A Carleman-Type Inequality in Elliptic Periodic Homogenization
aa r X i v : . [ m a t h . A P ] F e b A Carleman-Type Inequality in Elliptic PeriodicHomogenization
Yiping Zhang ∗ Academy of Mathematics and Systems Science, CAS;University of Chinese Academy of Sciences;Beijing 100190, P.R. China.
Abstract
In this paper, for a family of second-order elliptic equations with rapidly oscil-lating periodic coefficients, we are interested in a Carleman-type inequality for thesesolutions satisfying an additional growth condition in elliptic periodic homogeniza-tion, which implies a three-ball inequality without an error term at a macroscopicscale. Moreover, if we replace the additional growth condition by the doubling con-dition at a macroscopic scale, then the three-ball inequality without an error termholds at any scale. The proof relies on the convergence of H -norm for the solutionand the compactness argument. Since T. Carleman’s pioneer work [8], Carleman estimates have been indispensabletools for obtaining a three-ball (or three-cylinder) inequality and proving the unique con-tinuation property for partial differential equations. In general, the Carleman estimatesare weighted integral inequalities with suitable weight functions satisfying some convexityproperties. The three-ball inequality is obtained by applying the Carleman estimates bychoosing a suitable function. For Carleman estimates and the unique continuation prop-erties for the elliptic and parabolic operators, we refer readers to [3, 4, 10–12, 17, 21, 22]and their references therein for more results.Over the last forty years, there is a vast and rich mathematical literature on homog-enization. Most of these works are focused on qualitative results, such as proving theexistence of a homogenized equation. However, until recently, nearly all of the quantita-tive theory, such as the convergence rates in L and H , the W ,p -estimates, the Lipschitzestimates and the asymptotic expansion of the Green functions and fundamental solu-tions, were confined in periodic homogenization. There are many good expositions on ∗ Email:[email protected] L ε u ε =: − div ( A ( x/ε ) ∇ u ε ) = 0 , (1.1)where 1 > ε > A ( y ) = ( a ij ( y )) is a real symmetric d × d matrix-valued function in R d for d ≥
2. Assume that A ( y ) satisfies the following assumptions:(i) Ellipticity: For some 0 < µ < y ∈ R d , ξ ∈ R d , it holds that µ | ξ | ≤ A ( y ) ξ · ξ ≤ µ − | ξ | . (1.2)(ii) Periodicity: A ( y + z ) = A ( y ) for y ∈ R d and z ∈ Z d . (1.3)(iii)Lipschitz continuity: There exist constants ˜ M > | A ( x ) − A ( y ) | ≤ ˜ M | x − y | , for any x, y ∈ R d . (1.4)Let B ( x, r ) = (cid:8) y ∈ R d : | y − x | < r (cid:9) and B r = B (0 , r ). For positive constants M , N ≥ N ≥
1, let u ε ∈ H ( B ) be a solution of, and satisfies the following growthconditions, ˆ B | u ε | ≤ M max ((cid:18) ˆ B | u ε | (cid:19) N , (cid:18) ˆ B | u ε | (cid:19) /N ) , (1.5)then we could obtain the following result: Theorem 1.1.
Assume that the coefficient matrix A is symmetric and satisfies the con-ditions (1 . - (1 . , and let u ε ∈ H ( B ) be a solution of (1 . and satisfies the growthcondition (1 . . Then, there exists ε > , depending only on d , µ , ˜ M , M , N and N such that for < ε ≤ ε , there holds the following Carleman-type inequality, C ˆ B (cid:16) λ τ ϕ ( u ε η ) + λ τ ϕ (cid:12)(cid:12) ∇ (cid:2) ( u ε − εχ εj ∂ j u ) η (cid:3)(cid:12)(cid:12) (cid:17) e τϕ dx ≤ ˆ B (cid:2) L ε ( u ε η + εχ εj ∂ j ηu ε ) (cid:3) e τϕ dx, (1.6)2 ith ϕ = e − λ | x | and u given by Theorem 2.2, for all λ ≥ λ and τ ≤ τ ≤ τ + C ( λ , τ ) || u ε || L B || u ε || L B with λ and τ defined in Proposition 2.6, depending only on µ , where C ( λ , τ ) is a constant depending only on λ and τ , and could be specified in the proof ofCorollary 1.4 in Section 3. And η ∈ C ∞ ( B / \ B / ) is a fixed cutoff function such that ≤ η ≤ , η = 1 in B / \ B / . (1.7)Throughout this paper, we always assume that η is a fixed cutoff function defined inTheorem 1.1. We prove this theorem by compactness argument. For the compactnessmethod used in homogenization theory, we refer readers to [5, 6] for more details. Remark 1.2.
The growth condition (1 . allows that the function u ε grows at a speed ofpolynomials of any degree. For example, if u ε behaves like | x | k , for any k ∈ N + , then it iseasy to see that the conditions holds with M depending only on (cid:13)(cid:13) u ε / | x | k (cid:13)(cid:13) L ∞ ( B ) and forany N ≥ log 3 / log 2 .Due to the rapid oscillation of the coefficient matrix A ( x/ε ) , we could not expect aCarleman inequality in homogenization totally similar to the classical case. Moreover, itimplies that the growth condition (1 . is necessary in compactness argument to obtain theCarleman-type inequality (1 . in Example 3.2. (Meanwhile, one may use other methodsto derive a Carleman-type inequality in homogenization without the growth condition.) Remark 1.3.
The Carleman-type inequality (1 . continues to hold for the operator ˜ L ε = − div( A ε ∇ ) + b ε ∇ + c ε + λ with b ∈ L ∞ and c ∈ L ∞ being 1-periodic (the operator ˜ L ε is positive by adding a large constant λ ), since the L -norm as well as the H -normconvergence rates continue to hold for the solution u ε to the operator ˜ L ε [23], and theCarleman inequality as well as the unique continuation property continue to hold for thehomogenized operator ˜ L = − div( b A ∇ ) + M ( b ) ∇ + M ( c ) + λ (Remark 2.7), where wehave used the notations F ε = F ( x/ε ) for a function F and M ( G ) = ´ Y G ( y ) dy for a1-periodic function G . The most trivial application of Theorem 1.1 is the following three-ball inequality at amacroscopic scale without an error term.
Corollary 1.4.
Assume that the coefficient matrix A is symmetric and satisfies the con-ditions (1 . - (1 . , and let u ε ∈ H ( B ) be a solution of (1 . and satisfy the growthcondition (1 . . Then, for some constant C , depending only on d , µ , ˜ M , and M , thereholds the following three-ball inequality without an error term, || u ε || L ( B ) ≤ C || u ε || sL ( B ) || u ε || − sL ( B ) , (1.8) where s = αα + β with α = 1 − e − λ and β = 2( e − λ − e − λ ) for any λ ≥ λ with λ definedin Theorem 1.1. The Corollary above only implies the three-ball inequality at a macroscopic scale, inthe following theorem, we could obtain the three-ball inequality at every scale by usingCorollary 1.4 and the uniform doubling conditions proved in [18] in elliptic homogeniza-tion. 3 heorem 1.5.
Assume that A is symmetric and satisfies the conditions (1 . - (1 . . Let u ε ∈ H ( B ) be a solution to the equation L ε ( u ε ) = 0 in B , and for some positiveconstant M , u ε satisfies the following doubling condition at a macroscopic scale, B | u ε | ≤ M B √ µ | u ε | , then for any < r ≤ / , there holds the following three-sphere inequality without anerror term, || u ε || L ( B r ) ≤ C || u ε || sL ( B r ) || u ε || − sL ( B r ) , (1.9) with the same s defined in Corollary 1.4 and C depending only on d , µ , ˜ M and M . The first result about the approximate three-ball inequality was obtained by Kenigand Zhu in [15] with the help of the asymptotic behavior of Green functions and theLagrange interpolation technique, under the assumptions that the coefficient matrix A is only H¨older continuous. Later on, an improvement of a sharp exponential error term(in the sense that if A is only H¨older continuous, then the multiplicative factor mustbe at least exponential) in the error bound for the approximate three-ball inequality(under certain extra conditions) was discovered by Armstrong, Kuusi, and Smart in [2],as a consequence of the large-scale analyticity. Meanwhile, an approximate two-sphereone-cylinder inequality in parabolic periodic homogenization was obtained by the firstauthor in [24], under the assumptions tha the coefficient matrix A ( x, t ) is only H¨oldercontinuous, with the help of the asymptotic behavior of fundamental solutions and theLagrange interpolation technique.Recently, the three-ball inequality without an error term was discovered by Kenig,Zhu and Zhuge in [16] under the assumptions that A ∈ C , and u ε satisfies a doublingcondition at a macroscopic scale, with the help of the approximate three-ball inequalitywith a sharp exponential error term obtained in [2]. At this stage, we should comparethe three-ball inequality obtained in [16] with the result proved in Theorem 1.5 in thispaper. The result reads that: Theorem . Assume that the coefficient matrix A satisfies (1 . . τ > C > θ ∈ (0 , /
2) depending only on d , µ and λ such that if u ε is aweak solution of L ε ( u ε ) = 0 in B satisfying ˆ B | u ε | dx ≤ M ˆ B θ | u ε | dx, then for any r ∈ (0 , ˆ B θ | u ε | dx ≤ exp(exp( CM τ )) ˆ B θr | u ε | dx and ˆ B θr | u ε | dx ≤ exp(exp( CM τ )) ˆ B θ r | u ε | dx ! τ (cid:18) ˆ B r | u ε | dx (cid:19) − τ < τ <
1. It is clear that the authors in [16] have found an explicit estimatefor the constant C ( M ) in the doubling condition and in the three-ball inequality, withan unknown θ . However, in our Theorem 1.5, we could state θ explicitly and obtain thethree-ball inequality more directly with an unknown constant C ( M ). Throughout thispaper, with Y = [0 , d ∼ = R d / Z d , we use the following notation H m per ( Y ) =: (cid:26) f ∈ H m ( Y ) and f is 1-periodic with Y f dy = 0 (cid:27) , and we will write ∂ x i as ∂ i , F ε = F ( x/ε ) for a function F and M ( G ) = ´ Y G ( y ) dy for a1-periodic function G if the context is understand. Assume that A = A ( y ) satisfies the conditions (1 . . χ ( y ) ∈ H ( Y ; R d )denote the first order corrector for L ε , where χ j for j = 1 , · · · , d is the unique 1-periodicfunction in H ( Y ) such that ( L ( χ j ) = −L ( y j ) in Y, ffl Y χ j dy = 0 . (2.1)By the classical Schauder estimates, χ ∈ C ,α if A ∈ C ,α . The homogenized operator for L ε is given by L = − div( b A ∇ ), where b A = ( b a ij ) d × d and b a ij = Y (cid:20) a ij + a ik ∂∂y k ( χ j ) (cid:21) ( y ) dy. (2.2)It is well-known that the homogenized matrix b A also satisfies the ellipticity condition(1 .
2) with the same µ . What’s more, if A is symmetric, the same is also true for b A . Werefer the readers to [20] for the proofs.Denote the so-called flux correctors b ij by b ij ( y ) = b a ij − a ij ( y ) − a ik ( y ) ∂χ j ( y ) ∂y k , (2.3)where 1 ≤ i, j ≤ d . Lemma 2.1.
Suppose that A satisfies the conditions (1 . and (1 . . For ≤ i, j, k ≤ d ,there exists F ijk ∈ H per ( Y ) ∩ L ∞ ( Y ) such that b ij = ∂∂y k F kij and F kij = − F ikj . (2.4) Proof.
See [14, Remark 2.1].The following theorem states the existence of u in Theorem 1.1 and is used to controlthe second term on the left hand side of (1 . heorem 2.2. Suppose that A is symmetric and satisfies the conditions (1 . and (1 . .Let u ε ∈ H ( B ) be the weak solution of equation L ε ( u ε ) = 0 in B . Then there exists u ∈ H ( B / ) such that L ( u ) = 0 in B / , and || u ε − u || L ( B / ) ≤ C √ ε || u ε || L ( B ) , (2.5) where C depends only on d and µ .Proof. Due to the Caccioppoli’s inequality and the co-area formula, there exists r ∈ [ , ] such that ˆ ∂B r | u ε | dS + ˆ ∂B r |∇ u ε | dS ≤ C ˆ B | u ε | dx. (2.6)Then, we could consider the following Dirichlet problem, ( L ε ( u ε ) = 0 in B r u ε ∈ H ( ∂B r ) with || u ε || H ( ∂B r ) ≤ C || u ε || L ( B ) . (2.7)And let u satisfies the following equation, ( L ( u ) = 0 in B r u = u ε on ∂B r . (2.8)Since A is symmetric, therefore, it follows from the homogenization theory that thereholds (for the proof, see [20] for example) || u ε − u || L ( B / ) ≤ || u ε − u || L ( B r ) ≤ Cε ||∇ u || L ( B r ) + C √ ε || u ε || H ( ∂B r ) ≤ C √ ε || u ε || H ( ∂B r ) ≤ Cε || u ε || L ( B ) , (2.9)where we have used the H estimate for u and (2 .
6) in the third line in inequality (2 . Remark 2.3.
In Theorem 2.2, if we additionally assume that A is Lipschitz continuous,then there exists u ∈ H ( B / ) such that L ( u ) = 0 in B / , and || u ε − u || L ∞ ( B / ) ≤ Cε || u ε || L ( B ) . The main ideal of this proof is due to Lin and Shen [18], and we omit it here.
Next, we introduce the following well-known Div-Curl lemma whose proof may befound in [20]. 6 emma 2.4.
Let { u k } and { v k } be two bounded sequences in L (Ω; R d ) with Ω being abounded Lipschitz domain. Suppose that ( i ) u k ⇀ u and v k ⇀ v weakly in L (Ω; R d ) ; ( ii ) curl( u k ) = 0 in Ω and div( v k ) = f strongly in H − (Ω) .Then there holds ˆ Ω ( u k · v k ) ϕdx → ˆ Ω ( u · v ) ϕdx as k → ∞ , for any scalar function ϕ ∈ C (Ω) . The following interior Caccioppoli’s inequality with weights will be used in the proofof Theorem 1.1.
Lemma 2.5. (interior Caccioppoli’s inequality with weights) Assume that A satisfiesthe condition (1 . , and u ε ∈ H ( B ) is a weak solution of L ε ( u ε ) = 0 in B . Let ≤ s < s < s < s ≤ , then there holds ˆ B s \ B s |∇ u ε | e τϕ dx ≤ C (cid:18) s − s + 1 s − s (cid:19) ˆ B s \ B s | u ε | e τϕ dx + Cλ τ ˆ B s \ B s | x | | u ε | ϕ e τϕ dx, (2.10) where C depends only on d and µ and ϕ = e − λ | x | with λ and τ being positive constants.Proof. The proof is standard. Choose a cutoff function 0 ≤ ρ ( x ) ≤
1, such that ρ ( x ) = 1if x ∈ B s \ B s and ρ ( x ) = 0 if x / ∈ B s \ B s with |∇ ρ | ≤ C (cid:16) s − s + s − s (cid:17) . Then testingthe equation L ε ( u ε ) = 0 in B with u ε e τϕ ρ yields that ˆ B A ε ∇ u ε ∇ u ε e τϕ ρ − λτ ˆ B A ε ∇ u ε · xu ε ϕe τϕ ρ + 2 ˆ B A ε ∇ u ε ∇ ρu ε ρe τϕ = 0 . (2.11)Then, it follows from the Cauchy inequality that ˆ B |∇ u ε | ρ e τϕ ≤ Cλ τ ˆ B | x | | u ε | ϕ ρ e τϕ + C ˆ B | u ε | |∇ ρ | e τϕ . (2.12)Thus, we have completed this proof after noting the choice of ρ .At the end of this section, we introduce the following Carleman inequality for thehomogenized operator L = − div( b A ∇ ), whose proof may be found in [9]. Proposition 2.6. (Carleman inequality) Assume that A is symmetric and satisfies theconditions (1 . - (1 . , then there exist three positive constants C , λ and τ that candepend only on µ , such that C ˆ B (cid:0) λ τ ϕ ( v ˜ η ) + λ τ ϕ |∇ ( v ˜ η ) | (cid:1) e τϕ dx ≤ ˆ B [ L ( v ˜ η )] e τϕ dx, (2.13) with ϕ = e − λ | x | , for all v ∈ H ( B ) , ˜ η ∈ C ∞ ( B \ B / ) , λ ≥ λ and τ ≥ τ . emark 2.7. The Carleman inequality (2 . continues to hold for the operator ˜ L = − div( ˜ A ∇ ) + B · ∇ + c with symmetric ˜ A satisfying the ellipticity condition and beingLipschitz continuous, B ∈ L ∞ ( B ) d and c ∈ L ∞ (Ω) , where the constants C , λ and τ depends only on µ and the L ∞ ( B ) -norm of B and c . To proceed further, we first need to calculate the term L ε ( u ε η + εχ εj ∂ j ηu ε ) on the righthand side of (1 . Lemma 3.1.
Suppose that A is symmetric and satisfies the conditions (1 . - (1 . , andlet u ε ∈ H ( B ) be a solution to the equation L ε ( u ε ) = 0 in B , then there holds − L ε ( u ε η + εχ εj ∂ j ηu ε )=2 A ε ∇ u ε ∇ η + A ε ∇ ηu ε + A ε ∇ y χ εj ∇ ∂ j ηu ε + 2 A ε ∇ u ε ∇ y χ εj ∂ j η + div y ( A ε χ εj ) ∇ ∂ j ηu ε + εA ε χ εj ∇ ∂ j ηu ε + 2 εA ε χ εj ∇ ∂ j η ∇ u ε in L ( B ) . (3.1) Proof.
Since u ε ∈ H ( B ) satisfies L ε ( u ε ) = 0 in B , then it is easy to see that −L ε ( u ε η ) = div( A ε ∇ u ε η + A ε ∇ ηu ε )= 2 A ε ∇ u ε ∇ η + 1 ε ∂ y i a εij ∂ j ηu ε + A ε ∇ ηu ε . (3.2)In order to cancel out the term ε ∂ y i a εij ∂ j ηu ε , we need to consider the term εχ εj ∂ j ηu ε .Then, in view of the definition of the first order corrector χ j in (2 . −L ε ( εχ εj ∂ j ηu ε ) = div( A ε ∇ y χ εj ∂ j ηu ε + εA ε χ εj ∇ ∂ j ηu ε + εA ε χ εj ∂ j η ∇ u ε )= − ε ∂ y i a εij ∂ j ηu ε + A ε ∇ y χ εj ∇ ∂ j ηu ε + 2 A ε ∇ y χ εj ∂ j η ∇ u ε + div y ( A ε χ εj ) ∇ ∂ j ηu ε + εA ε χ εj ∇ ∂ j ηu ε + 2 εA ε χ εj ∇ ∂ j η ∇ u ε , (3.3)where we have used the following equalitydiv( εA ε χ εj ∂ j η ∇ u ε ) = εA ε χ εj ∇ ∂ j η ∇ u ε + A ε ∇ u ε ∇ y χ εj ∂ j η, in the above equation. Consequently, we have − L ε ( u ε η + εχ εj ∂ j ηu ε )=2 A ε ∇ u ε ∇ η + A ε ∇ ηu ε + A ε ∇ y χ εj ∇ ∂ j ηu ε + 2 A ε ∇ u ε ∇ y χ εj ∂ j η + div y ( A ε χ εj ) ∇ ∂ j ηu ε + εA ε χ εj ∇ ∂ j ηu ε + 2 εA ε χ εj ∇ ∂ j η ∇ u ε , (3.4)which completes the proof of Lemma 3.1. 8ow we are ready to give the proof of Theorem 1.1. Proof of Theorem 1.1 . We prove the result by contraction. Suppose that there existsequence { ε k } ⊂ R + , { A k } being symmetric and satisfying (1 . . { u k } ⊂ H ( B ), { u k, } ∈ H ( B / ) given by Theorem 2.2, { λ k } satisfying λ k ≥ λ and { τ k } satisfying τ ≤ τ k ≤ C ( λ , τ ) || u k || L B || u k || L B + 100 τ , such that ε k →
0, anddiv( A k ( x/ε k ) ∇ u k ) = 0 in B , (3.5) ˆ B | u k | ≤ M max ((cid:18) ˆ B | u k | (cid:19) N , (cid:18) ˆ B | u k | (cid:19) /N ) (3.6)div( c A k ∇ u k, ) = 0 in B / , (3.7)with || u k − u k, || L ( B / ) ≤ C √ ε k || u k || L ( B ) , (3.8)and C ˆ B (cid:16) λ k τ k ϕ k ( u k η ) + λ k τ k ϕ k (cid:12)(cid:12) ∇ (cid:2) ( u k − ε k χ ε k k,j ∂ j u k, ) η (cid:3)(cid:12)(cid:12) (cid:17) e τ k ϕ k dx> ˆ B (cid:2) div (cid:0) A ε k k ∇ ( u k η + ε k χ ε k k.j ∂ j ηu k ) (cid:1)(cid:3) e τ k ϕ k dx, (3.9)where χ k,j is the j -th corrector defined in (2 .
1) for the operator A k . Since c A k is symmetricand bounded in R d × d , we may assume that c A k → H (3.10)for some symmetric matrix H satisfying the ellipticity condition (1 . λ k ϕ k = λ k e − λ k | x | → λ k ϕ k = λ k e − λ k | x | → λ k → ∞ if | x | > /
2, then we could assumethat lim sup k →∞ λ k = λ ∞ < + ∞ . (3.11)By multiplying a constant to u k , we may assume that || u k || L ( B ) = 1 . (3.12)By Caccioppoli’s inequality, this implies that { u k } is bounded in H ( B r ) for any 0 < r <
3. Therefore, by passing to a subsequence still denoted by { u k } , we may further assumethat u k ⇀ u weakly in H ( B r ) , (3.13) u k → u strongly in L ( B r ) , (3.14) A k ( x/ε k ) ∇ u k ⇀ F weakly in L ( B r ) (3.15)for any 0 < r <
3, where u ∈ H ( B ) and F ∈ L ( B ). It follows from the theory ofhomogenization (see e.g. [20]) that F = H ∇ u anddiv( H ∇ u ) = 0 in B with || u || L ( B ) ≤ . (3.16)9f we write χ k,j ( y ) with j = 1 , · · · , d as the first order corrector for − div( A k ( x/ε k ) ∇ ),then it is easy to see thatdiv[ A k ( x/ε k ) ∇ ( u k − u − ε k χ k,j ( x/ε k ) ∂ j u )]= div( H ∇ u ) − div( A k ( x/ε k ) ∇ u ) − div( A k ( x/ε k ) ∇ χ k,j ( x/ε k ) ∂ j u ) − ε k div( A k ( x/ε k ) χ k,j ( x/ε k ) ∇ ∂ j u )= div(( H − c A k ) ∇ u ) − div(( A k ( x/ε k ) − c A k + A k ( x/ε k ) ∇ χ k ( x/ε k )) ∇ u ) − ε k div( A k ( x/ε k ) χ k,j ( x/ε k ) ∇ ∂ j u )= div(( H − c A k ) ∇ u ) − ε k ∂ x l (cid:8) F k,lij ∂ ij u (cid:9) − ε k div( A k ( x/ε k ) χ k,j ( x/ε k ) ∇ ∂ j u ) in B , (3.17)where F k,lij = − F k,ilj ∈ H ( Y ) ∩ L ∞ ( Y ) given by Lemma 2.1 after replacing the coeffi-cient matrix A by A k in this lemma.Consequently, it follows from the interior W ,p estimate [19], (3 .
10) and (3 .
14) that ||∇ ( u k − u − ε k χ k,j ( x/ε k ) ∂ j u ) || L p ( B r ) → , for any 1 < p < ∞ and r < . (3.18)Next, note that || u k, − u || L ( B / ) ≤ || u k, − u k || L ( B / ) + || u k − u || L ( B / ) → , as k → ∞ , (3.19)and u k, − u satisfiesdiv( c A k ∇ ( u k, − u )) = ( H − c A k ) ∇ u in B / . (3.20)Then, it follows from the interior H k estimates for harmonic functions and (3 .
19) that || u k, − u || H k ( B r ) → , ∀ k ∈ N + and r < / . (3.21)Therefore, there holds ||∇ ( u k − u − ε k χ k,j ( x/ε k ) ∂ j u k, ) || L p ( B r ) → r < / < p < ∞ . (3.22)In view of Lemma 3.1, we havediv ( A k ( x/ε k ) ∇ ( u k η + εχ k,j ( x/ε k ) ∂ j ηu k ))=2 A ε k k ∇ u k ∇ η + A ε k k ∇ ηu k + A ε k k ∇ y χ ε k k,j ∇ ∂ j ηu k + 2 A ε k k ∇ u k ∇ y χ ε k k,j ∂ j η + div y ( A ε k k χ ε k k,j ) ∇ ∂ j ηu k + ε k A ε k k χ ε k k,j ∇ ∂ j ηu k + 2 ε k A ε k k χ ε k k,j ∇ ∂ j η ∇ u k . (3.23)It is easy to see that 2 A ε k k ∇ u k ∇ η + A ε k k ∇ ηu k + A ε k k ∇ y χ ε k k,j ∇ ∂ j ηu k ⇀ H ∇ u ∇ η + H ∇ ηu weakly in L ( B ) . (3.24)Since ´ Y ∇ y χ k,j ( y ) dy = 0 and || A ε k k ∇ u k ∇ y χ ε k k,j ∂ j η || L ( B ) ≤ C with C independent of k ,then it follows from the so-called div-curve Lemma (Lemma 2.4) that A ε k k ∇ u k ∇ y χ ε k k,j ∂ j η ⇀ L ( B ) . (3.25)10eanwhile, we could easily obtain the following weak convergence,div y ( A ε k k χ ε k k,j ) ∇ ∂ j ηu k + ε k A ε k k χ ε k k,j ∇ ∂ j ηu k + 2 ε k A ε k k χ ε k k,j ∇ ∂ j η ∇ u k ⇀ L ( B ) . (3.26)Consequently, combining (3 . .
26) yields thatdiv ( A k ( x/ε k ) ∇ ( u k η + εχ k,j ( x/ε k ) ∂ j ηu k )) ⇀ H ∇ u ∇ η + H ∇ ηu = div( H ∇ ( uη )) weakly in L ( B ) . (3.27)To proceed, we first consider that, there exists some constant τ ∞ >
0, such thatlim sup k →∞ τ k = τ ∞ < + ∞ . (3.28)Then, letting k → ∞ along (3 .
11) and (3 .
28) on the both sides of (3 .
9) yields that C ˆ B (cid:0) λ ∞ τ ∞ ϕ ∞ ( uη ) + λ ∞ τ ∞ ϕ ∞ |∇ ( uη ) | (cid:1) e τ ∞ ϕ ∞ dx ≥ ˆ B [div( H ∇ ( uη ))] e τ ∞ ϕ ∞ dx ≥ C ˆ B (cid:0) λ ∞ τ ∞ ϕ ∞ ( uη ) + λ ∞ τ ∞ ϕ ∞ |∇ ( uη ) | (cid:1) e τ ∞ ϕ ∞ dx, (3.29)where we have used the Carleman inequality (Proposition 2.6) for the matrix coefficient H in the last inequality in (3 .
29) and ϕ ∞ = e − λ ∞ | x | . It follows from (3 .
29) and the uniquecontinuation for harmonic function that u ≡ B , (3.30)which contradicts to the conditions (3 . .
12) and (3 . k →∞ τ k → + ∞ . (3.31)In view of τ ≤ τ k ≤ C ( λ , τ ) || u k || L B || u k || L B + 100 τ , then it follows from (3 .
12) that || u k || L ( B ) → . Then, we could obtain u ≡ B , which implies that u ≡ B , due to the unique continuation for harmonic function. Thus leads to a contraction again.The growth condition (3 .
6) plays an important role in compactness argument. In thefollowing example, we could construct a counterexample without the growth condition(3 . xample 3.2. If we consider A k = ∆ for any k ≥ , then there exists a sequence ofharmonic functions { u k } such that ∆ u k = 0 in B , ˆ B | u k | dx = 1 and ˆ B | u k | dx → as k → ∞ . Actually, we could choose u k to be a harmonic polynomial of degree k with ´ B | u k | dx =1 , then it is easy to see that ∆ u k = 0 in R d and ´ B | u k | dx = 3 − k → as k →∞ . Consequently, this example shows that the growth condition (3 . is necessary inthe compactness argument to guarantee the Carleman-type inequality in elliptic periodichomogenization. However, we do not know that whether a Carleman-type inequality wouldhold without the growth condition (3 . . The proof of Corollary 1.4 is standard if we have obtained the Carleman-type inequal-ity (1 . Proof of Corollary 1.4 . We just need to consider the case 0 < ε ≤ ε , since thethree-ball inequality (1 .
8) continues to hold for u ε if ε ≥ ε without the growth condition(1 . .
6) and the choice of the cutoff function η , we have C ˆ B \ B λ τ ϕ | u ε | e τϕ dx ≤ ˆ B (cid:2) L ε ( u ε η + εχ εj ∂ j ηu ε ) (cid:3) e τϕ dx, (3.32)But, in view of Lemma 3.1, there holdsupp( L ε ( u ε η + εχ εj ∂ j ηu ε )) ⊂ { / ≤ | x | ≤ / } ∪ { / ≤ | x | ≤ / } (3.33)and ( L ε ( u ε η + εχ εj ∂ j ηu ε )) ≤ C ( | u ε | + |∇ u ε | ) . (3.34)Thus, we have ˆ B \ B λ τ ϕ | u ε | e τϕ dx ≤ C ˆ { B / \ B / } ∪ { B / \ B / } ( | u ε | + |∇ u ε | ) e τϕ dx ≤ C ˆ { B / \ B / } ∪ { B \ B / } (1 + λ τ ϕ ) | u ε | e τϕ dx, (3.35)where we have used the interior Caccioppoli’s inequality with weights in Lemma 2.5 inthe above inequality. Therefore, fixing λ and changing τ if necessary, (3 .
35) implies that,for τ ≤ τ k ≤ C ( λ , τ ) || u k || L B || u k || L B + 100 τ , C ˆ B | u ε | e τϕ dx ≤ ( C + 1) ˆ B | u ε | e τϕ dx + ˆ B \ B / | u ε | e τϕ dx. (3.36)In view of ϕ = exp {− λ | x | } , then it follows from that C ˆ B | u ε | dx ≤ e ατ ( C + 1) ˆ B | u ε | dx + e − βτ ˆ B | u ε | dx, (3.37)12here α = 1 − e − λ and β = 2 (cid:16) e − λ − e λ (cid:17) . (3.38)We now temporarily introduce the following notations, P = ( C + 1) ˆ B | u ε | dx, Q = C ˆ B | u ε | dx and R = ˆ B | u ε | dx. Then (3 .
37) becomes Q ≤ e ατ P + e − βτ R, for τ ≤ τ k ≤ C ( λ , τ ) || u k || L ( B ) || u k || L ( B ) + 100 τ . We could choose C ( λ , τ ) such that˜ τ = ln( R/P ) α + β ≤ C ( λ , τ ) || u k || L ( B ) || u k || L ( B ) . If ˜ τ ≥ τ , then τ = ˜ τ in yields that Q ≤ P αα + β R αα + β . (3.39)If ˜ τ < τ , R < e ( α + β ) τ P and then Q ≤ CR = CR αα + β R αα + β ≤ Ce ατ P αα + β R αα + β . (3.40)In conclusion, we find that in any case one of inequalities (3 .
39) and (3 .
40) holds. Thatis, in terms of the original notations, || u ε || L ( B ) ≤ C || u ε || sL ( B ) || u ε || − sL ( B ) , (3.41)with s = αα + β . Note that the constant C , in (3 . λ , then the constant C , in (3 . λ . However, if λ → ∞ , the inequality (1 . C = M / , follows directly from the growth condition (1 . C , in (1 . λ .We are now ready to give the proof of Theorem 1.5 with the help of Corollary 1.4 andthe uniform doubling conditions proved in [18]. Proof of Theorem 1.5 . It follows from [18, Thm 1.2] that there holds the followinguniform doubling condition for u ε , B r | u ε | dx ≤ ˜ C ( M ) B r/ | u ε | dx (3.42)for any 0 < r ≤
2, with ˜ C ( M ) depends only on d , ˜ M , µ and M .13t follows from Theorem 1.1 and Corollary 1.4 that there exists ε ( M ), depending onlyon d , ˜ M , µ and M , such that if v ∈ H ( B ) solves L ε ( v ) = 0 in B and v satisfies thefollowing doubling condition B | v | dx ≤ ˜ C ( M ) B | v | dx, then there holds the following three-sphere inequality, || v || L ( B ) ≤ C || v || sL ( B ) || v || − sL ( B ) , (3.43)with s = αα + β for α and β defined in (3 . . ε/ε ( M )) ≤ r ≤ /
2. Let v ε ( x ) = u ε ( rx ) , (3.44)then it is to check that v ε satisfiesdiv (cid:18) A (cid:18) xε/r (cid:19) ∇ v ε (cid:19) = 0 in B , with B | v ε | dx ≤ ˜ C ( N ) B | v ε | dx, and ( ε/r ) ≤ ε ( M ), then it follows from (3 .
43) and v ε ( x ) = u ε ( rx ) that || u ε || L ( B r ) ≤ C || u ε || sL ( B r ) || u ε || − sL ( B r ) . (3.45)Suppose now that 0 < r ≤ ( ε/ε ( M )), then εr ≥ ε ( M ) − . Therefore, it follows from theclassical theory for elliptic equation with Liphschitz coefficient matrix that (3 .
45) alsoholds true, with the same exponent s . Thus, we have completed this proof of Theorem1.5. Acknowledgements
The author thanks Prof. Zhongwei Shen and Yao Xu for helpful discussions.
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