Homogenization of accelerated Frenkel-Kontorova models with n types of particles
aa r X i v : . [ m a t h . A P ] J un Homogenization of accelerated Frenkel-Kontorovamodels with n types of particles N. Forcadel , C. Imbert , R. Monneau October 31, 2018
Abstract
We consider systems of ODEs that describe the dynamics of particles. Each particle satisfies a Newtonlaw (including a damping term and an acceleration term) where the force is created by the interactionswith other particles and with a periodic potential. The presence of a damping term allows the system tobe monotone. Our study takes into account the fact that the particles can be different.After a proper hyperbolic rescaling, we show that solutions of these systems of ODEs converge tosolutions of some macroscopic homogenized Hamilton-Jacobi equations.
AMS Classification:
Keywords: particle system, periodic homogenization, Frenkel-Kontorova models, Hamilton-Jacobi equations, hullfunction
The goal of this paper is to obtain homogenization results for the dynamics of accelerated Frenkel-Kontorovatype systems with n types of particles. The Frenkel-Kontorova model is a simple physical model used invarious fields: mechanics, biology, chemistry etc. The reader is referred to [4] for a general presentation ofmodels and mathematical problems. In this introduction, we start with the simplest accelerated Frenkel-Kontorova model where there is only one type of particle (see Eq. (1.2)). We then explain how to deal with n types of particles (see Eq. (1.6)). We finally present the general case, namely systems of ODEs of thefollowing form (for a fixed m ∈ N )(1.1) m d U i dτ + dU i dτ = F i ( τ, U i − m , . . . , U i + m )where U i ( τ ) denotes the position of the particle i ∈ Z at the time τ . Here, m is the mass of the particleand F i is the force acting on the particle i , which will be made precise later.Remark the presence of the damping term dU i dτ on the left hand side of the equation. If the mass m is assumed to be small enough, then this system is monotone. We will make such an assumption and themonotonicity of the system is crucial in our analysis.We recall that the case of fully overdamped dynamics, i.e. for m = 0, has already been treated in [10](for only one type of particles).Several results are related to our analysis. For instance in [5], homogenization results are obtained formonotone systems of Hamilton-Jacobi equations. Notice that they obtain a system at the limit while we willobtain a single equation. Techniques from dynamical systems are also used to study systems of ODEs; seefor instance [8, 18] and references therein. CEREMADE, UMR CNRS 7534, Universit´e Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris Cedex 16, France Universite Paris-Est, Cermics, Ecole des ponts, 6-8 avenue Blaise Pascal, 77455 Marne la Vallee Cedex 2, France. .1 The classical overdamped Frenkel-Kontorova model The classical Frenkel-Kontorova model describes a chain of classical particles evolving in a one dimensionalspace, coupled with their neighbours and subjected to a periodic potential. If τ denotes time and U i ( τ )denotes the position of the particle i ∈ Z , one of the simplest FK models is given by the following dynamics(1.2) m d U i dτ + dU i dτ = U i +1 − U i + U i − + sin (2 πU i ) + L where m denotes the mass of the particle, L is a constant driving force which can make the whole “trainof particles” move and the term sin (2 πU i ) describes the force created by a periodic potential whose periodis assumed to be 1. Notice that in the previous equation, we set to one physical constants in front of theelastic and the exterior forces (friction and periodic potential). The goal of our work is to describe whatis the macroscopic behaviour of the solution U of (1.2) as the number of particles per length unit goes toinfinity. As mentioned above, the particular case where m = 0 is referred to as the fully overdamped oneand has been studied in [10].We would like next to give the flavour of our main results. In order to do so, let us assume that at initialtime, particles satisfy U i (0) = ε − u ( iε ) dU i dτ (0) = 0for some ε > u ( x ) which satisfies the following assumption Initial gradient bounded from above and below (1.3) 0 < /K ≤ ( u ) x ≤ K on R for some fixed K > K − ε − , K ε − ).It is then natural to ask what is the macroscopic behaviour of the solution U of (1.2) as ε goes to zero, i.e. as the number of particles per length unit goes to infinity. To this end, we define the following functionwhich describes the rescaled positions of the particles(1.4) u ε ( t, x ) = εU ⌊ ε − x ⌋ ( ε − t )where ⌊·⌋ denotes the floor integer part. One of our main results states that the limiting dynamics as ε goesto 0 of (1.2) is determined by a first order Hamilton-Jacobi equation of the form(1.5) (cid:26) u t = F ( u x ) for ( t, x ) ∈ (0 , + ∞ ) × R ,u (0 , x ) = u ( x ) for x ∈ R where F is a continuous function to be determined. More precisely, we have the following homogenizationresult Theorem 1.1 ( Homogenization of the accelerated FK model).
There exists a critical value m c suchthat for all m ∈ ]0 , m c ] and all L ∈ R , there exists a continuous function F : R → R such that, underassumption (1.3) , the function u ε converges locally uniformly towards the unique viscosity solution u of (1.5) . Remark 1.2.
The critical mass m c is made precise in Assumption (A3) below.2 .2 Example of systems with n types of particles We now present the case of systems with n types of particles. Let us start with the typical problem we havein mind. Let n ∈ N \ { } be some integer and let us consider a sequence of real numbers ( θ i ) i ∈ Z such that θ i + n = θ i > i ∈ Z . It is then natural to consider the generalized FK model with n different types of particles that stay orderedon the real line. Then, instead of satisfying (1.2), we can assume that U i satisfies for τ ∈ (0 , + ∞ ) and i ∈ Z (1.6) m d U i d τ + dU i dτ = θ i +1 ( U i +1 − U i ) − θ i ( U i − U i − ) + sin (2 πU i ) + L Such a model is sketched on figure 1. As we shall see it, we can prove the same kind of homogenizationresults as Theorem 1.1. i−1 i i+1 i+2 periodic potential
Figure 1: The FK model with n = 2 type of particles (and of springs) and an interaction up to the m = 1neighboursAs we mentioned it before, it is crucial in our analysis to deal with monotone systems of ODEs. Inspiredof the work of Baesens and MacKay [2] and of Hu, Qin and Zheng [12], we introduce for all i ∈ Z thefollowing function Ξ i ( τ ) = U i ( τ ) + 2 m dU i dτ ( τ ) . Using this new function, the system of ODEs (1.6) can be rewritten in the following form: for τ ∈ (0 , + ∞ )and i ∈ Z , dU i dτ = m (Ξ i − U i ) d Ξ i dτ = 2 θ i +1 ( U i +1 − U i ) − θ i ( U i − U i − ) + 2 sin(2 πU i ) + 2 L + m ( U i − Ξ i ) . We point out that, in compare with [2, 12], our proof of the monotonicity of the system is simpler.It is convenient to introduce the following notation α = 12 m . Remark 1.3.
It would be also possible to consider more generally: Ξ i ( τ ) = U i ( τ ) + α dU i dτ ( τ ) with α > m .In order to simplify here the presentation, we choose α = 1 / (2 m ). Moreover, for the classical Frenkel-Kontorova model (1.2), the choice α = 1 / (2 m ) is optimal in the sense that the critical value m c for whichthe system is monotone is the best we can get. 3 .3 General systems with n types of particles More generally, we would like to study the generalized Frenkel-Kontorova model (1.1) with n types ofparticles. In order to do so, let us consider a general sequence of functions v = ( v j ( y )) j ∈ Z satisfying v j + n ( y ) = v j ( y + 1) . For m ∈ N , we set [ v ] j,m ( y ) = ( v j − m ( y ) , . . . , v j + m ( y )) . We are going to study a function ( u, ξ ) = (( u j ( τ, y )) j ∈ Z , ( ξ j ( τ, y )) j ∈ Z )satisfying the following system of equations: for all ( τ, y ) ∈ (0 , + ∞ ) × R and all j ∈ Z ,(1.7) (cid:26) ( u j ) τ = α ( ξ j − u j )( ξ j ) τ = 2 F j ( τ, [ u ( τ, · )] j,m ) + α ( u j − ξ j ) , (cid:26) u j + n ( τ, y ) = u j ( τ, y + 1) ξ j + n ( τ, y ) = ξ j ( τ, y + 1) . This system is referred to as the generalized Frenkel-Kontorova (FK for short) model . It is satisfied inthe viscosity sense (see Definition 2.1). Moreover, we will consider viscosity solutions which are possiblydiscontinuous.Let us now make precise the assumptions on the functions F j : R × R m +1 → R mapping ( τ, V ) to F j ( τ, V ). It is convenient to write V ∈ R m +1 as ( V − m , . . . , V m ).(A1) (Regularity) (cid:26) F j is continuous ,F j is Lipschitz continuous in V uniformly in τ and j . (A2) (Monotonicity in V i , i = 0 ) F j ( τ, V − m , ..., V m ) is non-decreasing in V i for i = 0 . (A3) (Monotonicity in V ) α + 2 ∂F j ∂V ≥ j ∈ Z . Keeping in mind the notation we chose above ( α = (2 m ) − ), this assumption can be interpreted as follows:the mass has to be small in comparison with the variations of the non-linearity, which means that the systemis sufficiently overdamped. This assumption guarantees that 2 F j ( τ, V ) + α V is non-decreasing in V for all j ∈ Z .(A4) (Periodicity) (cid:26) F j ( τ, V − m + 1 , ..., V m + 1) = F j ( τ, V − m , ..., V m ) ,F j ( τ + 1 , V ) = F j ( τ, V ) . (A5) (Periodicity of the type of particles) F j + n = F j for all j ∈ Z . n = 1, we explained in [10] that the system of ODEs can be embedded into a single partial differentialequation (more precisely, in a single ordinary differential equation with a real parameter x ). Here, takinginto account the “ n -periodicity” of the indices j , it can be embedded into n coupled systems of equations.The next assumption allows us to guarantee that the ordering property of the particles, i.e. u j ≤ u j +1 ,is preserved for all time.(A6) (Ordering) For all ( V − m , . . . , V m , V m +1 ) ∈ R m +2 such that V i +1 ≥ V i for all | i | ≤ m , we have2 F j +1 ( τ, V − m +1 , . . . , V m +1 ) + α V ≥ F j ( τ, V − m , . . . , V m ) + α V . Remark 1.4.
If, for all j ∈ { , . . . , n − } , we have F j +1 = F j then assumption (A6) is a direct consequenceof assumptions (A2) and (A3). Notice also that for n ≥
1, Condition (A6’) of Subsection 2.1 does not allowus to take α i = m i with different m i ’s. In particular, all the particles in our analysis have the same mass m . Example 1.
We see that Assumptions (A1)-(A5) are in particular satisfied for the FK system (1.6) with n types of particles ( θ n + j = θ j ), m = 1 and F j ( τ, V − , V , V ) = θ j +1 ( V − V ) − θ j ( V − V − ) + sin (2 πV ) + L for α ≥ θ j + θ j +1 ) + 4 π . To get (A6) we have to assume furthemore that α ≥ θ j + 4 π . We next rescale the generalized FK model: we consider for ε > u εj ( t, x ) = εu j (cid:18) tε , xε (cid:19) ξ εj ( t, x ) = εξ j (cid:18) tε , xε (cid:19) . The function ( u ε , ξ ε ) = (cid:16)(cid:0) u εj ( t, x ) (cid:1) j ∈ Z , (cid:0) ξ εj ( t, x ) (cid:1) j ∈ Z (cid:17) satisfies the following problem: for all j ∈ Z , t > x ∈ R (1.8) ( u εj ) t = α ξ εj − u εj ε ( ξ εj ) t = 2 F j (cid:18) tε , h u ε ( t, · ) ε i j,m (cid:19) + α u εj − ξ εj ε . (cid:26) u εj + n ( t, x ) = u εj ( t, x + ε ) ξ εj + n ( t, x ) = ξ εj ( t, x + ε )We impose the following initial conditions(1.9) (cid:26) u εj (0 , x ) = u (cid:0) x + jεn (cid:1) ξ εj (0 , x ) = ξ ε (cid:0) x + jεn (cid:1) . Finally, we assume that u and ξ ε satisfy (A0) (Gradient bound from below) There exist K > M > < /K ≤ ( u ) x ≤ K on R , < /K ≤ ( ξ ε ) x ≤ K on R , k u − ξ ε k ∞ ≤ M ε . Then we have the following homogenization result 5 heorem 1.5 ( Homogenization of systems with n types of particles). Assume that ( F j ) j satisfies (A1)-(A6) , and assume that the initial data u , ξ ε satisfy (A0) . Consider the solution (( u εj ) j ∈ Z , ( ξ εj ) j ∈ Z ) of (1.8) - (1.9) . Then, there exists a continuous function F : R R such that, for all integer j ∈ Z , the functions u εj and ξ εj converge uniformly on compact sets of (0 , + ∞ ) × R to the unique viscosity solution u of (1.5). Remark 1.6.
The reader can be surprised by the fact that we obtain, at the limit, only one equationto describe the evolution of the system. In fact, this essentially comes from Assumption (A6) and thedefinition of ξ εj . Indeed, it could be shown that assumption (A6) implies that the functions u ε and ξ ε arenon-decreasing with respect to j : u εj +1 ≥ u εj and ξ εj +1 ≥ ξ εj . Then, the system can be essentially sketched byonly two equations (one for the evolution of u and one for ξ ). But by the “microscopic definition” of ξ εj , wehave ξ εj = u εj + O ( ε ); hence only one equation is sufficient to describe the macroscopic evolution of all thesystem. Remark 1.7.
The case m = 0 corresponds to α = + ∞ . In this case, u ε ≡ ξ ε in (1.8) and Theorem 1.5still holds true.We will explain in the next subsection how the non-linearity ¯ F , known as the effective Hamiltonian, isdetermined. We will see that this has to do with the existence of solutions of (1.8), (1.9) of a specific form.They are constructed thanks to functions referred to as hull functions. In this subsection, we introduce the notion of hull function for System (1.7). More precisely, we look forspecial functions (( h j ( τ, z )) j ∈ Z , ( g j ( τ, z )) j ∈ Z such that ( u j ( τ, y ) , ξ j ( τ, y )) = ( h j ( τ, py + λτ ) , g j ( τ, py + λτ )) isa solution of (1.7) on Ω = ( −∞ , + ∞ ) × R = R . Here is a precise definition. Definition 1.8 ( Hull function for systems of n types of particles). Given ( F j ) j satisfying (A1)-(A6) , p ∈ (0 , + ∞ ) and a number λ ∈ R , we say that a family of functions (( h j ) j , ( g j ) j ) is a hull function for (1.7) if it satisfies for all ( τ, z ) ∈ R , j ∈ Z (1.10) ( h j ) τ + λ ( h j ) z = α ( g j − h j ) h j ( τ + 1 , z ) = h j ( τ, z ) h j ( τ, z + 1) = h j ( τ, z ) + 1 h j + n ( τ, z ) = h j ( τ, z + p ) h j +1 ( τ, z ) ≥ h j ( τ, z )( h j ) z ( τ, z ) ≥ ∃ C s . t . | h j ( τ, z ) − z | ≤ C ( g j ) τ + λ ( g j ) z = 2 F j ( τ, [ h ( τ, · )] j,m ( z )) + α ( h j − g j ) g j ( τ + 1 , z ) = g j ( τ, z ) g j ( τ, z + 1) = g j ( τ, z ) + 1 g j + n ( τ, z ) = g j ( τ, z + p ) g j +1 ( τ, z ) ≥ g j ( τ, z )( g j ) z ( τ, z ) ≥ ∃ C s . t . | g j ( τ, z ) − z | ≤ C .
In the case where the functions ( F j ) j do not depend on τ , we also require that the hull function (( h j ) j , ( g j ) j ) is independent on τ and we denote it by (( h j ( z )) j , ( g j ( z )) j ) . Remark 1.9.
The last line of (1.10) implies in particular that εh j ( τ, zε ) → z and εg j ( τ, zε ) → z as ε → p >
0, the following theorem explains how the effective Hamiltonian F ( p ) is determined by anexistence/non-existence result of hull functions as λ ∈ R varies. Theorem 1.10 ( Effective Hamiltonian and hull function).
Given ( F j ) j satisfying (A1)-(A6) and p ∈ (0 , + ∞ ) , there exists a unique real number λ for which there exists a hull function (( h j ) j , ( g j ) j ) (dependingon p ) satisfying (1.10) . Moreover the real number λ = F ( p ) , seen as a function of p , is continuous in (0 , + ∞ ) . We have moreover the following result 6 heorem 1.11 ( Qualitative properties of F ). Let ( F j ) j satisfying (A1)-(A6) . For any constant L ∈ R ,let F ( L, p ) denote the effective Hamiltonian given in Theorem 1.10 for p ∈ (0 , + ∞ ) , associated with ( F j ) j replaced by ( L + F j ) j .Then ( L, p ) F ( L, p ) is continuous and we have the following properties (i) (Bound) we have | F ( L, p ) − L | ≤ C p . (ii) (Monotonicity in L ) F ( L, p ) is non-decreasing in L .
In Section 2, we give some useful results concerning viscosity solutions for systems. In Section 3, we provethe convergence result assuming the existence of hull functions. The construction of hull functions is givenin Sections 4 and 5. Finally, Section 6 is devoted to the proof of the qualitative properties of the effectiveHamiltonian.
Given r, R > t ∈ R and x ∈ R , Q r,R ( t, x ) denotes the following neighbourhood of ( t, x ) Q r,R ( t, x ) = ( t − r, t + r ) × ( x − R, x + R ) . For V = ( V , . . . , V N ) ∈ R N , | V | ∞ denotes max j | V j | . Given a family of functions ( v j ( · )) j ∈ Z and twointegers j, m ∈ Z , [ v ] j,m denotes the function ( v j − m ( · ) , . . . , v j + m ( · )). This section is devoted to the definition of viscosity solutions for systems of equations such as (1.7), (1.8)and (1.10). In order to construct hull functions when proving Theorem 1.10, we will also need to considera perturbation of (1.7) with linear plus bounded initial data. For all these reasons, we define a viscositysolution for a generic equation whose Hamiltonian ( G j ) j satisfies proper assumptions.Before making precise assumptions, definitions and crucial results we will need later (such as stability,comparison principle, existence), we refer the reader to the user’s guide of Crandall, Ishii, Lions [7] and thebook of Barles [3] for an introduction to viscosity solutions and [6, 21, 16, 17] and references therein forresults concerning viscosity solutions for systems of weakly coupled partial differential equations. As we mentioned it before, we consider systems with general non-linearities ( G j ) j . Precisely, for 0 < T ≤ + ∞ , we consider the following Cauchy problem: for j ∈ Z , τ > y ∈ R ,(2.1) (cid:26) ( u j ) τ = α ( ξ j − u j )( ξ j ) τ = G j ( τ, [ u ( τ, · )] j,m , ξ j , inf y ′ ∈ R ( ξ j ( τ, y ′ ) − py ′ ) + py − ξ j ( τ, y ) , ( ξ j ) y ) (cid:26) u j + n ( τ, y ) = u j ( τ, y + 1) ξ j + n ( τ, y ) = ξ j ( τ, y + 1)submitted to the initial conditions(2.2) (cid:26) u j (0 , y ) = u ( y + jn ) := u ,j ( y ) ξ j (0 , y ) = ξ ( y + jn ) := ξ ,j ( y ) . xample 2. The most important example we have in mind is the following one G j ( τ, V − m , · · · , V m , r, a, q ) = 2 F j ( τ, V ) + α ( V − r ) + δ ( a + a ) q + for some constants δ ≥ , a , a, q ∈ R and where F j appears in (1.7) , (1.8) , (1.10) . In view of (2.1), it is clear that in the case where G j effectively depends on the variable a , solutions mustbe such that the infimum of ξ j ( τ, y ) − p · y is finite for all time τ . Hence, when G j does depend on a , we willonly consider solutions ξ j satisfying for some C ( T ) >
0: for all τ ∈ [0 , T ) and all y, y ′ ∈ R (2.3) | ξ j ( τ, y + y ′ ) − ξ j ( τ, y ) − py ′ | ≤ C . When T = + ∞ , we may assume that (2.3) holds true for all time T > C > (Initial condition) ( u , ξ ) satisfies (A0) (with ε = 1); it also satisfies (2.3) if G j depends on a for some j .As far as the ( G j ) j ’s are concerned, we make the following assumptions.(A1’) (Regularity) (i) G j is continuous.(ii) For all R >
0, there exists L = L ( R ) > τ, V, W, r, s, a, q , q , j , with a ∈ [ − R, R ], we have | G j ( τ, V, r, a, q ) − G j ( τ, W, s, a, q ) | ≤ L | V − W | ∞ + L | r − s | + L | q − q | . (iii) There exists L > V, a, b, τ, r, q , | G j ( τ, V, r, a, q ) − G j ( τ, V, r, b, q ) | ≤ L | a − b || q | . (A2’) (Monotonicity in V i , i = 0 ) G j ( τ, V − m , ..., V m , r, a, q ) is non-decreasing in V i for i = 0 . (A3’) (Monotonicity in a and V ) G j ( τ, V − m , ..., V m , r, a, q ) is non-decreasing in a and in V . (A4’) (Periodicity) For all ( τ, V, r, a, q ) ∈ R × R m +1 × R × R × R and j ∈ { , . . . , n } (cid:26) G j ( τ, V − m + 1 , ..., V m + 1 , r + 1 , a, q ) = G j ( τ, V − m , ..., V m , r, a, q ) ,G j ( τ + 1 , V, r, a, q ) = G j ( τ, V, r, a, q ) . (A5’) (Periodicity of the type of particles) G j + n = G j for all j ∈ Z . (A6’) (Ordering) For all ( V − m , . . . , V m , V m +1 ) ∈ R m +2 such that ∀ i, V i +1 ≥ V i , we have G j +1 ( τ, V − m +1 , . . . , V m +1 , r, a, q ) ≥ G j ( τ, V − m , . . . , V m , r, a, q ) . u ∗ and u ∗ , of a locallybounded function u . u ∗ ( τ, y ) = lim sup ( t,x ) → ( τ,y ) u ( t, x ) and u ∗ ( τ, y ) = lim inf ( t,x ) → ( τ,y ) u ( t, x ) . We can now define viscosity solutions for (2.1).
Definition 2.1 ( Viscosity solutions).
Let
T > and u : R → R and ξ : R → R be such that (A0’) issatisfied. For all j , consider locally bounded functions u j : R + × R → R and ξ j : R + × R → R . We denoteby Ω = (0 , T ] × R .– The function (( u j ) j , ( ξ j ) j ) is a sub-solution (resp. a super-solution ) of (2.1) on Ω if (2.3) holds truefor ξ j in the case where G j depends on a , and ∀ j, n, ∀ ( τ, y ) , u j + n ( τ, y ) = u j ( τ, y + 1) , ξ j + n ( τ, y ) = ξ j ( τ, y + 1) and for all j ∈ { , . . . , n } , u j and ξ j are upper semi-continuous (resp. lower semi-continuous), and forall ( τ, y ) ∈ Ω and any test function φ ∈ C (Ω) such that u j − φ attains a local maximum (resp. a localminimum) at the point ( τ, y ) , then we have (2.4) φ τ ( τ, y ) ≤ α ( ξ j ( τ, y ) − u j ( τ, y )) ( resp. ≥ ) and for all ( τ, y ) ∈ Ω and any test function φ ∈ C (Ω) such that ξ j − φ attains a local maximum (resp.a local minimum) at the point ( τ, y ) , then we have (2.5) φ τ ( τ, y ) ≤ G j ( τ, [ u ( τ, · )] j,m ( y ) , ξ j ( τ, y ) , inf y ′ ∈ R ( ξ j ( τ, y ′ ) − py ′ ) + py − ξ j ( τ, u ) , φ y ( τ, y ))( resp. ≥ ) . – The function (( u j ) j , ( ξ j ) j ) is a sub-solution (resp. super-solution ) of (2.1) , (2.2) if (( u j ) j , ( ξ j ) j ) is asub-solution (resp. super-solution) on Ω and if it satisfies moreover for all y ∈ R , j ∈ { , . . . , n } u j (0 , y ) ≤ u ( y + jn ) ( resp. ≥ ) ,ξ j (0 , y ) ≤ ξ ( y + jn ) ( resp. ≥ ) . – A function (( u j ) j , ( ξ j ) j ) is a viscosity solution of (2.1) (resp. of (2.1) , (2.2) ) if (( u ∗ j ) j , ( ξ ∗ j ) j ) is asub-solution and ((( u j ) ∗ ) j , (( ξ j ) ∗ ) j ) is a super-solution of (2.1) (resp. of (2.1) , (2.2) ). Sub- and super-solutions satisfy the following comparison principle which is a key property of the equation.
Proposition 2.2 ( Comparison principle).
Assume (A0’) and that ( G j ) j satisfy (A1’)-(A5’) . Let ( u j , ξ j ) (resp. ( v j , ζ j ) ) be a sub-solution (resp. asuper-solution) of (2.1) , (2.2) such that (2.3) holds true for ξ j and ζ j in the case where G j depends on a .We also assume that there exists a constant K > such that for all j ∈ { , . . . , n } and ( t, x ) ∈ [0 , T ] × R ,we have (2.6) u j ( t, x ) ≤ u ,j ( x ) + K (1 + t ) , ξ j ( t, x ) ≤ ξ ,j ( x ) + K (1 + t )(resp . − v j ( t, x ) ≤ − u ,j ( x ) + K (1 + t ) , − ζ j ( t, x ) ≤ − ξ ,j ( x ) + K (1 + t )) . If u j (0 , x ) ≤ v j (0 , x ) and ξ j (0 , x ) ≤ ζ j (0 , x ) for all j ∈ Z , x ∈ R , then u j ( t, x ) ≤ v j ( t, x ) and ξ j ( t, x ) ≤ ζ j ( t, x ) for all j ∈ Z , ( t, x ) ∈ [0 , T ] × R . emark 2.3. Even if it was not specified in [10], the Lipschitz continuity in q of G j is necessary to obtaina general comparison principle. Proof of Proposition 2.2.
In view of assumption (A1’)(i) and using the change of unknown functions ¯ u j ( t, x ) = e − λt u j ( t, x ) and ¯ ξ j ( t, x ) = e − λt ξ j ( t, x ), we classically assume, without loss of generality, that for all r ≥ s (2.7) G j ( τ, V, r, a, q ) − G j ( τ, V, s, a, q ) ≤ − L ′ ( r − s )for L ′ ≥ L > M = sup ( t,x ) ∈ (0 ,T ) × R max j ∈{ ,...,n } max ( u j ( t, x ) − v j ( t, x ) , ξ j ( t, x ) − ζ j ( t, x )) . The proof proceeds in several steps.
Step 1: The test function
We argue by contradiction by assuming that
M >
0. Classically, we duplicate the space variable by consid-ering for ε, α and η “small” positive parameters, the functions ϕ ( t, x, y, j ) = u j ( t, x ) − v j ( t, y ) − e At | x − y | ε − α | x | − ηT − tφ ( t, x, y, j ) = ξ j ( t, x ) − ζ j ( t, y ) − e At | x − y | ε − α | x | − ηT − t where A is a positive constant which will be chosen later. We also considerΨ( t, x, y, j ) = max( ϕ ( t, x, y, j ) , φ ( t, x, y, j )) . Using Inequalities (2.6) and Assumption (A0’), we get u j ( t, x ) − v j ( t, y ) ≤ u ,j ( x ) − u ,j ( y ) + 2 K (1 + T ) ≤ K | x − y | + 2 K (1 + T )and ξ j ( t, x ) − ζ j ( t, y ) ≤ K | x − y | + 2 K (1 + T ) . We then deduce that lim | x | , | y |→∞ ϕ ( t, x, y, j ) = lim | x | , | y |→∞ φ ( t, x, y, j ) = −∞ , Using also the fact that ϕ and φ are u.s.c, we deduce that Ψ reaches its maximum at some point (¯ t, ¯ x, ¯ y, ¯ j ).Let us assume that Ψ(¯ t, ¯ x, ¯ y, ¯ j ) = φ (¯ t, ¯ x, ¯ y, ¯ j ) (the other case being similar and even simpler). Using thefact that M >
0, we first remark that for α and η small enough, we haveΨ(¯ t, ¯ x, ¯ y, ¯ j ) =: M ε,α,η ≥ M > . In particular, ξ ¯ j (¯ t, ¯ x ) − ζ ¯ j (¯ t, ¯ y ) > . Step 2: Viscosity inequalities for ¯ t > a, b, ¯ p ∈ R such that a − b = η ( T − ¯ t ) + Ae A ¯ t | ¯ x − ¯ y | ε , ¯ p = e A ¯ t ¯ x − ¯ yε a ≤ G ¯ j (¯ t, [ u (¯ t, · )] ¯ j,m (¯ x ) , ξ ¯ j (¯ t, ¯ x ) , inf( ξ ¯ j (¯ t, y ′ ) − py ′ ) + p ¯ x − ξ ¯ j (¯ t, ¯ x ) , ¯ p + 2 α ¯ x ) b ≥ G ¯ j (¯ t, [ v (¯ t, · )] ¯ j,m (¯ y ) , ζ ¯ j (¯ t, ¯ y ) , inf( ζ ¯ j (¯ t, y ′ ) − py ′ ) + p ¯ y − ζ ¯ j (¯ t, ¯ y ) , ¯ p ) . Subtracting the two above inequalities, we get ηT + Ae A ¯ t | ¯ x − ¯ y | ε ≤ G ¯ j (¯ t, [ u (¯ t, · )] ¯ j,m (¯ x ) , ξ ¯ j (¯ t, ¯ x ) , inf( ξ ¯ j (¯ t, y ′ ) − py ′ ) + p ¯ x − ξ ¯ j (¯ t, ¯ x ) , ¯ p + 2 α ¯ x ) − G ¯ j (¯ t, [ v (¯ t, · )] ¯ j,m (¯ y ) , ζ ¯ j (¯ t, ¯ y ) , inf( ζ ¯ j (¯ t, y ′ ) − py ′ ) + p ¯ y − ζ ¯ j (¯ t, ¯ y ) , ¯ p ) =: ∆ G j . (2.8) Step 3: Estimate on u k (¯ t, ¯ x ) − v k (¯ t, ¯ y )If k ∈ { , . . . , n } , by the inequality ϕ (¯ t, ¯ x, ¯ y, k ) ≤ φ (¯ t, ¯ x, ¯ y, ¯ j ), we directly get that u k (¯ t, ¯ x ) − v k (¯ t, ¯ y ) ≤ ξ ¯ j (¯ t, ¯ x ) − ζ ¯ j (¯ t, ¯ y ) . If k
6∈ { , . . . , n } , let us define l k ∈ Z such that k − l k n = ˜ k ∈ { , . . . , n } . By periodicity, we then have u k (¯ t, ¯ x ) − v k (¯ t, ¯ y ) = u ˜ k + l k n (¯ t, ¯ x ) − v ˜ k + l k n (¯ t, ¯ y )= u ˜ k (¯ t, ¯ x + l k ) − v ˜ k (¯ t, ¯ y + l k ) ≤ ξ ¯ j (¯ t, ¯ x ) − ζ ¯ j (¯ t, ¯ y ) − α ( | ¯ x | − | ¯ x + l k | )where we have used the inequality ϕ (¯ t, ¯ x + l k , ¯ y + l k , ˜ k ) ≤ φ (¯ t, ¯ x, ¯ y, ¯ j ) to get the third line. Hence, for all k ∈ Z (and in particular for k ∈ { ¯ j − m, . . . , ¯ j + m } ), we finally deduce that(2.9) u k (¯ t, ¯ x ) − v k (¯ t, ¯ y ) ≤ ξ ¯ j (¯ t, ¯ x ) − ζ ¯ j (¯ t, ¯ y ) + α (cid:12)(cid:12) | ¯ x | − | ¯ x + l k | (cid:12)(cid:12) . Step 4: Estimate of ∆ G j in (2.8)Using successively (2.9) and (A1’)(ii), we obtain∆ G j ≤ G ¯ j (cid:18) ¯ t, (cid:2) v (¯ t, · ) + ξ ¯ j (¯ t, ¯ x ) − ζ ¯ j (¯ t, ¯ y ) + α (cid:12)(cid:12) | ¯ x | − | ¯ x + l · | (cid:12)(cid:12)(cid:3) ¯ j,m (¯ y ) , ξ ¯ j (¯ t, ¯ x ) , inf( ξ ¯ j (¯ t, y ′ ) − py ′ ) + p ¯ x − ξ ¯ j (¯ t, ¯ x ) , ¯ p + 2 α ¯ x (cid:19) − G ¯ j (cid:0) ¯ t, [ v (¯ t, · )] ¯ j,m (¯ y ) , ζ ¯ j (¯ t, ¯ y ) , inf( ζ ¯ j (¯ t, y ′ ) − py ′ ) + p ¯ y − ζ ¯ j (¯ t, ¯ y ) , ¯ p (cid:1) ≤ L ( ξ ¯ j (¯ t, ¯ x ) − ζ ¯ j (¯ t, ¯ y )) + L α max k ∈{ ¯ j − m,..., ¯ j + m } (cid:12)(cid:12) | ¯ x | − | ¯ x + l k | (cid:12)(cid:12) + G ¯ j (cid:0) ¯ t, [ v (¯ t, · )] ¯ j,m (¯ y ) , ξ ¯ j (¯ t, ¯ x ) , inf( ξ ¯ j (¯ t, y ′ ) − py ′ ) + p ¯ x − ξ ¯ j (¯ t, ¯ x ) , ¯ p + 2 α ¯ x (cid:1) − G ¯ j (cid:0) ¯ t, [ v (¯ t, · )] ¯ j,m (¯ y ) , ζ ¯ j (¯ t, ¯ y ) , inf( ζ ¯ j (¯ t, y ′ ) − py ′ ) + p ¯ y − ζ ¯ j (¯ t, ¯ y ) , ¯ p (cid:1) . G j ≤ L ( ξ ¯ j (¯ t, ¯ x ) − ζ ¯ j (¯ t, ¯ y )) + L α max k ∈{ ¯ j − m,..., ¯ j + m } (cid:12)(cid:12) | ¯ x | − | ¯ x + l k | (cid:12)(cid:12) − L ′ ( ξ ¯ j (¯ t, ¯ x ) − ζ ¯ j (¯ t, ¯ y ))(2.10) + G ¯ j (cid:0) ¯ t, [ v (¯ t, · )] ¯ j,m (¯ y ) , ζ ¯ j (¯ t, ¯ y ) , inf( ξ ¯ j (¯ t, y ′ ) − py ′ ) + p ¯ x − ξ ¯ j (¯ t, ¯ x ) , ¯ p + 2 α ¯ x (cid:1) − G ¯ j (cid:0) ¯ t, [ v (¯ t, · )] ¯ j,m (¯ y ) , ζ ¯ j (¯ t, ¯ y ) , inf( ζ ¯ j (¯ t, y ′ ) − py ′ ) + p ¯ y − ζ ¯ j (¯ t, ¯ y ) , ¯ p (cid:1) ≤ Lα max k ∈{ ¯ j − m,..., ¯ j + m } (2 | l k ¯ x | + l k )+ L (cid:18) inf( ξ ¯ j (¯ t, y ′ ) − py ′ ) + p ¯ x − ξ ¯ j (¯ t, ¯ x ) − inf( ζ ¯ j (¯ t, y ′ ) − py ′ ) − p ¯ y + ζ ¯ j (¯ t, ¯ y ) (cid:19) + | ¯ p | + G ¯ j (cid:0) ¯ t, [ v (¯ t, · )] ¯ j,m (¯ y ) , ζ ¯ j (¯ t, ¯ y ) , inf( ξ ¯ j (¯ t, y ′ ) − py ′ ) + p ¯ y − ξ ¯ j (¯ t, ¯ x ) , ¯ p + 2 α ¯ x (cid:1) − G ¯ j (cid:0) ¯ t, [ v (¯ t, · )] ¯ j,m (¯ y ) , ζ ¯ j (¯ t, ¯ y ) , inf( ξ ¯ j (¯ t, y ′ ) − py ′ ) + p ¯ y − ξ ¯ j (¯ t, ¯ x ) , ¯ p (cid:1) . Using the fact that α | ¯ x | → α →
0, we deduce that Lα max k (2 | l k ¯ x | + l k )+ G ¯ j (cid:0) ¯ t, [ v (¯ t, · )] ¯ j,m (¯ y ) , ζ ¯ j (¯ t, ¯ y ) , inf( ξ ¯ j (¯ t, y ′ ) − py ′ ) + p ¯ y − ξ ¯ j (¯ t, ¯ x ) , ¯ p + 2 α ¯ x (cid:1) − G ¯ j (cid:0) ¯ t, [ v (¯ t, · )] ¯ j,m (¯ y ) , ζ ¯ j (¯ t, ¯ y ) , inf( ξ ¯ j (¯ t, y ′ ) − py ′ ) + p ¯ y − ξ ¯ j (¯ t, ¯ y ) , ¯ p (cid:1) = o α (1)where we have used (2.3) to get a uniform bound R > ξ ¯ j (¯ t, y ′ ) − py ′ ) + p ¯ y − ξ ¯ j (¯ t, ¯ y ). Step 5: Passing to the limit
Using the fact that φ (¯ t, y ′ , y ′ , ¯ j ) ≤ φ (¯ t, ¯ x, ¯ y, ¯ j ), we deduce that ξ ¯ j (¯ t, y ′ ) − ξ ¯ j (¯ t, ¯ x ) ≤ ζ ¯ j (¯ t, y ′ ) − ζ ¯ j (¯ t, ¯ y ) + α | y ′ | . Combining this with the previous step, we get ηT + Ae A ¯ t | ¯ x − ¯ y | ε ≤ L (cid:18) inf( ζ ¯ j (¯ t, y ′ ) − py ′ − ζ ¯ j (¯ t, ¯ y ) + α | y ′ | )(2.11) − inf( ζ ¯ j (¯ t, y ′ ) − py ′ − ζ ¯ j (¯ t, ¯ y )) (cid:19) + | ¯ p | + p (¯ x − ¯ y ) | ¯ p | + o α (1) ≤ L (cid:18) inf( ζ ¯ j (¯ t, y ′ ) − py ′ + α | y ′ | ) − inf( ζ ¯ j (¯ t, y ′ ) − py ′ ) (cid:19) + | ¯ p | + pe A ¯ t | ¯ x − ¯ y | ε + o α (1) . Choosing A = 2 p , we finally get ηT ≤ o α (1) + (cid:0) inf( ζ ¯ j (¯ t, y ′ ) − py ′ + α | y ′ | ) − inf( ζ ¯ j (¯ t, y ′ ) − py ′ ) (cid:1) | ¯ p | . Using the fact that for ¯ p = O (1) when α → O (1) depends on ε which is fixed) and usingclassical arguments about inf-convolution, we get that (cid:0) inf( ζ ¯ j (¯ t, y ′ ) − py ′ + α | y ′ | ) − inf( ζ ¯ j (¯ t, y ′ ) − py ′ ) (cid:1) | ¯ p | = o α (1)and so ηT ≤ o α (1)12hich is a contradiction for α small enough. Step 6: Case ¯ t = 0We assume that there exists a sequence ε n → t = 0. In this case, we have0 < M ≤ M ε n ,α,η ≤ ξ (¯ x ) − ξ (¯ y ) − | ¯ x − ¯ y | ε n − α | x | ≤ ξ (¯ x ) − ξ (¯ y ) ≤ k Dξ k L ∞ | ¯ x − ¯ y | . Using the fact that | ¯ x − ¯ y | → ε n → τ , y ) ∈ (0 , T ) × R and for all r, R >
0, let us set Q r,R = ( τ − r, τ + r ) × ( y − R, y + R ) . We then have the following result which proof is similar to the one of Proposition 2.2
Proposition 2.4 ( Comparison principle on bounded sets).
Assume (A1’)-(A5’) and that G j ( τ, V, r, a, q ) does not depend on the variable a for each j . Assume that (( u j ) j , ( ξ j ) j ) is a sub-solution (resp. (( v j ) j , ( ζ ) j ) a super-solution) of (2.1) on the open set Q r,R ⊂ (0 , T ) × R .Assume also that for all j ∈ { , . . . , n } u j ≤ v j and ξ j ≤ ζ j on ( Q r,R + m \ Q r,R ) . Then u j ≤ v j and ξ j ≤ ζ j on Q r,R for j ∈ { , . . . , n } . We now turn to the existence issue. Classically, we need to construct barriers for (2.1). In view of(A1’)(ii) and (A4’), for K given in (A0), the following quantity(2.12) G = sup τ ∈ R , | q |≤ K , j ∈{ ,...,n } | G j ( τ, , , , q ) | is finite. Let us also denote L := L K . Hence, for all τ, a, b, r ∈ R , V ∈ R m +1 , q ∈ [ − K , K ] and j ∈ { , . . . , n } ,(2.13) | G j ( τ, V, r, a, q ) − G j ( τ, V, r, b, q ) | ≤ L | a − b | . Then we have the following lemma
Lemma 2.5 ( Existence of barriers).
Assume (A0’)-(A5’) . There exists a constant K > such that (( u + j ( τ, y )) j , ( ξ + j ( τ, y )) j ) = (( u ( y + jn ) + K τ ) j , ( ξ ( y + jn ) + K τ ) j ) and (( u − j ( τ, y )) j , ( ξ − j ( τ, y )) j ) = (( u ( y + jn ) − K τ ) j , ( ξ ( y + jn ) − K τ ) j ) are respectively super and sub-solution of (2.1) , (2.2) for all T > . Moreover, we can choose (2.14) K = max (cid:16) L C + L (cid:16) K mn + M (cid:17) + G, α M (cid:17) where C , ( K , M ) and G are respectively given in (2.3) , (A0’) and (2.12) . roof. We prove that (( u + j ( τ, y )) j , ( ξ + j ( τ, y )) j ) is a super-solution of (2.1), (2.2). In view of (A0) with ε = 1,we have for all j ∈ { , . . . , n } α ( ξ + j ( τ, y ) − u + j ( τ, y )) = α ( u ( y + jn ) − ξ ( y + jn )) ≤ α M ≤ K and G j (cid:18) τ, [ u + ( τ, · )] j,m ( y ) , ξ + j ( τ, y ) , inf y ′ ∈ R (cid:0) ξ + j ( τ, y ′ ) − py ′ (cid:1) + py − ξ + j ( τ, y ) , ( ξ + j ) y ( τ, y ) (cid:19) = G j (cid:18) τ, [ u + ( τ, · ) − ⌊ u + j ( τ, y ) ⌋ ] j,m ( y ) , ξ + j ( τ, y ) − ⌊ u + j ( τ, y ) ⌋ , inf y ′ ∈ R (cid:18) ξ ( y ′ + jn ) − py ′ (cid:19) + py − ξ ( y + jn ) , ( ξ ) y ( y + jn ) (cid:19) ≤ L C + L + L + G j (cid:18) τ, [ u + ( τ, · ) − u + j ( τ, y )] j,m ( y ) , ξ + j ( τ, y ) − u + j ( τ, y ) , , ( ξ ) y ( y + jn ) (cid:19) ≤ L C + L + L + L K mn + L M + G j (cid:18) τ, , . . . , , , , ( ξ ) y ( y + jn ) (cid:19) ≤ L C + 2 L + L K mn + L M + G where we have used the periodicity assumption (A4’) for the second line, assumptions (A0’) and (A1’)(ii)for the third line, the fact that | u ( y + j + kn ) − u ( y + jn ) | ≤ K mn for | k | ≤ m and assumption (A0’) for theforth line and | ( ξ + j ) y | ≤ K for the last line.When G j ( τ, V, r, a, q ) is independent on a , we can simply choose L = 0. This ends the proof of theLemma.By applying Perron’s method together with the comparison principle, we immediately get from theexistence of barriers the following result Theorem 2.6 ( Existence and uniqueness for (2.1) ). Assume (A0’)-(A5’) . Then there exists a uniquesolution (( u j ) j , ( ξ j ) j ) of (2.1) , (2.2) . Moreover the functions u j , ξ j are continuous for all j . We now claim that particles are ordered.
Proposition 2.7 ( Ordering of the particles).
Assume (A0’) and that the ( G j ) j ’s satisfy (A1’)-(A6’) .Let ( u j , ξ j ) be a solution of (2.1) - (2.2) such that (2.3) holds true for ξ j if G j depends on a . Assume alsothat the u j ’s are Lipschitz continuous in space and let L u denote a common Lipschitz constant. Then u j and ξ j are non-decreasing with respect to j .Proof of Proposition 2.7. The idea of the proof is to define ( v j , ζ j ) = ( u j +1 , ξ j +1 ). In particular, we have( v j (0 , y ) , ζ j (0 , y )) ≥ ( u j (0 , y ) , ξ j (0 , y )) . Moreover, (( v j ) j , ( ζ j ) j ) is a solution of (cid:26) ( v j ) τ = α ( ζ j − v j ) , ( ζ j ) τ = G j +1 ( τ, [ v ( τ, · )] j,m , ζ j , inf y ′ ∈ R ( ζ j ( τ, y ′ ) − py ′ ) + py − ζ j ( τ, y ) , ( ζ j ) y ) , (cid:26) v j + n ( τ, y ) = v j ( τ, y + 1) ,ζ j + n ( τ, y ) = ζ j ( τ, y + 1) (cid:26) v j (0 , y ) = u ( y + jn ) ,ζ j (0 , y ) = ξ ( y + jn ) . u j ≤ v j and ξ j ≤ ζ j . The arguments are essentially the same as those used in theproof of the comparison principle. The main difference is that (2.8) is replaced with ηT + Ae A ¯ t | ¯ x − ¯ y | ε ≤ G ¯ j (¯ t, [ u (¯ t, · )] ¯ j,m (¯ x ) , ξ ¯ j (¯ t, ¯ x ) , inf( ξ ¯ j (¯ t, y ′ ) − py ′ ) + p ¯ x − ξ ¯ j (¯ t, ¯ x ) , ¯ p + 2 α ¯ x ) − G ¯ j +1 (¯ t, [ v (¯ t, · )] ¯ j,m (¯ y ) , ζ ¯ j (¯ t, ¯ y ) , inf( ζ ¯ j (¯ t, y ′ ) − py ′ ) + p ¯ y − ζ ¯ j (¯ t, ¯ y ) , ¯ p ) ≤ G ¯ j (¯ t, [ u (¯ t, · )] ¯ j,m (¯ y ) , ξ ¯ j (¯ t, ¯ x ) , inf( ξ ¯ j (¯ t, y ′ ) − py ′ ) + p ¯ x − ξ ¯ j (¯ t, ¯ x ) , ¯ p + 2 α ¯ x ) − G ¯ j +1 (¯ t, [ v (¯ t, · )] ¯ j,m (¯ y ) , ζ ¯ j (¯ t, ¯ y ) , inf( ζ ¯ j (¯ t, y ′ ) − py ′ ) + p ¯ y − ζ ¯ j (¯ t, ¯ y ) , ¯ p ) + L L u | ¯ x − ¯ y | =: ∆ G j where we have used the Lipschitz continuity of u and Assumption (A1’).To obtain the desired contradiction, we have to estimate the right hand side of this inequality. First,using Step 3 of the proof of the comparison principle (with the same notation), we can define δ := ξ ¯ j (¯ t, ¯ x ) − ζ ¯ j (¯ t, ¯ y ) + L u | ¯ x − ¯ y | + α max k ∈{ ¯ j − m,..., ¯ j + m } (2 | l k ¯ x | + l k ) ≥ k ∈ { ¯ j − m, . . . , ¯ j + m } , we get from (2.9) the following estimate(2.15) u k (¯ t, ¯ y ) − v k (¯ t, ¯ y ) ≤ δ. Using Monotonicity Assumptions (A2’)-(A3’) together with (A1’), we get∆ G j ≤ G ¯ j (¯ t, [ u (¯ t, ¯ y ) + ( · − ¯ j ) δ ] ¯ j,m , ξ ¯ j (¯ t, ¯ x ) , inf( ξ ¯ j (¯ t, y ′ ) − py ′ ) + p ¯ x − ξ ¯ j (¯ t, ¯ x ) , ¯ p + 2 α ¯ x ) − G ¯ j +1 (¯ t, [ v (¯ t, ¯ y ) + ( · + 1) δ ] ¯ j,m , ζ ¯ j (¯ t, ¯ y ) , inf( ζ ¯ j (¯ t, y ′ ) − py ′ ) + p ¯ y − ζ ¯ j (¯ t, ¯ y ) , ¯ p )+ L (2 m + 1) δ + L L u | ¯ x − ¯ y | . Now we are going to use assumption (A6’). Remark first that we have for all k ∈ {− m, m − } v ¯ j + k (¯ t, ¯ y ) + ( m + k + 1) δ = u ¯ j + k +1 (¯ t, ¯ y ) + ( m + k + 1) δ and for k ∈ {− m, . . . , m } , (2.15) yields u ¯ j + k +1 (¯ t, ¯ y ) + ( m + k + 1) δ ≥ u ¯ j + k (¯ t, ¯ y ) + ( m + k ) δ . Thus (A6’) implies that(2.16) G ¯ j (¯ t, [ u (¯ t, · )] ¯ j,m (¯ y ) , ξ ¯ j (¯ t, ¯ x ) , inf( ξ ¯ j (¯ t, y ′ ) − py ′ ) + p ¯ x − ξ ¯ j (¯ t, ¯ x ) , ¯ p + 2 α ¯ x ) ≤ G ¯ j +1 (¯ t, [ v (¯ t, ¯ y ) + ( · + 1) δ ] ¯ j,m , ξ ¯ j (¯ t, ¯ x ) , inf( ξ ¯ j (¯ t, y ′ ) − py ′ ) + p ¯ x − ξ ¯ j (¯ t, ¯ x ) , ¯ p + 2 α ¯ x ) . Hence∆ G j ≤ G ¯ j +1 (¯ t, [ v (¯ t, ¯ y ) + ( · + 1) δ ] ¯ j,m , ξ ¯ j (¯ t, ¯ x ) , inf( ξ ¯ j (¯ t, y ′ ) − py ′ ) + p ¯ x − ξ ¯ j (¯ t, ¯ x ) , ¯ p + 2 α ¯ x ) − G ¯ j +1 (¯ t, [ v (¯ t, ¯ y ) + ( · + 1) δ ] ¯ j,m , ζ ¯ j (¯ t, ¯ y ) , inf( ζ ¯ j (¯ t, y ′ ) − py ′ ) + p ¯ y − ζ ¯ j (¯ t, ¯ y ) , ¯ p )+ L (2 m + 1)( ξ ¯ j (¯ t, ¯ x ) − ζ ¯ j (¯ t, ¯ y )) + 2( m + 1) L L u | ¯ x − ¯ y | + L (2 m + 1) α max k ∈{ ¯ j − m,..., ¯ j + m } (2 | l k ¯ x | + l k ) . Now, to obtain the desired contradiction, it suffices to follow the computation from (2.10); in particular,choose L ′ ≥ (2 m + 1) L in (2.7). Then we obtain ηT ≤ o α (1) + 2( m + 1) L L u | ¯ x − ¯ y | which is absurd for α and ε small enough (since | ¯ x − ¯ y | → ε → Convergence
This section is devoted to the proof of the main homogenization result (Theorem 1.5). The proof relieson the existence of hull functions (Theorem 1.10) and qualitative properties of the effective Hamiltonian(Theorem 1.11). As a matter of fact, we will use the existence of Lipschitz continuous sub- and super-hullfunctions (see Proposition 5.2). All these results are proved in the next sections.We start with some preliminary results. Through a change of variables, the following result is a straight-forward corollary of Lemma 2.5 and the comparison principle.
Lemma 3.1 ( Barriers uniform in ε ). Assume (A0)-(A5) . Then there is a constant
C > , such that forall ε > , the solution (( u εj ) j , ( ξ εj )) of (1.8) , (1.9) satisfies for all t > and x ∈ R | u εj ( t, x ) − u ( x + jεn ) | ≤ Ct and | ξ εj ( t, x ) − ξ ε ( x + jεn ) | ≤ Ct.
We also have the following preliminary lemma.
Lemma 3.2 ( ε -bounds on the gradient). Assume (A0)-(A5) . Then the solution (( u ε ) j , ( ξ εj ) j ) of (1.8) , (1.9) satisfies for all t > , x ∈ R , z > and j ∈ Z (3.1) ε (cid:22) zεK (cid:23) ≤ u εj ( t, x + z ) − u εj ( t, x ) ≤ ε (cid:24) zK ε (cid:25) and ε (cid:22) zεK (cid:23) ≤ ξ εj ( t, x + z ) − ξ εj ( t, x ) ≤ ε (cid:24) zK ε (cid:25) . Remark 3.3.
In particular we obtain that functions u εj ( t, x ) and ξ εj ( t, x ) are non-decreasing in x . Proof of Lemma 3.2.
We prove the bound from below (the proof is similar for the bound from above). Wefirst remark that (A0) implies that the initial condition satisfies for all j ∈ Z (3.2) u εj (0 , x + z ) = u ( x + z + jεn ) ≥ u ( x + jεn ) + z/K ≥ u εj (0 , x ) + kε with k = (cid:22) zεK (cid:23) and ξ εj (0 , x + z ) ≥ ξ εj (0 , x ) + kε . From (A4), we know that for ε = 1, the equation is invariant by addition of integers to solutions. Afterrescaling it, Equation (1.8) is invariant by addition of constants of the form kε , k ∈ Z . For this reason thesolution of (1.8) associated with initial data (( u εj (0 , x ) + kε ) j , ( ξ εj (0 , x ) + kε ) j ) is (( u εj + kε ) j , ( ξ εj + kε ) j ).Similarly the equation is invariant by space translations. Therefore the solution with initial data (( u εj (0 , x + z )) j , ( ξ εj (0 , x + z ) j ) is (( u εj ( t, x + z )) j , ( ξ εj ( t, x + z )) j ). Finally, from (3.2) and the comparison principle(Proposition 2.2), we get u εj ( t, x + z ) ≥ u εj ( t, x ) + kε and ξ εj ( t, x + z ) ≥ ξ εj ( t, x ) + kε which proves the bound from below. This ends the proof of the lemma.We now turn to the proof of Theorem 1.5. Proof of Theorem 1.5.
We only have to prove the result for all j ∈ { , . . . , n } . Indeed, using the fact that u εj + n ( t, x ) = u εj ( t, x + ε ) and ξ εj + n ( t, x ) = ξ εj ( t, x + ε ), we will get the complete result.For all j ∈ { , . . . , n } , we introduce the following half-relaxed limits u j = lim sup ε → ∗ u εj , ξ j = lim sup ε → ∗ ξ εj j = lim inf ε → ∗ u εj , ξ j = lim inf ε → ∗ ξ εj . These functions are well defined thanks to Lemma 3.1. We then define v = max j ∈{ ,...,n } max( u j , ξ j ) , v = min j ∈{ ,...,n } min( u j , ξ j ) . We get from Lemmas 3.1 and 3.2 that both functions w = v, v satisfy for all t > x, x ′ ∈ R , x ≤ x ′ (recallthat ξ ε → u as ε → | w ( t, x ) − u ( x ) | ≤ Ct ,K − | x − x ′ | ≤ w ( t, x ) − w ( t, x ′ ) ≤ K | x − x ′ | . (3.3)We are going to prove that v is a sub-solution of (1.5). Similarly, we can prove that v is a super-solution ofthe same equation. Therefore, from the comparison principle for (1.5), we get that u ≤ v ≤ v ≤ u . Andthen v = v = u , which shows the expected convergence of the full sequence u εj and ξ εj towards u for all j ∈ { , . . . , n } .We now prove in several steps that v is a sub-solution of (1.5). We classically argue by contradiction: weassume that there exists ( t, x ) ∈ (0 , + ∞ ) × R and a test function φ ∈ C such that(3.4) v ( t, x ) = φ ( t, x ) v ≤ φ on Q r, r ( t, x ) , with r > v ≤ φ − η on Q r, r ( t, x ) \ Q r,r ( t, x ) , with η > φ t ( t, x ) = F ( φ x ( t, x )) + θ, with θ > . Let p denote φ x ( t, x ). From (3.3), we get(3.5) 0 < /K ≤ p ≤ K . Combining Theorems 1.10 and 1.11, we get the existence of a hull function (( h i ) i , ( g i ) i ) associated with p such that λ = F ( p ) + θ F ( L, p ) with
L > . Indeed, we know from these results that the effective Hamiltonian is non-decreasing in L , continuous andgoes to ±∞ as L → ±∞ .We now apply the perturbed test function method introduced by Evans [9] in terms here of hull functionsinstead of correctors. Precisely, let us consider the following twisted perturbed test functions for i ∈ { , . . . , n } φ εi ( t, x ) = εh i (cid:18) tε , φ ( t, x ) ε (cid:19) , ψ εi ( t, x ) = εg i (cid:18) tε , φ ( t, x ) ε (cid:19) . Here the test functions are twisted in the same way as in [14]. We then define the family of perturbed testfunctions ( φ εi ) i ∈ Z , (( ψ εi ) i ∈ Z ) by using the following relation φ εi + kn ( t, x ) = φ εi ( t, x + εk ) , ψ εi + kn ( t, x ) = ψ εi ( t, x + εk ) . In order to get a contradiction, we first assume that the functions h i and g i are C and continuous in z uniformly in τ ∈ R , i ∈ { , . . . , n } . In view of the third line of (1.10), we see that this implies that h i and g i are uniformly continuous in z (uniformly in τ ∈ R , i ∈ { , . . . , n } ). For simplicity, and since we willconstruct approximate hull functions with such a (Lipschitz) regularity, we even assume that h i and g i areglobally Lipschitz continuous in z (uniformly in τ ∈ R , i ∈ { , . . . , n } ). We will next see how to treat thegeneral case. Case 1: h i and g i are C and globally Lipschitz continuous in z tep 1.1: (( φ εi ) i , ( ψ εi ) i ) is a super-solution of (1.8) in a neighbourhood of ( t, x )When h i and g i are C , it is sufficient to check directly the super-solution property of ( φ εi , ψ εi ) for ( t, x ) ∈ Q r,r ( t, x ). We begin by the equation satisfied by φ εi . We have, with τ = t/ε and z = φ ( t, x ) /ε ,( φ εi ) t ( t, x ) =( h i ) τ ( τ, z ) + φ t ( t, x )( h i ) z ( τ, z )=( φ t ( t, x ) − λ )( h i ) z ( τ, z ) + α ( g i ( τ, z ) − h i ( τ, z ))= (cid:18) φ t ( t, x ) − φ t ( t, x ) + θ (cid:19) ( h i ) z ( τ, z ) + α ε ( ψ εi ( t, x ) − φ εi ( t, x )) ≥ α ε ( ψ εi ( t, x ) − φ εi ( t, x ))(3.6)where we have used the equation satisfied by h i to get the second line and the non-negativity of h z , the factthat θ > φ is C , to get the last line on Q r,r ( t, x ) for r > ψ i . With the same notation, we have( ψ εi ) t ( t, x ) − F i τ, (cid:20) φ ε ( t, · ) ε (cid:21) i,m ( x ) ! − α ε ( φ εi − ψ εi )(3.7) =( g i ) τ ( τ, z ) + φ t ( t, x )( g i ) z ( τ, z ) − F i τ, (cid:20) φ ε ( t, · ) ε (cid:21) i,m ( x ) ! − α ( h i ( τ, z ) − g i ( τ, z ))=( φ t ( t, x ) − λ ) ( g i ) z ( τ, z ) + 2 L + 2 F i (cid:16) τ, [ h ( τ, · )] i,m ( z ) (cid:17) − F i τ, (cid:20) φ ε ( t, · ) ε (cid:21) i,m ( x ) !! ≥ ( φ t ( t, x ) − λ ) ( g i ) z ( τ, z ) + 2 L − L F (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ h ( τ, · )] i,m ( z ) − (cid:20) φ ε ( t, · ) ε (cid:21) i,m ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ where we have used that Equation (1.10) is satisfied by ( g i ) i to get the third line and (A1) to get the fourthone; here, L F denotes the largest Lipschitz constants of the F i ’s (for i ∈ { , . . . , n } ) with respect to V .Let us next estimate, for i ∈ { , . . . , n } , j ∈ {− m, . . . , m } and ε > I i,j = h i + j ( τ, z ) − φ εi + j ( t, x ) ε If i + j ∈ { , . . . , n } , then, by definition of φ i + j , we have I i,j = h i + j (cid:18) tε , φ ( t, x ) ε (cid:19) − φ εi + j ( t, x ) ε = 0 . If i + j
6∈ { , . . . , n } , let us define l such that 1 ≤ i + j − ln ≤ n . We then have I i,j = h i + j − ln ( τ, z + lp ) − φ εi + j − ln ( t, x + εl ) ε = h i + j − ln (cid:18) tε , φ ( t, x ) ε + lp (cid:19) − h i + j − ln (cid:18) tε , φ ( t, x + εl ) ε (cid:19) = h i + j − ln (cid:18) tε , φ ( t, x ) ε + lp (cid:19) − h i + j − ln (cid:18) tε , φ ( t, x ) ε + lp + o r (1) (cid:19) where o r (1) only depends on the modulus of continuity of φ x on Q r,r ( t, x ) (for ε small enough such that εl ≤ r with l uniformly bounded and then ( t, x + εl ) ∈ Q r, r ( t, x )). Hence, if h i are Lipschitz continuouswith respect to z uniformly in τ and i , we conclude that we can choose ε small enough so that(3.8) L − L F (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ h ( τ, · )] i,m ( z ) − (cid:20) φ ε ( t, · ) ε (cid:21) i,m ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ ≥ . ψ εi ) t ( t, x ) − F i τ, (cid:20) φ ε ( t, x ) ε (cid:21) i,m ( x ) ! + α ε ( φ εi − ψ εi ) ≥ ( φ t ( t, x ) − λ ) ( g i ) z ( τ, z ) ≥ (cid:18) θ φ t ( t, x ) − φ t ( t, x ) (cid:19) ( g i ) z ( τ, z )= (cid:18) θ o r (1) (cid:19) ( g i ) z ( τ, z ) ≥ . We used the non-negativity of ( g i ) z , the fact that θ > φ is C , to get the resulton Q r,r ( t, x ) for r > h i and g i are C and Lipschitz continuous on z uniformly in τ and i , (( φ εi ) i , ( ψ εi ) i ) is a viscosity super-solution of (1.8) on Q r,r ( t, x ). Step 1.2: getting the contradiction
By construction (see Remark 1.9), we have φ εi → φ and ψ εi → φ as ε → i ∈ { , . . . , n } , and thereforefrom the fact that u j ≤ ¯ v ≤ φ − η on Q r, r ( t, x ) \ Q r,r ( t, x ) (see (3.4)), we get for ε small enough u εi ≤ φ εi − η ≤ φ εi − εk ε on Q r, r ( t, x ) \ Q r,r ( t, x )with the integer k ε = ⌊ η/ε ⌋ . In the same way, we have ξ εi ≤ ψ εi − η ≤ ψ εi − εk ε on Q r, r ( t, x ) \ Q r,r ( t, x ) . Therefore, for mε ≤ r , we can apply the comparison principle on bounded sets to get(3.9) u εi ≤ φ εi − εk ε , ξ εi ≤ ψ εi − εk ε on Q r,r ( t, x ) . Passing to the limit as ε goes to zero, we get u i ≤ φ − η, ξ i ≤ φ − η on Q r,r ( t, x )which implies that v ≤ φ − η on Q r,r ( t, x ) . This gives a contradiction with v ( t, x ) = φ ( t, x ) in (3.4). Therefore v is a sub-solution of (1.5) on (0 , + ∞ ) × R and we get that u εj and ξ εj converges locally uniformly to u for j ∈ { , . . . , n } . This ends the proof of thetheorem. Case 2: general case for h In the general case, we can not check by a direct computation that (( φ εi ) i , ( ψ εi ) i ) is a super-solution on Q r,r ( t, x ). The difficulty is due to the fact that the h i and the g i may not be Lipschitz continuous in thevariable z .This kind of difficulties were overcome in [14] by using Lipschitz super-hull functions, i.e. functionssatisfying (1.10), except that the function is only a super-solution of the equation appearing in the first line.Indeed, it is clear from the previous computations that it is enough to conclude. In [14], such regular super-hull functions (as a matter of fact, regular super-correctors) were built as exact solutions of an approximateHamilton-Jacobi equation. Moreover this Lipschitz continuous hull function is a super-solution for the exactHamiltonian with a slightly bigger λ .Here we conclude using a similar result, namely Proposition 5.2. Notice that in Proposition 5.2 h i and g i are only Lipschitz continuous and not C . This is not a restriction, because the result of Step 1.1 can bechecked in the viscosity sense using test function (see [9] for further details). Comparing with [14], noticethat we do not have to introduce an additional dimension because here p > Ergodicity and construction of hull functions
In this section, we first study the ergodicity of the equation (2.1) by studying the associated Cauchy problem(Subsection 4.1). We then construct hull functions (Subsection 4.2).
In this subsection, we study the Cauchy problem associated with (2.1) with(4.1) G j ( τ, V, r, a, q ) = G δj ( τ, V, r, a, q ) = 2 F j ( τ, V ) + α ( V − r ) + δ ( a + a ) q + with δ ≥ a ∈ R and with initial data y py . We prove that there exists a real number λ (called the“slope in time” or “rotation number”) such that the solution ( u j , ξ j ) stays at a finite distance of the linearfunction λτ + py . We also estimate this distance and give qualitative properties of the solution.We begin by a regularity result concerning the solution of (2.1). Proposition 4.1 ( Bound on the gradient).
Assume (A1)-(A5) and p > . Let δ > , a ∈ R and ( u j , ξ j ) j be the solution of (2.1) , (2.2) with G j = G δj defined by (4.1) and u ( y ) = py . Assume that (2.3) holds true for ξ j . Then ( u j , ξ j ) j satisfies (4.2) 0 ≤ ( u j ) y ≤ p + 2 L F δ and ≤ ( ξ j ) y ≤ p + 2 L F δ where L F denotes the largest Lipschitz constant of the F i ’s for i = 1 , . . . , n .Proof. We first show that u j and ξ j are non-decreasing with respect to y . Since the equation (2.1) is invariantby translations in y and using the fact that for all b ≥
0, we have u ( y + b + jn ) ≥ u ( y + jn ) . We deduce from the comparison principle that u j ( τ, y + b ) ≥ u j ( τ, y ) and ξ j ( τ, y + b ) ≥ ξ j ( τ, y )which shows that u j and ξ j are non-decreasing in y .We now explain how to get the Lipschitz estimate. We would like to prove that M ≤ M = sup τ ∈ (0 ,T ) ,x,y ∈ R ,j ∈{ ,...,n } max (cid:26) u j ( τ, x ) − u j ( τ, y ) − L | x − y | − ηT − τ − α | x | ,ξ j ( τ, x ) − ξ j ( τ, y ) − L | x − y | − ηT − τ − α | x | (cid:27) as soon as L > p + L F δ > η, α >
0. We argue by contradiction by assuming that
M > L . We next exhibit a contradiction. The supremum defining M is attained since ξ j satisfies (2.3) and u j can be explicitly computed. Case 1.
Assume that the supremum is attained for the function u j at τ ∈ [0 , T ), j ∈ { , . . . , n } , x, y ∈ R .Since we have by assumption M >
0, this implies that τ > x = y . Hence we can obtain the two followingviscosity inequalities (by doubling the time variable and passing to the limit) a ≤ α ( ξ j ( τ, x ) − u j ( τ, x )) b ≥ α ( ξ j ( τ, y ) − u j ( τ, y ))with a − b = η ( T − τ ) . Subtracting these inequalities, we obtain η ( T − τ ) ≤ α ( { ξ j ( τ, x ) − ξ j ( τ, y ) } − { u j ( τ, x ) − u j ( τ, y ) } ) ≤ . We thus get η ≤ ase 2. Assume next that the supremum is attained for the function ξ j . By using the same notation andby arguing similarly, we obtain the following inequality η ( T − τ ) ≤ F j ( τ, u j − m ( τ, x ) , . . . , u j + m ( τ, x )) − F j ( τ, u j − m ( τ, y ) , . . . , u j + m ( τ, y ))+ α ( { u j ( τ, x ) − u j ( τ, y ) } − { ξ j ( τ, x ) − ξ j ( τ, y ) } )+ δ { p ( x − y ) − ( ξ j ( τ, x ) − ξ j ( τ, y )) } L sign + ( x − y ) + 2 αδ ( a + C ) | x | where sign + is the Heaviside function and where we have used (2.3). We now use– the fact that the supremum is attained for the function ξ j – the fact that ξ j ( τ, x ) > ξ j ( τ, y ) implies that x > y (remember that we already proved that ξ j isnon-decreasing with respect to y )– Assumption (A1); in the following, L F still denotes de largest Lipschitz constants of the F j ’s withrespect to V ;– the fact that αδ ( a + C ) | x | = o α (1)in order to get from the previous inequality the following one η ( T − τ ) ≤ L F sup l ∈{− m,...,m } | u j + l ( τ, x ) − u j + l ( τ, y ) | + δpL | x − y | − Lδ ( ξ j ( τ, x ) − ξ j ( τ, y )) + o α (1) . Using the same computation as the one of the proof of Proposition 2.2 Step 3, we getsup l ∈{− m,...,m } | u j + l ( τ, x ) − u j + l ( τ, y ) | = sup l ∈{− m,...,m } ( u j + l ( τ, x ) − u j + l ( τ, y )) ≤ ξ j ( τ, x ) − ξ j ( τ, y ) + Cα (1 + | x | )where C is a constant. Since Cα (1 + | x | ) = o α (1) and M >
0, we finally deduce that ηT ≤ L F ( ξ j ( τ, x ) − ξ j ( τ, y )) + δp ( ξ j ( τ, x ) − ξ j ( τ, y )) − Lδ ( ξ j ( τ, x ) − ξ j ( τ, y )) + o α (1)For α small enough, it is now sufficient to use once again that ξ j ( τ, x ) > ξ j ( τ, y ) and the fact that L > p + L F δ in order to get the desired contradiction in Case 2. The proof is now complete.We now claim that particles are ordered. Proposition 4.2 ( Ordering of the particles).
Assume (A0’) , (A1)-(A6) and let δ ≥ , a ∈ R and ( u δj , ξ δj ) j be the solution of (2.1) , (2.2) with G j = G δj defined by (4.1) . Assume that (2.3) holds true for ξ j if δ > . Then u δj and ξ δj are non-decreasing with respect to j .Proof. If δ >
0, the results is a straightforward consequence of Propositions 2.7 and 4.1. If δ = 0, the resultis obtained by stability of viscosity solution ( i.e. u δj → u j and ξ δj → ξ j as δ → Proposition 4.3 ( Ergodicity).
Let ≤ δ ≤ and a ∈ R . Assume (A0)-(A6) and let ( u j , ξ j ) j be asolution of (2.1) , (2.2) with G j defined in (4.1) and with initial data u ( y ) = ξ ( y ) = py with some p > .Then there exists λ ∈ R such that for all ( τ, y ) ∈ [0 , + ∞ ) × R , j ∈ { , . . . , n } (4.3) | u j ( τ, y ) − py − λτ | ≤ C and | ξ j ( τ, y ) − py − λτ | ≤ C and (4.4) | λ | ≤ C here C = 13 + 6 C α + 7 p + 2 K C = max (cid:18) α M , L F (2 + p ( m + n )) + sup τ | F ( τ, , . . . , | + ( p/ L F )( a + C ) (cid:19) (4.5) (where a is chosen equal to zero for δ = 0 ). Moreover we have for all τ ≥ , y, y ′ ∈ R , j ∈ { , . . . , n } (4.6) u j ( τ, y + 1 /p ) = u j ( τ, y ) + 1( u j ) y ( τ, y ) ≥ | u j ( τ, y + y ′ ) − u j ( τ, y ) − py ′ | ≤ u j +1 ( τ, y ) ≥ u j ( τ, y ) ξ j ( τ, y + 1 /p ) = ξ j ( τ, y ) + 1( ξ j ) y ( τ, y ) ≥ | ξ j ( τ, y + y ′ ) − ξ j ( τ, y ) − py ′ | ≤ ξ j +1 ( τ, y ) ≥ ξ j ( τ, y ) . In order to prove Proposition 4.3, we will need the following classical lemma from ergodic theory (see forinstance [19]).
Lemma 4.4.
Consider
Λ : R + → R a continuous function which is sub-additive, that is to say: for all t, s ≥ , Λ( t + s ) ≤ Λ( t ) + Λ( s ) . Then Λ( t ) t has a limit l as t → + ∞ and l = inf t> Λ( t ) t . We now turn to the proof of Proposition 4.3.
Proof of Proposition 4.3.
We perform the proof in three steps. We first recall that the fact that u j and ξ j are non-decreasing in y and j follows from Propositions 4.1 and 4.2. Step 1: control of the space oscillations.
We are going to prove the following estimate.
Lemma 4.5.
For all τ > , all y, y ′ ∈ R and all j ∈ { , . . . , n } , (4.7) | u j ( τ, y + y ′ ) − u j ( τ, y ) − py ′ | ≤ | ξ j ( τ, y + y ′ ) − ξ j ( τ, y ) − py ′ | ≤ . Proof.
We have u j (0 , y + 1 /p ) = ξ j (0 , y + 1 /p ) = ξ j (0 , y ) + 1 = u j (0 , y ) + 1 . Therefore from the comparison principle and from the integer periodicity of the Hamiltonian (see (A3’)), weget that u j ( τ, y + 1 /p ) = u j ( τ, y ) + 1 and ξ j ( τ, y + 1 /p ) = ξ j ( τ, y ) + 1 . Since u j ( τ, y ) is non-decreasing in y , we deduce that for all b ∈ [0 , /p ]0 ≤ u j ( τ, b ) − u j ( τ, ≤ y ∈ R , that we write py = k + a with k ∈ Z and a ∈ [0 , u j ( τ, y ) − u j ( τ,
0) = k + u j ( τ, a/p ) − u j ( τ, b ∈ [0 , /p ), u j ( τ, y ) − u j ( τ, − py = − a + u j ( τ, b ) − u j ( τ, τ > y ∈ R , | u j ( τ, y ) − u j ( τ, − py | ≤ . In the same way, we get | ξ j ( τ, y ) − ξ j ( τ, − py | ≤ . Finally, we obtain (4.7) by using the invariance by translations in y of the problem.22 tep 2: estimate on | u j ( τ, y ) − ξ j ( τ, y ) | .Lemma 4.6. For all j ∈ { , . . . , n } and ≤ δ ≤ , (4.8) k u j − ξ j k L ∞ ≤ C α where C is given by (4.5) .Proof. We recall that (( u j ) , ( ξ j )) is solution of(4.9) (cid:26) ( u j ) τ = α ( ξ j − u j )( ξ j ) τ ≤ F j ( τ, [ u ( τ, · )] j,m ) + α ( u j − ξ j ) + δ ( a + C )(( ξ j ) y ) + where we have used (2.3). Using Proposition 4.1, we deduce that (for δ ≤ δ ( a + C )(( ξ j ) y ) + ≤ ( a + C )( p + 2 L F ) . We now want to bound F j ( τ, [ u ( τ, · )] j,m ). We have F j ( τ, [ u ( τ, · )] j,m ( y )) = F j ( τ, [ u ( τ, · ) − ⌊ u j ( τ, y ) ⌋ ] j,m ( y )) ≤ L F + F j ( τ, [ u ( τ, · ) − u j ( τ, y )] j,m ( y )) ≤ L F + L F sup k ∈{ ,...,m } ( u j + k ( τ, y ) − u j ( τ, y )) + sup τ F ( τ, , . . . F for thesecond and third ones, and the fact that u l is non-decreasing with respect to l for the third line. Moreoverfor all i ∈ { , . . . , n } , k ∈ { , . . . m } , we have that0 ≤ u i + k ( τ, y ) − u i ( τ, y ) = u i + k − ⌈ kn ⌉ n ( τ, y + (cid:24) kn (cid:25) n ) − u i ( τ, y ) ≤ u i ( τ, y + (cid:24) kn (cid:25) n ) − u i ( τ, y ) ≤ p (cid:24) kn (cid:25) n ≤ p ( m + n )(4.12)where we have used the periodicity of u i for the first line, the monotonicity in i of u i for the second one andthe control of the oscillation (4.7) for the third one. We then deduce that F j ( τ, [ u j ( τ, · )] j,m ( y )) ≤ L F (2 + p ( m + n )) + sup τ F ( τ, , . . . . Combining this inequality with (4.9) and (4.10), we deduce that (cid:26) ( u j ) τ = α ( ξ j − u j )( ξ j ) τ ≤ C + α ( u j − ξ j )We now define for all j ∈ Z v j = ξ j − u j . Classical arguments from viscosity solution theory show that( v j ) τ ≤ C − α v j ) . We then deduce that v j ≤ C α . Using the same arguments with super-solution for ξ j , we get the desired result.23 tep 3: control of the time oscillations. We now explain how to control the time oscillations. The proof is inspired of [14]. Let us introduce thefollowing continuous functions defined for
T > λ u + ( T ) = sup j ∈{ ,...,n } sup τ ≥ u j ( τ + T, − u j ( τ, Tλ u − ( T ) = inf j ∈{ ,...,n } inf τ ≥ u j ( τ + T, − u j ( τ, T and λ ξ + ( T ) = sup j ∈{ ,...,n } sup τ ≥ ξ j ( τ + T, − ξ j ( τ, Tλ ξ − ( T ) = inf j ∈{ ,...,n } inf τ ≥ ξ j ( τ + T, − ξ j ( τ, T and λ + ( T ) = sup( λ u + ( T ) , λ ξ + ( T )) and λ − ( T ) = inf( λ u − ( T ) , λ ξ − ( T )) . In particular, these functions satisfy −∞ ≤ λ − ( T ) ≤ λ + ( T ) ≤ + ∞ .The goal is to prove that λ + ( T ) and λ − ( T ) have a common limit as T → ∞ . We would like to applyLemma 4.4.In view of the definition of λ u + and λ ξ + , we see that T T λ u + ( T ) and T T λ ξ + ( T ) are sub-additive.Analogously, T
7→ −
T λ u − ( T ) and T
7→ −
T λ ξ − ( T ) are also sub-additive. Hence, if we can prove that thesequantities λ u ± ( T ) , λ ξ ± ( T ) are finite, we will know that they converge. We will then have to prove that thelimits of λ + and λ − are the same. Step 3.1: first control on the time oscillations
We first prove that λ ± are finite. Lemma 4.7.
For all
T > , (4.13) − K − C T ≤ λ − ( T ) ≤ λ + ( T ) ≤ K + C T where C = C α + 3 + 2 p and K is defined in (2.14) .Proof. Consider j ∈ { , . . . , n } . Using the control of the space oscillations (4.7), we get that u j ( τ, y ) ≥ ∆ + py − ξ j ( τ, y ) ≥ ∆ + py − j ∈{ ,...,n } inf( u j ( τ, , ξ j ( τ, . Recalling (see Lemma 2.5) that ⌊ ∆ − p ⌋ + p ( y + jn ) − − K t is a sub-solution and using the comparisonprinciple on the time interval [ τ, τ + t ), we deduce that(4.14) u j ( τ + t, y ) ≥ ⌊ ∆ − p ⌋ + py + pjn − − K t and ξ j ( τ + t, y ) ≥ ⌊ ∆ − p ⌋ + py + pjn − − K t .
24e now want to estimate ∆ from below. Let us assume that the infimum in ∆ is reached for the index¯ j ∈ { , . . . , n } . Then ¯ j ≥ j − n since j ∈ { , . . . , n } . We then deduce that p + ⌊ ∆ − p ⌋ ≥ ∆ − ≥ u ¯ j ( τ, − C α − ≥ u j − n ( τ, − C α − ≥ u j ( τ, − − C α − ≥ u j ( τ, − C α − − p where we have used (4.8) for the second line, the fact that ( u j ) j is non-decreasing in j for the third line, theperiodicity of u j for the fourth line and (4.7) for the last one. In the same way, we get that p + ⌊ ∆ − p ⌋ ≥ ξ j ( τ, − C α − − p. Injecting this in (4.14), we get that(4.15) u j ( τ + t, y ) ≥ u j ( τ, − C + py − K t and ξ j ( τ + t, y ) ≥ ξ j ( τ, − C + py − K t. In the same way, we also get(4.16) u j ( τ + t, y ) ≤ u j ( τ,
0) + C + py + K t and ξ j ( τ + t, y ) ≤ ξ j ( τ,
0) + C + py + K t. Taking y = 0, we finally get (4.13). Step 3.2: Refined control on the time oscillations
We now estimate λ + − λ − in order to prove that they have the same limit. Lemma 4.8.
For all
T > , | λ + ( T ) − λ − ( T ) | ≤ C T where C = 6 + C α + 3 p + 2 C + 2 K .Proof. By definition of λ ± ( T ), for all ε >
0, there exists τ ± ≥ v ± ∈ { u , . . . u n , ξ , . . . ξ n } such that (cid:12)(cid:12)(cid:12)(cid:12) λ ± ( T ) − v ± ( τ ± + T, − v ± ( τ ± , T (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε. Consider j ∈ { , . . . , n } . We choose β ∈ [0 ,
1) such that τ + − τ − − β = k ∈ Z and we set∆ uj = u j ( τ + , − u j ( τ − + β, , ∆ ξj = ξ j ( τ + , − ξ j ( τ − + β, j ∈{ ,...,n } sup(∆ uj , ∆ ξj ) . u j ( τ + , y ) ≤ u j ( τ − + β, y ) + 2 + ⌈ ∆ ⌉ and ξ j ( τ + , y ) ≤ ξ j ( τ − + β, y ) + 2 + ⌈ ∆ ⌉ . Using the comparison principle, we then deduce that(4.17) u j ( τ + + T, y ) ≤ u j ( τ − + β + T, y ) + 2 + ⌈ ∆ ⌉ and ξ j ( τ + + T, y ) ≤ ξ j ( τ − + β + T, y ) + 2 + ⌈ ∆ ⌉ . We now want to estimate ⌈ ∆ ⌉ from above. Let us assume that the maximum in ∆ is reached for the index¯ j . We then have for all j ∈ { , . . . , n }⌈ ∆ ⌉ ≤ u ¯ j ( τ + , − u ¯ j ( τ − + β,
0) + 2 C α + 1 ≤ u j + n ( τ + , − u j − n ( τ − + β,
0) + 2 C α + 1 ≤ u j ( τ + , − u j ( τ − + β, −
1) + 2 C α + 1 ≤ u j ( τ + , − u j ( τ − + β,
0) + 2 C α + 3 + 2 p where we have used (4.8) for the first line, the fact that ( u j ) j is non-decreasing in j for the second line, theperiodicity of u j for the third line and (4.7) for the last one. In the same way, we also get ⌈ ∆ ⌉ ≤ ξ j ( τ + , − ξ j ( τ − + β,
0) + 2 C α + 3 + 2 p . Injecting this in (4.17), we get u j ( τ + + T, y ) ≤ u j ( τ − + β + T, y ) + 5 + 2 C α + 2 p + ∆ uj and ξ j ( τ + + T, y ) ≤ ξ j ( τ − + β + T, y ) + 5 + 2 C α + 2 p + ∆ ξj . Taking y = 0 and using (4.15) (with τ = τ − and t = β ) and (4.16) (with τ = τ − + T and t = β ), we get u j ( τ + + T, − u j ( τ + , ≤ u j ( τ − + T, − u j ( τ − ,
0) + 5 + 2 C α + 2 p + 2 C + 2 K . In the same way, we get ξ j ( τ + + T, − ξ j ( τ + , ≤ ξ j ( τ − + T, − ξ j ( τ − ,
0) + 5 + 2 C α + 2 p + 2 C + 2 K . Using also (4.8), (4.7) and the fact that ( u j ) j and ( ξ j ) j are non-decreasing in j , we finally get v + ( τ + + T, − v + ( τ + , ≤ v − ( τ − + T, − v − ( τ − ,
0) + C . The comparison of u j and ξ j makes appear the additional constant 2 C /α , and the comparison between u j and u k (and similarly between ξ j and ξ k ) creates an additional constant 1 + p . Indeed, we have u j ( τ, − u k ( τ,
0) = u j + n ( τ, − u k ( τ, ≤ u j + n ( τ, − u k ( τ,
0) + 1 + p ≤ p. This explains the value of the new constant C .This implies that T λ + ( T ) ≤ T λ − ( T ) + 2 ε + C . Since this is true for all ε >
0, the proof of the lemma is complete.26 tep 3.3: Conclusion
We now can conclude that lim T → + ∞ λ ± ( T ) are equal. If λ denotes the common limit, we also have, byLemma 4.4, that for every T > λ − ( T ) ≤ λ ≤ λ + ( T ) . Moreover, by Lemma 4.8, we have λ + ( T ) ≤ λ − ( T ) + C T and so λ − ( T ) ≤ λ ≤ λ − ( T ) + C T We finally deduce (using a similar argument for λ + ) that | λ ± ( T ) − λ | ≤ C T .
Combining this estimate and (4.7), we get with T = τ | u j ( τ, y ) − u j (0 , − py − λτ | ≤ C + 1and | ξ j ( τ, y ) − ξ j (0 , − py − λτ | ≤ C + 1 . This finally implies (4.3) with C = C + 1. In this subsection, we construct hull functions for a general Hamiltonian G j . As we shall see, this is astraightforward consequence of the construction of time-space periodic solutions of (4.18); see Proposition 4.9and Corollary 4.10 below. We will then prove that the time slope obtained in Proposition 4.3 is unique andthat the map p λ is continuous; see Proposition 4.11 below.Given p >
0, we consider the equation in R × R (4.18) (cid:26) ( u j ) τ = α ( ξ j − u j )( ξ j ) τ = G j ( τ, [ u ( τ, · )] j,m , ξ j , inf y ′ ∈ R ( ξ j ( τ, y ′ ) − py ′ ) + py − ξ j ( τ, y ) , ( ξ j ) y ) (cid:26) u j + n ( τ, y ) = u j ( τ, y + 1) ξ j + n ( τ, y ) = ξ j ( τ, y + 1) , where G j = G δj is given in (4.1) for δ ≥
0. Then we have the following result
Proposition 4.9. (Existence of time-space periodic solutions of (4.18) ) Let ≤ δ ≤ , a ∈ R and p > . Assume (A1)-(A6) . Then there exist functions (( u ∞ j ) j , ( ξ ∞ j ) j ) solving (4.18) on R × R and a real number λ ∈ R satisfying for all τ, y ∈ R , j ∈ { , . . . , n }| u ∞ j ( τ, y ) − py − λτ | ≤ ⌈ C ⌉ (4.19) | ξ ∞ j ( τ, y ) − py − λτ | ≤ ⌈ C ⌉ . Moreover (( u ∞ j ) j , ( ξ ∞ j ) j ) satisfies for j ∈ { , . . . , n } (4.20) u ∞ j ( τ, y + 1 /p ) = u ∞ j ( τ, y ) + 1 u ∞ j ( τ + 1 , y ) = u ∞ j ( τ, y + λ/p )( u ∞ j ) y ( τ, y ) ≥ u j +1 ( τ, y ) ≥ u j ( τ, y ) . ξ ∞ j ( τ, y + 1 /p ) = ξ ∞ j ( τ, y ) + 1 ξ ∞ j ( τ + 1 , y ) = ξ ∞ j ( τ, y + λ/p )( ξ ∞ j ) y ( τ, y ) ≥ ξ j +1 ( τ, y ) ≥ ξ j ( τ, y ) . Eventually, when the Hamiltonians G j are independent on τ , we can choose u ∞ j and ξ ∞ j independent on τ .
27y considering for all τ, z ∈ R (4.21) (cid:26) h j ( τ, z ) = u ∞ j ( τ, ( z − λτ ) /p ) if j ∈ { , . . . , n } h j + n ( τ, z ) = h j ( τ, z + p ) otherwiseand for all τ, z ∈ R ,(4.22) (cid:26) g j ( τ, z ) = ξ ∞ j ( τ, ( z − λτ ) /p ) if j ∈ { , . . . , n } g j + n ( τ, z ) = g j ( τ, z + p ) otherwisewe immediately get the following corollary Corollary 4.10. (Existence of hull functions)
Assume (A1)-(A6) . There exists a hull function (( h j ) j , ( g j ) j ) in the sense of Definition 1.8 satisfying | h j ( τ, z ) − z | ≤ ⌈ C ⌉ and | g j ( τ, z ) − z | ≤ ⌈ C ⌉ We now turn to the proof of Proposition 4.9.
Proof of Proposition 4.9.
The proof is performed in three steps. In the first one, we construct sub- andsuper-solutions of (4.18) in R × R with good translation invariance properties (see the first two lines of(4.20)). We next apply Perron’s method in order to get a (possibly discontinuous) solution satisfying thesame properties. Finally, in Step 3, we prove that if the functions G j do not depend on τ , then we canconstruct a solution in such a way that it does not depend on τ either. Step 1: global sub- and super-solution
By Proposition 4.3, we know that the solution ( u j , ξ j ) of (2.1), (2.2) with initial data u ( y ) = py = ξ ( y )satisfies on [0 , + ∞ ) × R (4.23) ( u j ) y ≥ , | u j ( τ, y ) − py − λτ | ≤ C , | u j ( τ, y + y ′ ) − u j ( τ, y ) − py ′ | ≤ ,u j +1 ( τ, y ) ≥ u j ( τ, y ) , ( ξ j ) y ≥ , | ξ j ( τ, y ) − py − λτ | ≤ C , | ξ j ( τ, y + y ′ ) − ξ j ( τ, y ) − py ′ | ≤ ,ξ j +1 ( τ, y ) ≥ ξ j ( τ, y ) . We first construct a sub-solution and a super-solution of (4.18) for τ ∈ R (and not only τ ≥
0) that alsosatisfy the first two lines of (4.20), i.e. satisfy for all k, l ∈ Z ,(4.24) U ( τ + k, y ) = U ( τ, y + λ kp ) and U ( τ, y + lp ) = U ( τ, y ) + l . To do so, we consider for j ∈ { , . . . , n } two sequences of functions (indexed by m ∈ N , m → ∞ ) u mj ( τ, y ) = u j ( τ + m, y ) − ⌊ λm ⌋ , ξ mj ( τ, y ) = ξ j ( τ + m, y ) − ⌊ λm ⌋ and consider u j = lim sup m → + ∞∗ u mj , ξ j = lim sup m → + ∞∗ ξ mj u j = lim inf m → + ∞∗ u mj , ξ j = lim inf m → + ∞∗ ξ mj . We first remark that thanks to (4.3), all these semi-limits are finite. We also remark that for all k, l ∈ Z ,( u j ( τ + k, y − kλ/p + l/p ) − l, ξ j ( τ + k, y − kλ/p + l/p ) − l )28s a sub-solution of (4.18). A similar remark can be done for the super-solutions ( u j , ξ j ) j .Now a way to construct sub-solution (resp. a super-solution) of (2.1) satisfying (4.24) is to consider(4.25) ( u ∞ j ( τ, y ) = (cid:0) sup k,l ∈ Z ( u j ( τ + k, y − kλ/p + l/p ) − l ) (cid:1) ∗ ,ξ ∞ j ( τ, y ) = (cid:0) sup k,l ∈ Z (cid:0) ξ j ( τ + k, y − kλ/p + l/p ) − l (cid:1)(cid:1) ∗ , and(4.26) u ∞ j ( τ, y ) = (cid:0) inf k,l ∈ Z (cid:0) u j ( τ + k, y − kλ/p + l/p ) − l (cid:1)(cid:1) ∗ ,ξ ∞ j ( τ, y ) = (cid:16) inf k,l ∈ Z (cid:16) ξ j ( τ + k, y − kλ/p + l/p ) − l (cid:17)(cid:17) ∗ . Notice that u ∞ j , u ∞ j , ξ ∞ j and ξ ∞ j satisfy moreover (4.23) on R × R . Therefore we have in particular u ∞ j ≤ u ∞ j + 2 ⌈ C ⌉ and ξ ∞ j ≤ ξ ∞ j + 2 ⌈ C ⌉ . Step 2: existence by Perron’s method
Applying Perron’s method we see that the lowest- ∗ super-solution (( u ∞ j ) j , ( ξ ∞ j ) j ) lying above (( u ∞ j ) j , ( ξ ∞ j ) j )is a (possibly discontinuous) solution of (4.18) on R × R and satisfies u ∞ j ≤ u ∞ j ≤ u ∞ j + 2 ⌈ C ⌉ and ξ ∞ j ≤ ξ ∞ j ≤ ξ ∞ j + 2 ⌈ C ⌉ . We next prove that u ∞ satisfies (4.20). For j ∈ { , . . . , n } , let us consider(4.27) ˜ u ∞ j ( τ, y ) = (cid:18) inf k,l ∈ Z (cid:0) u ∞ j ( τ + k, y − kλ/p + l/p ) − l (cid:1)(cid:19) ∗ ˜ ξ ∞ j ( τ, y ) = (cid:18) inf k,l ∈ Z (cid:0) ξ ∞ j ( τ + k, y − kλ/p + l/p ) − l (cid:1)(cid:19) ∗ By construction the family ((˜ u ∞ j ) j , ( ˜ ξ ∞ j ) j ) is a super-solution of (4.18) and is again above the sub-solution(( u ∞ j ) j , ( ξ ∞ j ) j ). Therefore from the definition of (( u ∞ j ) j , ( ξ ∞ j ) j ), we deduce that˜ u ∞ j = u ∞ j and ˜ ξ ∞ j = ξ ∞ j which implies that u ∞ j and ξ ∞ j satisfy (4.24), i.e the first two equalities of (4.20).Similarly, we can consider, for j ∈ { , . . . , n } ˆ u ∞ j ( τ, y ) = (cid:18) inf b ∈ [0 , + ∞ ) u ∞ j ( τ, y + b ) (cid:19) ∗ ˆ ξ ∞ j ( τ, y ) = (cid:18) inf b ∈ [0 , + ∞ ) ξ ∞ j ( τ, y + b ) (cid:19) ∗ which is again super-solution above the sub-solution (( u ∞ j ) j , ( ξ ∞ j ) j ). Thereforeˆ u ∞ j = u ∞ j and ˆ ξ ∞ j = ξ ∞ j which implies that u ∞ j and ξ ∞ j are non-decreasing in y , i.e. the third line of (4.20) is satisfied.Let us now prove that u ∞ j and ξ ∞ j are non-decreasing in j . We consider, for j ∈ { , . . . , n } ˇ u ∞ j ( τ, y ) = (cid:18) inf k ≥ u ∞ j + k ( τ, y ) (cid:19) ∗ = (cid:18) inf ≤ k Consider p > and assume (A1)-(A6) . Then– there exists a unique real number λ ∈ R such that there exists a solution (( u ∞ j ) j , ( ξ ∞ j ) j ) of (4.18) on R × R such that there exists C > such that for all τ , (4.28) | h j ( τ, z ) − z | ≤ C and | g j ( τ, z ) − z | ≤ C, where the h j and the g j are defined in (4.21) and (4.22) ; moreover, we can choose C = 2 ⌈ C ⌉ with C given in (4.5) ;– if λ is seen as a function G of p ( λ = G ( p ) ), then this function G : (0 , + ∞ ) → R is continuous. Before to prove this proposition, let us give the proof of Theorem 1.10. Proof of Theorem 1.10. Just apply Proposition 4.11 with G = F .30 roof of Proposition 4.11. The proof follows classical arguments. However, we give it for the reader’s con-venience. The proof is divided in two steps. Step 1: Uniqueness of λ Given some p ∈ (0 , + ∞ ), assume that there exist λ , λ ∈ R with their corresponding hull functions(( h j ) j , ( g j ) j ) , (( h j ) j , ( g j ) j ). Then define for i = 1 , j ∈ { , . . . , n } u ij ( τ, y ) = h ij ( τ, λ i τ + py ) and ξ ij ( τ, y ) = g ij ( τ, λ i τ + py )which are both solutions of equation (2.1) on [0 , + ∞ ) × R . By Corollary 4.10, we know that h j and g j satisfy(4.28). Then we have with C = 2 ⌈ C ⌉ u j (0 , y ) ≤ u j (0 , y ) + 2 C and ξ j (0 , y ) ≤ ξ j (0 , y ) + 2 C which implies (from the comparison principle) for all ( τ, y ) × [0 , + ∞ ) × R u j ( τ, y ) ≤ u j ( τ, y ) + 2 C and ξ j ( τ, y ) ≤ ξ j ( τ, y ) + 2 C . Using the fact that h ij ( τ + 1 , z ) = h ij ( τ, z ) and g ij ( τ + 1 , z ) = g ij ( τ, z ), we deduce that for τ = k ∈ N and y = 0 we have h j (0 , λ k ) ≤ h j (0 , λ k ) + 2 C and g j (0 , λ k ) ≤ g j (0 , λ k ) + 2 C which implies by (4.28) λ k ≤ λ k + 4 C . Because this is true for any k ∈ N , we deduce that λ ≤ λ . The reverse inequality is obtained exchanging (( h j ) j , ( g j ) j ) and (( h j ) j , ( g j ) j ). We finally deduce that λ = λ , which proves the uniqueness of the real λ , that we call G ( p ). Step 2: Continuity of the map p G ( p )Let us consider a sequence ( p m ) m such that p m → p > 0. Let λ m = G ( p m ) and (( h mj ) j , ( g mj ) j ) be thecorresponding hull functions. From Corollary 4.10, we can choose these hull functions such that for j ∈{ , . . . , n } | h mj ( τ, z ) − z | ≤ ⌈ C ⌉ , and | g mj ( τ, z ) − z | ≤ ⌈ C ⌉ and we have | λ m | ≤ C where we recall that C is defined in (4.5). Remark that both C and C depends on p m , but can be boundedfor p m in a neighbourhood of p . We deduce in particular that there exists a constant C > | h mj ( τ, z ) − z | ≤ C , | g mj ( τ, z ) − z | ≤ C and | λ m | ≤ C . Let us consider a limit λ ∞ of ( λ m ) m , and let us define h j = lim sup m → + ∞ ∗ h mj , and g j = lim sup m → + ∞ ∗ g mj . This family of functions (( h j ) j , ( g j ) j ) is such that the family(( u j ( τ, y )) j , ( ξ j ( τ, y )) j ) = (( h j ( τ, λ ∞ τ + py )) j , ( g j ( τ, λ ∞ τ + py )) j )is a sub-solution of (4.18) on R × R . On the other hand, if (( h j ) j , ( g j ) j ) denotes the hull function associatedwith p and λ = G ( p ), then(( u j ( τ, y )) j , ( ξ j ( τ, y )) j ) = (( h j ( τ, λτ + py )) j , ( g j ( τ, λτ + py )) j )31s a solution of (4.18) on R × R . Finally, as in Step 1, we conclude that λ ∞ ≤ λ . Similarly, considering h j = lim inf m → + ∞ ∗ h mj and g j = lim inf m → + ∞ ∗ g mj we can show that λ ∞ ≥ λ . Therefore λ ∞ = λ and this proves that G ( p m ) → G ( p ); the continuity of the map p G ( p ) follows and thisends the proof of the proposition. When proving the Convergence Theorem 1.5, we explained that, on the one hand, it is necessary to dealwith hull functions ( h, g ) = (( h j ( τ, z )) j , ( g j ( τ, z )) j ) that are uniformly continuous in z (uniformly in τ and j ) in order to apply Evans’ perturbed test function method; on the other hand, given some p > 0, we alsoknow some Hamiltonians F j , with corresponding effective Hamiltonian F ( p ), such that every correspondinghull function h j is necessarily discontinuous in z for α = + ∞ (see [1, 10]). Recall that a hull function ( h, g )solves in particular(5.1) (cid:26) ( h j ) τ + λ ( h j ) z = α ( g j − h j )( g j ) τ + λ ( g j ) z = 2 F j ( τ, [ h ( τ, · )] j,m ) + α ( h j − g j )with λ = F ( p ) and h j + n ( τ, z ) = h j ( τ, z + p ) , g j + n ( τ, z ) = g j ( τ, z + p ) . We overcome this difficulty as in [10] (see also [11, 14, 15]).We build approximate Hamiltonians G δ with corresponding effective Hamiltonians λ δ = G δ ( p ), andcorresponding hull functions ( h δ , g δ ), such that ( h δj , g δj ) is Lipschitz continuous with respect to z uniformly in τ and jG δ ( p ) → F ( p ) as δ → h δ , g δ ) is a sub-/super-solution of (5.1).We will show that it is enough to choose for δ ≥ G δj ( τ, V, r, a, q ) = 2 F j ( τ, V ) + α ( V − r ) + δ ( a + a ) q + with a ∈ R (in fact, we will consider a = ± Proposition 5.1 ( Existence of Lipschitz continuous approximate hull functions). Assume (A1)-(A3) . Given p > , < δ ≤ and a ∈ R , then there exists a family of Lipschitz continuousfunctions (( h j ) j , ( g j ) j ) satisfying for j ∈ { , . . . , n } (5.3) h j ( τ, z + 1) = h j ( τ, z ) + 1 h j ( τ + 1 , z ) = h j ( τ, z )0 ≤ ( h j ) z ≤ L F pδ g j ( τ, z + 1) = g j ( τ, z ) + 1 g j ( τ + 1 , z ) = g j ( τ, z )0 ≤ ( g j ) z ≤ L F pδ nd there exists λ ∈ R such that (5.4) ( h j ) τ + λ ( h j ) z = α ( g j − h j )( g j ) τ + λ ( g j ) z = 2 F j ( τ, [ h ( τ, · )] j,m ) + α ( h j − g j )+ δp { a + inf z ′ ∈ R ( h j ( τ, z ′ ) − z ′ ) + z − h j ( τ, z )) } ( h j ) z (cid:26) h j + n ( τ, z ) = h j ( τ, z + p ) g j + n ( τ, z ) = g j ( τ, z + p ) and for all τ, z, z ′ ∈ R (5.5) | h j ( τ, z ′ ) − z ′ + z − h j ( τ, z ) | ≤ and | g j ( τ, z ′ ) − z ′ + z − g j ( τ, z ) | ≤ . Moreover there exists a constant C > defined in (4.5) such that (5.6) | λ | ≤ C and for all ( τ, z ) ∈ R × R , (5.7) | h ( τ, z ) − z | ≤ C ( C , p, α , δ | a | p ) , | g ( τ, z ) − z | ≤ C ( C , p, α , δ | a | p ) . Moreover, when the F j do not depend on τ , we can choose the hull function (( h j ) j , ( g j ) j ) such that it doesnot depend on τ either.Proof of Proposition 5.1. The construction follows the one made in Proposition 4.3 and Proposition 4.9.However, Proposition 4.9 has to be adapted. Indeed, since we want to construct a Lipschitz continuousfunction with a precise Lipschitz estimate, we do not want to use Perron’s method. This is the reason whyhere we can use a space-time Lipschitz estimate of (( u j ) , ( ξ j )) to get enough compacity to pass to the limit.The space Lipschitz estimate comes from Proposition 4.1. The time Lipschitz estimate of the u j ’s followsfrom Lemma 4.6 and the equation satisfied by u j . The time Lipschitz estimate of the ξ j ’s is obtained inthe same way, using the fact that we can bound the right hand side of the equation satisfied by ξ j . Indeed,one can use the space oscillation estimate of u to bound F ( t, [ u ( t, · )] j,m ( x )) (as we did in (4.11)-(4.12)) andLemma 4.6 and Proposition 4.1 to bound remaining terms.We finally have Proposition 5.2 ( Sub- and super- Lipschitz continuous hull functions). We consider < δ ≤ andthe Lipschitz continuous hull function obtained in Proposition 5.1 for a = ± , that we call (( h δ, ± j ) j , ( g δ, ± j ) j ) ,and the corresponding value λ δ, ± of the effective Hamiltonian. Then we have ( h δ, + j ) τ + λ δ, + ( h δ, + j ) z = α ( g δ, + j − h δ, + j )( g δ, + j ) τ + λ δ, + ( g δ, + j ) z ≥ F j ( τ, [ h δ, + ( τ, · )] j,m ) + α ( h δ, + j − g δ, + j ) and λ ≤ λ δ, + → λ as δ → and ( h δ, − j ) τ + λ δ, − ( h δ, − j ) z = α ( g δ, − j − h δ, − j )( g δ, − j ) τ + λ δ, − ( g δ, − j ) z ≥ F j ( τ, [ h δ, − ( τ, · )] j,m ) + α ( h δ, − j − g δ, − j ) and λ ≥ λ δ, − → λ as δ → where λ = F ( p ) .Proof of Proposition 5.2. Inequalities ± λ δ, ± ≥ ± λ follow from the comparison principle. Remark thatbounds (5.6) and (5.7) on λ δ, ± and h δ, ± j are uniform as δ goes to zero. Hence the convergence λ δ, ± → λ holds true as δ → 0. Indeed, it suffices to adapt Step 2 of the proof of Proposition 4.11.33 Qualitative properties of the effective Hamiltonian Proof of Theorem 1.11. We recall that we have hull functions (( h j ) j , ( g j ) j ) solutions of (cid:26) ( h j ) τ + λ ( h j ) z = α ( g j − h j )( g j ) τ + λ ( g j ) z = 2 L + 2 F ( τ, [ h ( τ, · )] j,m ( z )) + α ( h j − g j )with λ = F ( L, p ).The continuity of the map ( L, p ) F ( L, p ) is easily proved as in step 2 of the proof of Proposition 4.11. (i) Bound This is a straightforward adaptation of the proof of (4.13). (ii) Monotonicity in L The monotonicity of the map L F ( L, p ) follows from the comparison principle on(( u j ( τ, y ) = h j ( τ, λτ + py )) j , ( ξ j ( τ, y ) = g j ( τ, λτ + py )) j where (( h j ) j , ( g j ) j ) is the hull function and λ = F ( L, p ). A An alternative proof of Proposition 4.1 In this section, we give an alternative proof of Proposition 4.1. We adapt here the method we used in [10]and we provide complementary details. A.1 Explanation of the estimate of Proposition 4.1 In this section, we formally explain how we derive the estimate obtained in Proposition 4.1.We can adapt the corresponding proof from [10]. For all η ≥ 0, we consider the following Cauchy problem(A.1) (cid:26) ( u j ) τ = α ( ξ j − u j )( ξ j ) τ = G δj ( τ, [ u ( τ, · )] j,m , ξ j ( τ, y ) , inf y ′ ∈ R ( ξ j ( τ, y ′ ) − py ′ ) + py − ξ j ( τ, y ) , ( ξ j ) y ) + η ( ξ j ) yy (cid:26) u j + n ( τ, y ) = u j ( τ, y + 1) ξ j + n ( τ, y ) = ξ j ( τ, y + 1) (cid:26) u j (0 , y ) = p (cid:0) y + jn (cid:1) ξ j (0 , y ) = p (cid:0) y + jn (cid:1) where G δj is given by G δj ( τ, V, r, a, q ) = 2 F j ( τ, V ) + α ( V − r ) + δ ( a + a ) q (remark that this is not exactly the function given by (5.2)). It is convenient to introduce the modifiedHamiltonian ˜ F i ( τ, V − m , . . . , V m ) = 2 F i ( τ, V − m , . . . , V m ) + α V so that G δj ( τ, V − m , . . . , V m , r, a, q ) = ˜ F j ( τ, V − m , . . . , V m ) − α r + δ ( a + a ) q . Hence, the Lipschitz constant of ˜ F j ( τ, V ) with respect to V is ˜ K = 2 L F + α . Case A: η > and F j ∈ C For η > 0, it is possible to show that there exists a unique solution (( u j ) j , ( ξ j ) j )of (A.1) in ( C α, α ) n for any α ∈ (0 , Step 1: bound from below on the gradient ζ j = ( ξ j ) y and v j = ( u j ) y , we can derive the previous equation in order to get the followingone(A.2) ( v j ) τ = α ( ζ j − v j )( ζ j ) τ − η ( ζ j ) yy = ( ˜ F j ) ′ V ( τ, [ u ( τ, · )] j,m ( y )) · [ v ( τ, · )] j,m ( y ) − α ζ j − δ ( ζ j − p ) ζ j + δ ( a + inf y ′ ∈ R ( ξ j ( τ, y ′ ) − py ′ ) + py − ξ j ( τ, y )) ( ζ j ) y v j + n ( τ, y ) = v j ( τ, y + 1) ζ j + n ( τ, y ) = ζ j ( τ, y + 1) v j (0 , y ) = ζ j (0 , y ) = p . Let us now define m v ( τ ) = inf j ∈{ ,...,n } inf y ∈ R v j ( τ, y ) and m ζ ( τ ) = inf j ∈{ ,...,n } inf y ∈ R ζ j ( τ, y ) . Then we have in the viscosity sense: ( m v ) τ ≥ α ( m ζ − m v )( m ζ ) τ ≥ ˜ L F min(0 , m v ) − α m ζ − δ ( m ζ − p ) m ζ m v (0) = m ζ (0) = p > L F min(0 , m v ) with˜ L F = 2 L F + α . The fact that (0 , 0) is a sub-solution of this monotone system of ODEs implies that, for j ∈ { , . . . , n } , v j ≥ m v ≥ ζ j ≥ m ζ ≥ . In particular, we see that ( u, ξ ) is a solution of (A.1) with G δj given by (5.2). Step 2: bound from above on the gradient Similarly we define m ζ ( τ ) = sup j ∈{ ,...,n } sup y ∈ R ζ j ( τ, y ) and m v ( τ ) = sup j ∈{ ,...,n } sup y ∈ R v j ( τ, y ) . Then we have in the viscosity sense ( m v ) τ ≤ α ( m ζ − m v )( m ζ ) τ ≤ (2 L F ) m v + α ( m v − m ζ ) − δ ( m ζ − p ) m ζ m v (0) = m ζ (0) = p > v j ≥ m v ≥ j ∈ { , . . . , n } . The constant function ( p +(2 L F ) δ − ) (for both components) is a super-solution of the previous monotone system of ODEs. Hence, theproof is complete in Case A. Case B: η = 0 and F general We can use an approximation argument as in [10]. This ends the proof of the proposition. A.2 Proof of the existence of a regular solution of (A.1) We just give the main idea.It can be useful to remark that u j + l can be rewritten as follows: for all l ∈ {− m, . . . , m } ,(A.3) u j + l ( τ, y ) = p · ( y + ( j + l ) /n ) e − α τ + α Z τ e α ( s − τ ) ξ j + l ( s, y ) ds . 35e set v j ( τ, y ) = ξ j ( τ, y ) − py . Then ( v j ) j is a solution of(A.4) ( v j ) t − η ( v j ) yy = F j ( t, [ v ( τ, · ) + p · ] j,m ( y )) + δ (1 + inf y ′ ( v ( τ, y ′ )) − v ( τ, y )) ( v y + p ) v j + n ( τ, y ) = v j ( τ, y + 1) + pv j (0 , y ) = p ( jn )where F j [ τ, [ ξ ( τ, · )] j,m ( y )] = 2 F j ( τ, [ u ( τ, · )] j,m ( y )) + α u j ( τ, y ) − ξ j ( τ, y ) with u given by (A.3) as a functionof the time integral of ξ . Since we attempt to get ξ j ( τ, y + p ) = ξ j ( τ, y ) + 1, we will look for functions v j which are periodic of period p . The basic idea is to use a fixed point argument. First, we “regularize” theright hand side of (A.4) by considering for some given K > F K,j ( τ, v ) = T K ( F j ( τ, [ v ( τ, · ) + p · ] j,m ( y ))) + δ (cid:18) T K (inf y ′ ( v ( τ, y ′ )) − v ( τ, y )) (cid:19) ( T K ( v y + p ))where T iK ∈ C ∞ b are truncature functions. In particular, F K,j ( τ, · ) ∈ W , ∞ uniformly in τ ∈ [0 , + ∞ ) and sofor all q > 1, there exists a solution w = ( w j ) j = A ( v ) ∈ W , q ([0 , T ] × [0 , p )) of( w j ) t − η ( w j ) yy = F K,j ( v )Now, we want to show that the operator A is a contraction. Let v , v ∈ W , q ([0 , T ] × [0 , p )). Standardparabolic estimates show that | A j ( v ) − A j ( v ) | W , q ([0 ,T ] × [0 , p )) ≤ C |F K,j ( τ, v ) − F K,j ( τ, v ) | L q ([0 ,T ] × [0 , p )) ≤ C (cid:16) | v − v | L q ([0 ,T ] × [0 , p )) + | inf( v ) − v − (inf( v ) − v ) | L q ([0 ,T ] × [0 , p )) + | ( v − v ) y | L q ([0 ,T ] × [0 , p )) (cid:17) ≤ CT β | v − v | W , q ([0 ,T ] × [0 , p )) for some β > T smallenough of a solution w j ∈ C α, α .While we have smooth solutions below the truncature, we can apply the arguments of Subsection A.1 andget estimates on the gradient of the solution which ensures that the solution is indeed below the truncature.Finally, a posteriori, the truncature can be completely removed because of our estimate on the gradient ofthe solution. Acknowledgements This work was partially supported by the ANR-funded project “MICA” (2006-2010). The second authorwas also partially supported by the ANR-funded project “Kam Faible” (2008-2012). References [1] S. 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