Convergence Rates and Hölder Estimates in Almost-Periodic Homogenization of Elliptic Systems
aa r X i v : . [ m a t h . A P ] J un Convergence Rates and H¨older Estimatesin Almost-Periodic Homogenizationof Elliptic Systems
Zhongwei Shen ∗ Abstract
For a family of second-order elliptic systems in divergence form with rapidly os-cillating almost-periodic coefficients, we obtain estimates for approximate correctorsin terms of a function that quantifies the almost periodicity of the coefficients. Theresults are used to investigate the problem of convergence rates. We also establishuniform H¨older estimates for the Dirichlet problem in a bounded C ,α domain. MSC2010:
Keywords: homogenization; almost periodic coefficients; approximate correctors;convergence rates.
In this paper we consider a family of second-order elliptic operators in divergence formwith rapidly oscillating almost-periodic coefficients, L ε = − div ( A ( x/ε ) ∇ ) = − ∂∂x i (cid:20) a αβij (cid:16) xε (cid:17) ∂∂x j (cid:21) , ε > . (1.1)We will assume that A ( y ) = (cid:0) a αβij ( y ) (cid:1) with 1 ≤ i, j ≤ d and 1 ≤ α, β ≤ m is real andsatisfies the ellipticity condition µ | ξ | ≤ a αβij ( y ) ξ αi ξ βj ≤ µ | ξ | for y ∈ R d and ξ = ( ξ αi ) ∈ R d × m , (1.2)where µ > A = A ( y ) is uniformly almost-periodic in R d ; i.e., A is the uniform limit ofa sequence of trigonometric polynomials in R d . ∗ Supported in part by NSF grant DMS-1161154. R d . Let u ε ∈ H (Ω; R m ) be the weak solutionof the Dirichlet problem: L ε ( u ε ) = F in Ω and u ε = g on ∂ Ω , (1.3)where F ∈ H − (Ω; R m ) and g ∈ H / ( ∂ Ω; R m ). Under the ellipticity condition (1.2) andalmost periodicity condition on A , it is known that u ε → u weakly in H (Ω; R m ) andthus strongly in L (Ω; R m ), as ε →
0. Furthermore, the function u is the solution of L ( u ) = F in Ω and u = g on ∂ Ω , (1.4)where L = − div (cid:0) b A ∇ (cid:1) is a second-order elliptic operator with constant coefficients,uniquely determined by A ( y ). As in the periodic case (see e.g. [5]), the constant ma-trix b A = (cid:0)b a αβij (cid:1) is called the homogenized matrix for A and L the homogenized operatorfor L ε . In this paper we shall be interested in quantitative homogenization results as wellas uniform estimates for solutions of (1.3).Homogenization of elliptic equations with rapidly oscillating almost-periodic or ran-dom coefficients was studied first by S. M. Kozlov in [22, 23] and by G.C. Papanicolaouand S.R.S. Varadhan in [25]. In particular, the o (1) convergence rate of u ε − u in C σ (Ω)for some σ > A ( y ), the sharp O ( ε ) rate in C (Ω) was proved in [22] foroperators with sufficiently smooth quasi-periodic coefficients. It is known that withoutadditional structure conditions on A ( y ), the O ( ε ) rate cannot be expected in general (see[7] for some interesting results in the 1-d case).In contrast to the periodic case, the equation for the exact correctors χ ( y ), − div (cid:0) A ( y ) ∇ χ ( y ) (cid:1) = div (cid:0) A ( y ) ∇ P ( y ) (cid:1) in R d , (1.5)may not be solvable in the almost-periodic (or random) setting for linear functions P ( y ).In [22] solutions χ ( y ) of (1.5) with sub-linear growth and almost-periodic gradient wereconstructed, and as a result, homogenization was obtained, for operators with trigonomet-ric polynomial coefficients by a lifting method. The homogenization result for the generalcase follows by an approximation argument. A different approach, which also gives thehomogenization of the second-order elliptic equations with random coefficients, is to for-mulate and solve an abstract auxiliary equation in a Hilbert space for ψ ( y ) = ∇ χ ( y ). Weoutline this approach in Section 2 and refer the reader to [20] for a detailed presentationand references.Another approach to homogenization involves the use of the so-called approximatecorrectors [25, 23]. Under certain mixing conditions, the approach has been employedsuccessfully to establish quantitative homogenization results for second-order linear el-liptic equations and systems in divergence form with random coefficients in [28, 26, 8].For nonlinear second-order elliptic equations and Hamilton-Jacobi equations, we referthe reader to [9, 1, 2] for recent advances and references on quantitative homogenization2esults. We point out that the almost-periodic case, which does not satisfy the mixingconditions generally imposed in the random case, is studied in [9, 1]. We also mentionthat sharp quantitative results were obtained recently in [15, 16, 17] for stochastic ho-mogenization of discrete linear elliptic equations in divergence form.In this paper we carry out a quantitative study of the approximate correctors χ T = (cid:0) χ βT,j (cid:1) for L ε in (1.1), where, for 1 ≤ j ≤ d and 1 ≤ β ≤ m , u = χ βT,j is defined by − div (cid:0) A ( y ) ∇ u (cid:1) + T − u = div (cid:0) A ( y ) ∇ P βj ( y ) (cid:1) in R d , (1.6)and P βj ( y ) = y j (0 , · · · , , · · · ,
0) with 1 in the β th position. Among other things, we willprove that for T ≥ σ ∈ (0 , T − k χ T k L ∞ ( R d ) ≤ C σ Θ σ ( T ) , (1.7) | χ T ( x ) − χ T ( y ) | ≤ C σ T − σ | x − y | σ for any x, y ∈ R d , (1.8)and for 0 < r ≤ T , sup x ∈ R d (cid:18) − ˆ B ( x,r ) |∇ χ T | (cid:19) / ≤ C σ (cid:18) Tr (cid:19) σ , (1.9)where C σ depends only on d , m , σ and A . The continuous function Θ σ ( T ), which isdecreasing and converges to zero as T → ∞ , is defined byΘ σ ( T ) = inf Suppose that A ( y ) = (cid:0) a αβij ( y ) (cid:1) satisfies the ellipticity condition (1.2) andis uniformly almost-periodic in R d . Let p > d , σ ∈ (0 , , and Ω be a bounded C ,α domainin R d for some α > . Then there exists a modulus η : (0 , → [0 , ∞ ) , which dependsonly on A and σ , such that lim t → η ( t ) = 0 and k u ε − u k C σ (Ω) ≤ C η ( ε ) k u k W ,p (Ω) (1.12) for ε ∈ (0 , , whenever u ε ∈ H (Ω) is the weak solution of (1.3) and u ∈ W ,p (Ω) thesolution of (1.4). Furthermore, we have k u ε − u − εχ T ( x/ε ) ∇ u k H (Ω) ≤ C η ( ε ) k u k W ,p (Ω) , (1.13) where T = ε − and χ T ( y ) denotes the approximate corrector defined by (1.6). The con-stants C in (1.12) and (1.13) depend only on Ω , p , σ and A . ρ ( R ) decays fastenough so that ´ ∞ ρ ( r ) r dr < ∞ . Theorem 1.2. Under the same assumptions as in Theorem 1.2, the following estimateshold: k u ε − u k L (Ω) ≤ C k u k W ,p (Ω) ( ˆ ∞ ε Θ σ ( r ) r dr + (cid:2) Θ ( ε − ) (cid:3) σ ) , (1.14) and k u ε − u − εχ T ( x/ε ) ∇ u k H (Ω) ≤ C k u k W ,p (Ω) ( ˆ ∞ ε Θ σ ( r ) r dr + (cid:2) Θ ( ε − ) (cid:3) σ ) (1.15) for any σ ∈ (0 , , where T = ε − and C depends only on Ω , A , p and σ . Remark 1.3. By taking R = √ T in (1.10), we obtain Θ σ ( T ) ≤ ρ ( √ T ) + T − σ for T ≥ ˆ ∞ ρ ( r ) r dr < ∞ , then ˆ ∞ Θ σ ( r ) r dr < ∞ (1.16)for any σ ∈ (0 , τ > C > ρ ( R ) ≤ C R − τ for all R ≥ . (1.17)Then, for T ≥ 1, Θ σ ( T ) ≤ C T − στσ + τ . It follows from (1.14) that k u ε − u k L (Ω) ≤ C ε στσ + τ k u k W ,p (Ω) . Since σ ∈ (0 , 1) is arbitrary, this gives k u ε − u k L (Ω) ≤ C γ ε γ k u k W ,p (Ω) for any 0 < γ < ττ + 1 . (1.18)Similarly, one may deduce from (1.15) that k u ε − u − εχ T ( x/ε ) ∇ u k H (Ω) ≤ C γ ε γ k u k W ,p (Ω) (1.19)for any 0 < γ < τ τ +1) . It is interesting to point out that if A is periodic, then ρ ( R ) = 0for R large and thus the condition (1.17) holds for any τ > 1. Consequently, estimates(1.18) and (1.19) yield convergence rates O ( ε − δ ) and O ( ε − δ ) for any δ > L (Ω) and H (Ω) respectively, which are near optimal. Also note that under the condition (1.17),our estimate (1.7) gives k χ T k L ∞ ≤ C δ T τ +1 + δ (1.20)for any δ > 0, while one has k χ T k L ∞ ≤ C , if A is periodic. Section 8 contains someexamples of quasi-periodic functions for which the condition (1.17) is satisfied.4n this paper we also establish the uniform H¨older estimates for the Dirichlet problem(1.3). Theorem 1.4. Suppose that A ( y ) = (cid:0) a αβij ( y ) (cid:1) satisfies the ellipticity condition (1.2) and isuniformly almost-periodic in R d . Let Ω be a bounded C ,α domain in R d for some α > .Let u ε be a weak solution of L ε ( u ε ) = F + div( f ) in Ω and u ε = g on ∂ Ω . (1.21) Then, for any σ ∈ (0 , , k u ε k C σ (Ω) ≤ C (cid:26) k g k C σ ( ∂ Ω) + sup x ∈ Ω0 1. In this case one has h f i = lim L →∞ − ˆ B (0 ,L ) f. It is known that if f, g ∈ B ( R d ), then f g has the mean value. Furthermore, under theequivalent relation that f ∼ g if k f − g k B = 0, the set B ( R d ) / ∼ is a Hilbert space withthe inner product defined by ( f, g ) = h f g i .A function f = ( f αi ) in Trig( R d ; R d × m ) is called potential if there exists g = ( g α ) ∈ Trig( R d ; R m ) such that f αi = ∂g α /∂x i . A function f = ( f αi ) in Trig( R d ; R d × m ) is calledsolenoidal if ∂f αi /∂x i = 0 for 1 ≤ α ≤ m . Let V (resp. V ) denote the closure of poten-tial (resp. solenoidal) trigonometric polynomials with mean value zero in B ( R d ; R d × m ).Then B ( R d ; R d × m ) = V ⊕ V ⊕ R d × m . (2.3)By the Lax-Milgram Theorem and the ellipticity condition (1.2), for any 1 ≤ j ≤ d and1 ≤ β ≤ m , there exists a unique ψ βj = ( ψ αβij ) ∈ V such that h a αγik ψ γβkj φ αi i = −h a αβij φ αi i for any φ = ( φ αi ) ∈ V . (2.4)Let b a αβij = h a αβij i + h a αγik ψ γβkj i . (2.5)and b A = (cid:0)b a αβij (cid:1) . Then µ | ξ | ≤ ˆ a αβij ξ αi ξ βj ≤ µ | ξ | (2.6)for any ξ = ( ξ αi ) ∈ R d × m , where µ depends only on d , m and µ . It is also known that c A ∗ = (cid:0) b A (cid:1) ∗ , where A ∗ denotes the adjoint of A , i.e., A ∗ = (cid:0) b αβij (cid:1) with b αβij = a βαji .As the following theorem shows, the homogenized operator for L ε is given by L = − div (cid:0) b A ∇ (cid:1) . Theorem 2.1. Let Ω be a bounded Lipschitz domain in R d and F ∈ H − (Ω; R m ) . Let u ε ∈ H (Ω; R m ) be a weak solution of L ε ( u ε ) = F in Ω . Suppose that u ε ⇀ u weaklyin H (Ω; R m ) . Then A ( x/ε ) ∇ u ε ⇀ b A ∇ u weakly in L (Ω; R dm ) . Consequently, if f ∈ H / ( ∂ Ω; R m ) and u ε is the unique weak solution in H (Ω; R m ) of the Dirichlet problem: L ε ( u ε ) = F in Ω and u ε = f on ∂ Ω , then, as ε → , u ε → u weakly in H (Ω; R m ) andstrongly in L (Ω; R m ) , where u is the unique weak solution in H (Ω; R m ) of the Dirichletproblem: L ( u ) = F in Ω and u = f on ∂ Ω . roof. See [20] for the single equation case ( m = 1). The proof for the case m > L ε ( u ε ) = div( f ) + F . This requires us to workwith a class of operators that are obtained from L A = − div (cid:0) A ( x ) ∇ (cid:1) through translationsand rotations of coordinates in R d . Observe that if L A ( u ) = F and x = Oy + z for somerotation O = ( O ij ) and z ∈ R d , then L B ( v ) = G , where v ( y ) = u ( Oy + z ), B = ( b αβij ( y ))with b αβij ( y ) = a αβℓk ( Oy + z ) O ℓi O kj , and G ( y ) = F ( Oy + z ). Thus, for each A = (cid:0) a αβij (cid:1) fixed, we shall consider the set of matrices, A = n B = (cid:0) b αβij ( y )) : b αβij ( y ) = a αβℓk ( Oy + z ) O ℓi O kj for some rotation O = ( O ij ) and z ∈ R d o . (2.7)Note that if B ( y ) = O t A ( Oy + z ) O ∈ A , where O t denotes the transpose of O , then thehomogenized matrix b B = O t b AO .The proof of Theorems 3.1 and 4.1 relies on the following extension of Theorem 2.1. Theorem 2.2. Let Ω be a bounded Lipschitz domain in R d and F ∈ H − (Ω; R m ) . Let u ℓ ∈ H (Ω; R m ) be a weak solution of − div (cid:0) A ℓ ( x/ε ℓ ) ∇ u ℓ ) = F in Ω , where ε ℓ → and A ℓ ∈ A . Suppose that u ℓ ⇀ u weakly in H (Ω; R m ) . Then u is a weak solution of − div (cid:0) e A ∇ u (cid:1) = F in Ω , where e A = O t b AO for some rotation O in R d .Proof. Suppose that A ℓ ( y ) = O tℓ A ( O ℓ y + z ℓ ) O ℓ for some rotations O ℓ and z ℓ ∈ R d . Bypassing to a subsequence we may assume that O ℓ → O , as ℓ → ∞ . Since A ( y ) is uniformlyalmost-periodic, { A ( y + z ℓ ) } ∞ ℓ =1 is pre-compact in C b ( R d ), the set of bounded continuousfunctions in R d . Thus, by passing to a subsequence, we may also assume that A ( y + z ℓ )converges uniformly in R d to an almost-periodic matrix B ( y ). Consequently, we obtain A ℓ ( y ) → e B ( y ) = O t B ( Oy ) O uniformly in R d . Note that be B = O t b BO = O t b AO .Now, let v ℓ ∈ H (Ω; R m ) be the weak solution of the Dirichlet problem: − div (cid:0) e B ( x/ε ℓ ) ∇ v ℓ (cid:1) = F in Ω and v ℓ = u ℓ on ∂ Ω . Using − div (cid:0) A ℓ ( x/ε ℓ ) ∇ ( u ℓ − v ℓ ) (cid:1) = div (cid:0) ( A ℓ ( x/ε ℓ ) − e B ( x/ε ℓ )) ∇ v ℓ (cid:1) in Ω and u ℓ − v ℓ = 0on ∂ Ω, we may use the energy estimates to deduce that k u ℓ − v ℓ k H (Ω) ≤ C k A ℓ − e B k L ∞ k∇ v ℓ k L (Ω) ≤ C k A ℓ − e B k L ∞ n k u ℓ k H (Ω) + k F k H − (Ω) o . It follows that u ℓ − v ℓ → H (Ω; R m ), as ℓ → ∞ .Finally, since v ℓ = v ℓ − u ℓ + u ℓ ⇀ u weakly in H (Ω; R m ), it follows from Theorem 2.1that e B ( x/ε ℓ ) ∇ v ℓ ⇀ e A ∇ u weakly in H (Ω; R d × m ), where e A = be B = O t b AO . As a result,we obtain − div (cid:0) e A ∇ u (cid:1) = F in Ω. This completes the proof.8 Uniform interior H¨older estimates The goal of this and next sections is to establish uniform interior and boundary H¨olderestimates for solutions of L ε ( u ε ) = f + div( g ). We will first use a compactness methodto deal with the special case L ε ( u ε ) = 0. The results are then used to establish size andH¨older estimates for fundamental solutions and Green functions for L ε . The general casefollows from the estimates for fundamental solutions and Green functions. Theorem 3.1. Let u ε ∈ H ( B ( x , r ); R m ) be a weak solution of div (cid:0) A ( x/ε ) ∇ u ε (cid:1) = 0 in B ( x , r ) , for some x ∈ R d and r > . Let σ ∈ (0 , . Then | u ε ( x ) − u ε ( y ) | ≤ C σ (cid:18) | x − y | r (cid:19) σ (cid:26) − ˆ B ( x , r ) | u ε | (cid:27) / (3.1) for any x, y ∈ B ( x , r ) , where C σ depends only on d , m , σ and A (not on ε , x , r ). Theorem 3.1 follows from Theorem 2.2 by a three-step compactness argument, similarto the periodic case in [3]. Lemma 3.2. Let < σ < . Then there exist constants ε > and θ ∈ (0 , / ,depending only on σ and A , such that − ˆ B ( y,θ ) | u ε − − ˆ B ( y,θ ) u ε | ≤ θ σ for any < ε < ε , (3.2) whenever u ε ∈ H ( B ( y, R m ) is a weak solution of div (cid:0) A ( x/ε ) ∇ u ε (cid:1) = 0 in B ( y, forsome y ∈ R d , and − ˆ B ( y, | u ε | ≤ . Proof. If div (cid:0) A ( x/ε ) ∇ u ε (cid:1) = 0 in B ( y, 1) and v ( x ) = u ε ( x + y ), then div (cid:0) B ( x/ε ) ∇ v (cid:1) = 0in B (0 , B ( x ) = A ( x + ε − y ) ∈ A . As a result, it suffices to establish estimate(3.2) for y = 0 and for solutions u ε of div (cid:0) B ( x/ε ) ∇ u ε ) = 0 in B (0 , B ∈ A .To this end, we first note that if w is a solution of a second-order elliptic system in B (0 , / 2) with constant coefficients satisfying the ellipticity condition (2.6), then − ˆ B (0 ,θ ) | w − − ˆ B (0 ,θ ) w | ≤ C θ − ˆ B (0 , / | w | for any 0 < θ < / , (3.3)where C depends only on d , m and µ . We now choose θ ∈ (0 , / 4) so small that2 d C θ < θ σ . (3.4)We claim that the estimate (3.2) with y = 0 holds for this θ and for some ε > 0, whichdepends only on A , whenever u ε is a weak solution of div (cid:0) B ( x/ε ) ∇ u ε (cid:1) = 0 in B (0 , 1) forsome B ∈ A . 9uppose this is not the case. Then there exist { ε ℓ } ⊂ R + , { B ℓ } ⊂ A , and { u ℓ } ⊂ H ( B (0 , R m ) such that ε ℓ → div (cid:0) B ℓ ( x/ε ℓ ) ∇ u ℓ (cid:1) = 0 in B (0 , , − ˆ B (0 , | u ℓ | ≤ , (3.5)and − ˆ B (0 ,θ ) | u ℓ − − ˆ B (0 ,θ ) u ℓ | > θ σ . (3.6)Since { u ℓ } is bounded in L ( B (0 , R m ), by Cacciopoli’s inequality, { u ℓ } is bounded in H ( B (0 , / R m ). By passing to a subsequence we may assume that u ℓ ⇀ u weaklyin H ( B (0 , / R m ) and in L ( B (0 , R m ). It follows from Theorem 2.2 that u is asolution of div (cid:0) e A ∇ u ) = 0 in B (0 , / e A = O t b AO for some rotation O in R d .Since the matrix O t b AO satisfies the ellipticity condition (2.6), estimate (3.3) holds for w = u . However, since u ℓ → u strongly in L ( B (0 , / R m ), we may deduce from (3.6)that θ σ ≤ − ˆ B (0 ,θ ) | u − − ˆ B (0 ,θ ) u | ≤ C θ − ˆ B (0 , / | u | ≤ d C θ − ˆ B (0 , | u | , (3.7)where we have used (3.3) for the second inequality.Finally, we note that the weak convergence of u ℓ in L ( B (0 , R m ) and the inequalityin (3.5) give − ˆ B (0 , | u | ≤ . In view of (3.7) we obtain θ σ ≤ d C θ , which contradicts (3.4). This completes theproof. Lemma 3.3. Fix < σ < . Let ε and θ be the constants given by Lemma 3.2. Let u ε ∈ H ( B ( y, R m ) be a weak solution of div( A ( x/ε ) ∇ u ε (cid:1) = 0 in B ( y, for some y ∈ R d . Then, if < ε < ε θ k − for some k ≥ , then − ˆ B ( y,θ k ) | u ε − − ˆ B ( y,θ k ) u ε | ≤ θ kσ − ˆ B ( y, | u ε | . (3.8) Proof. The lemma is proved by an induction argument on k , using Lemma 3.2 and therescaling property that if L ε ( u ε ) = 0 in B ( y, 1) and v ( x ) = u ε ( θ k x ), then L εθk ( v ) = 0 in B ( θ − k y, θ − k ) . See [3] for the periodic case. Proof of Theorem 3.1. By rescaling we may assume that r = 1. Suppose that u ε ∈ H ( B ( y, R m ) and div (cid:0) A ( x/ε ) ∇ u ε (cid:1) = 0 in B ( y, 2) for some y ∈ R d . We show that − ˆ B ( z,t ) | u ε − − ˆ B ( z,t ) u ε | ≤ C t σ − ˆ B ( z, | u ε | (3.9)10or any 0 < t < θ and z ∈ B ( y, θ ∈ (0 , / 4) is given by Lemma 3.2. Theestimate (3.1) follows from (3.9) by Campanato’s characterization of H¨older spaces.With Lemma 3.3 as our disposal, the proof of (3.9) follows the same line of argumentas in the periodic case. We refer the reader to [3] for details. We point out that theclassical local H¨older estimates for solutions of elliptic systems in divergence form withcontinuous coefficients are needed to handle the case ε ≥ θε and 0 < t < θ , as well asthe case 0 < ε < θε and 0 < t < ε/ε .It follows from (3.1) and Cacciopoli’s inequality that − ˆ B ( y,t ) |∇ u ε | ≤ C σ (cid:18) tr (cid:19) σ − ˆ B ( y,r ) |∇ u ε | for any 0 < t < r, (3.10)if div (cid:0) A ( x/ε ) ∇ u ε (cid:1) = 0 in B ( y, r ). Since A ∗ satisfies the same ellipticity and almostperiodicity conditions as A , estimate (3.16) also holds for solutions of div (cid:0) A ∗ ( x/ε ) ∇ u ε (cid:1) =0 in B ( y, r ). As a result, one may construct an m × m matrix of fundamental solutionsΓ ε ( x, y ) = (cid:0) Γ αβε ( x, y ) (cid:1) such that for each y ∈ R d , ∇ x Γ ε ( x, y ) is locally integrable and φ γ ( y ) = ˆ R d a αβij ( x/ε ) ∂∂x j n Γ βγε ( x, y ) o ∂φ α ∂x i dx (3.11)for any φ = ( φ α ) ∈ C ( R d , R m ) (see e.g. [18]). Moreover, if d ≥ 3, the matrix Γ ε ( x, y )satisfies | Γ ε ( x, y ) | ≤ C | x − y | − d (3.12)for any x, y ∈ R d and x = y , and | Γ ε ( x + h, y ) − Γ ε ( x, y ) | ≤ C σ | h | σ | x − y | d − σ , | Γ ε ( x, y + h ) − Γ ε ( x, y ) | ≤ C σ | h | σ | x − y | d − σ , (3.13)where x, y, h ∈ R d and 0 < | h | ≤ (1 / | x − y | . Since L ∗ ε (cid:0) Γ ε ( x, · ) (cid:1) = 0 in R d \ { x } , usingCacciopopli’s inequality and (3.12)-(3.13), we obtain (cid:26) − ˆ R ≤| y − x |≤ R |∇ y Γ ε ( x, y ) | dy (cid:27) / ≤ CR d − , (3.14)and (cid:26) − ˆ R ≤| y − x |≤ R |∇ y (cid:8) Γ ε ( x, y ) − Γ ε ( z, y ) | dy (cid:27) / ≤ C | x − z | σ R d − σ , (3.15)where x, z ∈ B ( x , r ) and R ≥ r . 11 heorem 3.4. Let u ε ∈ H ( B ( x , r ); R m ) be a weak solution of − div (cid:0) A ( x/ε ) ∇ u ε (cid:1) = f + div( g ) in B = B ( x , r ) . Let < σ < . Then, for any x, z ∈ B = B ( x , r ) , | u ε ( x ) − u ε ( z ) | ≤ C | x − z | σ (cid:26) r − σ (cid:18) − ˆ B | u ε | (cid:19) / + sup y ∈ B 3; the case d = 2 follows from the case d = 3 by adding a dummy variable(the method of ascending). We choose a cut-off function ϕ ∈ C ∞ ( B ( x , r/ ≤ ϕ ≤ ϕ = 1 in B ( x , r/ |∇ ϕ | ≤ Cr − . Since L ε ( u ε ) = f ϕ + div( gϕ ) − g ∇ ϕ − A ( x/ε ) ∇ u ε · ∇ ϕ − ∇ (cid:8) A ( x/ε ) u ε · ∇ ϕ (cid:9) , we obtain that for x ∈ B ( x , r ), u ε ( x ) = ˆ R d Γ ε ( x, y ) f ( y ) ϕ ( y ) dy − ˆ R d ∇ y Γ ε ( x, y ) g ( y ) ϕ ( y ) dy − ˆ R d Γ ε ( x, y ) g ( y ) ∇ ϕ ( y ) dy − ˆ R d Γ ε ( x, y ) A ( y/ε ) ∇ u ε ( y ) · ∇ ϕ ( y ) dy + ˆ R d ∇ y Γ ε ( x, y ) A ( y/ε ) u ε ( y ) ∇ ϕ ( y ) dy. (3.18)12t follows that for any x, z ∈ B ( x , r ), | u ε ( x ) − u ε ( z ) | ≤ C ˆ B | Γ ε ( x, y ) − Γ ε ( z, y ) | | f ( y ) | dy + C ˆ B |∇ y (cid:8) Γ ε ( x, y ) − Γ ε ( z, y ) (cid:9) | | g ( y ) | dy + C ˆ B | Γ ε ( x, y ) − Γ ε ( z, y ) | | g ( y ) | |∇ ϕ ( y ) | dy + C ˆ B | Γ ε ( x, y ) − Γ ε ( z, y ) | |∇ u ε ( y ) | |∇ ϕ ( y ) | dy + C ˆ B |∇ y Γ ε ( x, y ) − ∇ y Γ ε ( z, y ) | | u ε ( y ) | |∇ ϕ ( y ) | dy, (3.19)where 2 B = B ( x , r ). Since |∇ ϕ | = 0 in B ( x , r/ 2) and x, z ∈ B ( x , r ), the lastthree terms in the right hand side of (3.19) may be handled easily, using estimate (3.13),Cacciopoli’s inequality, and (3.15). They are bounded by C σ (cid:18) | y − z | r (cid:19) σ ((cid:18) − ˆ B | u ε | (cid:19) / + r (cid:18) − ˆ B | f | (cid:19) / + r (cid:18) − ˆ B | g | (cid:19) / ) , for any σ ∈ (0 , C ˆ B ( x, s ) | f ( y ) | dy | x − y | d − + C ˆ B ( z, s ) | f ( y ) | dy | z − y | d − + Cs σ ˆ B \ B ( x, s ) | f ( y ) | dy | x − y | d − σ , (3.20)where s = | x − z | and σ ∈ ( σ, B ( x, s ) as a union of sets { y : | y − x | ∼ j s } , it is not hard to verify that the first term in (3.20) is bounded by C s σ sup y ∈ B Suppose that − div (cid:0) A ( x/ε ) ∇ u ε (cid:1) = f in 2 B and f ∈ L p (2 B ; R m ) for some p ≥ 2, where 2 B = B ( x , r ). Assume d ≥ 3. Using (3.18) and Cacciopoli’s inequality,we may obtain that | u ε ( x ) | ≤ C ˆ B | f ( y ) || x − y | d − dy + C (cid:18) − ˆ B | u ε | (cid:19) / + Cr (cid:18) − ˆ B | f | (cid:19) / (3.22)for any x ∈ B = B ( x , r ). By the fractional integral estimates, this gives (cid:18) − ˆ B | u ε | q (cid:19) /q ≤ C (cid:18) − ˆ B | u ε | (cid:19) / + Cr (cid:18) − ˆ B | f | p (cid:19) /p , (3.23)where 0 < p − q ≤ d . For x ∈ ∂ Ω and 0 < r < r = diam(Ω), defineΩ r ( x ) = B ( x , r ) ∩ Ω and ∆ r ( x ) = B ( x , r ) ∩ ∂ Ω . (4.1) Theorem 4.1. Let Ω be a bounded C ,η domain in R d for some η > . Let u ε ∈ H (Ω r ( x ); R m ) be a weak solution of L ε ( u ε ) = 0 in Ω r ( x ) and u ε = 0 on ∆ r ( x ) ,for some x ∈ ∂ Ω and < r < r . Then, for any < σ < and x, y ∈ Ω r/ ( x ) , | u ε ( x ) − u ε ( y ) | ≤ C (cid:18) | x − y | r (cid:19) σ (cid:18) − ˆ Ω r ( x ) | u ε | (cid:19) / , (4.2) where C depends only on σ , A and Ω . Let φ : R d − → R be a C ,η function such that φ (0) = 0 , ∇ φ (0) = 0 , and k∇ φ k C ,η ( R d − ) ≤ M . (4.3)Let D ( r ) = D ( r, φ ) = (cid:8) ( x ′ , x d ) ∈ R d : | x ′ | < r and φ ( x ′ ) < x d < φ ( x ′ ) + 10( M + 1) r (cid:9) ,I ( r ) = I ( r, φ ) = (cid:8) ( x ′ , φ ( x ′ )) ∈ R d : | x ′ | < r (cid:9) . (4.4)By translation and rotation Theorem 4.1 may be reduced to the following.14 heorem 4.2. Let u ε ∈ H ( D ( r ); R m ) be a weak solution of div (cid:0) B ( x/ε ) ∇ u ε (cid:1) = 0 in D ( r ) and u ε = 0 on I ( r ) , for some r > and B ∈ A . Then, for any < σ < and x, y ∈ D ( r/ , | u ε ( x ) − u ε ( y ) | ≤ C (cid:18) | x − y | r (cid:19) σ (cid:18) − ˆ D r | u ε | (cid:19) / , (4.5) where C depends only on σ , A and ( η, M ) in (4.3). To prove Theorem 4.2 we need a homogenization result for a sequence of matrices inthe class A on a sequence of domains. Lemma 4.3. Let { B ℓ } be a sequence of matrices in A . Let { φ ℓ } be a sequence of C ,η functions satisfying (4.3). Suppose that div( B ℓ ( x/ε ℓ ) ∇ u ℓ ) = 0 in D ( r, φ ℓ ) and u ℓ = 0 on I ( r, φ ℓ ) for some r > , where ε ℓ → and k u ℓ k H ( D ( r,φ ℓ )) ≤ C . Then there existsubsequences of { φ ℓ } and { u ℓ } , which we still denote by { φ ℓ } and { u ℓ } respectively, anda function φ satisfying (4.3), u ∈ H ( D ( r, φ ); R m ) , and a constant matrix e B , such that ( φ ℓ → φ in C ( | x ′ | < r ) ,u ℓ ( x ′ , x d − φ ℓ ( x ′ )) ⇀ u ( x ′ , x d − φ ( x ′ )) weakly in H ( D ( r, R m ) , (4.6) and div (cid:0) e B ∇ u (cid:1) = 0 in D ( r, φ ) and u = 0 on I ( r, φ ) . (4.7) Moreover, the matrix e B , which is given by O t b AO for some rotation O in R d , satisfies theellipticity condition (2.6).Proof. Since k∇ φ ℓ k C ,η ( R d − ) ≤ M and k u ℓ k H ( D ( r,φ ℓ )) ≤ C , (4.6) follows by passing tosubsequences. Suppose that B ℓ ( y ) = O tℓ A ( O ℓ y + z ℓ ) O ℓ for some rotation O ℓ and z ℓ ∈ R d .By passing to a subsequence, we may assume that O ℓ → O . Since u ℓ → u weaklyin H (Ω; R m ) for any Ω ⊂⊂ D ( r, φ ), it follows from Theorem 2.2 that div (cid:0) e B ∇ u ) = 0 in D ( r, φ ), where e B = O t b AO . Finally, since v ℓ ( x ′ , x d ) = u ℓ ( x ′ , x d + φ ℓ ( x ′ )) ⇀ v ( x ′ , x d + φ ( x ′ ))weakly in H ( D ( r, v ℓ = 0 on I ( r, v = 0 on I ( r, u = 0 on I ( r, φ ). Proof of Theorem 4.2. With Lemma 4.3 at our disposal, Theorem 4.2 follows by thethree-step compactness argument, as in the periodic case. We refer the reader to [3] fordetails.With interior and boundary H¨older estimates in Theorems 3.1 and 4.1, one may con-struct an m × m matrix G ε ( x, y ) = (cid:0) G αβε ( x, y ) (cid:1) of Green functions for L ε for a bounded C ,η domain Ω. Moreover, if d ≥ | G ε ( x, y ) | ≤ C | x − y | − d (4.8)for any x, y ∈ Ω, and | G ε ( x, y ) − G ε ( z, y ) | ≤ C σ | x − z | σ | x − y | d − σ (4.9)15or any x, y, z ∈ Ω with | x − z | < (1 / | x − y | and for any 0 < σ < 1. Since G ε ( · , y ) = 0and G ε ( y, · ) = 0 on ∂ Ω, one also has | G ε ( x, y ) | ≤ C [ δ ( x )] σ (cid:2) δ ( y ) (cid:3) σ | x − y | d − σ + σ (4.10)for any x, y ∈ Ω and any 0 ≤ σ , σ < 1, where δ ( x ) = dist( x, ∂ Ω) and C depends only on A , Ω, σ and σ . Theorem 4.4. Let Ω be a bounded C ,η domain in R d for some η > . Suppose that L ε ( u ε ) = F in Ω and u ε = 0 on ∂ Ω . Then k u ε k C α (Ω) ≤ C α sup x ∈ Ω0 1) and use (4.9) to obtain ˆ Ω \ B ( x, t ) | G ε ( x, y ) − G ε ( z, y ) | | F ( y ) | dy ≤ C t β ˆ Ω \ B ( x, t ) | F ( y ) | dy | x − y | d − β ≤ C t α sup x ∈ Ω0 Let Ω be a bounded C ,η domain in R d for some η > . Suppose that L ε ( u ε ) = 0 in Ω and u ε = g on ∂ Ω . Then k u ε k C α (Ω) ≤ C α k g k C α ( ∂ Ω) (4.14) for any < α < , where C α depends only on A , Ω , and α .Proof. Without loss of generality we may assume that k g k C α ( ∂ Ω) = 1. Let v be theharmonic function in Ω such that v ∈ C (Ω) and v = g on ∂ Ω. It is well known that k v k C α (Ω) ≤ C α k g k C α ( ∂ Ω) = C α , where C α depends only on α and Ω. By interior estimatesfor harmonic functions, one also has |∇ v ( x ) | ≤ C α (cid:2) δ ( x ) (cid:3) α − (4.15)for any x ∈ Ω. Since L ε ( u ε − v ) = −L ε ( v ) in Ω and u ε − v = 0 on ∂ Ω, it follows that u ε ( x ) − v ( x ) = − ˆ Ω ∇ y G ε ( x, y ) A ( y/ε ) ∇ v ( y ) dy. This, together with (4.15), gives | u ε ( x ) − v ( x ) | ≤ C α ˆ Ω |∇ y G ε ( x, y ) | (cid:2) δ ( y ) (cid:3) α − dy. (4.16)17e will show that ˆ Ω |∇ y G ε ( x, y ) | (cid:2) δ ( y ) (cid:3) α − dy ≤ C α (cid:2) δ ( x ) (cid:3) α for any x ∈ Ω . (4.17)Assume (4.17) for a moment. Then | u ε ( x ) − v ( x ) | ≤ C α (cid:2) δ ( x ) (cid:3) α for any x ∈ Ω . (4.18)It follows that k u ε k L ∞ (Ω) ≤ k v k L ∞ (Ω) + C ≤ C . Let x, y ∈ Ω. To show | u ε ( x ) − u ε ( y ) | ≤ C | x − y | α , we consider three cases: (1) | x − y | < (1 / δ ( x ); (2) | x − y | < (1 / δ ( y ); (3) | x − y | ≥ max (cid:0) (1 / δ ( x ) , (1 / δ ( y ) (cid:1) . In the first case, since L ε ( u ε ) = 0 in Ω, we may usethe interior H¨older estimates in Theorem 3.1 to obtain | u ε ( x ) − u ε ( y ) | ≤ C α | x − y | α k u ε k L ∞ ( B ( x,δ ( x ) / ≤ C α | x − y | α . The second case is handled in the same manner. For the third case we use (4.18) andH¨older estimates for v to see that | u ε ( x ) − u ε ( y ) ≤ | u ε ( x ) − v ( x ) | + | v ( x ) − v ( y ) | + | v ( y ) − u ε ( y ) |≤ C (cid:2) δ ( x ) (cid:3) α + C | x − y | α + C (cid:2) δ ( y ) (cid:3) α ≤ C α | x − y | α . It remains to prove (4.17). To this end we fix x ∈ Ω and let r = δ ( x ) / 2. We first notethat ˆ B ( x,r ) |∇ y G ε ( x, y ) | (cid:2) δ ( y ) (cid:3) α − dy ≤ C r α − ˆ B ( x,r ) |∇ y G ε ( x, y ) | dy ≤ C r α , (4.19)where the last inequality follows from the first estimate in (4.13) by decomposing B ( x, r ) \{ } as ∪ ∞ j =0 (cid:8) B ( x, − j r ) \ B ( x, − j − r ) (cid:9) . To estimate the integral on Ω \ B ( x, r ), we observethat if Q is a cube in R d with the property that 3 Q ⊂ Ω \ { x } and ℓ ( Q ) ∼ dist( Q, ∂ Ω),then ˆ Q |∇ y G ε ( x, y ) | (cid:2) δ ( y ) (cid:3) α − dy ≤ C (cid:2) ℓ ( Q ) (cid:3) α − | Q | (cid:18) − ˆ Q |∇ y G ε ( x, y ) | dy (cid:19) / ≤ C (cid:2) ℓ ( Q ) (cid:3) α − | Q | (cid:18) − ˆ Q | G ε ( x, y ) | dy (cid:19) / ≤ C r α (cid:2) ℓ ( Q ) (cid:3) α + α − | Q | (cid:18) − ˆ Q dy | x − y | d − α + α ) (cid:19) / , (4.20)where α , α ∈ (0 , Q ⊂ Ω \ { x } ,we see that | x − y | ∼ | x − z | for any y, z ∈ Q . As a result, it follows from (4.20) that ˆ Q |∇ y G ε ( x, y ) | (cid:2) δ ( y ) (cid:3) α − dy ≤ C r α ˆ Q (cid:2) δ ( y ) (cid:3) α + α − | x − y | d − α + α dy. (4.21)18y decomposing Ω \ B ( x, r ) as a non-overlapping union of cubes Q with the said property(a Whitney type decomposition of Ω), we obtain ˆ Ω \ B ( x,r ) |∇ y G ε ( x, y ) | (cid:2) δ ( y ) (cid:3) α − dy ≤ C r α ˆ Ω (cid:2) δ ( y ) (cid:3) α + α − ( | x − y | + r ) d − α + α dy ≤ C r α ˆ R d + y α + α − d dy ( | r − y d | + r + | y ′ | ) d − α + α . (4.22)Finally, a direct computation shows that the integral on the right hand side of (4.22) isbounded by Cr α − α , provided that α > α and α > − α . This completes the proof. Proof of Theorem 1.4. This follows from Theorems 4.4, 4.5 and 4.6 by writing u ε = u (1) ε + u (2) ε + u (3) ε , where u (1) ε , u (2) ε , u (3) ε satisfy the conditions in Theorems 4.4, 4.5, 4.6,respectively. In this section we construct the approximate correctors χ T = (cid:0) χ βT,j (cid:1) = (cid:0) χ αβT,j (cid:1) and obtainsome preliminary estimates. Proposition 5.1. Let f ∈ L ( R d ; R m ) and g = ( g , . . . , g d ) ∈ L ( R d ; R d × m ) . Assumethat sup x ∈ R d ˆ B ( x, (cid:0) | f | + | g | (cid:1) < ∞ . Then, for T > , there exists a unique u ∈ H ( R d ; R m ) such that − div (cid:0) A ( x ) ∇ u (cid:1) + T − u = f + div( g ) in R d (5.1) and sup x ∈ R d ˆ B ( x, (cid:0) |∇ u | + | u | (cid:1) < ∞ . (5.2) Moreover, the solution u satisfies the estimate sup x ∈ R d − ˆ B ( x,T ) (cid:0) |∇ u | + T − | u | (cid:1) ≤ C sup x ∈ R d − ˆ B ( x,T ) (cid:0) | g | + T | f | (cid:1) , (5.3) where C depends only on d , m and µ .Proof. By rescaling we may assume that T = 1. The proof of the existence and estimate(5.3) may be found in [26]. It uses the fact that for f ∈ L ( R d ; R m ), g = ( g , . . . , g d ) ∈ L ( R d ; R d × m ) with compact support, there exists a constant λ > 0, depending only on d , m and µ , such that the solution of (5.1) in H ( R d ; R m ) satisfies ˆ R d e λ | x | (cid:8) |∇ u | + | u | (cid:9) dx ≤ C ˆ R d e λ | x | (cid:8) | f | + | g | (cid:9) dx. u ∈ H ( R d ; R m ) satisfies (5.2) and − div( A ( x ) ∇ u ) + u = 0 in R d . By Cacciopoli’s inequality, ˆ B (0 ,R ) |∇ u | + ˆ B (0 ,R ) | u | ≤ CR ˆ B (0 , R ) | u | for any R ≥ 1. It follows that ˆ B (0 ,R ) | u | ≤ CR d ˆ B (0 , d R ) | u | for any R ≥ 1. However, the condition (5.2) implies that ´ B (0 , d R ) | u | ≤ C u R d . Conse-quently, we obtain ´ B (0 ,R ) | u | ≤ C u R − d for any R ≥ u ≡ R d . Remark 5.2. The solution u of (5.1), given by Proposition 5.1, in fact satisfiessup x ∈ R d (cid:26) − ˆ B ( x,T ) |∇ u | p (cid:27) /p ≤ C sup x ∈ R d (cid:26) − ˆ B ( x,T ) | g | p (cid:27) /p + C sup x ∈ R d (cid:26) − ˆ B ( x,T ) T | f | (cid:27) / , (5.4)sup x ∈ R d (cid:26) − ˆ B ( x,T ) T − q | u | q (cid:27) /q ≤ C sup x ∈ R d (cid:26) − ˆ B ( x,T ) | g | p (cid:27) /p + C sup x ∈ R d (cid:26) − ˆ B ( x,T ) T | f | (cid:27) / (5.5)for some p > 2, depending only on d , m and µ , where (1 /q ) = (1 /p ) − (1 /d ) for d ≥ 3. If d = 2, the left hand side of (5.5) should be replaced by T − k u k L ∞ .To see (5.4), one uses the weak reverse H¨older estimate: if u is a weak solution of − div (cid:0) A ( x ) ∇ u ) = f + div( g ) in B r = B ( x , r ), then ( − ˆ B r/ |∇ u | p ) /p ≤ Cr (cid:26) − ˆ B r | u | (cid:27) / + C (cid:26) − ˆ B r | g | p (cid:27) /p + C r (cid:26) − ˆ B r | f | (cid:27) / for some p > 2, depending only on d , m and µ (see e.g. [14]). Estimate (5.5) follows from(5.4) by Sobolev imbedding.Let P βj ( x ) = x j e β , where 1 ≤ j ≤ d , 1 ≤ β ≤ m , and e β = (0 , . . . , , . . . , 0) with 1 inthe β th position. For T > 0, the approximate corrector is defined as χ T = (cid:0) χ αβT,j (cid:1) , where,for each 1 ≤ j ≤ d and 1 ≤ β ≤ m , u = χ βT,j = (cid:0) χ βT,j , . . . , χ mβT,j (cid:1) is the weak solution of − div (cid:0) A ( x ) ∇ u (cid:1) + T − u = div (cid:0) A ( x ) ∇ P βj (cid:1) in R d , (5.6)given by Proposition 5.1 It follows from (5.3) thatsup x ∈ R d − ˆ B ( x,T ) (cid:0) |∇ χ T | + T − | χ T | (cid:1) ≤ C, (5.7)where C depends only on d , m and µ . Clearly, this givessup x ∈ R d L ≥ T − ˆ B ( x,L ) (cid:0) |∇ χ T | + T − | χ T | (cid:1) ≤ C, (5.8)where C depends only on d , m and µ . 20 emma 5.3. Let x, y, z ∈ R d . Then (cid:26) − ˆ B ( x,T ) |∇ χ T ( t + y ) − ∇ χ T ( t + z ) | dt (cid:27) / ≤ C k A ( · + y ) − A ( · + z ) k L ∞ ( R d ) ,T − (cid:26) − ˆ B ( x,T ) | χ T ( t + y ) − χ T ( t + z ) | dt (cid:27) / ≤ C k A ( · + y ) − A ( · + z ) k L ∞ ( R d ) , (5.9) where C depends only on d , m and µ .Proof. Fix y, z ∈ R d and 1 ≤ j ≤ d , 1 ≤ β ≤ m . Let u ( t ) = χ βT,j ( t + y ) and v ( t ) = χ βT,j ( t + z ). Then w = u − v is a solution of − div (cid:0) A ( t + y ) ∇ w (cid:1) + T − w = div (cid:0) [ A ( t + y ) − A ( t + z )] ∇ P βj (cid:1) + div (cid:0) [ A ( t + y ) − A ( t + z )] ∇ v (cid:1) . In view of Proposition 5.1 we obtain − ˆ B ( x,T ) (cid:0) |∇ w | + T − | w | (cid:1) ≤ C sup x ∈ R d − ˆ B ( x,T ) | A ( t + y ) − A ( t + z ) | dt + C sup x ∈ R d − ˆ B ( x,T ) | A ( t + y ) − A ( t + z ) | |∇ v | dt ≤ C k A ( · + y ) − A ( · + z ) k L ∞ + C k A ( · + y ) − A ( · + z ) k L ∞ sup x ∈ R d − ˆ B ( x,T ) |∇ v | ≤ C k A ( · + y ) − A ( · + z ) k L ∞ , where we have used (5.7) in the last inequality. This completes the proof. Remark 5.4. For f ∈ L ( R d ), define k f k W = lim sup L →∞ sup x ∈ R d (cid:26) − ˆ B ( x,L ) | f | (cid:27) / . (5.10)Note that by (5.7), k∇ χ T k W + T − k χ T k W ≤ C, (5.11)where C depends only on d , m and µ . Moreover, by Lemma 5.3, for any τ ∈ R d , k∇ χ T ( · + τ ) − ∇ χ T ( · ) k W + T − k χ T ( · + τ ) − χ T ( · ) k W ≤ C k A ( · + τ ) − A ( · ) k L ∞ . (5.12)Since A is uniformly almost-periodic, for any ε > 0, the set (cid:8) τ ∈ R d : k A ( · + τ ) − A ( · ) k L ∞ ( R d ) < ε (cid:9) is relatively dense in R d . It follows that for any ε > 0, the set of τ for which the left handside of (5.12) is less than ε is also relatively dense in R d . By [6] this implies that ∇ χ T and χ T are limits of sequences of trigonometric polynomials with respect to the semi-norm k · k W in (5.10). In particular, ∇ χ T , χ T ∈ B ( R d ) for any T > emma 5.5. Let u T = χ βT,j for some T > , ≤ j ≤ d and ≤ β ≤ m . Then (cid:10) a αγik ∂u γT ∂x k ∂v α ∂x i (cid:11) + T − h u T · v i = − (cid:10) a αβij ∂v α ∂x i (cid:11) , (5.13) where v = ( v α ) ∈ H ( R d ; R m ) and v α , ∇ v α ∈ B ( R d ) .Proof. For any φ = ( φ α ) ∈ H ( R d ; R m ) with compact support, we have ˆ R d a αγik ∂u γT ∂x k · ∂φ α ∂x i + 1 T ˆ R d u T · φ = − ˆ R d a αβij ∂φ α ∂x i . (5.14)Let v = ( v α ) ∈ H ( R d ; R m ). Suppose that v α ∈ B ( R d ) and ∇ v α ∈ B ( R d ). Choose φ ( x ) = ϕ ( εx ) v ( x ) in (5.14), where ϕ ∈ C ∞ ( R d ). The desired result follows by a simplechange of variables x → x/ε in (5.14), multiplying both sides of the equation by ε d , andfinally letting ε → v be a constant in (5.13), we see that h χ βT,j i = 0 . (5.15)By taking v = χ βT,j , we obtain h A ∇ χ βT,j · ∇ χ βT,j i + T − h| χ βT,j | i = −h A ∗ ∇ χ βj,T i , (5.16)where A ∗ denotes the adjoint of A . This, in particular, implies that h|∇ χ T | i + T − h| χ T | i ≤ C, where C depends only on d , m and µ . Lemma 5.6. Let ψ = (cid:16) ψ αβij (cid:17) be defined by (2.4). Then, as T → ∞ , ∂∂x i (cid:16) χ αβT,j (cid:17) ⇀ ψ αβij weakly in B ( R d ) . (5.17) Proof. Fix 1 ≤ j ≤ d and 1 ≤ β ≤ m . Let e ψ βj = (cid:16) e ψ αβij (cid:17) ∈ B ( R d ; R dm ) be the weaklimit in B ( R d ) of a subsequence ∇ χ βT ℓ ,j , where T ℓ → ∞ . Since ∇ χ βT,j ∈ V , we see that e ψ βj ∈ V . Moreover, since T − h| χ T | i ≤ C , it follows by letting T → ∞ in (5.13) that (cid:10) a αγik e ψ γβkj ∂v α ∂x i (cid:11) = − (cid:10) a αβij ∂v α ∂x i (cid:11) for any v = ( v α ) ∈ Trig( R d ; R m ). This implies that e ψ βj is a solution of (2.4). By theuniqueness of the solution we obtain e ψ βj = ψ βj and hence (5.17).22 heorem 5.7. As T → ∞ , T − h| χ T | i → .Proof. Note that µ h| ψ − ∇ χ T | i ≤ (cid:10) a αγik (cid:26) ψ γβkj − ∂∂x k (cid:16) χ γβT,j (cid:17)(cid:27) (cid:26) ψ αβij − ∂∂x i (cid:16) χ αβT,j (cid:17)(cid:27) (cid:11) = h a αβik ψ γβkj ψ αβij i − (cid:10) a αγik ∂∂x k (cid:16) χ γβT,j (cid:17) ψ αβij (cid:11) − T − h| χ T | i , where we have used equations (2.4) and (5.13). In view of Lemma 5.6 this implies thatas T → ∞ , T − h| χ T | i → 0, and k ψ − ∇ χ T k B → . (5.18) Remark 5.8. For T > 0, let b a αβT,ij = h a αβij i + (cid:10) a αγik ∂∂x k (cid:16) χ γβT,j (cid:17) (cid:11) (5.19)be the approximate homogenized coefficients. Then | b a αβij − b a αβT,ij | = | (cid:10) a αγik (cid:26) ψ γβkj − ∂∂x k (cid:16) χ γβT,j (cid:17)(cid:27) (cid:11) |≤ C k ψ − ∇ χ T k B . (5.20) In this section we will establish sharp estimates for approximate correctors χ T . The proofrelies on the uniform L ∞ and H¨older estimates obtained in Section 3 for solutions of L ε ( u ε ) = f + div( g ). Lemma 6.1. For T ≥ , k χ T k L ∞ ( R d ) ≤ C T, (6.1) where C is independent of T . Moreover, for any < σ < and | x − y | ≤ T , | χ T ( x ) − χ T ( y ) | ≤ C σ T − σ | x − y | σ , (6.2) where C σ depends only on σ and A .Proof. We consider the case d ≥ 3. The 2-d case follows by the method of ascending.Let 1 ≤ j ≤ d and 1 ≤ β ≤ m . Fix z ∈ R d and consider the function u ( x ) = χ βT,j ( x ) + P βj ( x − z ) . (6.3)23t follows from (5.7) that (cid:26) − ˆ B ( z, T ) | u | (cid:27) / ≤ C T. (6.4)Since div (cid:0) A ( x ) ∇ u (cid:1) = T − χ βT,j in R d , (6.5)we may apply the estimate (3.23) repeatedly to show that (cid:26) − ˆ B ( z, T ) | u | p (cid:27) /p ≤ C p T (6.6)for any 2 < p < ∞ , where C p depends only on p and A . This, together with (3.17), gives k u k L ∞ ( B ( z,T )) ≤ C T. Hence, | χ βT,j ( z ) | ≤ C T for any z ∈ R d . Finally, estimate (6.2) follows from (6.1) and theH¨older estimate (3.16). Lemma 6.2. Let σ , σ ∈ (0 , and < p < ∞ . Then, for any ≤ r ≤ T , sup x ∈ R d (cid:18) − ˆ B ( x,r ) |∇ χ T | p (cid:19) /p ≤ C T σ (cid:18) Tr (cid:19) σ , (6.7) where C depends only on p , σ , σ , and A .Proof. Let u be the same as in the proof of Lemma 6.1. By Cacciopoli’s inequality, − ˆ B ( z,r ) |∇ u | ≤ Cr − − ˆ B ( z, r ) | u − u ( z ) | + Cr k T − χ T k L ∞ , where 0 < r ≤ T . In view of (6.1) and (6.2), this givessup z ∈ R d (cid:18) − ˆ B ( z,r ) |∇ χ T | (cid:19) / ≤ C σ (cid:18) Tr (cid:19) σ (6.8)for any σ ∈ (0 , 1) and 0 < r ≤ T . Since A is uniformly continuous in R d , by the local W ,p estimates for elliptic systems in divergence form, it follows from (6.5) that (cid:18) − ˆ B ( z, |∇ u | p (cid:19) /p ≤ C p (cid:18) − ˆ B ( z, |∇ u | (cid:19) / + C T − k χ T k L ∞ , for any z ∈ R d and 2 < p < ∞ , where C p depends only on p and A . This, together with(6.8), yields sup z ∈ R d (cid:18) − ˆ B ( z, |∇ χ T | p (cid:19) /p ≤ C p,σ T σ σ ∈ (0 , 1) and p ∈ (2 , ∞ ). Consequently, for any 1 ≤ r ≤ T and σ ∈ (0 , z ∈ R d (cid:18) − ˆ B ( z,r ) |∇ χ T | p (cid:19) /p ≤ C p,σ T σ . (6.9)The desired estimate (6.7) now follows from (6.8) and (6.9) by a simple interpolation of L p norms. Theorem 6.3. Let T ≥ . The approximate corrector χ T is uniformly almost-periodic in R d . Moreover, for any y, z ∈ R d , k χ T ( · + y ) − χ T ( · + z ) k L ∞ ( R d ) ≤ C T k A ( · + y ) − A ( · + z ) k L ∞ ( R d ) , (6.10) where C is independent of T and y, z .Proof. We assume d ≥ 3. The case d = 2 follows from the case d = 3 by the method ofascending. Fix y, z ∈ R d and 1 ≤ j ≤ d , 1 ≤ β ≤ m . Let u ( x ) = χ βT,j ( x + y ) − χ βT,j ( x + z ) . Note that − div (cid:0) A ( x + y ) ∇ u (cid:1) = − T − u + div (cid:2)(cid:0) A ( x + y ) − A ( x + z ) (cid:1) ∇ P βj (cid:3) + div (cid:2)(cid:0) A ( x + y ) − A ( x + z ) (cid:1) ∇ v (cid:3) , (6.11)where v ( x ) = χ βT,j ( x + z ). Let B = B ( x , T ). As in the proof of Theorem 3.4, we choose acut-off function ϕ ∈ C ∞ ( B ( x , T / ϕ = 1 in B ( x , T / 2) and |∇ ϕ | ≤ C T − .Using the representation formula by fundamental solutions and (6.11), we obtain, for any x ∈ B , | u ( x ) | ≤ C T − ˆ B | Γ y ( x, t ) | | u ( t ) | dt + C k A ( · + y ) − A ( · + z ) k L ∞ ˆ B |∇ t (cid:0) Γ y ( x, t ) ϕ ( t ) (cid:1) | dt + C k A ( · + y ) − A ( · + z ) k L ∞ ˆ B |∇ v ( t ) | |∇ t (cid:0) Γ y ( x, t ) ϕ ( t ) (cid:1) | dt + C T (cid:18) − ˆ B |∇ u | (cid:19) / + C (cid:18) − ˆ B | u | (cid:19) / , (6.12)where we have used Γ y ( x, t ) = Γ( x + y, t + y ) to denote the matrix of fundamental solutionsfor the operator − div (cid:0) A ( · + y ) ∇ (cid:1) in R d . By Lemma 5.3 the last two terms in the righthand side of (6.12) are bounded by the right hand side of (6.10). Using the size estimate(3.12) and Cacciopoli’s inequality, it is also not hard to see that the second term in theright hand side of (6.12) is bounded by the right hand side of (6.10).25o treat the third term in the right hand side of (6.12), we note that ˆ B |∇ v ( t ) | |∇ t (cid:0) Γ y ( x, t ) ϕ ( t ) (cid:1) | dt ≤ C ∞ X ℓ =0 (cid:18) − ˆ | t − x |∼ − ℓ T |∇ v ( t ) | dt (cid:19) / (cid:18) − ˆ | t − x |∼ − ℓ T |∇ t (cid:0) Γ y ( x, t ) ϕ (cid:1) | dt (cid:19) / (2 − ℓ T ) d ≤ C ∞ X ℓ =0 (2 ℓ ) σ · (2 − ℓ T ) − d · (2 − ℓ T ) d ≤ C T, where σ ∈ (0 , 1) and we have used (6.8) to estimate the integral involving |∇ v ( t ) | for thesecond inequality. As a result, we have proved that for any x ∈ B , | u ( x ) | ≤ C T − ˆ B | u ( t ) || x − t | d − dt + C T k A ( · + y ) − A ( · + z ) k L ∞ . (6.13)By the fractional integral estimates, this implies that (cid:18) − ˆ B | u | q (cid:19) /q ≤ C (cid:18) − ˆ B | u | p (cid:19) /p + C T k A ( · + y ) − A ( · + z ) k L ∞ , where 1 < p < q ≤ ∞ and (1 /p ) − (1 /q ) < (2 /d ). Since (cid:18) − ˆ B | u | (cid:19) / ≤ C T k A ( · + y ) − A ( · + z ) k L ∞ by Lemma 5.3, a simple iteration argument shows that k u k L ∞ ( B ) ≤ C T k A ( · + y ) − A ( · + z ) k L ∞ . This completes the proof. Remark 6.4. Let u ( x ) = χ T ( x + y ) − χ T ( x + z ), as in the proof of Theorem 6.3. Then | u ( t ) − u ( s ) | ≤ C σ (cid:18) | t − s | T (cid:19) σ T k A ( · + y ) − A ( · + z ) k L ∞ , (6.14)for any σ ∈ (0 , 1) and t, s ∈ R d , where C σ depends only on σ and A . This follows from(6.11), (6.10) and (3.16). By Cacciopoli’s inequality and (6.14) we may deduce thatsup x ∈ R d (cid:18) − ˆ B ( x,r ) |∇ u | (cid:19) / ≤ C σ (cid:18) Tr (cid:19) σ k A ( · + y ) − A ( · + z ) k L ∞ (6.15)for any σ ∈ (0 , heorem 6.5. Let T ≥ . Then T − k χ T k L ∞ ( R d ) ≤ C σ (cid:26) ρ ( R ) + (cid:18) RT (cid:19) σ (cid:27) (6.16) for any R > and σ ∈ (0 , , where C σ depends only on σ and A . In particular, T − k χ T k L ∞ ( R d ) → , as T → ∞ .Proof. Let y, z ∈ R d . Suppose | z | ≤ R . Then | χ T ( y ) − χ T (0) | ≤ | χ T ( y ) − χ T ( z ) | + | χ T ( z ) − χ T (0) |≤ C T k A ( · + y ) − A ( · + z ) k L ∞ ( R d ) + C σ T − σ R σ , where we have used Theorem 6.3 and Lemma 6.1. It follows thatsup y ∈ R d T − | χ T ( y ) − χ T (0) | ≤ C ρ ( R ) + C σ (cid:18) RT (cid:19) σ (6.17)for any R > | χ T (0) | ≤ (cid:12)(cid:12)(cid:12)(cid:12) − ˆ B (0 ,L ) (cid:8) χ T ( y ) − χ T (0) (cid:9) dy (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) − ˆ B (0 ,L ) χ T ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup y ∈ R d | χ T ( y ) − χ T (0) | + (cid:12)(cid:12)(cid:12)(cid:12) − ˆ B (0 ,L ) χ T ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) . Since h χ T i = 0, we may let L → ∞ in the estimate above to obtain | χ T (0) | ≤ sup y ∈ R d | χ T ( y ) − χ T (0) | . This, together with (6.17), yields the estimate (6.16).For T ≥ σ > 0, defineΘ σ ( T ) = inf 1) in (6.18), we see thatΘ σ ( T ) ≤ ρ ( T α ) + T − σ (1 − α ) . (6.20)This, in particular, implies thatif ˆ ∞ ρ ( r ) r dr < ∞ , then ˆ ∞ Θ σ ( r ) r dr < ∞ . heorem 6.6. Let T ≥ . Then (cid:0)(cid:10) | ψ − ∇ χ T | (cid:11)(cid:1) / ≤ C σ ˆ ∞ T/ Θ σ ( r ) r dr (6.21) for σ ∈ (0 , , where C σ depends only on σ and A .Proof. Fix 1 ≤ j ≤ d and 1 ≤ β ≤ m . Let u = χ βT,j , v = χ β T,j , and w = u − v . It followsfrom Lemma 5.5 that h A ∇ w · ∇ ϕ i = 14 T h v · ϕ i − T h u · ϕ i for any ϕ ∈ H ( R d , R m ) with ϕ, ∇ ϕ ∈ B ( R d ). By taking ϕ = w , we obtain h|∇ w | i ≤ C T − (cid:8) h| u | i + h| v | i (cid:9) ≤ C σ (cid:8) Θ σ ( T ) + Θ σ (2 T ) (cid:9) , (6.22)where we have used (6.19) for the second inequality. Hence, we have proved that (cid:0)(cid:10) |∇ χ T − ∇ χ T | (cid:11)(cid:1) / ≤ C σ ˆ TT/ Θ σ ( r ) r dr, where we have used the fact that Θ σ ( r ) is decreasing. Consequently, ∞ X ℓ =0 (cid:0) h|∇ χ ℓ T − ∇ χ ℓ +1 T | i (cid:1) / ≤ C σ ˆ ∞ T/ Θ σ ( r ) r dr. (6.23)Recall that by (5.18), h| ψ − ∇ χ T | i → T → ∞ . The estimate (6.21) now followsfrom (6.23). Remark 6.7. Suppose that there exist C > τ > ρ ( R ) ≤ C/R τ for R ≥ . (6.24)By taking R = T στ + σ in (6.16), we obtain T − k χ T k L ∞ ≤ C Θ σ ( T ) ≤ C T − τστ + σ . Since σ ∈ (0 , 1) is arbitrary, this shows that T − k χ T k L ∞ ≤ C δ T − ττ +1 + δ (6.25)for any δ ∈ (0 , C δ depends only on δ and A . Under the condition (6.24), byTheorem 6.6, we also obtain (cid:0) h| ψ − ∇ χ T | i (cid:1) / ≤ C δ T − ττ +1 + δ (6.26)for any δ ∈ (0 , Convergence rates In this section we give the proof of Theorems 1.1 and 1.2. Lemma 7.1. Let h ∈ L ( R d ) and T > . Suppose that there exists σ ∈ (0 , such that sup x ∈ R d (cid:18) − ˆ B ( x,r ) | h | (cid:19) / ≤ (cid:18) Tr (cid:19) − σ for any < r ≤ T. (7.1) Let u ∈ H ( R d ) be the solution of − ∆ u + T − u = h in R d , (7.2) given by Proposition 5.1. Then k u k L ∞ ≤ C T , k∇ u k L ∞ ≤ C T, (7.3) and |∇ u ( x ) − ∇ u ( y ) | ≤ C T − σ | x − y | σ for any x, y ∈ R d , (7.4) where C depends only on d and σ . Furthermore, u ∈ H ( R d ) and sup x ∈ R d (cid:18) − ˆ B ( x,T ) |∇ u | (cid:19) / ≤ C. (7.5) Proof. By rescaling we may assume T = 1. It follows from Proposition 5.1 and (7.1) thatsup x ∈ R d (cid:18) − ˆ B ( x, | u | (cid:19) / ≤ C and sup x ∈ R d (cid:18) − ˆ B ( x, |∇ u | (cid:19) / ≤ C , (7.6)where C depends only on d . Fix x ∈ R d and let φ ∈ C ∞ ( B ( x , φ = 1 in B ( x , uφ as an integral and using the fundamentalsolution for − ∆, the desired estimates follow from (7.1) by a standard procedure. Weleave the details to the reader.Under additional almost periodicity conditions on h , the next lemma gives muchsharper estimates for the solution u of (7.2). Lemma 7.2. Let h ∈ L ( R d ) and T > . Suppose that there exists σ ∈ (0 , such that sup x ∈ R d (cid:18) − ˆ B ( x,r ) | h | (cid:19) / ≤ C (cid:18) Tr (cid:19) − σ , sup x ∈ R d (cid:18) − ˆ B ( x,r ) | h (( t + y ) − h ( t + z ) | dt (cid:19) / ≤ C (cid:18) Tr (cid:19) − σ k A ( · + y ) − A ( · + z ) k L ∞ (7.7) for any < r ≤ T and y, z ∈ R d . Let u ∈ H ( R d ) be the solution of (7.2), given byProposition 5.1. Then T − k u k L ∞ ≤ C Θ ( T ) + |h h i| ,T − k∇ u k L ∞ ≤ C Θ σ ( T ) , (7.8) where Θ σ ( T ) is defined by (6.18) and C depends at most on d , σ and C . roof. By applying Lemma 7.1 to the function (cid:0) u ( x + y ) − u ( x + z ) (cid:1) / (cid:0) C k A ( · + y ) − A ( · + z ) k L ∞ (cid:1) , with y, z fixed, we obtain k u ( · + y ) − u ( · + z ) k L ∞ ≤ C T k A ( · + y ) − A ( · + z ) k L ∞ , k∇ u ( · + y ) − ∇ u ( · + z ) k L ∞ ≤ C T k A ( · + y ) − A ( · + z ) k L ∞ , (7.9)where C depends only on d , C and σ . This shows that u and ∇ u are uniformly almost-periodic. In particular, u and ∇ u have mean values and h∇ u i = 0. Also, note thatcondition (7.7) implies that h ∈ B ( R d ) and hence has the mean value h h i . It is easy todeduce from the equation (7.2) that h u i = T h h i .Note that for any y ∈ R d and z ∈ R d with | z | ≤ R ≤ T , T − | u ( y ) − u (0) | ≤ T − | u ( y ) − u ( z ) | + T − | u ( z ) − u (0) |≤ C k A ( · + y ) − A ( · + z ) k L ∞ + C T − R, where we have used (7.9) and k∇ u k L ∞ ≤ C T for the second inequality. It follows fromthe definition of ρ ( R ) thatsup y ∈ R d T − | u ( y ) − u (0) | ≤ C n ρ ( R ) + T − R o for any 0 < R ≤ T. By the definition of Θ , this givessup y ∈ R d T − | u ( y ) − u (0) | ≤ C Θ ( T ) . (7.10)Using | T − u (0) | ≤ T − (cid:12)(cid:12)(cid:12)(cid:12) − ˆ B (0 ,L ) { u ( y ) − u (0) } dy (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) − ˆ B (0 ,L ) u ( x ) (cid:12)(cid:12)(cid:12)(cid:12) for any L > L → ∞ , | T − u (0) | ≤ C Θ ( T ) + T − |h u i| = C Θ ( T ) + |h h i| . (7.11)The first inequality in (7.8) now follows from (7.10) and (7.11).Finally, we point out that the second inequality in (7.8) follows in the same manner,using (7.9) and (7.4) as well as the fact that the mean value of ∇ u is zero.We are now ready to estimate the rates of convergence of u ε to u . Theorem 7.3. Let u ε ( ε ≥ be the weak solution of L ε ( u ε ) = F in Ω and u ε = g on ∂ Ω . Suppose that u ∈ W , (Ω) . Let w ε ( x ) = u ε ( x ) − u ( x ) − εχ T,j ( x/ε ) ∂u ∂x j + v ε , (7.12)30 here T = ε − and v ε ∈ H (Ω; R m ) is the weak solution of the Dirichlet problem: L ε ( v ε ) = 0 in Ω and v ε = εχ T,j ( x/ε ) ∂u ∂x j on ∂ Ω . (7.13) Then k w ε k H (Ω) ≤ C σ n Θ σ ( T ) + h| ψ − ∇ χ T |i o k u k W , (Ω) (7.14) for any σ ∈ (0 , , where C σ depends only on σ , A and Ω .Proof. With loss of generality we may assume that k u k W , (Ω) = 1 . (7.15)A direct computation shows that L ε ( w ε ) = − div (cid:0) B T ( x/ε ) ∇ u (cid:1) + ε div (cid:8) A ( x/ε ) χ T ( x/ε ) ∇ u (cid:9) , (7.16)where B T ( y ) = (cid:0) b αβT,ij ( y ) (cid:1) is given by b αβT,ij ( y ) = b a αβij − a αβij ( y ) − a αγik ( y ) ∂∂y k n χ γβT,j ( y ) o . (7.17)Since w ε ∈ H (Ω; R m ), it follows from (7.16) that c ˆ Ω |∇ w ε | dx ≤ (cid:12)(cid:12)(cid:12)(cid:12) ˆ Ω div (cid:0) B T ( x/ε ) ∇ u (cid:1) · w ε dx (cid:12)(cid:12)(cid:12)(cid:12) + ˆ Ω | εχ T ( x/ε ) | |∇ u ||∇ w ε | dx = I + I . (7.18)It suffices to show that I + I ≤ C σ n Θ σ ( T ) + h| ψ − ∇ χ T |i o k w ε k H (Ω) (7.19)for any σ ∈ (0 , I ≤ C ε k χ T k L ∞ k∇ w ε k L (Ω) ≤ C Θ σ ( T ) k∇ w ε k L (Ω) (7.20)for any σ ∈ (0 , I , we let h ( y ) = h T ( y ) = B T ( y ) − h B T i and solve the equation(7.2). More precisely, let h = (cid:0) h αβij (cid:1) and f = ( f αβij ), where f αβij ∈ H ( R d ) sovles − ∆ f αβij + T − f αβij = h αβij in R d . (7.21)By (6.8) and (6.15), the function h satisfies the condition (7.7) for any σ ∈ (0 , h h i = 0, it follows from Lemma 7.2 that T − k f k L ∞ ≤ C Θ ( T ) ,T − k∇ f k L ∞ ≤ C Θ σ ( T ) (7.22)31or any σ ∈ (0 , I in (7.18) by (cid:12)(cid:12)(cid:12)(cid:12) ˆ Ω div (cid:8) ∆ f ( x/ε ) ∇ u (cid:9) · w ε dx (cid:12)(cid:12)(cid:12)(cid:12) + T − ˆ Ω | f ( x/ε ) | |∇ u | |∇ w ε | dx + C h| ψ − ∇ χ T |ik w ε k L (Ω) , (7.23)where we have used the fact |h B T i| ≤ C h| ψ − ∇ χ T |i . Note that by (7.22), the second termin (7.23) is bounded by C Θ ( T ) k∇ w ε k L (Ω) .It remains to estimate the first term in (7.23), which we denote by I . To this endwe writediv { ∆ f ( x/ε ) ∇ u } · w ε = ∂∂x i ( ∆ f αβij ( x/ε ) ∂u β ∂x j ) · w αε = ∂∂x i ( ∂∂x k ( ∂f αβij ∂x k − ∂f αβkj ∂x i ) ( x/ε ) ∂u β ∂x j ) · w αε + ∂∂x i ( ∂ f αβkj ∂x k ∂x i ( x/ε ) ∂u β ∂x j ) · w αε = − ∂∂x i ( ε ( ∂f αβij ∂x k − ∂f αβkj ∂x i ) ( x/ε ) ∂ u β ∂x k ∂x j ) · w αε + ∂∂x i ( ∂ f αβkj ∂x k ∂x i ( x/ε ) ∂u β ∂x j ) · w αε , where we have used the product rule and the fact that ∂ ∂x i ∂x k (" ∂f αβij ∂x k − ∂f αβkj ∂x i x/ε (cid:1) ∂u β ∂x j ) = 0 . It then follows from an integration by parts that I ≤ Cε ˆ Ω |∇ f ( x/ε ) | |∇ u | |∇ w ε | dx + C X j,α,β ˆ Ω (cid:12)(cid:12) ∇ ∂f αβkj ∂x k ( x/ε ) (cid:12)(cid:12) |∇ u | |∇ w ε | dx = I (1)11 + I (2)11 . (7.24)In view of (7.22) we have I (1)11 ≤ C ε k∇ f k L ∞ k∇ w ε k L (Ω) ≤ C Θ σ ( T ) k∇ w ε k L (Ω) (7.25)for any σ ∈ (0 , I (2)11 , we note that by the definition of χ T , ∂h αβij ∂y i = ∂∂y i n b αβT,ij o = − T χ αβT,j . It follows that − ∆ ∂f αβij ∂y i ! + 1 T ∂f αβij ∂y i ! = − T χ αβT,j . Observe that the function T − χ T satisfies the assumption on h in Lemma 7.2 with σ = 1.As a result, we obtain (cid:13)(cid:13)(cid:13) ∇ ∂f αβij ∂x i (cid:13)(cid:13)(cid:13) L ∞ ≤ C σ Θ σ ( T )for any σ ∈ (0 , I (2)11 by C σ Θ σ ( T ) k∇ w ε k L (Ω) , and completesthe proof.The next lemma gives an estimate for the norm of v ε in H (Ω). Lemma 7.4. Let v ε be the weak solution of (7.13) with T = ε − . Then k v ε k H (Ω) ≤ C σ (cid:0) T − k χ T k L ∞ (cid:1) − σ n k∇ u k L ∞ (Ω) + k∇ u k L (Ω) o (7.26) for any σ ∈ (0 , / , where C σ depends only on A , Ω , and σ .Proof. We may assume that k∇ u k L ∞ (Ω) + k∇ u k L (Ω) = 1. We may also assume that δ = T − k χ T k L ∞ > 0, and is small, since δ → T → ∞ . Choose a cut-off function η δ ∈ C ∞ ( R d ) so that 0 ≤ η δ ≤ η δ ( x ) = 1 if dist( x, ∂ Ω) < δ , η δ ( x ) = 0 if dist( x, ∂ Ω) ≥ δ ,and |∇ η δ | ≤ C δ − . Note that k v ε k H (Ω) ≤ C ε k χ T ( x/ε ) ∇ u k H / ( ∂ Ω) ≤ C ε k η δ χ T ( x/ε ) ∇ u k H (Ω) ≤ C ( k χ T k L ∞ δ − / ε + (cid:18) ˆ Ω δ |∇ χ T ( x/ε ) | dx (cid:19) / ) , (7.27)where Ω δ = (cid:8) x ∈ Ω : dist( x, ∂ Ω) ≤ δ (cid:9) . Since k χ T k L ∞ δ − / ε = δ / , we only need toestimate the integral of |∇ χ T ( x/ε ) | over Ω δ .To this end, we cover Ω δ with cubes { Q j } of side length δ so that P j | Q j | ≤ C δ . Itfollows that ˆ Ω δ |∇ χ T ( x/ε ) | dx ≤ X j ˆ Q j |∇ χ T ( x/ε ) | dx ≤ X j | Q j | − ˆ ε Q j |∇ χ T | ≤ C δ sup ℓ ( Q )= δT − ˆ Q |∇ χ T | ≤ C σ δ − σ (7.28)for any σ ∈ (0 , Proof of Theorem 1.1. It follows from Theorem 7.3 and Lemma 7.4 that for any σ ∈ (0 , 1) and δ ∈ (0 , / k u ε − u − εχ T ( x/ε ) ∇ u k H (Ω) ≤ C n Θ σ ( T ) + h| ψ − ∇ χ T |i o k u k W , (Ω) + C (cid:2) Θ σ ( T ) (cid:3) − δ n k∇ u k L ∞ (Ω) + k∇ u k L (Ω) o ≤ C n h| ψ − ∇ χ T |i + (cid:2) Θ σ ( T ) (cid:3) − δ o k u k W ,p (Ω) ≤ C n h| ψ − ∇ χ T |i + (cid:2) Θ ( T ) (cid:3) σ ( − δ ) o k u k W ,p (Ω) , (7.29)where T = ε − and we have used the Sobolev imbedding k∇ u k L ∞ (Ω) ≤ C k u k W ,p (Ω) for p > d . This implies that k u ε − u k L (Ω) ≤ k εχ T ( x/ε ) ∇ u k L (Ω) + C n h| ψ − ∇ χ T |i + (cid:2) Θ ( T ) (cid:3) o k u k W ,p (Ω) ≤ C n h| ψ − ∇ χ T |i + (cid:2) Θ ( T ) (cid:3) o k u k W ,p (Ω) , where C depends only on A and Ω. Since h| ψ − ∇ χ T |i + (cid:2) Θ ( T ) (cid:3) → T → ∞ , onemay find a modulus η on (0 , A , such that η (0+) = 0 and h| ψ − ∇ χ T |i + (cid:2) Θ ( T ) (cid:3) ≤ η ( T − )for T ≥ 1. As a result, we obtain k u ε − u − εχ T ( x/ε ) ∇ u k H (Ω) ≤ C η ( ε ) k u k W ,p (Ω) , k u ε − u k L (Ω) ≤ C η ( ε ) k u k W ,p (Ω) . Finally, we observe that by Theorem 1.4, for any σ ∈ (0 , k u ε k C σ (Ω) ≤ C n k g k C σ ( ∂ Ω) + k F k L d (Ω) o ≤ C n k u k C σ (Ω) + k∇ u k L d (Ω) o ≤ C k u k W ,d (Ω) . It follows by interpolation that for any σ ∈ (0 , k u ε − u k C σ (Ω) ≤ C e η ( ε ) k u k W ,p (Ω) , where e η is a modulus function depending only on A and σ , and e η (0+) = 0. This completethe proof. 34 roof of Theorem 1.2. Estimate (1.15) follows directly from (7.29) and Theorem 6.6.To see (1.14) we use k u ε − u k L (Ω) ≤ k u ε − u − εχ T ( x/ε ) ∇ u + v ε k L (Ω) + k v ε k L (Ω) ≤ C σ n Θ σ ( T ) + h| ψ − ∇ ψ T |i o k u k W , (Ω) + k v ε k L (Ω) , (7.30)where v ε is defined in Theorem 7.3. By Theorem 1.4 we obtain k v ε k L (Ω) ≤ C k v ε k L ∞ (Ω) ≤ C k εχ T ( x/ε ) ∇ u k C σ ( ∂ Ω) ≤ C n ε − σ k χ T k C ,σ + Θ σ ( T ) o k∇ u k C σ ( ∂ Ω) ≤ C n T σ − k χ T k C ,σ + Θ σ ( T ) o k u k W ,p (Ω) , where p > d , σ ∈ (0 , 1) and 0 < σ < − dp . Since T − k χ T k L ∞ ≤ C σ Θ σ ( T ) and | χ T ( x ) − χ T ( y ) | ≤ C α T − α | x − y | α for any α ∈ (0 , T σ − k χ T k C ,σ ≤ C [Θ σ ( T )] − σ for any σ > σ . Hence, k v ε k L (Ω) ≤ C (cid:2) Θ σ ( T ) (cid:3) − δ k u k W ,p (Ω) ≤ C (cid:2) Θ ( T ) (cid:3) σ (1 − δ ) k u k W ,p (Ω) for any δ, σ ∈ (0 , 1) and p > d , where C depends only on δ , p , σ , A and Ω. This, togetherwith (7.30) and Theorem 6.6, gives k u ε − u k L (Ω) ≤ C n h| ψ − ∇ χ T |i + (cid:2) Θ ( T ) (cid:3) σ o k u k W ,p (Ω) ≤ C ( ˆ ∞ ε Θ σ ( r ) r dr + (cid:2) Θ ( ε − ) (cid:3) σ ) k u k W ,p (Ω) for any σ ∈ (0 , In this section we consider the case where A ( x ) is quasi-periodic and continuous. Moreprecisely, without loss of generality, we will assume that ( A ( x ) = B ( j λ ( x )) ,B is 1-periodic and continuous in R M , (8.1)where M = m + m + · · · + m d , and for x = ( x , x , . . . , x d ) ∈ R d , j λ ( x ) = ( λ x , λ x , . . . , λ m x , λ x , . . . , λ m x , . . . , λ d x d , . . . , λ m d d x d ) ∈ R M . i = 1 , . . . , d , the set { λ i , . . . , λ m i i } is assumed to be linearly independentover Z . Under these conditions it is known that A ( x ) is uniformly almost-periodic. Weshall be interested in conditions on λ = (cid:0) λ ji (cid:1) that implies the power decay of ρ ( R ) as R → ∞ . For convenience we consider ρ ( R ) = sup y ∈ R d inf z ∈ R d k z k ∞ ≤ R k A ( · + y ) − A ( · + z ) k L ∞ , (8.2)where k z k ∞ = max( | z | , . . . , | z d | ) for z = ( z , . . . , z d ). It is easy to see that ρ ( √ dR ) ≤ ρ ( R ) ≤ ρ ( R ).Let ω ( δ ) = sup (cid:8) | B ( x ) − B ( y ) | : k x − y k ∞ ≤ δ (cid:9) , δ > B ( x ). For x ∈ R , write x = [ x ]+ < x > , where [ x ] ∈ Z and < x > ∈ [ − / , / x = ( x , . . . , x M ) ∈ R M , we define [ x ] = ([ x ] , . . . , [ x N ]) and < x > = ( < x >, . . . , < x M > ). It is easy to see that k < x > k ∞ gives the distance from x to Z M with respect to the norm k · k ∞ . Lemma 8.1. Let ρ ( R ) be defined by (8.2). Then, for any R > , ρ ( R ) ≤ ω ( θ λ ( R )) ,where θ λ ( R ) = sup x ∈ [ − / , / M inf z ∈ R d k z k ∞ ≤ R k x − < j λ ( z ) > k ∞ . (8.3) Proof. Note that, since B is 1-periodic, | B ( x ) − B ( y ) | = | B ( y + [ x − y ]+ < x − y > ) − B ( y ) | = | B ( y + < x − y > ) − B ( y ) |≤ ω ( k < x − y > k ∞ )for any x, y ∈ R M . It follows that | A ( x + y ) − A ( x + z ) | ≤ ω ( k < j λ ( y ) − j λ ( z ) > k ∞ ) , for any x, y, z ∈ R d . This implies that ρ ( R ) ≤ sup y ∈ R d inf z ∈ R d k z k ∞ ≤ R ω ( k < j λ ( y ) − j λ ( z ) > k ∞ ) . Using k < j λ ( y ) − j λ ( z ) > k ∞ = k << j λ ( y ) > − < j λ ( z ) >> k ∞ ≤ k < j λ ( y ) > − < j λ ( z ) > k ∞ , we obtain ρ ( R ) ≤ sup y ∈ R d inf z ∈ R d k z k ∞ ≤ R ω ( k < j λ ( y ) > − < j λ ( z ) > k ∞ ) ≤ ω (cid:0) θ λ ( R ) (cid:1) , where we have used the continuity of ω ( δ ) for the second inequality.36et λ i = ( λ i , λ i , . . . , λ m i i ) ∈ R m i for each 1 ≤ i ≤ d , and j λ i ( t ) = ( λ i t, λ i t, . . . , λ m i i t ) ∈ R m i for t ∈ R . Thus, for z = ( z , z , . . . , z d ) ∈ R d , j λ ( z ) = ( j λ ( z ) , j λ ( z ) , . . . , j λ d ( z d )) . It follows that k x − < j λ ( z ) > k ∞ = max ≤ i ≤ d k x i − < j λ i ( z i ) > k ∞ , where x = ( x , x , . . . , x d ) ∈ R M and x i ∈ R m i . This implies that θ λ ( R ) = max ≤ i ≤ d θ λ i ( R ) , (8.4)where θ λ i ( R ) = sup x ∈ [ − / , / mi inf t ∈ R | t |≤ R k x − < j λ i ( t ) > k ∞ . (8.5)Note that if m i = 1, then θ λ i ( R ) = 0 for R large. We will use the Erd¨os-Turan-Koksmainequality in the discrepancy theory to estimate the function θ λ i ( R ), defined by (8.5), for m i ≥ P = P N = { x , x , . . . , x N } be a finite subset of [ − / , / m . The discrepancy of P is defined as D N ( P ) = sup B (cid:12)(cid:12)(cid:12)(cid:12) A ( B ; P ) N − | B | (cid:12)(cid:12)(cid:12)(cid:12) , where the supremum is taken over all rectangular boxes B = [ a , b ] × · · · [ a m , b m ] ⊂ [ − / , / m , and A ( B ; P ) denotes the number of elements of P in B . It follows from theErd¨os-Turan-Koksma inequality that D N ( P ) ≤ C H + X n ∈ Z m < k n k ∞ ≤ H | n | ) · · · (1 + | n m | ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X x ∈ P e πi ( n · x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (8.6)for any H ≥ 1, where C depends only on m (see e.g. [10, p.15]). It is not hard to see thatmax y ∈ [ − / , / m min z ∈ P N k y − z k ∞ ≤ 12 [ D N ( P N )] m . (8.7) Lemma 8.2. Let R ≥ and ℓ ≥ be two positive integers. We divide the interval [ − R, R ] into Rℓ subintervals of length /ℓ . Let N = 2 Rℓ and P N = (cid:26) x = < j λ ( t ) > ∈ [ − / , / m : t = j + kℓ , − R ≤ j ≤ R − and ≤ k ≤ ℓ − (cid:27) , here λ = ( λ , . . . , λ m ) ∈ R m and m ≥ . Suppose that there exist c > and τ > suchthat | n · λ | ≥ c | n | − τ for any n ∈ Z m \ { } . (8.8) Then D N ( P N ) ≤ C n R − τ +1 (log R ) m − + N − R τ +1 (log R ) m − o , (8.9) where C depends only on m , c , | λ | and τ .Proof. Let f ( t ) = e πi ( n · λ ) t and I n = 1 N X x ∈ P N e πi ( n · x ) = 1 N X j,k f ( t jk ) , (8.10)where n ∈ Z m \ { } , j = − R, . . . R − k = 0 , . . . , ℓ − t jk = j + kℓ . Using (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) R ˆ R − R f ( t ) dt − N X j,k f ( t jk ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ℓ − k f ′ k ∞ , we obtain | I n | ≤ Cℓ − k f ′ k ∞ + (cid:12)(cid:12)(cid:12)(cid:12) R ˆ R − R f ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ℓ − | n · λ | + CR | n · λ |≤ C (cid:8) ℓ − | n | + R − | n | τ (cid:9) , where we have used the assumption (8.8). In view of (8.6), we obtain D N ( P N ) ≤ C H + X n ∈ Z m < k z k ∞ ≤ H ℓ − | n | + R − | n | τ (1 + | n | ) · · · (1 + | n m | ) ≤ C (cid:26) H + ˆ | x |≤ C H ℓ − | x | + R − | x | τ (1 + | x | ) · · · (1 + | x m | ) dx (cid:27) ≤ C (cid:26) H + RN − H (log H ) m − + R − H τ (log H ) m − (cid:27) for any H ≥ 2. By taking H = R τ +1 , we obtain the estimate (8.9). Theorem 8.3. Let λ = ( λ , . . . , λ d ) with λ i = ( λ i , . . . , λ m i i ) ∈ R m i for ≤ i ≤ d . Supposethat there exist c > and τ > such that for each ≤ i ≤ d with m i ≥ , | n · λ i | ≥ c | n | − τ for any n ∈ Z m i \ { } . (8.11) Then, for any R ≥ , θ λ ( R ) ≤ C R − e m ( τ +1) (log R ) − e m , (8.12) where e m = max { m , . . . , m d } and C depends only on d , e m , c and τ . roof. Suppose m i ≥ 2. Let P = P N be same as in Lemma 8.2. It follows from (8.7) andLemma 8.2 that θ λ i ( R ) ≤ C n R − τ +1 (log R ) m i − + N − R τ +1 (log R ) m i − o mi ≤ C R − mi ( τ +1) (log R ) − mi , where we have taken N = C R τ +1 . This, together with (8.4), gives (8.12). Remark 8.4. Suppose that A ( x ) = B ( j λ ( x )) and B ( y ) is 1-periodic. Also assume that λ satisfies the condition (8.11) and B ( y ) is H¨older continuous of order α for some α ∈ (0 , ρ ( R ) ≤ C R − α e m ( τ +1) (log R ) α (1 − e m ) (8.13)for R ≥ 1. In view of Remark 1.3 this leads to k u ε − u k L (Ω) ≤ C γ ε γ k u k W ,p (Ω) for any 0 < γ < αα + e m ( τ +1) . We point out that if A ( y ) satisfies the condition (8.11) and issufficiently smooth, the sharp estimate k u ε − u k L (Ω) = O ( ε ) was obtained in [22]. References [1] S. 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