Homogenization and Orowan's law for anisotropic fractional operators of any order
aa r X i v : . [ m a t h . A P ] O c t HOMOGENIZATION AND OROWAN’S LAWFOR ANISOTROPIC FRACTIONAL OPERATORS OF ANY ORDER
STEFANIA PATRIZI AND ENRICO VALDINOCI
Abstract.
We consider an anisotropic L´evy operator I s of any order s ∈ (0 ,
1) and weconsider the homogenization properties of an evolution equation.The scaling properties and the effective Hamiltonian that we obtain is different ac-cording to the cases s < / s > / Introduction
In this paper we study an evolutionary problem run by a fractional and possiblyanisotropic operator of elliptic type.These type of equations arise natural in crystallography, in which the solution of theequation has the physical meaning of the atom dislocation inside the crystal structure,see e.g. the detailed discussion of the Pierls-Nabarro crystal dislocation model in [12].Due to their mathematical interest and in view of the concrete applications in physicalmodels, these problems have been extensively studied in the recent literature, also usingnew methods coming from the analysis of fractional operators, see for instance [10, 11, 7,5, 4] and references therein.In particular, here we study an homogenization problem, related to long-time behaviorsof the system at a macroscopic scale. The scaling of the system and the results obtainedwill be different according to the fractional parameter s ∈ (0 , s > / s < /
2, the non-local features are predominant and theeffective Hamiltonian will involve the fractional operator of order s . That is, roughlyspeaking, for any s ∈ (0 , { s, } ,which reveals the stronger non-local effects present in the case s < / s < / s ∈ (0 , Mathematics Subject Classification.
Key words and phrases.
Crystal dislocation, homogenization, fractional operators.The authors have been supported by the ERC grant 277749 “EPSILON Elliptic Pde’s and Symmetryof Interfaces and Layers for Odd Nonlinearities”.
We now recall in further detail the state of the art for the homogenization of fractionalproblems in crystal dislocation, then we introduce the formal setting that we deal withand present in details our results.In [10] Monneau and the first author study an homogenization problem for the evolutivePierls-Nabarro model, which is a phase field model describing dislocation dynamics. Theyconsider the following equation(1.1) ( ∂ t u ǫ = I [ u ǫ ( t, · )] − W ′ (cid:0) u ǫ ǫ (cid:1) + σ (cid:0) tǫ , xǫ (cid:1) in R + × R N u ǫ (0 , x ) = u ( x ) on R N , where W is a periodic potential and I is an anisotropic L´evy operator of order 1, whichincludes as particular case the operator − ( − ∆) , and they prove that the solution u ǫ of(1.1) converges as ǫ → u of the following homogenized problem(1.2) ( ∂ t u = H ( ∇ x u, I [ u ( t, · )]) in R + × R N u (0 , x ) = u ( x ) on R N . For ǫ = 1, the solution u ǫ has the physical meaning of an atom dislocation along a slipplane (the rest position of the atom lies on the lattice that is prescribed by the periodicityof the potential W ). The number ǫ describes the ratio between the microscopic scaleand the macroscopic scale and then it is a small number. After a suitable rescalingone gets equation (1.1). The limit u can be interpreted as a macroscopic plastic strainsatisfying the macroscopic plastic flow rule (1.2). The function H , usually called effectiveHamiltonian, is determined, as usual in homogenization, by a cell problem, which is inthis case, for p ∈ R N and L ∈ R , the following:(1.3) ( λ + ∂ τ v = I [ v ( τ, · )] + L − W ′ ( v + λτ + p · y ) + σ ( τ, y ) in R + × R N v (0 , y ) = 0 on R N . For any p ∈ R N and L ∈ R , the quantity λ = λ ( p, L ) is the unique number for whichthere exists a solution v of (1.3) which is bounded in R + × R N . Therefore, the func-tion H ( p, L ) := λ ( p, L ) is well defined, and, in addition, this function turns out to becontinuous and non-decreasing in L .In a second paper [11], the authors consider, as a particular case, the one in which N =1, I = − ( − ∆) is the half Laplacian and σ ≡
0, and they study the behavior of H ( p, L )for small p and L . In this regime they recover the Orowan’s law, which claims that H ( p, L ) ∼ c | p | L for some constant of proportionality c >
0. To show this last result, estimates on thelayer solution associated to − ( − ∆) , i.e. on the solution φ of(1.4) − ( − ∆) φ = W ′ ( φ ) in R φ ′ > R lim x →−∞ φ ( x ) = 0 , lim x → + ∞ φ ( x ) = 1 , φ (0) = 12 , are needed. Such estimates were proved in [7] under suitable assumptions on W , whilethe existence of a unique solution φ of (1.4) was proved in [3].Recently, these kind of estimates have been proved for layer solutions associated to thefractional Laplacian − ( − ∆) s for s ∈ (0 ,
1) by Palatucci, Savin and the second author
OMOGENIZATION AND OROWAN’S LAW 3 in [13]. More general results on φ were obtained by Dipierro, Palatucci and the secondauthor in [5] for the case s ∈ (cid:2) , (cid:1) . See also [2] for related results.In this paper, in view of these new estimates, we want to extend the results of [10] and[11] to the case where the non-local operator in (1.1) is an anisotropic L´evy operator ofany order s ∈ (0 , ϕ ∈ C ( R N ) ∩ L ∞ ( R N ), let us define(1.5) I s [ ϕ ]( x ) := P V Z R N ϕ ( x + y ) − ϕ ( x ) | y | N +2 s g (cid:18) y | y | (cid:19) dy, where P V stands for the principal value of the integral and the function g satisfies(H1) g ∈ C ( S N − ) , g > g even.When g ≡ C ( N, s ) with C ( N, s ) suitable constant depending on the dimension N and onthe exponent s , then (1.5) is the integral representation of − ( − ∆) s .In addition to (H1) we make the following assumptions:(H2) W ∈ C , ( R ) and W ( v + 1) = W ( v ) for any v ∈ R ;(H3) σ ∈ C , ( R + × R N ) and σ ( t + 1 , x ) = σ ( t, x ), σ ( t, x + k ) = σ ( t, x ) for any k ∈ Z N and ( t, x ) ∈ R + × R N ;(H4) u ∈ W , ∞ ( R N ).For s > we consider the following homogenization problem:(1.6) ( ∂ t u ǫ = ǫ s − I s [ u ǫ ( t, · )] − W ′ (cid:0) u ǫ ǫ (cid:1) + σ (cid:0) tǫ , xǫ (cid:1) in R + × R N u ǫ (0 , x ) = u ( x ) on R N , and for s < :(1.7) ( ∂ t u ǫ = I s [ u ǫ ( t, · )] − W ′ (cid:0) u ǫ ǫ s (cid:1) + σ (cid:0) tǫ s , xǫ (cid:1) in R + × R N u ǫ (0 , x ) = u ( x ) on R N . Remark that the scalings for s > and s < are different. They formally coincide when s = . We prove that the solution u ǫ of (1.6) converges as ǫ → u of thehomogenized problem(1.8) ( ∂ t u = H ( ∇ x u ) in R + × R N u (0 , x ) = u ( x ) on R N , with an effective Hamiltonian H which does not depend on I s anymore, while the solution u ǫ of (1.7) converges as ǫ → u solution of the following(1.9) ( ∂ t u = H ( I s [ u ]) in R + × R N u (0 , x ) = u ( x ) on R N , with an effective Hamiltonian H not depending on the gradient. As we will see, thefunctions H and H are determined by the following cell problem:(1.10) ( λ + ∂ τ v = I s [ v ( τ, · )] + L − W ′ ( v + λτ + p · y ) + σ ( τ, y ) in R + × R N v (0 , y ) = 0 on R N , that is H and H are determined by the unique λ for which (1.10) possesses a boundedsolution (according to the cases s > and s < , respectively). We observe that thesolutions of (1.8) and (1.9) may have quite different behaviors, since ∇ u and I s [ u ] may STEFANIA PATRIZI AND ENRICO VALDINOCI be very different at a given point, even in dimension 1 and when s is close to (see forinstance [6]). Following [10], in order to solve (1.10), we show for any p ∈ R N and L ∈ R the existence of a unique solution of(1.11) ( ∂ τ w = I s [ w ( τ, · )] + L − W ′ ( w + p · y ) + σ ( τ, y ) in R + × R N w (0 , y ) = 0 on R N , and we look for some λ such that w − λτ is bounded. Precisely we have: Theorem 1.1 (Ergodicity) . Assume (H1)-(H4). For L ∈ R and p ∈ R N , there exists aunique viscosity solution w ∈ C b ( R + × R N ) of (1.11) and there exists a unique λ ∈ R such that w satisfies: w ( τ, y ) τ converges towards λ as τ → + ∞ , locally uniformly in y .The real number λ is denoted by H ( p, L ) . The function H ( p, L ) is continuous on R N × R and non-decreasing in L . Once the cell problem is solved, we can prove the following convergence results:
Theorem 1.2 (Convergence for s > ) . Assume (H1)-(H4). The solution u ǫ of (1.6) converges towards the solution u of (1.8) locally uniformly in ( t, x ) , where H ( p ) := H ( p, and H ( p, L ) is defined in Theorem 1.1. Theorem 1.3 (Convergence for s < ) . Assume (H1)-(H4). The solution u ǫ of (1.7) converges towards the solution u of (1.9) locally uniformly in ( t, x ) , where H ( L ) := H (0 , L ) and H ( p, L ) is defined in Theorem 1.1. We point out that the effective Hamiltonians H and H represent the speed of prop-agation of the dislocation dynamics according to (1.8) and (1.9). In particular, due toTheorems 1.2 and 1.3, such speed only depends on the slope of the dislocation in theweakly non-local setting s > and only on an operator of order s of the dislocation inthe strongly non-local setting s < .We will next consider the case: N = 1, I s = − ( − ∆) s and σ ≡
0, and we will make thefurther following assumptions on the potential W :(1.12) W ∈ C ,β ( R ) for some 0 < β < W ( v + 1) = W ( v ) for any v ∈ R W = 0 on Z W > R \ Z α = W ′′ (0) > W is even if s ∈ (cid:0) , (cid:1) . OMOGENIZATION AND OROWAN’S LAW 5
Under assumption (1.12), it is known, see [2] and [13], that there exists a unique function φ solution of(1.13) I s [ φ ] = W ′ ( φ ) in R φ ′ > R lim x →−∞ φ ( x ) = 0 , lim x → + ∞ φ ( x ) = 1 , φ (0) = 12 . Then we can prove the following extension of the Orowan’s law:
Theorem 1.4.
Assume (1.12) and let p , L ∈ R with p = 0 . Then the function H defined in Theorem 1.1 satisfies (1.14) H ( δp , δ s L ) δ s → c | p | L as δ → + with c = (cid:18)Z R ( φ ′ ) (cid:19) − . We notice that (1.14) can be rephrased using the notation p := δp and L := δL , bysaying H ( p, L ) = c | p | L + higher order terms,which in particular shows that H has a linear growth close to the origin. We observethat assumption (1.12) is stronger than (H2), since it requires the minima to be non-degenerate, it assumes further smoothness on the potential and the even property in thecase s < . This last property is natural for physical applications, since typically the effectof a dislocation in a given direction compensates with the one in the opposite direction(in particular it is satisfied in the classical Peierls-Nabarro model in which W ( u ) = 1 − cos(2 πu )). From the technical point of view, this property is needed only in the stronglynon-local case s < since the first order asymptotic decay of the layer solution (1.13) liesbelow a critical threshold (the even property allows us to deduce a useful second orderapproximation).The rest of the paper is organized as follows. First we recall some definitions and basicfact about viscosity solutions. Then, in Section 2 we imbed our problem into one in onedimension more, to keep track of all the homogenized quantities, and we state the ansatzon the solution we look for. The corrector equation will be studied in Section 3, whereTheorem 1.1 will be proved. Thus, we will prove Theorems 1.2 and 1.3 in Sections 4and 5, respectively. Then we present the extension of the Orowan’s law and the proof ofTheorem 1.4 in Section 6.1.1. Notations and definition of viscosity solution.
We denote by B r ( x ) the ball ofradius r centered at x . The cylinder ( t − τ, t + τ ) × B r ( x ) is denoted by Q τ,r ( t, x ). ⌊ x ⌋ and ⌈ x ⌉ denote respectively the floor and the ceiling integer part functions of a realnumber x .It is convenient to introduce the singular measure defined on R N \ { } by µ ( dz ) = 1 | z | N +2 s g (cid:18) z | z | (cid:19) dz, and to denote I ,rs [ ϕ, x ] = Z | z | r ( ϕ ( x + z ) − ϕ ( x ) − ∇ ϕ ( x ) · z ) µ ( dz ) , I ,rs [ ϕ, x ] = Z | z | >r ( ϕ ( x + z ) − ϕ ( x )) µ ( dz ) . STEFANIA PATRIZI AND ENRICO VALDINOCI
For a function u defined on (0 , T ) × R N , 0 < T + ∞ , for 0 < α < < u > αx the seminorm defined by < u > αx := sup ( t,x ′′ ) , ( t,x ′ ) ∈ (0 ,T ) × R Nx ′′6 = x ′ | u ( t, x ′′ ) − u ( t, x ′ ) || x ′′ − x ′ | α and by C αx ((0 , T ) × R N ) the space of continuous functions defined on (0 , T ) × R N that arebounded and with bounded seminorm < u > αx .Finally, we denote by U SC b ( R + × R N ) (resp., LSC b ( R + × R N )) the set of upper (resp.,lower) semicontinuous functions on R + × R N which are bounded on (0 , T ) × R N for any T > C b ( R + × R N ) := U SC b ( R + × R N ) ∩ LSC b ( R + × R N ).Let us conclude by recalling the definition of viscosity solution for a general first ordernon-local equation with associated initial condition:(1.15) ( u t = F ( t, x, u, Du, I s [ u ]) in R + × R N u (0 , x ) = u ( x ) on R N , where F ( t, x, u, p, L ) is continuous and non-decreasing in L . The definition relies on thefollowing observation: if ϕ is a bounded C function, then for any r > I s [ ϕ, x ] = Z | z | r ( ϕ ( x + z ) − ϕ ( x ) − ∇ ϕ ( x ) · z ) µ ( dz ) + Z | z | >r ( ϕ ( x + z ) − ϕ ( x )) µ ( dz )= I ,rs [ ϕ, x ] + I ,rs [ ϕ, x ]and this expression is independent of r because of the antisymmetry of ∇ ϕ ( x ) · zµ ( dz ). Definition 1.1 (viscosity solution) . A function u ∈ U SC b ( R + × R N ) (resp., u ∈ LSC b ( R + × R N ) ) is a viscosity subsolution (resp., supersolution) of (1.15) if u (0 , x ) ( u ) ∗ ( x ) (resp., u (0 , x ) > ( u ) ∗ ( x ) ) and for any ( t , x ) ∈ R + × R N , any τ ∈ (0 , t ) and any test function ϕ ∈ C ( R + × R N ) such that u − ϕ attains a local maximum (resp., minimum) at the point ( t , x ) on Q ( τ,r ) ( t , x ) , then we have ∂ t ϕ ( t , x ) − F ( t , x , u ( t , x ) , ∇ x ϕ ( t , x ) , I ,rs [ ϕ ( t , · ) , x ] + I ,rs [ u ( t , · ) , x ]) (resp., > , for a positive number r . A function u ∈ C b ( R + × R N ) is a viscosity solution of (1.15) ifit is a viscosity sub and supersolution of (1.15) . One can prove that Definition 1.1 does not depend on r and if the inequality above issatisfied for a given r >
0, then it is satisfied for any r >
0, see [10] and the referencestherein. 2.
The Ansatz
As explained in [10], because of the presence of the term W ′ (cid:0) u ǫ ǫ (cid:1) in (1.6) and (1.7),in order to get the homogenization results, we need to imbed our problems into higherdimensional ones. Let us first assume s > . Then we will consider:(2.1) ( ∂ t U ǫ = ǫ s − I s [ U ǫ ( t, · , x N +1 )] − W ′ (cid:0) U ǫ ǫ (cid:1) + σ (cid:0) tǫ , xǫ (cid:1) in R + × R N +1 U ǫ (0 , x, x N +1 ) = u ( x ) + x N +1 on R N +1 OMOGENIZATION AND OROWAN’S LAW 7 and we will prove that U ǫ converges as ǫ → U ( t, x, x N +1 ) = u ( t, x ) + x N +1 with u the solution of (1.8). We remark that U satisfies:(2.2) ( ∂ t U = H ( ∇ x U ) in R + × R N +1 U (0 , x, x N +1 ) = u ( x ) + x N +1 on R N +1 . The convergence of U ǫ to U will imply the converge of u ǫ to u . In order to prove thisresult, we introduce the higher dimensional cell problem: for P = ( p, ∈ R N +1 and L ∈ R :(2.3) (cid:26) λ + ∂ τ V = L + I s [ V ( τ, · , y N +1 )] − W ′ ( V + P · Y + λτ ) + σ ( τ, y ) in R + × R N +1 V (0 , Y ) = 0 on R N +1 . Here we use the notation Y = ( y, y N +1 ). The right Ansatz for U ǫ solution of (2.1), turnsout to be(2.4) U ǫ ( t, x, x N +1 ) ≃ ˜ U ǫ ( t, x, x N +1 ) := U ( t, x, x N +1 ) + ǫV (cid:18) tǫ , xǫ , U ( t, x, x N +1 ) − λt − p · xǫ (cid:19) with V the bounded solution of (2.3), for suitable values of p and L . Let us verify it.Fix P = ( t , x , x N +1 ) ∈ R + × R N +1 and let ˜ U ǫ ( t, x, x N +1 ) be defined as in (2.4). Letus denote(2.5) λ = ∂ t U ( P ) , p = ∇ x U ( P ) , and F ( t, x, x N +1 ) = U ( t, x, x N +1 ) − λt − p · x, τ = tǫ , y = xǫ , y N +1 = F ( t, x, x N +1 ) ǫ . We remark that P = ( p,
1) = ∇ ( x,x N +1 ) U ( P ) and˜ U ǫ ( t, x, x N +1 ) ǫ = V ( τ, y, y N +1 ) + λτ + p · y + y N +1 = V ( τ, Y ) + P · Y + λτ. Here we assume for simplicity that U and V are smooth. The proof of convergenceconsists in showing that ˜ U ǫ is a solution of (2.1) in a cylinder ( t − r, t + r ) × B r ( x , x N +1 )for r > r → + . This will allow us tocompare U ǫ with ˜ U ǫ and, thanks to the boundedness of V , to conclude that U ǫ convergesto U as ǫ → U ǫ into (2.1), we find the equation λ + ∂ τ V ( τ, Y ) = ǫ s − I s [ U ( t, · , x N +1 ) , x ] + I s [ V ( τ, · , y N +1 ) , y ] − W ′ ( V + P Y + λτ ) + σ ( τ, y ) + θ r , where θ r = ( ∂ t U ( P ) − ∂ t U ( t, x, x N +1 ))( ∂ y N +1 V ( τ, Y ) + 1)+ ǫ s I s (cid:20) V (cid:18) tǫ , · ǫ , F ( t, · , x N +1 ) ǫ (cid:19) , x (cid:21) − I s [ V ( τ, · , y N +1 ) , y ] . If V is solution of (2.3) with p as in (2.5) and L = 0, and U satisfies ∂ t U ( P ) = λ = H ( ∇ x U ( P ) , U ǫ will be a solution of (2.1) up to small errors ǫ s − I s [ U ( t, · , x N +1 ) , x ] = STEFANIA PATRIZI AND ENRICO VALDINOCI o ǫ (1) as ǫ → θ r = o r (1) as r → + . As we will see in Section 4, this last propertyholds true if the corrector V satisfies: | V | , | ∂ y N +1 V | C in R + × R N +1 for some C > ∂ y N +1 V ( τ, · , · ) is H¨older continuous, uniformly in time.Since in (2.3) the quantity I s [ V ( τ, · , y N +1 )] is computed only in the y variable, we cannotexpect this kind of regularity for the correctors. Nevertheless, following [10], we are ableto construct regular approximated sub and supercorrectors, i.e., sub and supersolutionsof approximate N + 1-dimensional cell problems, and this is enough to conclude.Similarly for s < , we will consider:(2.6) ( ∂ t U ǫ = I s [ U ǫ ( t, · , x N +1 )] − W ′ (cid:0) U ǫ ǫ s (cid:1) + σ (cid:0) tǫ s , xǫ (cid:1) in R + × R N +1 U ǫ (0 , x, x N +1 ) = u ( x ) + x N +1 on R N +1 , and we will show that U ǫ converges as ǫ → U ( t, x, x N +1 ) = u ( t, x ) + x N +1 with u the solution of (1.9). Here U is solution of(2.7) ( ∂ t U = H ( I s [ U ( t, · , x N +1 )]) in R + × R N +1 U (0 , x, x N +1 ) = u ( x ) + x N +1 on R N +1 . In this case, the right Ansatz turns out to be U ǫ ( t, x, x N +1 ) ≃ U ( t, x, x N +1 ) + ǫ s V (cid:18) tǫ s , xǫ , U ( t, x, x N +1 ) − λtǫ s (cid:19) where V is the bounded solution of (2.3) for p = 0 and L = I s [ U ( t, · , x N +1 ) , x ] . Correctors
In this section we prove Theorem 1.1 and the existence of smooth approximated suband supersolutions of the higher dimensional cell problem (2.3) introduced in Section 2which are needed to show the convergence Theorems 1.2 and 1.3. The proof of theseresults is given in [10] for the case s = 1 and it is essentially based on the comparisonprinciple and invariance under integer translations. Therefore it can be easily extendedto the case s ∈ (0 ,
1) and for this reason, here we only give a sketch of it.
Step 1: Lipschitz correctors.
One introduces the problem: for η > a , L ∈ R , p ∈ R N and P = ( p, ∂ τ U = L + I s [ U ( τ, · , y N +1 )] − W ′ ( U + P · Y ) + σ ( τ, y )+ η [ a + inf Y ′ U ( τ, Y ′ ) − U ( τ, Y )] | ∂ y N +1 U + 1 | in R + × R N +1 U (0 , Y ) = 0 on R N +1 , and show the existence of the viscosity solution U η ∈ C b ( R + × R N +1 ). When η > y N +1 with − ∂ y N +1 U η ( τ, Y ) k W ′′ k ∞ η . See the proof of Propositions 6.2, 6.3 and 6.4 in [10] for details about the existence andregularity of the solution of (3.1). As we will explain in Step 5, choosing conveniently
OMOGENIZATION AND OROWAN’S LAW 9 the number a in (3.1), we obtain sub and supersolutions of the N + 1-dimensional cellproblem (2.3) which are Lipschitz continuous in y N +1 . Step 2: Ergodicity.
Using the comparison principle, and the periodicity of σ and W , one can prove thefollowing ergodic result: Proposition 3.1 (Ergodic properties) . There exists a unique λ η = λ η ( p, L ) such that theviscosity solution U η ∈ C b ( R + × R N +1 ) of (3.1) with η > , satisfies: (3.2) | U η ( τ, Y ) − λ η τ | C for all τ > , Y ∈ R N +1 , with C independent of η . Moreover (3.3) L − k W ′ k ∞ − k σ k ∞ + ηa λ η L + k W ′ k ∞ + k σ k ∞ + ηa . Proposition 3.1 can be proved like Proposition 6.4 in [10].
Step 3: Proof of Theorem 1.1.
Let U be the solution of (3.1) with η = 0, then the function w ( τ, y ) := U ( τ, y, λ such that(3.4) | w ( τ, y ) − λτ | C. This property implies that λ is the unique number such that w ( τ, y ) /τ converges towards λ as τ → + ∞ , and Theorem 1.1 is proved.The next two steps are only needed in the proof of Theorems 1.2 and 1.3. We firststate some properties of the effective Hamiltonian, then in Step 5, we construct approx-imate sub and supersolutions of (2.3) which are smooth also in the additional variable y N +1 . This further regularity property is needed to control the error when we comparethe solution U ǫ of (2.1) and (2.6) with the corresponding ansatz, as explained in Section 2. Step 4: Properties of the effective Hamiltonian
We have
Proposition 3.2 (Properties of the effective Hamiltonian) . Let p ∈ R N and L ∈ R . Let H ( p, L ) be the constant defined by Theorem 1.1, then H : R N × R → R is a continuousfunction with the following properties: (i) H ( p, L ) → ±∞ as L → ±∞ for any p ∈ R N ; (ii) H ( p, · ) is non-decreasing on R for any p ∈ R N ; (iii) If σ ( τ, y ) = σ ( τ, − y ) then H ( p, L ) = H ( − p, L );(iv) If W ′ ( − s ) = − W ′ ( s ) and σ ( τ, − y ) = − σ ( τ, y ) then H ( p, − L ) = − H ( p, L ) . For the proof of Proposition 3.2 see Proposition 5.4 in [10].
Step 5: Construction of smooth approximate sub and supercorrectors.
The ergodic property (3.1) of U η implies that there exists C > C + inf Y ′ U η ( τ, Y ′ ) − U η ( τ, Y ) > , for any η >
0. Then, one take U + η to be the solution of (3.1) with a = C and U − η to bethe solution of (3.1) with a = 0. We remark that U + η and U − η are respectively super andsubsolution of ∂ τ U = L + I s [ U ( τ, · , y N +1 )] − W ′ ( U + P · Y ) + σ ( τ, y ) . Let λ + η = lim τ → + ∞ U + η ( τ, Y ) τ and λ − η = lim τ → + ∞ U − η ( τ, Y ) τ , whose the existence is guaranteed byProposition 3.1. Stability results and the ergodic property (3.2) imply that λ + η , λ − η → λ as η →
0, with λ given by Theorem 1.1.Next, one set W + η ( τ, Y ) := U + η ( τ, Y ) − λ + η τ and W − η ( τ, Y ) := U − η ( τ, Y ) − λ − η τ. Then W + η and W − η are respectively super and subsolution of (2.3) with respectively λ = λ + η and λ = λ − η , and are Lipschitz continuous in the variable y N +1 . One can in addition showthat these functions are of class C α with respect to y uniformly in y N +1 , for 0 < α < min { , s } . This comes from Proposition 4.7 in [10] that can be easily adapted to thecase s ∈ (0 , W + η and W − η are not enough in order to prove the conver-gence results, Theorems 1.2 and 1.3, as pointed out in Section 2. Therefore, one introducesa positive smooth function ρ : R → R , with support in B (0) and mass 1 and defines asequence of mollifiers ( ρ δ ) δ by ρ δ ( r ) = δ ρ (cid:0) rδ (cid:1) , r ∈ R . Then, one finally defines V ± η,δ ( t, y, y N +1 ) := W ± η ( t, y, · ) ⋆ ρ δ ( · ) = Z R W ± η ( t, y, z ) ρ δ ( y N +1 − z ) dz. Choosing properly δ = δ ( η ), one can prove the following result: Proposition 3.3 (Smooth approximate correctors) . Let λ be the constant defined byTheorem 1.1. For any fixed p ∈ R N , P = ( p, , L ∈ R and η > small enough, thereexist real numbers λ + η ( p, L ) , λ − η ( p, L ) , a constant C > (independent of η, p and L ) andbounded super and subcorrectors V + η , V − η , i.e. respectively a super and a subsolution of (3.5) λ ± η + ∂ τ V ± η = L + I s [ V ± η ( τ, · , y N +1 )] − W ′ ( V ± η + P · Y + λ ± η τ ) + σ ( τ, y ) ∓ o η (1) in R + × R N +1 V ± η (0 , Y ) = 0 on R N +1 , where o η (1) → as η → + , such that (3.6) lim η → + λ + η ( p, L ) = lim η → + λ − η ( p, L ) = λ ( p, L ) , locally uniformly in ( p, L ) , λ ± η satisfy (i) and (ii) of Proposition 3.2 and for any ( τ, Y ) ∈ R + × R N +1 (3.7) | V ± η ( τ, Y ) | C. OMOGENIZATION AND OROWAN’S LAW 11
Moreover V ± η are of class C w.r.t. y N +1 , and for any < α < min { , s } (3.8) − ∂ y N +1 V ± η k W ′′ k ∞ η , (3.9) k ∂ y N +1 y N +1 V ± η k ∞ C η , < ∂ y N +1 V ± η > αy , C η,α . Proof of Theorem 1.2
To prove Theorem 1.2, as explained in Section 2, we introduce the higher dimensionalproblem (2.1) and we prove the convergence of the solution U ǫ to the solution U of (2.2).Let us first state the following Proposition 4.1.
For ǫ > there exists U ǫ ∈ C b ( R + × R N +1 ) (unique) viscosity solutionof (2.1) . Moreover, there exists a constant C > independent of ǫ such that (4.1) | U ǫ ( t, x, x N +1 ) − u ( x ) − x N +1 | Ct.
Proposition 4.1 as well as the existence of a unique solution of problems (1.6), (1.8)and (2.2) is a consequence of the Perron’s method and the comparison principle for theseequations, see [10] and references therein. Let us exhibit the link between the problem in R N and the problem in R N +1 . Lemma 4.2 (Link between the problems on R N and on R N +1 ) . If u ǫ and U ǫ denoterespectively the solution of (1.6) and (2.1) , then we have (cid:12)(cid:12)(cid:12) U ǫ ( t, x, x N +1 ) − u ǫ ( t, x ) − ǫ j x N +1 ǫ k(cid:12)(cid:12)(cid:12) ǫ, (4.2) U ǫ (cid:16) t, x, x N +1 + ǫ j aǫ k(cid:17) = U ǫ ( t, x, x N +1 ) + ǫ j aǫ k for any a ∈ R . This lemma follows from the comparison principle for (2.1) and the invariance by ǫ -translations w.r.t. x N +1 . Lemma 4.3.
Let u and U be respectively the solutions of (1.8) and (2.2) . Then, wehave U ( t, x, x N +1 ) = u ( t, x ) + x N +1 . Lemma 4.3 is a consequence of the comparison principle for (2.2) and the invariance bytranslations w.r.t. x N +1 .Let us proceed with the proof of Theorem 1.2. In what follows we will use the notation X = ( x, x N +1 ). By (4.1), we know that the family of functions { U ǫ } ǫ> is locally bounded,then U + ( t, X ) := lim sup ǫ → ∗ U ǫ ( t, X ) := lim sup ǫ → t ′ ,X ′ ) → ( t,X ) U ǫ ( t ′ , X ′ )is everywhere finite, so it becomes classical to prove that U + is a subsolution of (2.2).Similarly, we can prove that U − ( t, X ) := lim inf ǫ → ∗ U ǫ ( t, X ) := lim inf ǫ → t ′ ,X ′ ) → ( t,X ) U ǫ ( t ′ , X ′ )is a supersolution of (2.2). Moreover U + (0 , X ) = U − (0 , X ) = u ( x ) + x N +1 . The compar-ison principle for (2.2) then implies that U + U − . Since the reverse inequality U − U + always holds true, we conclude that the two functions coincide with U , the unique vis-cosity solution of (2.2).By Lemmata 4.2 and 4.3, the convergence of U ǫ to U proves in particular that u ǫ converges towards u viscosity solution of (1.8).To prove that U + is a subsolution of (2.2), we argue by contradiction. We consider atest function φ such that U + − φ attains a zero maximum at ( t , X ) with t > X = ( x , x N +1 ). Without loss of generality we may assume that the maximum is strictand global. Suppose that there exists θ > ∂ t φ ( t , X ) = H ( ∇ x φ ( t , X )) + θ. By Proposition 3.2, we know that there exists L > H ( ∇ x φ ( t , X )) + θ = H ( ∇ x φ ( t , X ) ,
0) + θ = H ( ∇ x φ ( t , X ) , L ) . By Propositions 3.3 and 3.2, we can consider a sequence L η → L as η → + , such that λ + η ( ∇ x φ ( t , X ) , L η ) = λ ( ∇ x φ ( t , X ) , L ). We choose η so small that L η − o η (1) > L / >
0, where o η (1) is defined in Proposition 3.3. Let V + η be the approximate supercorrectorgiven by Proposition 3.3 with p = ∇ x φ ( t , X ) , L = L η and λ + η = λ + η ( p, L η ) = λ ( p, L ) = ∂ t φ ( t , X ) . For simplicity of notations, in the following we denote V = V + η . We consider the function F ( t, X ) = φ ( t, X ) − p · x − λ + η t, and as in [10] we introduce the “ x N +1 -twisted perturbed test function” φ ǫ defined by:(4.3) φ ǫ ( t, X ) := ( φ ( t, X ) + ǫV (cid:16) tǫ , xǫ , F ( t,X ) ǫ (cid:17) + ǫk ǫ in ( t , t ) × B ( X ) U ǫ ( t, X ) outside , where k ǫ ∈ Z will be chosen later.We are going to prove that φ ǫ is a supersolution of (2.1) in Q r,r ( t , X ) for some r < properly chosen and such that Q r,r ( t , X ) ⊂ ( t , t ) × B ( X ). First, we observe thatsince U + − φ attains a strict maximum at ( t , X ) with U + − φ = 0 at ( t , X ) and V isbounded, we can ensure that there exists ǫ = ǫ ( r ) > ǫ ǫ (4.4) U ǫ ( t, X ) φ ( t, X )+ ǫV (cid:18) tǫ , xǫ , F ( t, X ) ǫ (cid:19) − γ r , in (cid:18) t , t (cid:19) × B ( X ) \ Q r,r ( t , X )for some γ r = o r (1) >
0. Hence choosing k ǫ = ⌈ − γ r ǫ ⌉ we get U ǫ φ ǫ outside Q r,r ( t , X ) . Let us next study the equation satisfied by φ ǫ . For this, we observe that aǫ − j aǫ k aǫ OMOGENIZATION AND OROWAN’S LAW 13 and so, from (4.2), we deduce that U ǫ ( t, x, x N +1 ) + a − ǫ U ǫ (cid:16) t, x, x N +1 + ǫ j aǫ k(cid:17) U ǫ ( t, x, x N +1 ) + a. Consequently, passing to the limit, we obtain that U + ( t, x, x N +1 + a ) = U + ( t, x, x N +1 ) + a for any a ∈ R .From this, we derive that ∂ x N +1 F ( t , X ) = ∂ x N +1 φ ( t , X ) = 1. Then, there exists r > Id × F : Q r ,r ( t , X ) −→ U r ( t, x, x N +1 ) ( t, x, F ( t, x, x N +1 ))is a C -diffeomorphism from Q r ,r ( t , X ) onto its range U r . Let G : U r → R be themap such that Id × G : U r −→ Q r ,r ( t , X )( t, x, ξ N +1 ) ( t, x, G ( t, x, ξ N +1 ))is the inverse of Id × F . Let us introduce the variables τ = t/ǫ , Y = ( y, y N +1 ) with y = x/ǫ and y N +1 = F ( t, X ) /ǫ . Let us consider a test function ψ such that φ ǫ − ψ attainsa global zero minimum at ( t, X ) ∈ Q r ,r ( t , X ) and defineΓ ǫ ( τ, Y ) = 1 ǫ [ ψ ( ǫτ, ǫy, G ( ǫτ, ǫy, ǫy N +1 )) − φ ( ǫτ, ǫy, G ( ǫτ, ǫy, ǫy N +1 ))] − k ǫ . Then ψ ( t, X ) = φ ( t, X ) + ǫ Γ ǫ (cid:18) tǫ , xǫ , F ( t, X ) ǫ (cid:19) + ǫk ǫ and Γ ǫ is a test function for V :(4.5) Γ ǫ ( τ , Y ) = V ( τ , Y ) and Γ ǫ ( τ, Y ) V ( τ, Y ) for all ( ǫτ, ǫY ) ∈ Q r ,r ( t , X ) , where τ = t/ǫ , y = x/ǫ, y N +1 = F ( t, X ) /ǫ , Y = ( y, y N +1 ). From Proposition 3.3, weknow that V is Lipschitz continuous w.r.t. y N +1 with Lipschitz constant M η dependingon η . This implies that(4.6) | ∂ y N +1 Γ ǫ ( τ , Y ) | M η . Simple computations yield with P = ( p, ∈ R N +1 :(4.7) ( λ + η + ∂ τ Γ ǫ ( τ , Y ) = ∂ t ψ ( t, X ) + (cid:0) ∂ y N +1 Γ ǫ ( τ , Y ) (cid:1) ( ∂ t φ ( t , X ) − ∂ t φ ( t, X )) ,λ + η τ + P · Y + V ( τ , Y ) = φ ǫ ( t,X ) ǫ − k ǫ . Using (4.7) and (4.6), equation (3.5) yields for any ρ > ∂ t ψ ( t, X ) + o r (1) > L η + I ,ρs [Γ ǫ ( τ , · , y N +1 ) , y ] + I ,ρs [ V ( τ , · , y N +1 ) , y ] − W ′ (cid:18) φ ǫ ( t, X ) ǫ (cid:19) + σ (cid:18) tǫ , xǫ (cid:19) − o η (1) . (4.8)Now, to complete the proof of Theorem 1.2, we state the following lemma (which will beproved in the next subsection): Lemma 4.4. (Supersolution property for φ ǫ ) For ǫ ǫ ( r ) < r r , we have ∂ t ψ ( t, X ) > ǫ s − (cid:0) I , s (cid:2) ψ ( t, · , x N +1 ) , x (cid:3) + I , s (cid:2) φ ǫ ( t, · , x N +1 ) , x (cid:3)(cid:1) − W ′ (cid:18) φ ǫ ( t, X ) ǫ (cid:19) + σ (cid:18) tǫ , xǫ (cid:19) − o η (1) + o r (1) + L η . (4.9)The proof of Lemma 4.4 is postponed to the next subsection, for the convenience ofthe reader, so we complete now the proof of Theorem 1.2. For this, let r r be so smallthat o r (1) > − L /
4. Then, recalling that L η − o η (1) > L /
2, for ǫ ǫ ( r ) we have ∂ t ψ ( t, X ) > ǫ s − (cid:0) I , s (cid:2) ψ ( t, · , x N +1 ) , x (cid:3) + I , s (cid:2) φ ǫ ( t, · , x N +1 ) , x (cid:3)(cid:1) − W ′ (cid:18) φ ǫ ( t, X ) ǫ (cid:19) + σ (cid:18) tǫ , xǫ (cid:19) + L , and therefore φ ǫ is a supersolution of (2.1) in Q r,r ( t , X ).Since U ǫ φ ǫ outside Q r,r ( t , X ), by the comparison principle, we conclude that U ǫ ( t, X ) φ ( t, X ) + ǫV (cid:18) tǫ , xǫ , F ( t, X ) ǫ (cid:19) + ǫk ǫ in Q r,r ( t , X )and we obtain the desired contradiction by passing to the upper limit as ǫ → t , X )using the fact that U + ( t , X ) = φ ( t , X ): 0 − γ r .This ends the proof of Theorem 1.2.4.1. Proof of Lemma 4.4.
The result will follow from (4.8) and the following inequality(4.10) I ,ρs [Γ ǫ ( τ , · , y N +1 ) , y ] + I ,ρs [ V ( τ , · , y N +1 ) , y ] > ǫ s − (cid:0) I , s (cid:2) ψ ( t, · , x N +1 ) , x (cid:3) + I , s (cid:2) φ ǫ ( t, · , x N +1 ) , x (cid:3)(cid:1) + o r (1)Keep in mind that y N +1 = F ( t,X ) ǫ . Since ψ ( t, X ) = φ ( t, X ) + ǫ Γ ǫ (cid:16) tǫ , xǫ , F ( t,X ) ǫ (cid:17) + ǫk ǫ , wehave I , s (cid:2) ψ ( t, · , x N +1 ) , x (cid:3) = I + I , (4.11)where I = Z | x | ǫ Γ ǫ (cid:16) tǫ , x + xǫ , F ( t,x + x,x N +1 ) ǫ (cid:17) − Γ ǫ ( τ , Y ) −∇ y Γ ǫ ( τ , Y ) · xǫ − ∂ y N +1 Γ ǫ ( τ , Y ) ∇ x F ( t, X ) · xǫ ! µ ( dx ) ,I = Z | x | (cid:0) φ ( t, x + x, x N +1 ) − φ ( t, X ) − ∇ φ ( t, X ) · x (cid:1) µ ( dx ) . In order to show (4.10), we show successively in Steps 1, 2 and 3: ǫ s − I I ,ρs [Γ ǫ ( τ , · , y N +1 ) , y ] + I ,ρs [ V ( τ , · , y N +1 ) , y ] + o r (1) + C ǫ ρ − s ǫ s − I o r (1) ǫ s − I , s (cid:2) φ ǫ ( t, · , x N +1 ) , x (cid:3) o r (1) OMOGENIZATION AND OROWAN’S LAW 15
Because the expressions are non linear and non-local and with a singular kernel, thereis no simple computation and we have to carefully check those inequalities sometimessplitting terms in easier parts to estimate.
Step 1:
We can choose ǫ so small that for any ǫ ǫ and any ρ > ǫ s − I I ,ρs [Γ ǫ ( τ , · , y N +1 ) , y ] + I ,ρs [ V ( τ , · , y N +1 ) , y ] + o r (1) + C ǫ ρ − s . Take ρ > δ > ρ small and R > ǫR <
1. Since g is even, we canwrite I = I + I + I + I , where I = Z | x | ǫρ ǫ (cid:18) Γ ǫ (cid:18) tǫ , x + xǫ , F ( t, x + x, x N +1 ) ǫ (cid:19) − Γ ǫ ( τ , Y ) − ∇ y Γ ǫ ( τ , Y ) · xǫ − ∂ y N +1 Γ ǫ ( τ , Y ) ∇ x F ( t, X ) · xǫ (cid:17) µ ( dx ) ,I = Z ǫρ | x | ǫδ ǫ (cid:18) Γ ǫ (cid:18) tǫ , x + xǫ , F ( t, x + x, x N +1 ) ǫ (cid:19) − Γ ǫ ( τ , Y ) (cid:19) µ ( dx ) ,I = Z ǫδ | x | ǫR ǫ (cid:18) Γ ǫ (cid:18) tǫ , x + xǫ , F ( t, x + x, x N +1 ) ǫ (cid:19) − Γ ǫ ( τ , Y ) (cid:19) µ ( dx ) ,I = Z ǫR | x | ǫ (cid:18) Γ ǫ (cid:18) tǫ , x + xǫ , F ( t, x + x, x N +1 ) ǫ (cid:19) − Γ ǫ ( τ , Y ) (cid:19) µ ( dx ) . Moreover I ,ρs [ V ( τ , · , y N +1 ) , y ] = J + J + J , where J = Z ρ< | z | δ ( V ( τ , y + z, y N +1 ) − V ( τ , Y )) µ ( dz ) ,J = Z δ< | z | R ( V ( τ , y + z, y N +1 ) − V ( τ , Y )) µ ( dz ) ,J = Z | z | >R ( V ( τ , y + z, y N +1 ) − V ( τ , Y )) µ ( dz ) . STEP 1.1:
Estimate of ǫ s − I and I ,ρs [Γ ǫ ( τ , · , y N +1 ) , y ] . Since Γ ǫ is of class C , we have(4.12) | ǫ s − I | , |I ,ρs [Γ ǫ ( τ , · , y N +1 ) , y ] | C ǫ ρ − s , where C ǫ depends on the second derivatives of Γ ǫ . Notice that if we knew that V is smoothin y too, we could choose ρ = 0.STEP 1.2 Estimate of ǫ s − I − J . Using (4.5) and the fact that g is even, we can estimate ǫ s − I − J as follows ǫ s − I − J Z ρ< | z | δ (cid:20) V (cid:18) τ , y + z, F ( t, x + ǫz, x N +1 ) ǫ (cid:19) − V (cid:18) τ , y + z, F ( t, X ) ǫ (cid:19)(cid:21) µ ( dz )= Z ρ< | z | δ (cid:26)(cid:20) V (cid:18) τ , y + z, F ( t, x + ǫz, x N +1 ) ǫ (cid:19) − V (cid:18) τ , y + z, F ( t, X ) ǫ (cid:19) − ∂ y N +1 V (cid:18) τ , y + z, F ( t, X ) ǫ (cid:19) ∇ x F ( t, X ) · z (cid:21) + (cid:2) ∂ y N +1 V ( τ , y + z, y N +1 ) − ∂ y N +1 V ( τ , Y ) (cid:3) ∇ x F ( t, X ) · z (cid:9) µ ( dz ) . Next, using (3.9), we get(4.13) ǫ s − I − J C Z | z | δ ( | z | + | z | α ) µ ( dz ) Cδ α +1 − s , for 2 s − < α < Estimate of ǫ s − I − J . If M η is the Lipschitz constant of V w.r.t. y N +1 , then ǫ s − I − J Z δ< | z | R (cid:18) V (cid:18) τ , y + z, F ( t, x + ǫz, x N +1 ) ǫ (cid:19) − V (cid:18) τ , y + z, F ( t, X ) ǫ (cid:19)(cid:19) µ ( dz ) M η Z δ< | z | R (cid:12)(cid:12)(cid:12)(cid:12) F ( t, x + ǫz, x N +1 ) ǫ − F ( t, X ) ǫ (cid:12)(cid:12)(cid:12)(cid:12) µ ( dz ) M η Z δ< | z | R sup | z | R |∇ x F ( t, x + ǫz, x N +1 ) || z | µ ( dz ) . Then(4.14) ǫ s − I − J C sup | z | R |∇ x F ( t, x + ǫz, x N +1 ) | (cid:18) δ s − − R s − (cid:19) . STEP 1.4:
Estimate of ǫ s − I and J . Since V is uniformly bounded on R + × R N +1 , we have ǫ s − I Z R< | z | ǫ (cid:18) V (cid:18) τ , y + z, F ( t, x + ǫz, x N +1 ) ǫ (cid:19) − V ( τ , Y ) (cid:19) µ ( dz ) Z | z | >R k V k ∞ µ ( dz ) CR s . (4.15)Similarly(4.16) | J | CR s . Now, from (4.12), (4.13), (4.14), (4.15) and (4.16), we infer that ǫ s − I I ,ρs [Γ ǫ ( τ , · , y N +1 ) , y ] + I ,ρs [ V ( τ , · , y N +1 ) , y ] + 2 C ǫ ρ − s + Cδ α +1 − s + C sup | z | R |∇ x F ( t, x + ǫz, x N +1 ) | (cid:18) δ s − − R s − (cid:19) + CR s . OMOGENIZATION AND OROWAN’S LAW 17
We remark that, from the definition of F , we havesup | z | R |∇ x F ( t, x + ǫz, x N +1 ) | sup | z | R |∇ φ ( t, x + ǫz, x N +1 ) − ∇ φ ( t , X ) | sup | z | R |∇ φ ( t, x + ǫz, x N +1 ) − ∇ φ ( t, X ) | + |∇ φ ( t, X ) − ∇ φ ( t , X ) | C ( ǫR + r ) . Now, we choose R = R ( r ) such R → + ∞ as r → + , ǫ = ǫ ( r ) such that Rǫ ( r ) r and δ = δ ( r ) > δ → r → + and r/δ s − → r → + . With this choice,for any ǫ ǫ and any ρ < δCδ α +1 − s + C sup | z | R |∇ x F ( t, x + ǫz, x N +1 ) | (cid:18) δ s − − R s − (cid:19) + CR s = o r (1) as r → + , and Step 1 is proved.The next two steps are trivial. Step 2: ǫ s − I Cǫ s − .Step 3: ǫ s − I , s (cid:2) φ ǫ ( t, · , x N +1 ) , x (cid:3) Cǫ s − . Finally Steps 1, 2 and 3 give ǫ s − I , s (cid:2) ψ ( t, · , x N +1 ) , x (cid:3) + ǫ s − I , s (cid:2) φ ǫ ( t, · , x N +1 ) , x (cid:3) I ,ρs [Γ ǫ ( τ , · , y N +1 ) , y ] + I ,ρs [ V ( τ , · , y N +1 ) , y ] + o r (1) + C ǫ ρ − s . from which, using inequality (4.8) and letting ρ → + , we get (4.9).5. Proof of Theorem 1.3
The proof of Theorem 1.3 is similar to the proof of Theorem 1.2, therefore we only givea sketch of it. As in Theorem 1.2, we argue by contradiction, assuming that there is atest function φ such that U + − φ attains a strict zero maximum at ( t , X ) with t > X = ( x , x N +1 ), and ∂ t φ ( t , X ) = H ( L ) + θ for some θ >
0, where L = Z | x | ( φ ( t , x + x, x N +1 ) − φ ( t , X ) − ∇ x φ ( t , X ) · x ) µ ( dx )+ Z | x | > ( U + ( t , x + x, x N +1 ) − U + ( t , X )) µ ( dx ) . (5.1)Then, we choose L > L η → L as η → + , such that λ + η (0 , L η + L ) = λ (0 , L + L ) = λ (0 , L ) + θ = H ( L ) + θ. Let V be the approximate supercorrector given by Proposition 3.3 with p = 0 , L = L + L η and λ + η = λ + η (0 , L + L η ) = ∂ t φ ( t , X ) . Let us introduce the “ x N +1 -twisted perturbed test function” φ ǫ defined by: φ ǫ ( t, X ) := ( φ ( t, X ) + ǫ s V (cid:16) tǫ s , xǫ , F ( t,X ) ǫ s (cid:17) + ǫ s k ǫ in ( t , t ) × B ( X ) U ǫ ( t, X ) outside , where F ( t, X ) = φ ( t, X ) − λ + η t and k ǫ ∈ Z is opportunely chosen. As in Section 4, wecan prove that φ ǫ is a supersolution of (2.6) in a neighborhood Q r,r ( t , X ) of ( t , X ), forsome small r properly chosen. Moreover U ǫ φ ǫ outside Q r,r ( t , X ) . The contradiction follows by comparison.6.
Proof of Theorem 1.4
In this section we restrict ourself to the case: N = 1, I s = − ( − ∆) s and σ ≡
0. Forfixed p, L ∈ R , let us introduce the corrector u ( τ, y ) := w ( τ, y ) + py where w is the solution of (1.11) given by Theorem 1.1. Then u is solution of(6.1) ( ∂ τ u = L + I s [ u ( τ, · )] − W ′ ( u ) in R + × R u (0 , y ) = py on R , and by the ergodic property (3.4) it satisfies(6.2) | u ( τ, y ) − py − λτ | C. The idea underlying the proof of Theorem 1.4 is related to a fine asymptotics of equa-tion (6.1). We want to show that if u solves (6.1) with p = δ | p | and L = δ s L , i.e.(6.3) ∂ τ u = δ s L + I s [ u ( τ, · )] − W ′ ( u )and u (0 , y ) = δp y , then u ( τ, y ) ∼ δp y + λτ + bounded with λ ∼ δ s c | p | L . We deduce that we should have u ( τ, y ) τ → λ = δ s c | p | L as τ → + ∞ . We see that this λ = H ( δp , δ s L ) is exactly the one we expect asymptotically in Theo-rem 1.4.Following the idea of [11], one may expect to find particular solutions u of (6.3) thatwe can write u ( τ, y ) = h ( δp y + λτ )for some λ ∈ R and a function h (called hull function) satisfying | h ( z ) − z | C. This means that h solves λh ′ = δ s L + δ s | p | s I s [ h ] − W ′ ( h ) . Then it is natural to introduce the non-linear operator:(6.4)
N L λL [ h ] := λh ′ − δ s L − δ s | p | s I s [ h ] + W ′ ( h ) OMOGENIZATION AND OROWAN’S LAW 19 and for the ansatz for λ : λ L δ = δ s c | p | L it is natural to look for an ansatz h L δ for h . We define (see Proposition 6.1) h L δ ( x ) = lim n → + ∞ s L δ,n ( x )where for s > and for all p = 0, L ∈ R , δ > n ∈ N we define the sequence offunctions { s L δ,n ( x ) } n by(6.5) s L δ,n ( x ) = δ s L α + n X i = − n φ (cid:18) x − iδ | p | (cid:19) − n + δ s n X i = − n ψ (cid:18) x − iδ | p | (cid:19) where α = W ′′ (0) > φ is the solution of (1.13). The corrector ψ is the solution ofthe following problem(6.6) I s [ ψ ] = W ′′ ( φ ) ψ + L W ′′ (0) ( W ′′ ( φ ) − W ′′ (0)) + cφ ′ in R lim x → + − ∞ ψ ( x ) = 0 c = L R R ( φ ′ ) . For s < , the function ψ defined above may not decay fast enough so that the sequence n X i = − n ψ (cid:18) x − iδ | p | (cid:19) converges. Therefore, in this case we define(6.7) s L δ,n ( x ) = δ s L α + n X i = − n φ (cid:18) x − iδ | p | (cid:19) − n + δ s n X i = − n ψ (cid:18) x − iδ | p | (cid:19) τ (cid:18) x − iδ | p | (cid:19) where τ = τ R , is a smooth function satisfying(6.8) τ ( x ) x ∈ R τ R ( x ) = 1 if | x | Rτ R ( x ) = 0 if | x | > R. The number R is a large parameter that will be chosen depending on δ . Proposition 6.1. (Good ansatz)
Assume (1.12) and R = δ | p | in (6.8) . Then, for any L ∈ R , δ > and x ∈ R , thereexists the finite limit h Lδ ( x ) = lim n → + ∞ s Lδ,n ( x ) . Moreover h Lδ has the following properties: (i) h Lδ ∈ C ( R ) and satisfies (6.9) N L λ Lδ L [ h Lδ ]( x ) = o ( δ s ) , where lim δ → o ( δ s ) δ s = 0 , uniformly for x ∈ R and locally uniformly in L ∈ R ; Here λ Lδ = δ s c | p | L and N L λL is defined in (6.4). (ii) There exists a constant
C > such that | h Lδ ( x ) − x | C for any x ∈ R . Proof of Theorem 1.4.
We will show that Theorem 1.4 follows from Proposition 6.1and the comparison principle.Fix η > L = L − η . By (i) of Proposition 6.1, there exists δ = δ ( η ) > δ ∈ (0 , δ ) we have(6.10) N L λ Lδ L [ h Lδ ] = N L λ Lδ L [ h Lδ ] − δ s η < R . Let us consider the function e u ( τ, y ), defined by e u ( τ, y ) = h Lδ ( δp y + λ Lδ τ ) . By (ii) of Proposition 6.1, we have(6.11) | e u ( τ, y ) − δp y − λ Lδ τ | ⌈ C ⌉ , where ⌈ C ⌉ is the ceil integer part of C . Moreover, by (6.10) and (6.11), e u satisfies (cid:26) e u τ δ s L + I s [ e u ] − W ′ ( e u ) in R + × R e u (0 , y ) δp y + ⌈ C ⌉ on R . Let u ( τ, y ) be the solution of (6.1), with p = δp and L = δ s L , whose existence isensured by Theorem 1.1. Then from the comparison principle and the periodicity of W ,we deduce that e u ( τ, y ) u ( τ, y ) + ⌈ C ⌉ . By the previous inequality and (6.11), we get λ Lδ τ u ( τ, y ) − δp y + 2 ⌈ C ⌉ , and dividing by τ and letting τ go to + ∞ , we finally obtain δ s c | p | ( L − η ) = λ Lδ H ( δp , δ s L ) . Similarly, it is possible to show that H ( δp , δ s L ) δ s c | p | ( L + η ) . We have proved that for any η > δ = δ ( η ) > δ ∈ (0 , δ )we have (cid:12)(cid:12)(cid:12)(cid:12) H ( δp , δ s L ) δ s − c | p | L (cid:12)(cid:12)(cid:12)(cid:12) c | p | η, i.e. (1.14), as desired. OMOGENIZATION AND OROWAN’S LAW 21
Preliminary results.
Under the assumptions (1.12) on W , there exists a unique solution of (1.13) which isof class C ,β , as shown in [2], see also [13]. When s < we suppose in addition that W is even. This implies that the function φ − C ,βloc ( R ) ∩ L ∞ ( R ) of the problem (6.6) is provedin [13]. Actually, the regularity of W implies that φ ∈ C ,β ( R ) and ψ ∈ C ,β ( R ).To prove Proposition 6.1 we need several preliminary results. We first state the followingtwo lemmata about the behavior of the functions φ and ψ at infinity. We denote by H ( x )the Heaviside function defined by H ( x ) = ( x >
00 for x < . Then we have
Lemma 6.2 (Behavior of φ ) . Assume (1.12) . Let φ be the solution of (1.13) , then thereexists a constant K > such that (6.12) (cid:12)(cid:12)(cid:12)(cid:12) φ ( x ) − H ( x ) + 12 sα x | x | s (cid:12)(cid:12)(cid:12)(cid:12) K | x | s , for | x | > , if s > , (6.13) | φ ( x ) − H ( x ) | K | x | s , for | x | > , if s < , and for any x ∈ R , s ∈ (0 , , (6.14) 0 < φ ′ ( x ) K | x | s , (6.15) | φ ′′ ( x ) | K | x | s , (6.16) | φ ′′′ ( x ) | K | x | s . Proof.
Estimate (6.12) is proved in [5], while estimates (6.13) and (6.14) are proved in[13].Since the proof of (6.15) and (6.16) is an adaptation of the one given in [11] for thesame estimates in the case s = , we only sketch it.To get (6.15), as in the proof of Lemma 3.1 in [11] one looks to the equations satisfiedby φ := φ ′′ − Cφ ′ a ( x ), where φ ′ a ( x ) := φ ′ (cid:0) xa (cid:1) , a > I s [ φ ] − W ′′ ( φ ) φ = Cφ ′ a (cid:18) W ′′ ( φ ) − a s W ′′ ( φ a ) (cid:19) + W ′′′ ( φ )( φ ′ ) . For a and R large enough, we can prove that in R \ [ − R , R ] we have I s [ φ ] − W ′′ ( φ ) φ > W ′′ ( φ ) > . Choosing C so large that φ − R , R ], the comparison principle implies φ R , therefore φ ′′ Cφ ′ a ( x ) in R . Similarly one can prove that φ ′′ > − Cφ ′ a ( x ) in R , andusing (6.14), (6.15) follows.In the same way, comparing φ ′′′ with Cφ ′ a ( x ), we get estimate (6.16). (cid:3) Lemma 6.3 (Behavior of ψ ) . Assume (1.12) . Let ψ be the solution of (6.6) , then forany L ∈ R there exist K and K > , depending on L such that (6.17) (cid:12)(cid:12)(cid:12)(cid:12) ψ ( x ) − K x | x | s (cid:12)(cid:12)(cid:12)(cid:12) K | x | s , for | x | > , if s > and for any s ∈ (0 , and x ∈ R (6.18) | ψ ′ ( x ) | K | x | s , (6.19) | ψ ′′ ( x ) | K | x | s . Proof.
We follow the proof of Lemma 3.2 in [11]. Let us start with the proof of (6.17).Since we want to point out where we use s > , we give it in the details. For a > φ a ( x ) := φ (cid:0) xa (cid:1) , which is solution of I s [ φ a ] = 1 a s W ′ ( φ a ) in R . In what follows, we denote e φ ( x ) = φ ( x ) − H ( x ) . Let a and b be positive numbers, thenmaking a Taylor expansion of the derivatives of W (remind W ′ (0) = 0), we get I s [ ψ − ( φ a − φ b )] = W ′′ ( φ ) ψ + Lα ( W ′′ ( φ ) − W ′′ (0)) + cφ ′ + (cid:18) b s W ′ ( φ b ) − a s W ′ ( φ a ) (cid:19) = W ′′ ( φ )( ψ − ( φ a − φ b )) + W ′′ ( e φ )( φ a − φ b ) + Lα ( W ′′ ( e φ ) − W ′′ (0))+ cφ ′ + (cid:18) b s W ′ ( e φ b ) − a s W ′ ( e φ a ) (cid:19) = W ′′ ( φ )( ψ − ( φ a − φ b )) + W ′′ (0)( φ a − φ b ) + Lα W ′′′ (0) e φ + cφ ′ + W ′′ (0) (cid:18) b s e φ b − a s e φ a (cid:19) + ( φ a − φ b ) O ( e φ ) + O ( e φ ) + O ( e φ a ) + O ( e φ b ) . Then the function ψ = ψ − ( φ a − φ b ) satisfies I s [ ψ ] − W ′′ ( φ ) ψ = α ( φ a − φ b ) + Lα W ′′′ (0) e φ + cφ ′ + α (cid:18) b s e φ b − a s e φ a (cid:19) + ( φ a − φ b ) O ( e φ ) + O ( e φ ) + O ( e φ a ) + O ( e φ b ) . We want to estimate the right-hand side of the last equality. By Lemma 6.2, for | x | > max { , | a | , | b |} we have α ( φ a − φ b ) + Lα W ′′′ (0) e φ > − x s | x | s (cid:20) ( a s − b s ) + Lα W ′′′ (0) (cid:21) − K α | x | s (cid:18) a s + b s + | L | α | W ′′′ (0) | (cid:19) . OMOGENIZATION AND OROWAN’S LAW 23
Choose a, b > a s − b s ) + Lα W ′′′ (0) = 0, then α ( φ a − φ b ) + Lα W ′′′ (0) e φ > − C | x | s , for | x | > max { , | a | , | b |} . Here and in what follows, as usual C denotes various positiveconstants. From Lemma 6.2 we also derive that α (cid:18) b s e φ b − a s e φ a (cid:19) > − C | x | s , and cφ ′ > − C | x | s . Moreover, since s > , we have( φ a − φ b ) O ( e φ ) + O ( e φ ) + O ( e φ a ) + O ( e φ b ) > − C | x | s > − C | x | s , for | x | > max { , | a | , | b |} . Then we conclude that there exists R > | x | > R we have I s [ ψ ] − W ′′ ( φ ) ψ > − C | x | s . Now, let us consider the function φ ′ d ( x ) = φ ′ (cid:0) xd (cid:1) , d >
0, which is solution of I s [ φ ′ d ] = 1 d s W ′′ ( φ d ) φ ′ d in R , and denote ψ = ψ − e Cφ ′ d , with e C >
0. Then, for | x | > R we have I s [ ψ ] > W ′′ ( φ ) ψ − e Cd s W ′′ ( φ d ) φ ′ d − C | x | s = W ′′ ( φ ) ψ + e Cφ ′ d (cid:18) W ′′ ( φ ) − d s W ′′ ( φ d ) (cid:19) − C | x | s . Let us choose d > R > R such that (cid:26) W ′′ ( φ ) − d s W ′′ ( φ d ) > W ′′ (0) > R \ [ − R , R ]; W ′′ ( φ ) > R \ [ − R , R ] , then from (6.14), for e C large enough we get I s [ ψ ] − W ′′ ( φ ) ψ > R \ [ − R , R ] . Choosing e C such that moreover ψ < − R , R ] , we can ensure that ψ R . Indeed, assume by contradiction that there exists x ∈ R \ [ − R , R ] such that ψ ( x ) = sup R ψ > . Then I s [ ψ, x ] I s [ ψ, x ] − W ′′ ( φ ( x )) ψ ( x ) > W ′′ ( φ ( x )) > , from which ψ ( x ) , a contradiction. Therefore, ψ R which implies, with together (6.12) and (6.14), ψ K x | x | s + K | x | s for | x | > . Looking at the function ψ − ( φ a − φ b ) + e Cφ ′ d , we conclude similarly that ψ > K x | x | s − K | x | s for | x | > , and (6.17) is proved.Now let us turn to (6.18). By deriving the first equation in (6.6), we see that thefunction ψ ′ which is bounded and of class C ,β , is a solution of I s [ ψ ′ ] = W ′′ ( φ ) ψ ′ + W ′′′ ( φ ) φ ′ ψ + Lα W ′′′ ( φ ) φ ′ + cφ ′′ in R . Then the function ψ ′ = ψ ′ − Cφ ′ a , satisfies I s [ ψ ′ ] − W ′′ ( φ ) ψ ′ = Cφ ′ a (cid:18) W ′′ ( φ ) − a s W ′′ ( φ a ) (cid:19) + W ′′′ ( φ ) φ ′ ψ + Lα W ′′′ ( φ ) φ ′ + cφ ′′ = Cφ ′ a (cid:18) W ′′ ( φ ) − a s W ′′ ( φ a ) (cid:19) + O (cid:18)
11 + | x | s (cid:19) , by (6.14) and (6.15) and as before we deduce that for C and a large enough ψ ′ R ,which implies that ψ ′ K | x | s . The inequality ψ ′ > − K | x | s is obtained similarly byproving that ψ ′ + Cφ ′ a > R .Similarly, estimate (6.19) is obtained by comparing ψ ′′ with Cφ ′ a for some large a and C and using (6.14), (6.15) and (6.16). (cid:3) Proof of Proposition 6.1.
For simplicity of notation we denote (for the rest of the paper) x i = x − iδ | p | , e φ ( z ) = φ ( z ) − H ( z ) . Then we have the following six claims (whose proofs are postponed to the end of thesection).
Claim 1:
Let x = i + γ , with i ∈ Z and γ ∈ (cid:0) − , (cid:3) , then there exist numbers θ i ∈ ( − , such that n X i = − ni = i x − i | x − i | s → − sγ + ∞ X i =1 ( i − θ i γ ) s − ( i + γ ) s ( i − γ ) s as n → + ∞ , OMOGENIZATION AND OROWAN’S LAW 25 i − X i = − n | x − i | s → + ∞ X i =1 i + γ ) s as n → + ∞ , n X i = i | x − i | s → + ∞ X i =1 i − γ ) s as n → + ∞ . We remark that the three series on the right hand side above converge uniformly for γ ∈ (cid:0) − , (cid:3) and θ i ∈ ( − ,
1) since behave like the series + ∞ X i =1 i s . Claim 2:
Assume s < . Let x = i + γ , with i ∈ Z and γ ∈ (cid:0) − , (cid:3) , then (6.20) lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i = − ni = i [ e φ ( x i )] k − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ckδ s (2 k − | γ | and (6.21) n X i = − ni = i ,i ± |I s [ τ, x i ] | Cδ s . Claim 3:
For any x ∈ R the sequence { s Lδ,n ( x ) } n converges as n → + ∞ . Claim 4:
The sequence { ( s Lδ,n ) ′ } n converges on R as n → + ∞ , uniformly on compact sets. Claim 5:
The sequence { ( s Lδ,n ) ′′ } n converges on R as n → + ∞ , uniformly on compactsets. Claim 6:
For any x ∈ R the sequence n X i = − n I s [ s Lδ,n , x i ] converges as n → + ∞ . With these claims, we are in the position of completing the proof of Proposition 6.1,by arguing as follows.
Proof of ii)
When s > , (ii) is a consequence of (6.50) in the proof of Claim 3. Next, let us assume s < . Let x = i + γ with i ∈ Z and γ ∈ (cid:0) − , (cid:3) . For n > | i | ,we have i = n X i = − n φ ( x i ) − n − x = i = n X i = − n φ ( x i ) − n − i − γ = i − X i = − n ( φ ( x i ) −
1) + φ ( x i ) + n X i = i +1 φ ( x i ) − γ = i = n X i = − ni = i e φ ( x i ) + φ ( x i ) − γ. Then by (6.20) with k = 1(6.22) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i = n X i = − n φ ( x i ) − n − x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C, with C independent of x . Finally, for i = i − , i , i + 1 and R = δ | p | | x i | = | i + γ − i | δ | p | > δ | p | > R, therefore τ ( x i ) = 0. This implies that n X i = − n ψ ( x i ) τ ( x i ) is actually the sum of only threeterms and therefore(6.23) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i = n X i = − n ψ ( x i ) τ ( x i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k ψ k ∞ . Estimates (6.22) and (6.23) imply (ii).
Proof of i)
The function h Lδ ( x ) = lim n → + ∞ s Lδ,n ( x ) is well defined for any x ∈ R by Claim 3. Moreover,by Claims 4 and 5 and classical analysis results, it is of class C on R with( h Lδ ) ′ ( x ) = lim n → + ∞ ( s Lδ,n ) ′ ( x ) , ( h Lδ ) ′′ ( x ) = lim n → + ∞ ( s Lδ,n ) ′′ ( x ) , and the convergence of { s Lδ,n } n , { ( s Lδ,n ) ′ } n and { ( s Lδ,n ) ′′ } n is uniform on compact sets.Finally, as in [11] (see Section 4), we have for any x ∈ R (6.24) I s [ h Lδ , x ] = lim n → + ∞ I s [ s Lδ,n , x ] . To conclude the proof of Proposition 6.1, we only have to prove (6.9), which is aconsequence of the estimates above and the following lemma.
Lemma 6.4. (First asymptotics)
We have lim n → + ∞ N L λ Lδ L [ s Lδ,n ]( x ) = o ( δ s ) as δ → where lim δ → o ( δ s ) δ s = 0 , uniformly for x ∈ R . OMOGENIZATION AND OROWAN’S LAW 27
Now we can conclude the proof of (i). Indeed, by Claim 3, Claim 4 and (6.24), for any x ∈ R N L λ Lδ L [ h Lδ ]( x ) = lim n → + ∞ N L λ Lδ L [ s Lδ,n ]( x ) , and Lemma 6.4 implies that N L λ Lδ L [ h Lδ ]( x ) = o ( δ s ) , as δ → , where lim δ → o ( δ s ) δ s = 0, uniformly for x ∈ R . Proof of Lemma 6.4.
Let us first assume s > . Step 1: First computation
Fix x ∈ R , let i ∈ Z and γ ∈ (cid:0) − , (cid:3) be such that x = i + γ , let δ | p | > n > | i | .Then we have A := N L λ Lδ L [ s Lδ,n ]( x )= λ Lδ δ | p | n X i = − n (cid:2) φ ′ ( x i ) + δ s ψ ′ ( x i ) (cid:3) − n X i = − n (cid:2) I s [ φ, x i ] + δ s I s [ ψ, x i ] (cid:3) + W ′ Lδ s α + n X i = − n (cid:2) φ ( x i ) + δ s ψ ( x i ) (cid:3)! − δ s L where we have used the definitions and the periodicity of W . Using the equation (1.13)satisfied by φ , we can rewrite it as A = λ Lδ δ | p | φ ′ ( x i ) + δ s ψ ′ ( x i ) + n X i = − ni = i (cid:2) φ ′ ( x i ) + δ s ψ ′ ( x i ) (cid:3) − n X i = − ni = i W ′ ( e φ ( x i )) − δ s n X i = − ni = i I s [ ψ, x i ] − δ s I s [ ψ, x i ]+ W ′ Lδ s α + n X i = − n h e φ ( x i ) + δ s ψ ( x i ) i! − W ′ ( e φ ( x i )) − δ s L. Using the Taylor expansion of W ′ (remind that W ′ (0) = 0) and the definition of λ Lδ , weget A = δ s c L φ ′ ( x i ) + δ s ψ ′ ( x i ) + n X i = − ni = i (cid:2) φ ′ ( x i ) + δ s ψ ′ ( x i ) (cid:3) − W ′′ (0) n X i = − ni = i e φ ( x i ) − δ s n X i = − ni = i I s [ ψ, x i ] − δ s I s [ ψ, x i ]+ W ′′ ( φ ( x i )) Lδ s α + δ s ψ ( x i ) + n X i = − ni = i h e φ ( x i ) + δ s ψ ( x i ) i − δ s L + E with the error term E = E + E , where E = − n X i = − ni = i W ′ ( e φ ( x i )) + W ′′ (0) n X i = − ni = i e φ ( x i )and E = O Lδ s α + δ s ψ ( x i ) + n X i = − ni = i h e φ ( x i ) + δ s ψ ( x i ) i . Simply reorganizing the terms, we get with c = c L : A = δ s c δ s ψ ′ ( x i ) + n X i = − ni = i (cid:2) φ ′ ( x i ) + δ s ψ ′ ( x i ) (cid:3) − W ′′ (0) n X i = − ni = i e φ ( x i ) − δ s n X i = − ni = i I s [ ψ, x i ]+ W ′′ ( φ ( x i )) n X i = − ni = i h e φ ( x i ) + δ s ψ ( x i ) i + δ s (cid:16) − I s [ ψ, x i ] + W ′′ ( φ ( x i )) ψ ( x i ) + Lα W ′′ ( φ ( x i )) − L + cφ ′ ( x i ) (cid:17) + E. Using equation (6.6) satisfied by ψ , we get A = δ s c δ s ψ ′ ( x i ) + n X i = − ni = i (cid:2) φ ′ ( x i ) + δ s ψ ′ ( x i ) (cid:3) + ( W ′′ ( φ ( x i )) − W ′′ (0)) n X i = − ni = i e φ ( x i ) − δ s n X i = − ni = i I s [ ψ, x i ] + W ′′ ( φ ( x i )) δ s n X i = − ni = i ψ ( x i ) + E. OMOGENIZATION AND OROWAN’S LAW 29
Let us bound the second term of the last equality, uniformly in x . Step 2: Bound on n X i = − ni = i (cid:2) φ ′ ( x i ) + δ s ψ ′ ( x i ) (cid:3) From (6.14) and (6.18) it follows that − δ s | p | s K n X i = − ni = i | x − i | s n X i = − ni = i (cid:2) φ ′ ( x i ) + δ s ψ ′ ( x i ) (cid:3) δ s | p | s ( K + δ s K ) n X i = − ni = i | x − i | s , and then by Claim 1 we get(6.25) − Cδ s lim n → + ∞ n X i = − ni = i (cid:2) φ ′ ( x i ) + δ s ψ ′ ( x i ) (cid:3) Cδ s . Here and henceforth, C denotes various positive constants independent of x . Step 3: Bound on ( W ′′ ( φ ( x i )) − W ′′ (0)) n X i = − ni = i e φ ( x i )Let us prove that(6.26) lim n → + ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( W ′′ ( φ ( x i )) − W ′′ (0)) n X i = − ni = i e φ ( x i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Cδ s . By (6.12) we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i = − ni = i e φ ( x i ) + δ s | p | s sα n X i = − ni = i x − i | x − i | s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K δ s | p | s n X i = − ni = i | x − i | s . (6.27)If | γ | > δ | p | then | x i | = | γ | δ | p | > | e φ ( x i ) + δ s | p | s sα γ | γ | s | K δ s | p | s | γ | s which implies that | W ′′ ( e φ ( x i )) − W ′′ (0) | | W ′′′ (0) e φ ( x i ) | + O ( e φ ( x i )) C δ s | γ | s + C δ s | γ | s . By the previous inequality, (6.27) and Claim 1 we deduce thatlim n → + ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( W ′′ ( φ ( x i )) − W ′′ (0)) n X i = − ni = i e φ ( x i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C (cid:18) δ s | γ | s + δ s | γ | s (cid:19) ( δ s | γ | + δ s ) Cδ s where C is independent of γ .Finally, if | γ | < δ | p | , from (6.27) and Claim 1 we conclude that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) lim n → + ∞ ( W ′′ ( φ ( x i )) − W ′′ (0)) n X i = − ni = i e φ ( x i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Cδ s | γ | + Cδ s Cδ s , and (6.26) is proved. Step 4: Bound on δ s n X i = − ni = i I s [ ψ, x i ]We compute I s [ ψ ] = W ′′ ( e φ ) ψ + Lα ( W ′′ ( e φ ) − W ′′ (0)) + cφ ′ = W ′′ (0) ψ + Lα W ′′′ (0) e φ + O ( e φ ) ψ + O ( e φ ) + cφ ′ . (6.28)Estimates (6.12) and (6.17) implies that the sequences n X i = − ni = i O ( e φ ( x i )) ψ ( x i ) , n X i = − ni = i O ( e φ ( x i )) behave like the series ∞ X i =1 i s , therefore they are convergent since s > . Moreover, by(6.28), (6.12), (6.14), (6.17) and Claim 1, we have(6.29) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) lim n → + ∞ δ s n X i = − ni = i I s [ ψ, x i ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C ( δ s + δ s ) Cδ s . Step 5: Bound on W ′′ ( φ ( x i )) δ s n X i = − ni = i ψ ( x i )Similarly, from (6.17) and Claim 1 we get(6.30) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) lim n → + ∞ W ′′ ( φ ( x i )) δ s n X i = − ni = i ψ ( x i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Cδ s . OMOGENIZATION AND OROWAN’S LAW 31
Step 6: Bound on the error E Finally, again from (6.12), (6.17) and Claim 1 it follows that(6.31) (cid:12)(cid:12)(cid:12)(cid:12) lim n → + ∞ E (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) lim n → + ∞ O Lδ s α + δ s ψ ( x i ) + n X i = − ni = i h e φ ( x i ) + δ s ψ ( x i ) i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Cδ s Cδ s . Next, let us estimate E . From (6.12) and using s > , we have(6.32) | E | n X i = − ni = i | W ′ ( e φ ( x i )) − W ′′ (0) e φ ( x i ) | = n X i = − ni = i | O ( e φ ( x i )) | Cδ s Cδ s . Step 7: Conclusion
Therefore, from (6.25), (6.26), (6.29), (6.30), (6.31) and (6.32) we conclude that − Cδ s lim n → + ∞ N L λ Lδ L [ s Lδ,n ] Cδ s with C independent of x and Lemma 6.4 for s > is proved.Now, let us turn to the case s < . Step 1’: First computation
Making computations like in Step 1, we get A := N L λ Lδ L [ s Lδ,n ]( x )= δ s c δ s ( ψτ ) ′ ( x i ) + n X i = − ni = i (cid:2) φ ′ ( x i ) + δ s ( ψτ ) ′ ( x i ) (cid:3) + ( W ′′ ( φ ( x i )) − W ′′ (0)) n X i = − ni = i e φ ( x i ) − δ s n X i = − ni = i I s [ ψτ, x i ] + W ′′ ( φ ( x i )) δ s n X i = − ni = i ( ψτ )( x i )+ δ s (cid:16) − I s [ ψτ, x i ] + W ′′ ( φ ( x i ))( ψτ )( x i ) + Lα W ′′ ( φ ( x i )) − L + cφ ′ ( x i ) (cid:17) + E, where again E = E + E with E the error term coming from in the approximation of n X i = − ni = i W ′ ( e φ ( x i )) with W ′′ (0) n X i = − ni = i e φ ( x i ), and E = O Lδ s α + δ s ψ ( x i ) + n X i = − ni = i h e φ ( x i ) + δ s ψ ( x i ) τ ( x i ) i . To control the term I s [ ψτ, x i ], we use the following formula which can be found for instancein [1] page 7:(6.33) I s [ ψτ, x i ] = τ ( x i ) I s [ ψ, x i ] + ψ ( x i ) I s [ τ, x i ] − B ( ψ, τ )( x i ) , where B ( ψ, τ )( x i ) = C ( s ) Z R ( ψ ( y ) − ψ ( x i ))( τ ( y ) − τ ( x i )) | x i − y | s dy. Therefore the quantity A can be rewritten in the following way: A = δ s c δ s ( ψτ ) ′ ( x i ) + n X i = − ni = i (cid:2) φ ′ ( x i ) + δ s ( ψτ ) ′ ( x i ) (cid:3) + ( W ′′ ( φ ( x i )) − W ′′ (0)) n X i = − ni = i e φ ( x i ) − δ s n X i = − ni = i I s [ ψτ, x i ] + W ′′ ( φ ( x i )) δ s n X i = − ni = i ( ψτ )( x i )+ δ s (cid:16) − τ ( x i ) I s [ ψ, x i ] + W ′′ ( φ ( x i ))( ψτ )( x i ) + Lα W ′′ ( φ ( x i )) − L + cφ ′ ( x i ) (cid:17) + δ s B ( ψ, τ )( x i ) − δ s ψ ( x i ) I s [ τ, x i ] + E. Now, we remark that | x i | = | γ | δ | p | δ | p | = R, then by (6.8) τ ( x i ) = 1. Therefore, using the equation satisfied by ψ (6.6), we get − τ ( x i ) I s [ ψ, x i ] + W ′′ ( φ ( x i ))( ψτ )( x i ) + Lα W ′′ ( φ ( x i )) − L + cφ ′ ( x i ) = 0and consequently A = δ s c δ s ( ψτ ) ′ ( x i ) + n X i = − ni = i (cid:2) φ ′ ( x i ) + δ s ( ψτ ) ′ ( x i ) (cid:3) + ( W ′′ ( φ ( x i )) − W ′′ (0)) n X i = − ni = i e φ ( x i ) − δ s n X i = − ni = i I s [ ψτ, x i ] + W ′′ ( φ ( x i )) δ s n X i = − ni = i ( ψτ )( x i )+ δ s B ( ψ, τ )( x i ) − δ s ψ ( x i ) I s [ τ, x i ] + E. Let us proceed to the estimate of A . OMOGENIZATION AND OROWAN’S LAW 33
Step 2’: Bound on n X i = − ni = i (cid:2) φ ′ ( x i ) + δ s ( ψτ ) ′ ( x i ) (cid:3) As in Step 2, using (6.14) and Claim 1, we get(6.34) 0 lim n → + ∞ n X i = − ni = i φ ′ ( x i ) Cδ s . Next, for i = i − , i , i + 1, and R = δ | p | | x i | = | i + γ − i | δ | p | > δ | p | > R, therefore τ ( x i ) = τ ′ ( x i ) = 0. Then, using (6.18) and the fact that lim x →±∞ ψ ( x ) = 0, we get δ s n X i = − ni = i ( ψτ ) ′ ( x i ) = δ s ( ψτ ) ′ ( x i − ) + δ s ( ψτ ) ′ ( x i +1 )= δ s ( ψτ ) ′ (cid:18) − γδ | p | (cid:19) + δ s ( ψτ ) ′ (cid:18) γδ | p | (cid:19) = o ( δ s ) . (6.35) Step 3’: Bound on ( W ′′ ( φ ( x i )) − W ′′ (0)) n X i = − ni = i e φ ( x i )From (6.20) with k = 1 we know thatlim n → + ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i = − ni = i e φ ( x i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Cδ s | γ | . As in Step 3 if | γ | > δ | p | , then (6.13) implies | e φ ( x i ) | C δ s | γ | s , and so, using that W ′′′ (0) = 0 | W ′′ ( e φ ( x i )) − W ′′ (0) | C δ s | γ | s . Then we havelim n → + ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( W ′′ ( φ ( x i )) − W ′′ (0)) n X i = − ni = i e φ ( x i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C δ s | γ | s δ s | γ | Cδ s . Finally, if | γ | < δ | p | , then lim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i = − ni = i e φ ( x i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Cδ s | γ | Cδ s . We conclude that(6.36) lim n → + ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( W ′′ ( φ ( x i )) − W ′′ (0)) n X i = − ni = i e φ ( x i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Cδ s . Step 4’: Bound on δ s n X i = − ni = i I s [ ψτ, x i ]Using formula (6.33), we see that(6.37) δ s n X i = − ni = i I s [ ψτ, x i ] = δ s n X i = − ni = i { τ ( x i ) I s [ ψ, x i ] + ψ ( x i ) I s [ τ, x i ] − B ( ψ, τ )( x i ) } . As we have already pointed out in Step 2’, for i = i − , i , i + 1, τ ( x i ) = 0, therefore δ s n X i = − ni = i τ ( x i ) I s [ ψ, x i ] = δ s τ ( x i − ) I s [ ψ, x i − ] + δ s τ ( x i +1 ) I s [ ψ, x i +1 ] . We point out that x i − = − γδ | p | → −∞ as δ → x i +1 = 1 + γδ | p | → + ∞ as δ → . Then from the equation (6.6), estimates (6.12), (6.14) and lim x →±∞ ψ ( x ) = 0, we deducethat I s [ ψ, x i − ] and I s [ ψ, x i +1 ] are o (1) as δ →
0, this implies that(6.38) δ s n X i = − ni = i τ ( x i ) I s [ ψ, x i ] = o ( δ s ) as δ → . Similarly, from the behavior of ψ at infinity we infer that δ s ψ ( x i − ) I s [ τ, x i − ] , δ s ψ ( x i +1 ) I s [ τ, x i +1 ] are o ( δ s ) as δ → . This and (6.21) imply that(6.39) δ s n X i = − ni = i ψ ( x i ) I s [ τ, x i ] = o ( δ s ) as δ → . OMOGENIZATION AND OROWAN’S LAW 35
Let us now consider the term δ s n X i = − ni = i B ( ψ, τ )( x i ) . For i = i − , i , i + 1, using that τ ( x i ) = 0, we have | B ( ψ, τ )( x i ) | C ( s ) Z R | ψ ( y ) − ψ ( x i ) | τ ( y ) | x i − y | s dy C Z R τ ( y ) | x i − y | s dy = C I s [ τ, x i ] . Therefore, from (6.21) we infer that(6.40) δ s n X i = − ni = i ,i ± | B ( ψ, τ )( x i ) | Cδ s . Next, we remark that for γ ∈ (cid:0) − , (cid:3) and R = δ | p | , either x i − ∈ [ − R, − R ] or x i +1 ∈ ( R, R ]. Suppose for instance that x i − ∈ [ − R, − R ] (i.e. 0 γ ). We have B ( ψ, τ )( x i − ) = C ( s ) Z R ( ψ ( y ) − ψ ( x i − ))( τ ( y ) − τ ( x i − )) | x i − − y | s dy = C ( s ) Z R − R ( ψ ( y ) − ψ ( x i − )) τ ( y ) | x i − − y | s dy − Cτ ( x i − )) I s [ ψ, x i − ] . We have already pointed out that I s [ ψ, x i − ] = o (1) as δ →
0. Let us consider the firstterm of the right-hand side of the last equality. Using that R = δ | p | , x i − = − γ δ | p | ∈ [ − R, − R ] and estimate (6.18), we get (cid:12)(cid:12)(cid:12)(cid:12)Z R − R ( ψ ( y ) − ψ ( x i − )) τ ( y ) | x i − − y | s dy (cid:12)(cid:12)(cid:12)(cid:12) max [ − R, − R/ ψ ′ Z − R − R | x i − − y | s dy + C Z R − R | x i − − y | s dy = C max [ − R, − R/ ψ ′ " ( x i − + 2 R ) − s + (cid:18) − R − x i − (cid:19) − s + C " R − x i − ) s − − R − x i − ) s Cδ s . We conclude that(6.41) δ s B ( ψ, τ )( x i − ) = o ( δ s ) as δ → . Similarly we can prove that(6.42) δ s B ( ψ, τ )( x i +1 ) = o ( δ s ) as δ → . Estimates (6.40), (6.41) and (6.42) imply(6.43) δ s n X i = − ni = i B ( ψ, τ )( x i ) = o ( δ s ) as δ → . In conclusion, putting together (6.37), (6.38), (6.39) and (6.43) we get(6.44) δ s n X i = − ni = i I s [ ψτ, x i ] = o ( δ s ) as δ → . Step 5’: Bound on W ′′ ( φ ( x i )) δ s n X i = − ni = i ( ψτ )( x i )As in Step 2’, using that τ ( x i ) = 0 for i = i − , i , i + 1 and that lim x →±∞ ψ ( x ) = 0, weget(6.45) δ s n X i = − ni = i ( ψτ )( x i ) = δ s ( ψτ )( x i − ) + δ s ( ψτ )( x i +1 ) = o ( δ s ) as δ → . Step 6’: Bound on δ s B ( ψ, τ )( x i ) − δ s ψ ( x i ) I s [ τ, x i ] Remember that x i = γδ | p | , | γ | . Let us first assume | γ | , then | x i | δ | p | = R , and |I s [ τ, x i ] | = C (cid:12)(cid:12)(cid:12)(cid:12)Z R τ ( y ) − | y − x i | s dy (cid:12)(cid:12)(cid:12)(cid:12) = C (cid:12)(cid:12)(cid:12)(cid:12)Z | y | >R τ ( y ) − | y − x i | s dy (cid:12)(cid:12)(cid:12)(cid:12) C Z | y | >R | y − x i | s dy = C ( x i + R ) s + C ( R − x i ) s CR s = Cδ s . Then δ s | ψ ( x i ) I s [ τ, x i ] | Cδ s . Now let us assume | γ | > . In this case ψ ( x i ) = o (1) as δ →
0, with o (1) independent of γ and then δ s ψ ( x i ) I s [ τ, x i ] = o ( δ s ) as δ →
0. We conclude that for any γ ∈ (cid:0) − , (cid:3) we have(6.46) δ s ψ ( x i ) I s [ τ, x i ] = o ( δ s ) as δ → . OMOGENIZATION AND OROWAN’S LAW 37
Finally, let us consider the term δ s B ( ψ, τ )( x i ). Again, if | γ | , then δ s | B ( ψ, τ )( x i ) | = δ s C ( s ) (cid:12)(cid:12)(cid:12)(cid:12)Z R ( ψ ( y ) − ψ ( x i ))( τ ( y ) − | y − x i − | s dy (cid:12)(cid:12)(cid:12)(cid:12) δ s C Z | y | >R | y − x i | s dy Cδ s . If | γ | > , then either x i ∈ (cid:2) − R, − R (cid:3) or x i ∈ (cid:2) R , R (cid:3) . Suppose for instance x i ∈ (cid:2) − R, − R (cid:3) , then computations similar to those done in Step 5’ for B ( ψ, τ )( x i − ), showthat B ( ψ, τ )( x i ) = o (1) as δ →
0. We conclude that for any γ ∈ (cid:0) − , (cid:3) we have(6.47) δ s B ( ψ, τ )( x i ) = o ( δ s ) as δ → . Step 6”: Bound on the error E From (6.20) with k = 1, and the fact that τ ( x i ) = 0 for i = i − , i , i + 1 it followsthat(6.48) (cid:12)(cid:12)(cid:12)(cid:12) lim n → + ∞ E (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) lim n → + ∞ O Lδ s α + δ s ψ ( x i ) + n X i = − ni = i h e φ ( x i ) + δ s ψ ( x i ) τ ( x i ) i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Cδ s . Next, let us estimate E . Remember that for s < we assume W even, this implies W k − (0) = 0 for any integer k >
1. Therefore − E = n X i = − ni = i W ′ ( e φ ( x i )) − W ′′ (0) e φ ( x i )= W IV (0) n X i = − ni = i ( e φ ( x i )) + W V I (0) n X i = − ni = i ( e φ ( x i )) + ... + W k (0) n X i = − ni = i ( e φ ( x i )) k − + n X i = − ni = i O (( e φ ( x i )) k +1 ) . Fix k such that 2 s (2 k + 1) >
1, then by (6.13) the sequence n X i = − ni = i O (( e φ ( x i )) k +1 ) isconvergent since behaves like the series ∞ X i =1 i s (2 k +1) and n X i = − ni = i | O (( e φ ) k +1 ) | Cδ s (2 k +1) . This estimate, together with (6.20) imply that(6.49) | E | Cδ s . Step 7’: Conclusion
Therefore, from (6.34), (6.35), (6.36), (6.44), (6.45), (6.46), (6.47), (6.48) and (6.49) weconclude that lim n → + ∞ N L λ Lδ L [ s Lδ,n ] = o ( δ s ) as δ → δ → o ( δ s ) δ s = 0, uniformly for x ∈ R and Lemma 6.4 for s < is proved.6.4. Proof of Claims 1-6.Proof of Claim 1.
We have for n > | i | n X i = − ni = i x − i | x − i | s = i − X i = − n i + γ − i ( i + γ − i ) s + n X i = i +1 i + γ − i ( i − i − γ ) s = n + i X i =1 i + γ ) s − n − i X i =1 i − γ ) s Using that, for some θ i ∈ ( − , i − γ ) s − ( i + γ ) s ( i + γ ) s ( i − γ ) s = 4 sγ ( i − θ i γ ) s − ( i + γ ) s ( i − γ ) s , we get n X i = − ni = i x − i | x − i | s = n X i =1 sγ ( i − θ i γ ) s − ( i + γ ) s ( i − γ ) s , if i = 0 n − i X i =1 sγ ( i − θ i γ ) s − ( i + γ ) s ( i − γ ) s + n + i X i = n − i +1 i + γ ) s , if i > n + i X i =1 sγ ( i − θ i γ ) s − ( i + γ ) s ( i − γ ) s − n − i X i = n + i +1 i − γ ) s , if i < → − + ∞ X i =1 sγ ( i − θ i γ ) s − ( i + γ ) s ( i − γ ) s as n → + ∞ . Let us prove the second limit of the claim. i − X i = − n | x − i | s = n + i X i =1 i + γ ) s → + ∞ X i =1 i + γ ) s as n → + ∞ . Finally n X i = i +1 | x − i | s = n − i X i =1 i − γ ) s → + ∞ X i =1 i − γ ) s as n → + ∞ , and the claim is proved. OMOGENIZATION AND OROWAN’S LAW 39
Proof of Claim 2.
When s < , we assume that W is even and this implies that the function φ ( x ) − φ satisfies φ ( − x ) = − φ ( x ) + 1 , and therefore for any integer k > φ ( − x )] k − = [ − φ ( x ) + 1] k − = − [ φ ( x ) − k − . For simplicity, let us assume i >
0. We have n X i = − ni = i [ e φ ( x i )] k − = i − X i = − n [ φ ( x i ) − k − + n X i = i +1 [ φ ( x i )] k − = n + i X i =1 (cid:20) φ (cid:18) i + γδ | p | (cid:19) − (cid:21) k − + n − i X i =1 (cid:20) φ (cid:18) − i − γδ | p | (cid:19)(cid:21) k − = n + i X i =1 (cid:20) φ (cid:18) i + γδ | p | (cid:19) − (cid:21) k − − n − i X i =1 (cid:20) φ (cid:18) i − γδ | p | (cid:19) − (cid:21) k − = n − i X i =1 ((cid:20) φ (cid:18) i + γδ | p | (cid:19) − (cid:21) k − − (cid:20) φ (cid:18) i − γδ | p | (cid:19) − (cid:21) k − ) + n + i X i = n − i +1 (cid:20) φ (cid:18) i + γδ | p | (cid:19) − (cid:21) k − = n − i X i =1 (cid:26)(cid:20) φ (cid:18) i + γδ | p | (cid:19) − φ (cid:18) i − γδ | p | (cid:19)(cid:21) · k − X l =0 (cid:18) φ (cid:18) i + γδ | p | (cid:19) − (cid:19) l (cid:18) φ (cid:18) i − γδ | p | (cid:19) − (cid:19) k − − l ) + n + i X i = n − i +1 (cid:20) φ (cid:18) i + γδ | p | (cid:19) − (cid:21) k − = n − i X i =1 φ ′ (cid:18) i + θ i γδ | p | (cid:19) γδ | p | k − X l =0 (cid:18) φ (cid:18) i + γδ | p | (cid:19) − (cid:19) l (cid:18) φ (cid:18) i − γδ | p | (cid:19) − (cid:19) k − − l + n + i X i = n − i +1 (cid:20) φ (cid:18) i + γδ | p | (cid:19) − (cid:21) k − for some θ i ∈ ( − , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i = − ni = i [ e φ ( x i )] k − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Cδ s (2 k − | γ | n − i X i =1 i − | γ | ) s k − X l =0 i − | γ | ) s (2 k − + C n + i X i = n − i +1 | i + γ | s (2 k − Ckδ s (2 k − | γ | n − i X i =1 i − | γ | ) s (2 k − + C n + i X i = n − i +1 | i + γ | s (2 k − . Passing to the limit as n → + ∞ , we get (6.20)Next, let us turn to the proof of (6.21). For i = i − , i , i + 1, and R = δ | p | | x i | = | i + γ − i | δ | p | > δ | p | > R, therefore τ ( x i ) = 0 and0 I s [ τ, x i ] = Z R τ ( y ) | x i − y | s dy = Z R − R τ ( y ) | x i − y | s dy Z R − R | x i − y | s dy = Z | x i | +2 R | x i |− R y s dy = 2 s (cid:20) | x i | − R ) s − | x i | + 2 R ) s (cid:21) = 16 s R ( | x i | + 2 Rθ i ) s − ( | x i | − R ) s ( | x i | + 2 R ) s , OMOGENIZATION AND OROWAN’S LAW 41 for some θ i ∈ ( − , R = δ | p | we have0 n X i = − ni = i ,i ± I s [ τ, x i ] s δ s | p | s i − X i = − n ( i + γ − i + θ i ) s − ( i + γ − i − s ( i + γ − i + 1) s + 8 s δ s | p | s n X i = i +2 ( − i − γ + i + θ i ) s − ( − i − γ + i − s ( − i − γ + i + 1) s = Cδ s n + i X i =2 ( i + γ + θ i ) s − ( i + γ − s ( i + γ + 1) s + Cδ s n − i X i =2 ( i − γ + θ i ) s − ( i − γ − s ( i − γ + 1) s Cδ s n + | i | X i =2 (cid:0) i + (cid:1) s (cid:0) i − (cid:1) , which implies (6.20). Proof of Claim 3.
Fix x ∈ R and let i ∈ Z be the closest integer to x such that x = i + γ , with γ ∈ (cid:0) − , (cid:3) and | x − i | > for i = i . Let δ be so small that δ | p | >
2, then | x − i | δ | p | > i = i . Letus first assume s > . Then, for n > | i | using (6.12) and (6.17) we get s Lδ,n ( x ) = φ ( x i ) + δ s ψ ( x i ) + i + i − X i = − n (cid:2) φ ( x i ) − δ s ψ ( x i ) (cid:3) + n X i = i +1 (cid:2) φ ( x i ) + δ s ψ ( x i ) (cid:3) C + i − (cid:18) sα − δ s K (cid:19) δ s | p | s n X i = − ni = i x − i | x − i | s + ( K + δ s K ) δ s | p | s n X i = − ni = i | x − i | s , and s Lδ,n ( x ) > C + i − (cid:18) sα − δ s K (cid:19) δ s | p | s n X i = − ni = i x − i | x − i | s − ( K + δ s K ) δ s | p | s n X i = − ni = i | x − i | s . Then from Claim 1 we conclude that the sequence { s Lδ,n ( x ) } n is convergent as n → + ∞ ,moreover for x = i + γ , we have (6.50) | s Lδ,n ( x ) − x | C. When s < , the convergence of n X i = − n φ ( x i ) − n follows from (6.20) for k = 1. The sum n X i = − n ψ ( x i ) τ ( x i ) is actually the sum of only three terms, since as we have seen in the proofof Claim 2, τ ( x i ) = 0 for i = i − , i , i + 1. This concludes the proof of Claim 3. Proof of Claim 4.
To prove the uniform convergence, it suffices to show that { ( s Lδ,n ) ′ ( x ) } n is a Cauchy se-quence uniformly on compact sets. Let us consider a bounded interval [ a, b ] and let x ∈ [ a, b ]. Let us first assume s > . For δ | p | > k > m > / {| a | , | b |} , by(6.14) and (6.18) we have( s Lδ,k ) ′ ( x ) − ( s Lδ,m ) ′ ( x ) = 1 δ | p | − m − X i = − k (cid:2) φ ′ ( x i ) + δ s ψ ′ ( x i ) (cid:3) + 1 δ | p | k X i = m +1 (cid:2) φ ′ ( x i ) + δ s ψ ′ ( x i ) (cid:3) ( K + δ s K ) δ s | p | s " − m − X i = − k | x − i | s + k X i = m +1 | x − i | s ( K + δ s K ) δ s | p | s " − m − X i = − k | a − i | s + k X i = m +1 | b − i | s , and ( s Lδ,k ) ′ ( x ) − ( s Lδ,m ) ′ ( x ) > − K δ s | p | s " − m − X i = − k | a − i | s + k X i = m +1 | b − i | s . Then by Claim 1 sup x ∈ [ a,b ] | ( s Lδ,k ) ′ ( x ) − ( s Lδ,m ) ′ ( x ) | → k, m → + ∞ . When s < , the convergence of n X i = − ni = i φ ′ ( x i ) is again consequence of estimate (6.14), andthe convergence of n X i = − ni = i ( ψτ ) ′ ( x i ) comes from the fact that this is actually the sum of threeterms, being τ ( x i ) = τ ′ ( x i ) = 0 for i = i − , i , i + 1. Claim 4 is therefore proved. Proof of Claim 5.
Claim 5 can be proved like Claim 4, using (6.15), (6.19) and the properties of τ . Proof of Claim 6.
Let us first assume s > . We have I s [ φ ] = W ′ ( φ ) = W ′ ( e φ ) = W ′′ (0) e φ + O ( e φ ) . OMOGENIZATION AND OROWAN’S LAW 43
We note that, since s > , if | x | > φ ( x )) C | x | s C | x | s . Let x = i + γ with γ ∈ (cid:0) − , (cid:3) , and n > | i | . From (6.12) we get n X i = − n I s [ φ, x i ] = I [ φ, x i ] + n X i = − ni = i I s [ φ, x i ]= I [ φ, x i ] + n X i = − ni = i [ α e φ ( x i ) + O ( e φ ( x i )) ] I [ φ, x i ] − δ s | p | s s n X i = − ni = i x − i | x − i | s + C n X i = − ni = i | x − i | s , for some C > n X i = − n I s [ φ, x i ] > I [ φ, x i ] − δ s | p | s s n X i = − ni = i x − i | x − i | s − C n X i = − ni = i | x − i | s . Then, by Claim 1 n X i = − n I s [ φ, x i ] converges as n → + ∞ .Let us consider now n X i = − n I s [ ψ, x i ]. From the following estimate I s [ ψ ] = W ′′ ( e φ ) ψ + Lα ( W ′′ ( e φ ) − W ′′ (0)) + cφ ′ = W ′′ (0) ψ + Lα W ′′′ (0) e φ + O ( e φ ) ψ + O ( e φ ) + cφ ′ , (6.12), (6.14) and (6.17) we get n X i = − n I s [ ψ, x i ] I [ ψ, x i ] + e C n X i = − ni = i x − i | x − i | s + C n X i = − ni = i | x − i | s , and n X i = − n I s [ ψ, x i ] > I [ ψ, x i ] + e C n X i = − ni = i x − i | x − i | s − C n X i = − ni = i | x − i | s , for some e C ∈ R and C >
0, which ensures the convergence of n X i = − n I s [ ψ, x i ].Now, let us assume s < . Fix k such that 2 s (2 k + 1) >
1. Since W is even, W k +1 (0) = 0 for any integer k >
1. Then I s [ φ ] = W ′ ( e φ ) = W ′′ (0) e φ + W IV (0)( e φ ) + ... + W k (0)( e φ ) k − + O (( e φ ) k +1 ) . Therefore, for x = i + γ n X i = − n I s [ φ, x i ] = I s [ φ, x i ] + n X i = − ni = i I s [ φ, x i ]= I s [ φ, x i ] + W ′′ (0) n X i = − ni = i e φ ( x i ) + W IV (0) n X i = − ni = i ( e φ ( x i )) + ... + W k (0) n X i = − ni = i ( e φ ( x i )) k − + n X i = − ni = i O (( e φ ( x i )) k +1 ) . The sequence n X i = − ni = i O (( e φ ( x i )) k +1 ) is convergent since, by (6.13) behaves like n X i =1 i s (2 k +1) which is convergent being the exponent 2 s (2 k + 1) greater than 1. The convergence ofthe remaining sequences is assured by (6.20).Finally, let us consider n X i = − n I s [ ψτ, x i ]. The following formula, which can be found forinstance in [1] page 7, holds true I s [ ψτ, x i ] = τ ( x i ) I s [ ψ, x i ] + ψ ( x i ) I s [ τ, x i ] − B ( ψ, τ )( x i ) , where B ( ψ, τ )( x i ) = C ( s ) Z R ( ψ ( y ) − ψ ( x i ))( τ ( y ) − τ ( x i )) | x − y | s dy. We remark that | B ( ψ, τ )( x i ) | C Z R | τ ( y ) − τ ( x i ) || x − y | s dy = C Z R τ ( y ) | x − y | s dy = C I s [ τ, x i ] , for i = i − , i , i + 1. Indeed, as we have already pointed out τ ( x i ) = 0 for these in-dices. Therefore the sequences n X i = − n ψ ( x i ) I s [ τ, x i ] and n X i = − n B ( ψ, τ )( x i ) converge by (6.21).Also n X i = − n τ ( x i ) I s [ ψ, x i ] is the sum of only three terms and then we can conclude that n X i = − n I s [ ψτ, x i ] is convergent as n → + ∞ . This concludes the proof of Claim 6. References [1]
B. Barrios, I. Peral, F. Soria and E. Valdinoci , A Widder’s type theorem for the heatequation with nonlocal diffusion,
Arch. Ration. Mech. Anal. , to appear.[2]
X. Cabr´e and Y. Sire , Nonlinear equations for fractional Laplacians II: existence, uniqueness,and qualitative properties of solutions.
Trans. Amer. Math. Soc. , to appear.[3]
X. Cabr´e and J. Sol`a-Morales , Layer solutions in a half-space for boundary reactions,
Comm.Pure Appl. Math. , (2005) no. 12, 1678-1732.[4] S. Dipierro, A. Figalli and E. Valdinoci , Strongly nonlocal dislocation dynamics in crystals,
Comm. Partial Differential Equations , to appear.
OMOGENIZATION AND OROWAN’S LAW 45 [5]
S. Dipierro, G. Palatucci and E. Valdinoci , Dislocation dynamics in crystals: a macroscopictheory in a fractional Laplace setting,
Comm. Math. Phys. , to appear.[6]
S. Dipierro, O. Savin and E. Valdinoci , All functions are locally s -harmonic up to a smallerror, preprint .[7] M. Gonz´alez and R. Monneau , Slow motion of particle systems as a limit of a reaction-diffusionequation with half-Laplacian in dimension one,
Discrete Contin. Dyn. Syst. , (2012), no. 4, 1255-1286.[8] C. Imbert and R. Monneau , Homogenization of first order equations with u/ǫ -periodic Hamilto-nians. Part I: local equations,
Arch. Ration. Mech. Anal. , (2008), no. 1, 49-89.[9] C. Imbert, R. Monneau and E. Rouy , Homogenization of first order equations with u/ǫ -periodicHamiltonians. Part II: application to dislocations dynamics,
Comm. Partial Differential Equations , (2008), no. 1-3, 479-516.[10] R. Monneau and S. Patrizi , Homogenization of the Peierls-Nabarro model for dislocation dy-namics,
J. Differential Equations , (2012), no. 7, 2064-2015.[11] R. Monneau and S. Patrizi , Derivation of the Orowan’s law from the Peierls-Nabarro model,
Comm. Partial Differential Equations , (2012), no. 10, 1887-1911.[12] F. R. N. Nabarro , Fifty-year study of the Peierls-Nabarro stress,
Mat. Sci. Eng. A , (1997), 67-76.[13] G. Palatucci, O. Savin and E. Valdinoci , Local and global minimizers for a variational energyinvolving a fractional norm,
Ann. Mat. Pura Appl. , (4) (2013), no. 4, 673-718.
Weierstraß Institut f¨ur Angewandte und Stochastik, Mohrenstrasse 39, D-10117Berlin, Germany
E-mail address : [email protected]
E-mail address ::