1 + V ȳ + z F = Jz E 1 X ( U t x + z x + Jz N +1 t X V Y E 2 X = M 1 + M 2 with E 1 X = z N x + z x N +1 B 1 X 0 B 2 0 ) ( t x + z x N +1 / t X V Y) E 2 X = z N x + z x N +1 B 1 X 0 N \B N \B R0 0 because r<1/2 and t X r. Hence, M 2 U t x + x x N +1 t X V Ydx x 1/2 U L x N +1 xn V dx ) x 1/2 Moreover using the fact that for = 1 x N +1 xn , a = x x 0, we have E 1 X = B / a/ and we remark that it contains B / with 2 = 2 x 0 x 2 0. Now we write: M 1 = JzV ȳ + z F t x + z x N +1 / V Y dz E 1 X + + B / a/\b / Jz B / Jz = M 11 + M 12 + M 13 ( ) t x + z xn +1 t X dz ( ) t x + z xn +1 t X x t x x N +1 z dz

24 502 Imbert et al. We have M 11 2V, M 13 D 2 xx z / M B / a/\b / dz C z 2 2 Jzdz CD2 xx and Collecting all the estimates, we get the desired bound on M [ t x N +1 ] x. Step 2. More precise estimate from above. Let us now estimate more precisely from above the quantity: [ ] M t x N +1 x = ( t x + z x Jz N +1 t X ) dz Choose such that R 0 2/ where R 0 is defined in (2), use (37), the definition (36) of, and the fact that J is even to get: [ ] M ( ) t x + z xn Jz +1 + V t X V Ȳ dz z 2/ + z>2/ ( ) t x + z xn Jz +1 t x x N +1 x t x z z R 0 ( ) t x + z xn + Jz +1 t x x N +1 x t x x N +1 z dz R 0 z 2/ ( ( + Jz V ȳ + z F t x + z x ) ( N +1 V ȳ F t x x )) N +1 dz z 2/ ( ) U t x + z x + Jz N +1 t X dz = T z>2/ 1 + T 2 + T 3 + T 4 where ( ) t x + z xn T 1 = Jz +1 t x x N +1 t x z dz z R 0 T 2 = t x + x x N +1 t x x N +1 x t x x N +1 xdx R 0 x 2 ( ( T 3 = Jz V ȳ + z F t x + z x ) ( N +1 V ȳ F t x x )) N +1 dz z 2/ T 4 = U t x + x x N +1 t X V Y k dx x>2 Estimate of T 1. T 1 Jz D2 z 2 dz = z R 0 2 Estimate of T 2. We claim that: T 2 = x 2 t 0 x 0 + x x 0 N +1 t 0x 0 x 0 N +1 t 0x 0 x 0 N +1 xdx + o r1

25 Remark first that: Homogenization of First Order Equations 503 x R 0 t x + x x N +1 t x x N +1 x t x x N +1 xdx = o 1 = o r 1 Now use the continuity of the map t x t x + z x N +1 t X t X z dz z 2 it follows from the continuity of the integrand and of the bound: t x + z x N +1 t X x t X z D2 z 2 2 Estimate of T 3. We claim that T 3 = MV ȳ N +1 ȳ + o r 1 Let us define ( ( T 3 = Jz V ȳ + z F t x + z x ) ( N +1 V ȳ F t x x )) N +1 dz N Then, on the one hand we have T 3 T 3 2V On the other hand (for R r > 0), z 2/ Jz dz = o 1 T 3 MV ȳ N +1ȳ 4V N \B Rr D + z R r t x + z x N +1 t x x N +1 p z Jz dz D o r 1 + z R r t x + z x N +1 t x x N +1 x t x x N +1 z + CrzJz dz o r 1 + CR r 2 + rr r for R r +as r 0. Now choose r 2 and R r such that rr r 0. We conclude that: T 3 MV ȳ N +1ȳ o r 1 Estimate of T 4. Remark that for all x B R (with R chosen later): U t x + x x N +1 t X V Y U + t 0 x 0 + x x 0 N +1 t 0X 0 + o 1 + o r 1

26 504 Imbert et al. eeping in mind that t 0 X 0 = U + t 0 X 0, we can choose R big enough so that: T 4 + z R z R x>2 ( U + t 0 x 0 + x x 0 N +1 U + t 0 X 0 ) dx + Combining all these estimates yields (41) with L 0 given in (35). 5. Approximate Cell Problems In this section, we approximate the Hamiltonian of (1) in such a way that: first, the modified solutions are approximate solutions and are sub- or supersolutions of the exact problem; secondly, they are bounded uniformly with respect to the approximation; thirdly they are Lipschitz continuous with respect to Y where the Lipschitz constant depend on the approximation. Our first technical choice is to consider the non-local normal speed of the dislocation (see Section 2.1) and the correction term as variables c and h. Next, in order to ensure that modified solutions are good approximate solutions, H + is constructed so that it coincides with the exact Hamiltonian on a ball of radius centered at the origin. To ensure the Lipschitz continuity of the solution, the Hamiltonian is modified in such a way that it is constant outside a starshaped compact set +. To finish with, in order to be sure that the solution of the approximate problem is a supersolution of the exact one, the approximate Hamiltonian H + is constructed above the exact one on +. We modify Hc h q = cq+h in order to ensure that it is coercive w.r.t. Q = q q N +1. Lemma 4 (Approximate Hamiltonian). There exist a 0-starshaped compact set + N +1 containing the ball B 0 and two piecewise linear nondecreasing functions T (truncature functions) with 0, such that the approximate Hamiltonian: H + c h Q = T H c Q + h where H c Q = c + qq+q N +1 satisfies for all c C C and h G G, = cq+q N +1 +h if Q H + c h Q H c Q + h cq+h if Q + = M + if Q + (42) for some constant M +. Proof. First, we consider two other constants R 0 R1 > 0, we define for c r : 0 if r R 0 r = r R 0 if R 0 r R1 R 1 R0 if r R1

27 and we introduce for >0: Homogenization of First Order Equations 505 H c q = c + qq and H c Q = H c q + q N +1 We choose R 0 and R1 as follows: R 0 = and R1 = R0 + 2C (43) so that, on the one hand, H c q is coercive w.r.t. q uniformly with respect to c C C and, on the other hand, q = 0ifq. Next, define for r 0 and : and where h r = suph c q + hc C C h G G q r r = infr 0 q r H c q + h>for any c h C C G G h r = h r + r ( H + c Q = T H c Q + h) (44) T = h + + h 1 + h + if h if h 2h if 2h with + 1 to be fixed later. Now we can introduce the following compact set of N +1 : + = { Q N +1 q r 2h q N +1 q N +1 2h inf H + G } with inf H = inf c C q N H c q C (we used (43)) and we can check that B 0 { Q N +1 c h C C G G H c Q + h 2h } + Eventually, we define: M + = sup h q + q N +1 Q + Remark that M + 2h inf H + G 2h by choosing q = 0 and q N +1 =2h inf H + G. Let us check that if we choose + 1 so that: we are sure that (42) holds true h + = M

28 506 Imbert et al. proved. M + First, if Q, then H c Q + h h and the first property of (42) is Secondly, if Q +, then H and we are done; or H c Q + h 2h that T for 2h. To finish with, consider Q + inf H + G. In this case, c Q + h M + and: either H + c h Q = and in this case, we use the fact. Suppose first that q N +1 > 2h H c Q + h q N +1+inf H G>2h and we are done. In the other case, q >r 2h q N +1 and we conclude, by definition of r that for all c h C C G G, and we can conclude in this case too. H c q + h>2h q N +1 We can now consider the approximate cell problem: + V = H + cy + L + MV y N +1 y h + P Y + V p + y V P + V + in + N +1 (45) V0Y= 0 on N +1 and state the existence of Lipschitz continuous approximate supercorrectors. Proposition 6 (Lipschitz Continuous Approximate Supercorrectors). Let p N and P = p 1 N. Let us consider the truncated Hamiltonian H + defined by (44) for > 1 +p 2 large enough. For all, there exist real numbers +, and solutions V + of (45) with = + satisfying (26) and W 1 a Priori Bounds on the Correctors. continuous correctors with: + Lp c p+g (46) We can construct bounded Lipschitz V + Y 6C + (47) with C + = 2c +G +1 c and c0 = inf 0 d 01/2 N dz minjz + d Jz d > 0 and N ( P + V + Y ) + + V Y 2h inf H +G y N +1 (48) with inf H = inf c C q H c q < 0. N Further Properties of the Correctors. The correctors satisfy { V + V + y y N = V + y y N +1 y + k y N +1 V + y y N +1 1 for every k N (49)

29 Homogenization of First Order Equations 507 If p = P/Q with P N and Q \0, then following Imbert and Monneau (2008), it is possible to build solutions on the whole space-time satisfying V + y + Qk y N +1 = V + y y N +1 for every k N Proof. To construct such approximate correctors, we first consider the solution of the following initial value problem: W = H + cy + L + MW y N +1 y hp Y + W p + y W P + W + W0Y= 0 in + N +1 on N +1 We perform the proof in two steps. We first construct barriers and get gradient estimates on W for short times; secondly, we control the oscillations of W with respect to space for long times and finally, we control the oscillations with respect to time. Step 1. Existence of a solution and gradient estimates. We first construct barriers for (50) of the form W ± Y = + Lp+ ± C 1. Since P <, one can easily check that if C 1 =c p+g, then W ± is a supersolution (resp. subsolution) of (50). It follows that there exists a unique solution W of (50) (by Perron s method, as we did in Theorem 3 for instance). Let us now recall that we constructed the approximate Hamiltonian H + for speeds c C C. Moreover, we remark that: MW y N +1 osc W with osc W = max Y N +1 W Y min Y N +1 W Y and osc W0 = 0. This is the reason why we introduce, the first time such that osc W +c +L C. We remark that U Y = W Y + P Y satisfies U = H + cy + L + MU y N +1 p y hu y U U + U0Y= P Y in + N +1 on N +1 (50) (51) Remark that U0Y= P B 0 + that is starshaped with respect to the origin and compact. Hence, for all N +1, there exists M such that Q>M + H + c Q = M +