Homogenization for non-self-adjoint locally periodic elliptic operators
HHOMOGENIZATION FOR NON-SELF-ADJOINTLOCALLY PERIODIC ELLIPTIC OPERATORS
NIKITA N. SENIK
Abstract.
We study the homogenization problem for matrix strongly ellipticoperators on L ( R d ) n of the form A ε = − div A ( x, x/ε ) ∇ . The function A isLipschitz in the first variable and periodic in the second. We do not requirethat A ∗ = A , so A ε need not be self-adjoint. In this paper, we provide, forsmall ε , two terms in the uniform approximation for ( A ε − µ ) − and a firstterm in the uniform approximation for ∇ ( A ε − µ ) − . Primary attention ispaid to proving sharp-order bounds on the errors of the approximations. Introduction
Homogenization dates back to the late 1960s, and for more than fifty years ithas become a well-established theory. In the simplest case, homogenization dealswith asymptotic properties of solutions to differential equations with oscillatingcoefficients. Given a periodic (with period in each variable) uniformly boundedand uniformly positive definite function A : R d → C d × d , consider the differentialequation(1.1) − div A ( ε − x ) ∇ u ε − µu ε = f, where ε > , µ ∈ C \ R + and f ∈ L ( R d ) . The coefficients of the equation are ε -periodic and hence rapidly oscillate if ε is small. In homogenization theory oneis interested in studying the asymptotic behavior of u ε as ε becomes smaller. It isa basic fact that, after passing to a subsequence if necessary, u ε converges to thesolution u of the differential equation(1.2) − div A ∇ u − µu = f with constant A . Since, in applications, the elliptic operator on the left side of (1.1)usually describes a physical process in a highly heterogeneous medium, this meansthat, in certain aspects, the process evolves very similar to that in a homogeneousmedium.It is a basic fact about homogenization theory that u ε converges to u in L ( R d ) ;we refer the reader to [BLP78], [BP84] or [ZhKO93] for the details. Stated differ-ently, the resolvent of − div A ( ε − x ) ∇ converges in the strong operator topologyto the resolvent of − div A ∇ . In [BSu01] (see also [BSu03]), Birman and Suslinaproved that, in fact, the resolvent converges in norm. Moreover, they found asharp-order bound on the rate of convergence. Since that time there have been a Mathematics Subject Classification.
Primary 35B27; Secondary 35J15, 35J47.
Key words and phrases. homogenization, operator error estimates, locally periodic operators,effective operator, corrector.The author was partially funded by Young Russian Mathematics award, Rokhlin grant andRFBR grant 16-01-00087. a r X i v : . [ m a t h . A P ] M a y NIKITA N. SENIK number of interesting further results in this direction – see [Gri04], [Gri06], [Zh05],[ZhP05], [B08], [KLS12], [Su13 ], [Su13 ], [ChC16] and [ZhP16], to name a few.Here we focus on a more general problem than the periodic one in (1.1). Let A = { A kl } with A kl : R d × R d → C n × n being uniformly bounded functions that areLipschitz in the first variable and periodic in the second (see Section 3 for a precisedefinition). Consider the operator A ε on the complex space L ( R d ) n given by A ε = − div A ( x, ε − x ) ∇ = − d (cid:88) k,l =1 ∂ k A kl ( x, ε − x ) ∂ l . The coefficients now depend not only on the “fast” variable, ε − x , but also on the“slow” one, x . Assume that, for all ε in some neighborhood of , the operator A ε iscoercive and furthermore the constants in the coercivity bound are independent of ε .Then A ε is strongly elliptic for such ε and there is a sector containing the spectrumof A ε . In this paper, we will obtain approximations for ( A ε − µ ) − and ∇ ( A ε − µ ) − (with µ outside the sector) in the operator norm and prove that (cid:107) ( A ε − µ ) − − ( A − µ ) − (cid:107) L → L ≤ Cε, (1.3) (cid:107) ( A ε − µ ) − − ( A − µ ) − − ε C εµ (cid:107) L → L ≤ Cε (1.4)and(1.5) (cid:107)∇ ( A ε − µ ) − − ∇ ( A − µ ) − − ε ∇K εµ (cid:107) L → L ≤ Cε, the estimates being sharp with respect to the order (see Theorems 6.1 and 6.2).The effective operator A is of the same form as A ε , but its coefficients depend onlyon the slow variable. In contrast, the correctors K εµ and C εµ involve rapidly oscillatingfunctions as well. The first of these plays the role of the traditional corrector anddiffers from the latter in that it involves a smoothing operator. The idea of usinga smoothing to regularize the traditional corrector is due to Griso, see [Gri02].The other corrector has no analogue in classical theory and was first presentedin [BSu05] for purely periodic operators. Assume for simplicity that A ∗ = A . Then C εµ has the form C εµ = ( K εµ − L µ ) − M εµ + ( K εµ − L µ ) ∗ (see Section 5). What is interesting here is that an analog of C εµ for periodic opera-tors, while looking similar to this one, does not include the term M εµ , see [Se17 ].In fact, one cannot remove M εµ from C εµ if the estimate (1.4) is to remain true, seeRemark 5.9 for examples. So this term is a special feature of non-periodic problems.The results of the present paper extend the author’s work [Se17 ] on periodicelliptic problems, where we studied non-self-adjoint scalar operators whose coeffi-cients were periodic in some variables and Lipschitz in the others. Put differently,the fast and slow variables were separated in the sense that A ε ( x ) = A ( x , ε − x ) ,where x = ( x , x ) . We proved analogs of the estimates (1.3)–(1.5), yet the correc-tors were slightly different, see Remark 6.3 below. It should be pointed out that theoperators in [Se17 ] were allowed to involve lower-order terms with quite generalcoefficients.Previous results on uniform approximations for locally periodic elliptic opera-tors are due to, on the one hand, Borisov and, on the other hand, Pastukhovaand Tikhomirov. In [B08] Borisov established the estimates (1.3) and (1.5) forcertain matrix self-adjoint operators with smooth coefficients. In the paper [PT07]of Pastukhova and Tikhomirov, similar results were proved for scalar self-adjoint OMOGENIZATION FOR LOCALLY PERIODIC ELLIPTIC OPERATORS 3 operators with rough coefficients (although their techniques also apply to non-self-adjoint problems). As far as I know, the estimate (1.4) in the locally periodicsettings was not obtained even for the simplest cases.To prove the estimates, we develop the ideas of [Se17 ]. In the first step weestablish a variant of the resolvent identity that involves the resolvents of the orig-inal and the effective operators and a corrector (see Section 7). This combinationcomes as no surprise, for it is well known that the effective operator and a correctorform a first approximation to the original operator (see, e.g., [BLP78] or [ZhKO93]).When this is done, all the desired estimates will follow at once. However, we cannotuse the same technique as in [Se17 ], so the identity is proved by different means.The point is that the technique depends heavily on the smoothing operator thathas been chosen. In the case of periodic operators, the smoothing was based on theGelfand transform; but it is not as convenient now. To my knowledge, no naturalsmoothing for operators with locally periodic coefficients is known, so we choose theSteklov smoothing operator, which is the most simple and has proved to be quiteuseful; see [Zh05] and [ZhP05], where that smoothing first appeared in the contextof homogenization, as well as [PT07], [Su13 ] and [Su13 ]. We remark that a verysimilar smoothing had been used earlier in [Gri02] and [Gri04] (see also [Gri06]).Our technique is strongly influenced by all these works.I believe that the same method can be of use for locally periodic problems ondomains with Dirichlet or Neumann boundary conditions as well.It is also worth noting that, once the estimates (1.3)–(1.5) are verified, a lim-iting argument will give similar results for operators whose coefficients are Höldercontinuous in the first variable, see Remark 6.6. These results, together with theresults stated here, have been announced in [Se17 ].The plan of the paper is as follows. Section 2 contains basic definitions andnotation. In Section 3 we introduce the original operator. We study the effectiveoperator in Section 4 and correctors in Section 5. Section 6 states the main results.Section 7 is the core of the paper, where we first prove the identity and thencomplete the proofs. 2. Notation
The symbol (cid:107) · (cid:107) U will stand for the norm on a normed space U . If U and V are Banach spaces, then B ( U, V ) is the Banach space of bounded linear operatorsfrom U to V . When U = V , the space B ( U ) = B ( U, U ) becomes a Banach algebrawith the identity map I . The norm and the inner product on C n are denoted by | · | and (cid:104) · , · (cid:105) , respectively. We shall often identify B ( C n , C m ) and C m × n .Let Σ be a domain in R d and U a Banach space. The space C , ( ¯Σ; U ) consistsof those uniformly continuous functions u : Σ → U for which (cid:107) u (cid:107) C , (¯Σ; U ) = (cid:107) u (cid:107) C (¯Σ; U ) + [ u ] C , (¯Σ; U ) < ∞ , where (cid:107) u (cid:107) C (¯Σ; U ) = sup x ∈ Σ (cid:107) u ( x ) (cid:107) U and [ u ] C , (¯Σ; U ) = sup x ,x ∈ Σ ,x (cid:54) = x (cid:107) u ( x ) − u ( x ) (cid:107) U | x − x | . We will use the notation (cid:107) · (cid:107) C , , (cid:107) · (cid:107) C and [ · ] C , as shorthand for (cid:107) · (cid:107) C , (¯Σ; U ) , (cid:107) · (cid:107) C (¯Σ; U ) and [ · ] C , (¯Σ; U ) when the context makes clear which Σ and U are meant. NIKITA N. SENIK
The symbol L p (Σ; U ) stands for the L p -space of strongly measurable functionson Σ with values in U . In case U = C n , we write (cid:107) · (cid:107) p, Σ for the norm on L p (Σ) n and ( · , · ) Σ for the inner product on L (Σ) n . We let W mp (Σ) n denote the usualSobolev space of C n -valued functions on Σ and ( W mp (Σ) n ) ∗ , its dual space underthe pairing ( · , · ) Σ . If C ∞ c (Σ) n is dense in W mp (Σ) n , then W − mp + (Σ) n = ( W mp (Σ) n ) ∗ ,where p + is the exponent conjugate to p .Let Q be the closed cube in R d with center and side length , sides being parallelto the axes. Then ˜ W mp ( Q ) n denotes the completion of ˜ C m ( Q ) n in the W mp -norm.Here ˜ C m ( Q ) is the class of m -times continuously differentiable functions on Q whoseperiodic extension to R d enjoys the same smoothness. Notice that ˜ L p ( Q ) n coincideswith the space of all periodic functions in L p, loc ( R d ) n . The spaces ˜ W mp ( R d × Q ) n and ˜ C m ( R d × Q ) n are defined in a similar fashion. If p = 2 , we write H m for W mp , H − m for W − mp , etc. The symbol ˜ H m ( Q ) n will stand for the subspace of functionsin ˜ H m ( Q ) n with mean value zero. Any u ∈ ˜ H ( Q ) n satisfies the Poincaré inequality(2.1) (cid:107) u (cid:107) ,Q ≤ (2 π ) − (cid:107) Du (cid:107) ,Q , as can be seen by using Fourier series. Here and below, D = − i ∇ .We will often use the notation α (cid:46) β to mean that that there is a constant C ,depending only on some fixed parameters (these are listed in Theorems 6.1 and 6.2),such that α ≤ Cβ . 3. Original operator
Let each A kl be a function in C , ( ¯ R d ; ˜ L ∞ ( Q )) n × n . Then A = { A kl } may bethought of as a bounded mapping A : R d × R d → B ( C d × n ) that is Lipschitz inthe first variable and periodic in the second. As is well known, for any func-tion u : R d × R d → L ( Q ) satisfying the Carathéodory condition (i.e., the require-ment of continuity with respect to the first variable and measurability with respectto the second) the map τ ε u : R d → L ( Q ) defined for x ∈ R d and z ∈ Q by(3.1) τ ε u ( x, z ) = u ( x, ε − x, z ) , is measurable (here ε > ). Notice that, if v is another function from R d × R d to L ( Q ) , then τ ε ( uv ) = ( τ ε u )( τ ε v ) . We adopt the notation u ε = τ ε u .Consider the matrix operator A ε : H ( R d ) n → H − ( R d ) n given by(3.2) A ε = D ∗ A ε D. It is easy to see that A ε is bounded, with bound C (cid:91) = (cid:107) A (cid:107) C :(3.3) (cid:107)A ε u (cid:107) − , , R d ≤ C (cid:91) (cid:107) Du (cid:107) , R d for all u ∈ H ( R d ) n . Now we impose a condition that will render A ε elliptic.Namely, we assume that A ε is coercive uniformly in ε ∈ (cid:69) , where (cid:69) = (0 , ε ] with ε ∈ (0 , , that is, there are c A > and C A ≥ such that(3.4) Re( A ε Du, Du ) R d + C A (cid:107) u (cid:107) , R d ≥ c A (cid:107) Du (cid:107) , R d for every u ∈ H ( R d ) n . It follows that A ε is m -sectorial with sector (cid:83) = (cid:8) z ∈ C : | Im z | ≤ c − A C (cid:91) (Re z + C A ) (cid:9) OMOGENIZATION FOR LOCALLY PERIODIC ELLIPTIC OPERATORS 5 independent of ε . Whenever µ / ∈ (cid:83) , the operator A εµ = A ε − µ is an isomorphismand hence is invertible; moreover, for any f ∈ H − ( R d ) n we have(3.5) (cid:107) ( A εµ ) − f (cid:107) , , R d (cid:46) (cid:107) f (cid:107) − , , R d . Before proceeding, we make a few remarks about the coercivity condition. Itfollows from (3.4) (via Lemma 4.1) that A satisfies the Legendre–Hadamard condi-tion(3.6) Re (cid:104) A ( · ) ξ ⊗ η, ξ ⊗ η (cid:105) ≥ c A | ξ | | η | , ξ ∈ R d , η ∈ C n , so A ε is strongly elliptic for all ε > . The Legendre–Hadamard condition doesnot generally imply (3.4). If we restrict our attention to the real-valued case, thenfor scalar operators the two statements are equivalent. But this is no longer truefor matrix operators, let alone the complex-valued case. A necessary and sufficientalgebraic condition on A that would guarantee (3.4) is not known.It is worthwhile to point out that we have to be able to verify the coercivitybound for all ε in some interval (0 , ε ] , which may be rather difficult. A sufficientcondition not involving ε is that the operator D ∗ A ( x, · ) D is strongly coercive on H ( R d ) n and furthermore there is c > so that for any x ∈ R d and u ∈ H ( R d ) n (3.7) Re( A ( x, · ) Du, Du ) R d ≥ c (cid:107) Du (cid:107) , R d . This can be seen by noticing that, by change of variable, the above inequalityremains true with A ( x, ε − y ) in place of A ( x, y ) . Then a partition of unity argumentwill do the job, since A is uniformly continuous in the first variable.As an example of A satisfying (3.7), let b ( D ) be a matrix first-order differentialoperator with symbol ξ (cid:55)→ b ( ξ ) = d (cid:88) k =1 b k ξ k , where b k ∈ C m × n . Suppose that the symbol has the property that, for some α > , b ( ξ ) ∗ b ( ξ ) ≥ α | ξ | , ξ ∈ R d . Let g be a function in C , ( ¯ R d ; ˜ L ∞ ( Q )) m × m with Re g uniformly positive definite.Now if we take A kl = b ∗ k gb l , then application of the Fourier transform will yield Re( A ( x, · ) Du, Du ) R d = Re( g ( x, · ) b ( D ) u, b ( D ) u ) R d ≥ α (cid:107) (Re g ) − / (cid:107) − C (cid:107) Du (cid:107) , R d . Homogenization for self-adjoint operators of this type was studied by Birman andSuslina in the purely periodic setting (see, e.g., [BSu01], [BSu03], [BSu05], [BSu06],[Su13 ] and [Su13 ]) and by Borisov in the locally periodic setting (see [B08]).Observe that the more restrictive Legendre condition, which amounts to theuniform positive definiteness of Re A , does ensure coercivity, but excludes somestrongly elliptic operators with important applications – such as certain elasticityoperators. 4. Effective operator
Given ξ ∈ C d × n and x ∈ R d , we let N ξ ( x, · ) be the weak solution of(4.1) D ∗ A ( x, · )( DN ξ ( x, · ) + ξ ) = 0 NIKITA N. SENIK in ˜ H ( Q ) n . The function N ξ is well defined, since D ∗ A ( x, · ) ξ is a continuous linearfunctional on ˜ H ( Q ) n and the operator D ∗ A ( x, · ) D is strongly coercive on ˜ H ( Q ) n ,as we shall now see. Lemma 4.1.
For any x ∈ R d and all u ∈ ˜ H ( Q ) n , we have (4.2) Re( A ( x, · ) Du, Du ) Q ≥ c A (cid:107) Du (cid:107) ,Q . Proof.
Fix u ε = εu ε ϕ with u ∈ ˜ C ( Q ) n and ϕ ∈ C ∞ c ( R d ) . We substitute u ε into(3.4) and let ε tend to 0. Then, because u ε and Du ε − ( Du ) ε ϕ converge in L to , lim ε → Re (cid:90) R d (cid:104) A ε ( x )( Du ) ε ( x ) , ( Du ) ε ( x ) (cid:105)| ϕ ( x ) | dx ≥ lim ε → c A (cid:90) R d | ( Du ) ε ( x ) | | ϕ ( x ) | dx. It is well known that if f ∈ C c ( R d ; ˜ L ∞ ( Q )) , then lim ε → (cid:90) R d f ε ( x ) dx = (cid:90) R d (cid:90) Q f ( x, y ) dx dy (see, for instance, [A92, Lemmas 5.5 and 5.6]). As a result, Re (cid:90) R d (cid:90) Q (cid:104) A ( x, y ) Du ( y ) , Du ( y ) (cid:105)| ϕ ( x ) | dx dy ≥ c A (cid:90) R d (cid:90) Q | Du ( y ) | | ϕ ( x ) | dx dy. Since ϕ is an arbitrary function in C ∞ c ( R d ) and since A is continuous in the firstvariable, we conclude that, for any x ∈ R d , Re (cid:90) Q (cid:104) A ( x, y ) Du ( y ) , Du ( y ) (cid:105) dy ≥ c A (cid:90) Q | Du ( y ) | dy. (cid:3) It is clear from Lemma 4.1 and Poincaré’s inequality (2.1) that
Re( A ( x, · ) Du, Du ) Q (cid:38) (cid:107) u (cid:107) , ,Q for every u ∈ ˜ H ( Q ) n . Thus, the definition of N ξ makes good sense.Denote by N the map sending ξ to N ξ . Evidently, N ξ depends linearly on ξ ,so N is simply an operator of multiplication by a function (still denoted by N ).The next lemma shows that N has the same regularity in the first variable as A . Remark . In what follows, we denote differentiation in the first variable by D and differentiation in the second variable by D . When no confusion can arise, weomit the subscript and write D , as we did before. Lemma 4.3.
We have N ∈ C , ( ¯ R d ; ˜ H ( Q )) .Proof. The identity (4.1), together with Lemma 4.1, yields c A (cid:107) DN ξ ( x, · ) (cid:107) ,Q ≤ (cid:107) A ( x, · ) (cid:107) ∞ ,Q | ξ | , whence(4.3) (cid:107) D N (cid:107) L ∞ ( R d ; L ( Q )) ≤ c − A (cid:107) A (cid:107) C . Next, by (4.1) again, for any x , x ∈ R d and v ∈ ˜ H ( Q ) n (cid:0) A ( x , · )( DN ξ ( x , · ) − DN ξ ( x , · )) , Dv (cid:1) Q = − (cid:0) ( A ( x , · ) − A ( x , · ))( ξ + DN ξ ( x , · )) , Dv (cid:1) Q . Taking v = N ξ ( x , · ) − N ξ ( x , · ) and using Lemma 4.1, we obtain c A (cid:107) DN ξ ( x , · ) − DN ξ ( x , · ) (cid:107) ,Q ≤ (cid:107) A ( x , · ) − A ( x , · ) (cid:107) ∞ ,Q (cid:107) ξ + DN ξ ( x , · ) (cid:107) ,Q . OMOGENIZATION FOR LOCALLY PERIODIC ELLIPTIC OPERATORS 7
It now follows from (4.3) that [ D N ] C , (¯ R d ; L ( Q )) ≤ c − A (1 + c − A (cid:107) A (cid:107) C )[ A ] C , . We have proved that D N ∈ C , ( ¯ R d ; L ( Q )) . But then Poincaré’s inequality (2.1)implies that N ∈ C , ( ¯ R d ; L ( Q )) as well. (cid:3) Let A : R d → B ( C d × n ) be given by(4.4) A ( x ) = (cid:90) Q A ( x, y )( I + D N ( x, y )) dy. Since A and D N are continuous in the first variable, so is A . In fact, we have A ∈ C , ( ¯ R d ) . Indeed, the estimate (cid:107) A (cid:107) C (¯ R d ) ≤ (cid:107) A (cid:107) C (cid:107) I + D N (cid:107) C is immediate from the definition of A , and that [ A ] C , (¯ R d ) ≤ (cid:107) A (cid:107) C [ D N ] C , + [ A ] C , (cid:107) I + D N (cid:107) C follows by an easy calculation. Hence, (cid:107) A (cid:107) C , (¯ R d ) is finite.Now we define the effective operator A : H ( R d ) n → H − ( R d ) n by setting(4.5) A = D ∗ A D. Observe that A is bounded and coercive (recall Gårding’s inequality) and thus m -sectorial. It can be proved that A satisfies an estimate similar to (3.4) withexactly the same constants, however the bound on its norm may be differentfrom (3.3). Nevertheless, the sector for A remains the same as for A ε . We brieflysketch the argument; see [Se17 , Section 2.3] for a related proof. First consider thetwo-scale effective system as in [A92] and check that the associated form, which isdefined on H ( R d ) n ⊕ L ( R d ; ˜ H ( Q )) n by u ⊕ U (cid:55)→ ( A ( D u + D U ) , D u + D U ) R d × Q , is m -sectorial with sector (cid:83) . We only remark that the coercivity is obtained bysubstituting u + εU ε (with sufficiently smooth u and U ) into (3.4) for u and letting ε tend to ; cf. the proof of Lemma 4.1. Then notice that ( A u, u ) R d = ( A ( D u + D U ) , D u + D U ) R d × Q provided U = N D u (which is definitely in L ( R d ; ˜ H ( Q )) n ). The claim is proved.Thus, we see that the operator A µ = A − µ is an isomorphism as long as µ is outside (cid:83) . In addition, standard regularity theory for strongly elliptic systems(see, e.g., [McL00, Theorem 4.16]) implies that the pre-image of L ( R d ) n under A µ is all of H ( R d ) n and for any f ∈ L ( R d ) n (4.6) (cid:107) ( A µ ) − f (cid:107) , , R d (cid:46) (cid:107) f (cid:107) , R d . Let us return to our discussion of coercivity at the end of the previous section.As we have seen, (4.2) follows from (3.4), which in turn is a consequence of (3.7).On the other hand, (4.2) does not generally imply (3.7), and there are examples(for n > , of course) where (4.2) holds, but (3.7) is false, see [BF15]. In suchcases, a subsequence of ( A εµ ) − may still converge in the weak operator topologyto ( A µ ) − , but A will fail to be strongly elliptic, i.e., A will not satisfy theLegendre–Hadamard condition. NIKITA N. SENIK Correctors
Let the operator K µ : L ( R d ) n → ˜ H ( R d × Q ) n be given by(5.1) K µ = N D ( A µ ) − . Lemma 4.3, combined with the estimate (4.6), readily implies that K µ is continuous:(5.2) (cid:107)K µ f (cid:107) , , R d × Q (cid:46) (cid:107) f (cid:107) , R d . The very same argument shows that D D K µ is bounded on L ( R d ) n as well:(5.3) (cid:107) D D K µ f (cid:107) , R d × Q (cid:46) (cid:107) f (cid:107) , R d . Since we do not impose any extra assumptions on the coefficients, the traditionalcorrector τ ε K µ will not even map L ( R d ) n into itself. So we must first appropriatelyregularize the traditional corrector, and a smoothing operator is used for exactlythis purpose.5.1. Smoothing.
Let T ε : L ( R d × Q ) → L ( R d × Q ; L ( Q )) be the translationoperator(5.4) T ε u ( x, y, z ) = u ( x + εz, y ) , where ( x, y ) ∈ R d × Q and z ∈ Q . Certainly, for any u, v ∈ L ( R d × Q ) satisfying uv ∈ L ( R d × Q ) we have T ε uv = ( T ε u )( T ε v ) . Next, the adjoint of T ε is given by ( T ε ) ∗ u ( x, y ) = (cid:90) Q u ( x − εz, y, z ) dz. Note that ( T ε ) ∗ is defined on L ( R d × Q ) and L ( R d ) as well, by way of identifyingthese spaces with the corresponding subspaces of L ( R d × Q ; L ( Q )) . We define theSteklov smoothing operator S ε : L ( R d × Q ) → L ( R d × Q ) to be the restriction of ( T ε ) ∗ to L ( R d × Q ) . In other words,(5.5) S ε u ( x, y ) = (cid:90) Q T ε u ( x, y, z ) dz. The operator S ε is plainly self-adjoint.Here we collect some facts about T ε and S ε . Lemma 5.1.
The restriction of τ ε T ε to ˜ L ( R d × Q ) is an isometry.Proof. By change of variable, (cid:107) τ ε T ε u (cid:107) , R d × Q = (cid:90) R d (cid:90) Q | u ( x, ε − x − z ) | dx dz. But since u is periodic in the second variable, this equals (cid:107) u (cid:107) , R d × Q . (cid:3) A related result for S ε is the following. Lemma 5.2.
The restriction of τ ε S ε to ˜ L ( R d × Q ) is bounded, with bound atmost .Proof. This is immediate from Cauchy’s inequality and Lemma 5.1. (cid:3)
It is easy to see that both T ε and S ε converge in the strong operator topology tothe identity operator, yet they do not converge in norm. The uniform convergencewill, however, take place if we restrict them to certain Sobolev spaces. OMOGENIZATION FOR LOCALLY PERIODIC ELLIPTIC OPERATORS 9
Lemma 5.3.
For any u ∈ C ∞ c ( R d × Q ) we have (5.6) (cid:107) ( T ε − I ) u (cid:107) , R d × Q × Q (cid:46) ε (cid:107) D u (cid:107) , R d × Q . Proof.
Notice that u ( x + εz, y ) − u ( x, y ) = εi (cid:90) (cid:104) D u ( x + εtz, y ) , z (cid:105) dt. Hence, (cid:107) ( T ε − I ) u ( · , y, z ) (cid:107) , R d ≤ εr Q (cid:107) D u ( · , y ) (cid:107) , R d , where r Q = 1 / Q . Integrating out the y and z variables then yields (5.6). (cid:3) Lemma 5.4.
For any u ∈ C ∞ c ( R d × Q ) we have (cid:107) ( S ε − I ) u (cid:107) , R d × Q (cid:46) ε (cid:107) D u (cid:107) , R d × Q , (5.7) (cid:107) ( S ε − I ) u (cid:107) , R d × Q (cid:46) ε (cid:107) D D u (cid:107) , R d × Q . (5.8) Proof.
The inequality (5.7) comes from (5.6). To prove (5.8), notice that u ( x + εz, y ) − u ( x, y ) = εi (cid:104) D u ( x, y ) , z (cid:105) − ε (cid:90) (1 − t ) (cid:104) D D u ( x + εtz, y ) z, z (cid:105) dt. The first term on the right-hand side has mean value zero for a.e. x and y (because Q is centered at the origin), so (cid:107) ( S ε − I ) u ( · , y ) (cid:107) , R d ≤ ε r Q (cid:107) D D u ( · , y ) (cid:107) , R d . Integrating over Q completes the proof. (cid:3) Now we can prove the following result.
Lemma 5.5.
For any u ∈ ˜ C ∞ c ( R d × Q ) we have (cid:107) τ ε T ε u − τ ε S ε u (cid:107) , R d × Q (cid:46) ε (cid:107) D u (cid:107) , R d × Q . Proof.
We write τ ε T ε u − τ ε S ε u = τ ε T ε ( I − S ε ) u + τ ε S ε ( T ε − I ) u (here S ε T ε is understood to be defined as S ε T ε = T ε S ε , that is, we apply S ε to T ε u regarding the new variable resulting from the operator T ε as a parameter).Then, it follows from Lemmas 5.1 and 5.4 that (cid:107) τ ε T ε ( I − S ε ) u (cid:107) , R d × Q (cid:46) ε (cid:107) D u (cid:107) , R d × Q , while Lemmas 5.2 and 5.3 imply that (cid:107) τ ε S ε ( T ε − I ) u (cid:107) , R d × Q (cid:46) ε (cid:107) D u (cid:107) , R d × Q . These observations combine to give the desired estimate. (cid:3)
Remark . We note that the results of Lemmas 5.1–5.5 persist if we replace the L -norms by the L p -norms with p ∈ [1 , ∞ ] . This will play a role in what follows. Correctors.
We define the first corrector K εµ : L ( R d ) n → H ( R d ) n by(5.9) K εµ = τ ε S ε K µ . More explicitly, K εµ f ( x ) = (cid:90) Q N ( x + εz, ε − x ) D ( A µ ) − f ( x + εz ) dz. Because of the smoothing S ε , this corrector is bounded with (cid:107)K εµ f (cid:107) , R d (cid:46) (cid:107) f (cid:107) , R d , (5.10) (cid:107) D K εµ f (cid:107) , R d (cid:46) ε − (cid:107) f (cid:107) , R d . (5.11)Indeed, using Lemma 5.2, we see that (cid:107)K εµ f (cid:107) , R d ≤ (cid:107)K µ f (cid:107) , R d × Q , (cid:107) D K εµ f (cid:107) , R d ≤ (cid:107) D K µ f (cid:107) , R d × Q + ε − (cid:107) D K µ f (cid:107) , R d × Q . The estimates (5.10) and (5.11) then follow from (5.2).While the L -norm of K εµ f is merely uniformly bounded, the L -norm of S ε K εµ f turns out to be of order ε . Lemma 5.7.
For any ε ∈ (cid:69) and f ∈ L ( R d ) n we have (cid:107)S ε K εµ f (cid:107) , R d (cid:46) ε (cid:107) f (cid:107) , R d . Proof.
By definition of S ε and K εµ , S ε K εµ f ( x ) = (cid:90) Q (cid:90) Q T ε K µ f ( x + εw, ε − x + z, z ) dw dz. Since K µ f ( x, · ) is periodic and has mean value zero, we have (cid:90) Q (cid:90) Q K µ f ( x + εw, ε − x + z ) dw dz = 0 , and hence S ε K εµ f ( x ) = (cid:90) Q (cid:90) Q ( T ε − I ) K µ f ( x + εw, ε − x + z, z ) dw dz. Changing variables and keeping in mind that K µ f is periodic in the second variable,we find that (cid:107)S ε K εµ f (cid:107) , R d ≤ (cid:107) ( T ε − I ) K µ f (cid:107) , R d × Q × Q . The result is therefore immediate from Lemma 5.3 and the estimate (5.2). (cid:3)
To describe the second corrector, we need some additional notation. Let ( A εµ ) + be the adjoint of A εµ . Then we construct the effective operator ( A µ ) + , the cor-rector ( K εµ ) + and the other objects (which will be marked with “ + ” as well) for ( A εµ ) + just as we did for A εµ . (It may be noted in passing that ( A µ ) + is the adjointof A µ .) Of course, all results for A εµ will transfer to ( A εµ ) + . We shall not explicitlyformulate these results here, but refer to them by the numbers of the correspondingstatements for A εµ with “ + ” following the reference (for example, Lemma 5.7 + andthe estimate (5.10) + ).Define L µ : L ( R d ) n → L ( R d ) n by(5.12) L µ = ( D K + µ ) ∗ A (cid:0) D ( A µ ) − + D K µ (cid:1) OMOGENIZATION FOR LOCALLY PERIODIC ELLIPTIC OPERATORS 11 and M εµ : L ( R d ) n → L ( R d ) n by(5.13) M εµ = ε − (cid:0) τ ε T ε (cid:0) D (( A µ ) + ) − + D K + µ (cid:1)(cid:1) ∗ τ ε [ A, T ε ] (cid:0) D ( A µ ) − + D K µ (cid:1) . A more convenient way of dealing with these operators is to look at their forms. Ifwe set u = ( A µ ) − f , U = K µ f and u +0 = (( A µ ) + ) − g , U + = K + µ g , then ( L µ f, g ) R d = (cid:0) A ( D u + D U ) , D U + (cid:1) R d × Q and ( M εµ f, g ) R d = ε − (cid:0) τ ε [ A, T ε ]( D u + D U ) , τ ε T ε ( D u +0 + D U + ) (cid:1) R d × Q . Both L µ and M εµ are bounded. Indeed, | ( L µ f, g ) R d | ≤ (cid:107) A ( D u + D U ) (cid:107) , R d × Q (cid:107) D U + (cid:107) , R d × Q , and so, according to the estimates (4.6), (5.2) and (5.2) + ,(5.14) (cid:107)L µ f (cid:107) , R d (cid:46) (cid:107) f (cid:107) , R d . Likewise, observing that τ ε [ A, T ε ] = τ ε ( I − T ε ) A · τ ε T ε (by the multiplicativity of τ ε and T ε ), we conclude that | ( M εµ f, g ) R d | ≤ r Q [ A ] C , (cid:107) τ ε T ε ( D u + D U ) (cid:107) , R d × Q (cid:107) τ ε T ε ( D u +0 + D U + ) (cid:107) , R d × Q . This, together with Lemma 5.1 and the estimates (4.6), (5.2) and (4.6) + , (5.2) + ,yields that(5.15) (cid:107)M εµ f (cid:107) , R d (cid:46) (cid:107) f (cid:107) , R d . Now we introduce the second corrector C εµ : L ( R d ) n → L ( R d ) n by(5.16) C εµ = ( K εµ − L µ ) − M εµ + (( K εµ ) + − L + µ ) ∗ . Then (5.10), (5.14), (5.15) and (5.10) + , (5.14) + imply that C εµ is continuous:(5.17) (cid:107)C εµ f (cid:107) , R d (cid:46) (cid:107) f (cid:107) , R d . Remark . From (5.15) we know that the operator norm of M εµ is boundeduniformly in ε . In some situations, we can go further and prove that(5.18) (cid:107)M εµ f (cid:107) , R d (cid:46) ε (cid:107) f (cid:107) , R d . The term M εµ can then be removed from C εµ , because, in the context where thecorrector C εµ is needed (see Theorem 6.2 below), this term will be absorbed to theerror.The estimate (5.18) is true, for instance, if A ∈ C , ( ¯ R d ; ˜ L ∞ ( Q )) . To see this, wenotice that if u, v ∈ ˜ H ( R d × Q ) d × n , then uv ∈ ˜ W ( R d × Q ) and, by an L -variantof Lemma 5.5 (see Remark 5.6), (cid:12)(cid:12)(cid:0) τ ε ( I − T ε ) A · τ ε T ε u, τ ε T ε v (cid:1) R d × Q − (cid:0) τ ε ( I − S ε ) A · τ ε T ε u, τ ε T ε v (cid:1) R d × Q (cid:12)(cid:12) ≤ εr Q [ A ] C , (cid:107) τ ε T ε u ¯ v − τ ε S ε u ¯ v (cid:107) , R d × Q (cid:46) ε (cid:107) D u ¯ v (cid:107) , R d × Q (we have reversed the order of integration to pass from S ε to T ε in the secondterm on the left). This means that we may replace the function τ ε ( I − T ε ) A in M εµ by τ ε ( I − S ε ) A with error being of order ε . But since A ∈ C , ( ¯ R d ; ˜ L ∞ ( Q )) ,an L ∞ -variant of the second estimate in Lemma 5.4 (again see Remark 5.6) willimply that M εµ itself is of order ε . Another example when (5.18) holds is the casewhere the fast and slow variables are separated, that is, A ε ( x ) = A ( x , ε − x ) with x = ( x , x ) . Since only the rapid oscillations must be regularized, we maychoose T ε to be the translation operator in the variable x : T ε u ( x, y )( z ) = u ( x , x + εz , y ) . Then ( I − T ε ) A is identically zero. Operators with such coefficients have beenstudied in [Se17 ]. Remark . Given the previous remark, it may be tempting to conjecture that(5.18) holds for all A ∈ C , ( ¯ R d ; ˜ L ∞ ( Q )) . However, this is not the case, as thefollowing example shows. Define χ ( x ) = (cid:88) k ∈ N k − cos 2 k πx. Then χ is uniformly continuous, but does not satisfy a Hölder condition of any orderat all points (see [H16, Section 4] for details). Let A be a uniformly positive definiteLipschitz function on R whose derivative equals χ on (0 , and is off (0 , , and let A ( y ) = 4 π / (2 + sin 2 πy ) − . Set A ( x, y ) = A ( x ) A ( y ) . Select an f ∈ L ( R ) insuch a way that | Du | = 1 on (0 , . It is a straightforward, yet tedious, calculationto see that ( M ε k µ f, f ) R = (log ε − k ) − + O ( ε k ) , k → ∞ , where ε k = 2 − k . In fact, for any monotone function ζ ∈ C ([0 , that satisfies ζ (0) = 0 and ζ ( ε ) ≥ ε , we can construct a uniformly elliptic operator A ε on H ( R ) and find a sequence { ε k } k ∈ N converging to such that ( M ε k µ f, f ) R = ζ ( ε k ) + O ( ε k ) , k → ∞ , for some f ∈ L ( R ) . The idea is to adjust gaps in the Fourier series for χ . Remark . We observe that L µ can be written in the form L µ = ( A µ ) − D ∗ L D ( A µ ) − , where L : H ( R d ) n → L ( R d ) n is a first-order differential operator with boundedcoefficients: L = (cid:90) Q N + ( · , y ) ∗ D ∗ A ( · , y )( I + D N ( · , y )) dy (cf. [Se17 , Remark 2.6]). Likewise, we can write M εµ as M εµ = ( A µ ) − D ∗ M ε D ( A µ ) − where M ε is the bounded function given by M ε ( x ) = ε − (cid:90) Q ( I + D N + ( x, ε − x + z )) ∗ ∆ εz A ( x, ε − x + z )( I + D N ( x, ε − x + z )) dz with ∆ εz A ( x, y ) = A ( x + εz, y ) − A ( x, y ) .6. Main results
Now we formulate the main results of the paper.
Theorem 6.1. If µ / ∈ (cid:83) , then for any ε ∈ (cid:69) and f ∈ L ( R d ) n we have (cid:107) ( A εµ ) − f − ( A µ ) − f (cid:107) , R d (cid:46) ε (cid:107) f (cid:107) , R d , (6.1) (cid:107) D ( A εµ ) − f − D ( A µ ) − f − εD K εµ f (cid:107) , R d (cid:46) ε (cid:107) f (cid:107) , R d . (6.2) OMOGENIZATION FOR LOCALLY PERIODIC ELLIPTIC OPERATORS 13
The estimates are sharp with respect to the order, and the constants depend onlyon the parameters d , n , µ , the norm (cid:107) A (cid:107) C , and the constants c A and C A in thecoercivity bound. Theorem 6.2. If µ / ∈ (cid:83) , then for any ε ∈ (cid:69) and f ∈ L ( R d ) n it holds that (6.3) (cid:107) ( A εµ ) − f − ( A µ ) − f − ε C εµ f (cid:107) , R d (cid:46) ε (cid:107) f (cid:107) , R d . The estimate is sharp with respect to the order, and the constant depends only onthe parameters d , n , µ , the norm (cid:107) A (cid:107) C , and the constants c A and C A in thecoercivity bound.Remark . These results should be compared with those in [Se17 ]. Supposethat A ε is periodic, that is, A ε ( x ) = A ( x , ε − x ) , where x = ( x , x ) . In [Se17 ]we proved estimates similar to (6.1)–(6.3), but with different correctors in (6.2)and (6.3). The difference stems from the smoothing operator. As mentioned earlier,in the periodic case we may reduce T ε to the translation operator in the variable x .Then S ε will involve averaging over εQ , with Q being the basic cell for the latticeof periods (not necessarily of full rank). The Gelfand transform provides anothersmoothing that is, in a sense, dual to the first one and involves averaging over thedual cell ε − Q ∗ in the reciprocal space. (Here Q ∗ is the Wigner–Seitz cell in thedual lattice.) It is this last smoothing that appeared in [Se17 ]. One can verifydirectly that either of these may be used in the corrector K εµ . As for L µ and M εµ ,the former does not depend on smoothing and is just the same as in [Se17 ], andthe latter is zero by the choice of T ε (see Remark 5.8). Remark . Theorems 6.1 and 6.2 can be extended to allow all µ / ∈ spec A , thoughit may be necessary to replace (cid:69) by a smaller set (cid:69) µ depending on µ . Indeed, theproofs of the theorems go over without change to the case µ / ∈ spec A provided weestablish estimates similar to (3.5) and (4.6). By the first resolvent identity, thisamounts to checking that A εµ as an operator on L ( R d ) n has a uniformly boundedinverse. Suppose that µ ∈ (cid:83) (otherwise (cid:69) µ = (cid:69) ). We know from Theorem 6.1 thatif ν / ∈ (cid:83) , then (cid:107) ( A εν ) − f − ( A ν ) − f (cid:107) , R d ≤ C ν ε (cid:107) f (cid:107) , R d for all ε ∈ (cid:69) and f ∈ L ( R d ) n . Therefore, using the identity ( A εµ ) − − ( A µ ) − = (cid:16) I − ( µ − ν ) A ν ( A µ ) − (cid:0) ( A εν ) − − ( A ν ) − (cid:1)(cid:17) − × A ν ( A µ ) − (cid:0) ( A εν ) − − ( A ν ) − (cid:1) A ν ( A µ ) − , we see that ( A εµ ) − is bounded on L ( R d ) n uniformly in ε ≤ ε µ,ν ∧ ε , where ε µ,ν < dist( µ, spec A ) C ν | µ − ν | (cid:0) dist( µ, spec A ) + | µ − ν | (cid:1) . It follows that we can set (cid:69) µ = (0 , ε µ,ν ∧ ε ] . Remark . We note that the operator D ( A εµ ) − converges in the uniform topologyif and only if D ∗ A ( x, · ) ξ = 0 on ˜ H ( Q ) n for every x ∈ R d and ξ ∈ C d × n , in whichcase N is zero and hence so is K εµ . Notice also that the effective coefficients arethen obtained by ordinary averaging over Q . Remark . By keeping track of [ A ] C , in estimates, we can find that the constantson the right of (6.1) and (6.2) depend linearly on [ A ] C , , while the constant on the right of (6.3), quadratically. These observations play a role in proving resultssimilar to Theorems 6.1 and 6.2 when the coefficients are Hölder continuous, oreven continuous, in the slow variable. The key idea is to use mollification to replace A with a function A δ that is Lipschitz in the first variable. In the case of Höldercontinuous coefficients, we are able to control both the convergence rate of A δ to A in a Hölder seminorm and the growth rate of [ A δ ] C , in terms of δ as δ → .In the end, this allows us to obtain the desired operator estimates. However, ifthe coefficients are only continuous, such an approach yields the convergence of theresolvent, but not the rate. These results have been announced in [Se17 ]; detailedproofs will appear elsewhere.7. Proof of the main results
Our first task is to obtain an identity involving ( A εµ ) − , ( A µ ) − and K εµ that willplay a crucial role in the proofs.Fix f ∈ L ( R d ) n and g ∈ H − ( R d ) n . Let u = ( A µ ) − f , U = K µ f , U ε = K εµ f and u + ε = (( A εµ ) + ) − g . Then we have(7.1) (cid:0) ( A εµ ) − f − ( A µ ) − f − ε K εµ f, g (cid:1) R d = ( A u , u + ε ) R d − ( A ε ( S ε u + εU ε ) , u + ε ) R d − ( A ε ( I − S ε ) u , u + ε ) R d + εµ ( U ε , u + ε ) R d . Let us look at the first two terms on the right. By the definition of the effectivecoefficients, ( A u , u + ε ) R d = (cid:0) A ( D u + D U ) , D u + ε (cid:1) R d × Q . Then Lemma 5.1 yields that(7.2) ( A u , u + ε ) R d = (cid:0) τ ε T ε A ( D u + D U ) , T ε D u + ε (cid:1) R d × Q (notice here that u + ε does not depend on the second variable). On the other hand,(7.3) ( A ε ( S ε u + εU ε ) , u + ε ) R d = (cid:0) τ ε A T ε ( D u + D U ) , D u + ε (cid:1) R d × Q + ε (cid:0) τ ε A T ε D U, D u + ε (cid:1) R d × Q . Commuting T ε past A in the first term on the right and combining the resultingidentity with (7.2), we conclude that(7.4) ( A u , u + ε ) R d − ( A ε ( S ε u + εU ε ) , u + ε ) R d = (cid:0) τ ε T ε A ( D u + D U ) , ( T ε − I ) D u + ε (cid:1) R d × Q − (cid:0) τ ε [ A, T ε ]( D u + D U ) , D u + ε (cid:1) R d × Q − ε (cid:0) τ ε A T ε D U, D u + ε (cid:1) R d × Q . We would like to be able to prove that the norm of the operator correspondingto the left-hand side is of order ε . It is clear from the previous discussion that thelast two terms on the right satisfy the desired estimate. The same would be truefor the first term if we could integrate by parts and transfer D from ( T ε − I ) u + ε to A ( D u + D U ) . The following technical result will be useful for this purpose. Lemma 7.1.
Let F ∈ C , ( ¯ R d ; ˜ L ( Q )) d × n be such that D ∗ F ( x, · ) = 0 on ˜ H ( Q ) n for each x ∈ R d . Then D ∗ τ ε T ε F = τ ε T ε D ∗ F on C c ( R d ; C ( Q )) n for any ε > . OMOGENIZATION FOR LOCALLY PERIODIC ELLIPTIC OPERATORS 15
Proof.
It suffices to check the assertion for ε = 1 , because the general result willthen follow from this special case applied to the function ( x, y ) (cid:55)→ F ( εx, y ) . Aftera change of variables, we must show that, for any ϕ ∈ C c ( R d ; C ( Q )) n ,(7.5) (cid:90) R d (cid:90) Q (cid:104) F ( x, x + y ) , D ϕ ( x, y ) (cid:105) dx dy = (cid:90) R d (cid:90) Q (cid:104) D ∗ F ( x, x + y ) , ϕ ( x, y ) (cid:105) dx dy. Were F smooth, this would be nothing but the usual integration by parts formula.But we can find a sequence of divergence free smooth functions that converges, ina certain sense, to the function F , which will yield the desired conclusion.If e k ( y ) = e πi (cid:104) y,k (cid:105) , where k ∈ Z d , then we let F K ( x, · ) denote the partial sumof the Fourier series for F ( x, · ) : F K ( x, · ) = (cid:88) | k |≤ K ˆ F k ( x ) e k . By hypothesis, D ∗ F ( x, · ) = 0 on ˜ H ( Q ) n , so (cid:104) ˆ F k ( x ) , k (cid:105) = (2 π ) − (cid:90) Q (cid:104) F ( x, y ) , De k ( y ) (cid:105) dy = 0 for each k ∈ Z d . An integration by parts then gives(7.6) (cid:90) R d (cid:90) Q (cid:104) F K ( x, x + y ) , D ϕ ( x, y ) (cid:105) dx dy = (cid:90) R d (cid:90) Q (cid:104) D ∗ F K ( x, x + y ) , ϕ ( x, y ) (cid:105) dx dy (notice here that D ˆ F k ( x ) are exactly the Fourier coefficients of D F ( x, · ) ).Our goal now is to pass from (7.6) to (7.5). Let f be a function in C , ( ¯ R d ; ˜ L ( Q )) and let f K ( x, · ) be the partial sum of the Fourier series for f ( x, · ) . We claim that f K → f in the weak- ∗ topology on C c ( R d × Q ) ∗ as K → ∞ . Indeed, given any ψ ∈ C c ( R d × Q ) , the sequence of functions x (cid:55)→ ( f K ( x, · ) , ψ ( x, · )) Q convergespointwise to the function x (cid:55)→ ( f ( x, · ) , ψ ( x, · )) Q , because f K ( x, · ) → f ( x, · ) in L .In addition, all the functions in the sequence are supported in a compact set andare uniformly bounded, since | ( f K ( x, · ) , ψ ( x, · )) Q | ≤ (cid:107) f ( x, · ) (cid:107) ,Q (cid:107) ψ ( x, · ) (cid:107) ,Q ≤ (cid:107) f (cid:107) C (cid:107) ψ (cid:107) C . We see that ( f K , ψ ) R d × Q → ( f, ψ ) R d × Q by the Lebesgue dominated convergencetheorem, and the claim follows.The proof is completed now by letting K → ∞ in (7.6). (cid:3) By definition, we have A ( D u + D U ) = A ( I + D N ) Du . Assume for themoment that u , u + ε ∈ C ∞ c ( R d ) n . We recall that, for each x ∈ R d and ξ ∈ C d × n , D ∗ A ( x, · )( I + D N ( x, · )) ξ = 0 on ˜ H ( Q ) n , so Lemma 7.1 applies to show that(7.7) (cid:0) τ ε T ε A ( D u + D U ) , ( T ε − I ) D u + ε (cid:1) R d × Q = (cid:0) τ ε T ε D ∗ A ( D u + D U ) , ( T ε − I ) u + ε (cid:1) R d × Q for every u , u + ε ∈ C ∞ c ( R d ) n . Moreover, since the form ( u , u + ε ) (cid:55)→ (cid:0) τ ε T ε A ( D u + D U ) , ( T ε − I ) D u + ε (cid:1) R d × Q is continuous on H ( R d ) n × H ( R d ) n and since the form ( u , u + ε ) (cid:55)→ (cid:0) τ ε T ε D ∗ A ( D u + D U ) , ( T ε − I ) u + ε (cid:1) R d × Q is continuous on H ( R d ) n × L ( R d ) n , the last equality holds for any u ∈ H ( R d ) n and u + ε ∈ H ( R d ) n .Now that we have this result, (7.4) becomes(7.8) ( A u , u + ε ) R d − ( A ε ( S ε u + εU ε ) , u + ε ) R d = (cid:0) τ ε T ε D ∗ A ( D u + D U ) , ( T ε − I ) u + ε (cid:1) R d × Q − (cid:0) τ ε [ A, T ε ]( D u + D U ) , D u + ε (cid:1) R d × Q − ε (cid:0) τ ε A T ε D U, D u + ε (cid:1) R d × Q . Putting (7.8) into (7.1), we finally obtain the desired identity:(7.9) (cid:0) ( A εµ ) − f − ( A µ ) − f − ε K εµ f, g (cid:1) R d = (cid:0) τ ε T ε D ∗ A ( D u + D U ) , ( T ε − I ) u + ε (cid:1) R d × Q − (cid:0) τ ε [ A, T ε ]( D u + D U ) , D u + ε (cid:1) R d × Q − ε (cid:0) τ ε A T ε D U, D u + ε (cid:1) R d × Q − ( A ε ( I − S ε ) u , u + ε ) R d + εµ ( U ε , u + ε ) R d . We are now in a position to prove the theorems.
Proof of Theorem . We estimate each term in (7.9). By Lemmas 5.1 and 5.3,(7.10) (cid:12)(cid:12)(cid:0) τ ε T ε D ∗ A ( D u + D U ) , ( T ε − I ) u + ε (cid:1) R d × Q (cid:12)(cid:12) ≤ (cid:107) τ ε T ε D ∗ A ( D u + D U ) (cid:107) , R d × Q (cid:107) ( T ε − I ) u + ε (cid:107) , R d × Q (cid:46) ε (cid:0) (cid:107) Du (cid:107) , , R d + (cid:107) D D U (cid:107) , R d × Q + (cid:107) D U (cid:107) , R d × Q (cid:1) (cid:107) Du + ε (cid:107) , R d . Using Lemma 5.1 again, we see that(7.11) (cid:12)(cid:12)(cid:0) τ ε [ A, T ε ]( D u + D U ) , D u + ε (cid:1) R d × Q (cid:12)(cid:12) ≤ εr Q [ A ] C , (cid:107) τ ε T ε ( D u + D U ) (cid:107) , R d × Q (cid:107) D u + ε (cid:107) , R d × Q (cid:46) ε (cid:0) (cid:107) Du (cid:107) , R d + (cid:107) D U (cid:107) , R d × Q (cid:1) (cid:107) Du + ε (cid:107) , R d (recall that r Q = 1 / Q ) and(7.12) ε (cid:12)(cid:12)(cid:0) τ ε A T ε D U, D u + ε (cid:1) R d × Q (cid:12)(cid:12) ≤ ε (cid:107) A (cid:107) C (cid:107) τ ε T ε D U (cid:107) , R d × Q (cid:107) D u + ε (cid:107) , R d × Q (cid:46) ε (cid:107) D U (cid:107) , R d × Q (cid:107) Du + ε (cid:107) , R d . Next, it follows from the estimate (3.3) and Lemma 5.4 that | ( A ε ( I − S ε ) u , u + ε ) R d | (cid:46) (cid:107) ( I − S ε ) u (cid:107) , , R d (cid:107) u + ε (cid:107) , , R d (cid:46) ε (cid:107) Du (cid:107) , , R d (cid:107) u + ε (cid:107) , , R d . Finally, Lemma 5.2 yields ε | ( U ε , u + ε ) R d | ≤ ε (cid:107) U ε (cid:107) , R d (cid:107) u + ε (cid:107) , R d (cid:46) ε (cid:107) U (cid:107) , R d × Q (cid:107) u + ε (cid:107) , R d . In summary, we have found that (cid:12)(cid:12)(cid:0) ( A εµ ) − f − ( A µ ) − f − ε K εµ f, g (cid:1) R d (cid:12)(cid:12) (cid:46) ε (cid:0) (cid:107) Du (cid:107) , , R d + (cid:107) D D U (cid:107) , R d × Q + (cid:107) U (cid:107) , , R d × Q (cid:1) (cid:107) u + ε (cid:107) , , R d . Now suppose that g ∈ L ( R d ) n . Then from (4.6), (5.2), (5.3), (5.10) and (3.5) + , (cid:12)(cid:12)(cid:0) ( A εµ ) − f − ( A µ ) − f, g (cid:1) R d (cid:12)(cid:12) (cid:46) ε (cid:107) f (cid:107) , R d (cid:107) g (cid:107) , R d , OMOGENIZATION FOR LOCALLY PERIODIC ELLIPTIC OPERATORS 17 which proves (6.1). On the other hand, setting g = D ∗ h where h ∈ L ( R d ) d × n andusing (4.6), (5.2), (5.3) and (3.5) + , we obtain (cid:12)(cid:12)(cid:0) ( A εµ ) − f − ( A µ ) − f − ε K εµ f, D ∗ h (cid:1) R d (cid:12)(cid:12) (cid:46) ε (cid:107) f (cid:107) , R d (cid:107) h (cid:107) , R d , which proves (6.2). (cid:3) Proof of Theorem . Let u +0 = (( A µ ) + ) − g , U + = K + µ g and U + ε = ( K εµ ) + g .As a first step, we rewrite the corrector C εµ dropping, as we may, terms with operatornorm of order ε .By the very definition of C εµ , ( C εµ f, g ) R d = ( K εµ f, g ) R d − ( L µ f, g ) R d − ( M εµ f, g ) R d + ( f, ( K εµ ) + g ) R d − ( f, L + µ g ) R d . We claim that(7.13) − ε ( L µ f, g ) R d − ε ( M εµ f, g ) R d + ε ( f, ( K εµ ) + g ) R d − ε ( f, L + µ g ) R d ≈ (cid:0) τ ε T ε D ∗ A ( D u + D U ) , ( T ε − I )( u +0 + εU + ε ) (cid:1) R d × Q − (cid:0) τ ε [ A, T ε ]( D u + D U ) , D ( u +0 + εU + ε ) (cid:1) R d × Q − ε (cid:0) τ ε A T ε D U, D ( u +0 + εU + ε ) (cid:1) R d × Q , where the symbol ≈ is used to indicate equality up to terms that will eventually beabsorbed into the error.Indeed, by Lemma 5.1 we have ( L µ f, g ) R d = (cid:0) τ ε T ε D ∗ A ( D u + D U ) , τ ε T ε U + (cid:1) R d × Q . Now observe that τ ε T ε U + may be replaced by τ ε S ε U + . This is so because (cid:12)(cid:12)(cid:0) τ ε T ε D ∗ A ( D u + D U ) , τ ε T ε U + − τ ε S ε U + (cid:1) R d × Q (cid:12)(cid:12) ≤ (cid:107) τ ε T ε D ∗ A ( D u + D U ) (cid:107) , R d × Q (cid:107) τ ε T ε U + − τ ε S ε U + (cid:107) , R d × Q , whence, by Lemmas 5.1 and 5.5 and the estimates (4.6), (5.2), (5.3) and (5.2) + , (cid:12)(cid:12)(cid:0) τ ε T ε D ∗ A ( D u + D U ) , τ ε T ε U + − τ ε S ε U + (cid:1) R d × Q (cid:12)(cid:12) (cid:46) ε (cid:107) f (cid:107) , R d (cid:107) g (cid:107) , R d . Recalling that U + ε = τ ε S ε U + , we see that(7.14) ( L µ f, g ) R d ≈ (cid:0) τ ε T ε D ∗ A ( D u + D U ) , U + ε (cid:1) R d × Q . We next want to show that(7.15) ( f, L + µ g ) R d ≈ (cid:0) τ ε A T ε D U, D ( u +0 + εU + ε ) (cid:1) R d × Q . According to Lemma 5.1, ( f, L + µ g ) R d = (cid:0) τ ε T ε AD U, τ ε T ε ( D u +0 + D U + ) (cid:1) R d × Q . We commute T ε through A and use Lemma 5.1 and the estimates (5.2) and(4.6) + , (5.2) + to get ( f, L + µ g ) R d ≈ (cid:0) τ ε A T ε D U, τ ε T ε ( D u +0 + D U + ) (cid:1) R d × Q (notice here that τ ε [ A, T ε ] = τ ε ( I − T ε ) A · τ ε T ε ). A similar argument usingLemma 5.3 shows that τ ε T ε D u +0 (which is, of course, equal to T ε D u +0 ) may bereplaced by D u +0 . With a little extra care we can pass from τ ε T ε D U + to εD U + ε ,as well. Indeed, εD U + ε = ετ ε S ε D U + + τ ε S ε D U + , where ετ ε S ε D U + createsanother error term and τ ε S ε D U + is handled exactly as above, by Lemma 5.5.Hence (7.15) is proved. Repeating these last arguments for τ ε T ε ( D u +0 + D U + ) , we find also that(7.16) ( M εµ f, g ) R d ≈ ε − (cid:0) τ ε [ A, T ε ]( D u + D U ) , D ( u +0 + εU + ε ) (cid:1) R d × Q . Let us turn to the term involving ( K εµ ) + . By the definition of u and U + ε , ( f, ( K εµ ) + g ) R d = ( A u , U + ε ) R d − µ ( u , U + ε ) R d . Applying Lemmas 5.4 and 5.7 + and the estimates (4.6) and (5.10) + yields | ( u , U + ε ) R d | ≤ | (( S ε − I ) u , U + ε ) R d | + | ( u , S ε U + ε ) R d | (cid:46) ε (cid:107) f (cid:107) , R d (cid:107) g (cid:107) , R d , so ( f, ( K εµ ) + g ) R d ≈ ( A u , U + ε ) R d . Thus, from Lemma 5.1 and the definition of the effective coefficients, we have(7.17) ( f, ( K εµ ) + g ) R d ≈ (cid:0) τ ε T ε D ∗ A ( D u + D U ) , T ε U + ε (cid:1) R d × Q . To summarize: by (7.14)–(7.17), (7.13) reduces to showing that(7.18) (cid:12)(cid:12)(cid:0) τ ε T ε D ∗ A ( D u + D U ) , ( T ε − I ) u +0 (cid:1) R d × Q (cid:12)(cid:12) (cid:46) ε (cid:107) f (cid:107) , R d (cid:107) g (cid:107) , R d . Let us prove (7.18). From Lemma 7.1, we know that (cid:0) τ ε T ε D ∗ A ( D u + D U ) , ( T ε − I ) u +0 (cid:1) R d × Q = (cid:0) τ ε T ε A ( D u + D U ) , ( T ε − I ) D u +0 (cid:1) R d × Q (cf. (7.7)). Lemmas 5.3 and 5.5 and the estimates (4.6), (5.2), (5.3) and (4.6) + enable us to replace τ ε T ε A ( D u + D U ) with τ ε S ε A ( D u + D U ) . Reversingthe order of integration to switch T ε and S ε and again using Lemma 7.1, we get (cid:0) τ ε T ε D ∗ A ( D u + D U ) , ( T ε − I ) u +0 (cid:1) R d × Q ≈ (cid:0) τ ε T ε D ∗ A ( D u + D U ) , ( S ε − I ) u +0 (cid:1) R d × Q . It then follows from Lemmas 5.1 and 5.4 and the estimates (4.6), (5.2), (5.3)and (4.6) + that (cid:12)(cid:12)(cid:0) τ ε T ε D ∗ A ( D u + D U ) , ( S ε − I ) u +0 (cid:1) R d × Q (cid:12)(cid:12) (cid:46) ε (cid:107) f (cid:107) , R d (cid:107) g (cid:107) , R d . We have verified (7.18), and therefore the claim is established.Now we subtract (7.13) from (7.9) to obtain (cid:0) ( A εµ ) − f − ( A µ ) − f − ε C εµ f, g (cid:1) R d ≈ (cid:0) τ ε T ε D ∗ A ( D u + D U ) , ( T ε − I )( u + ε − u +0 − εU + ε ) (cid:1) R d × Q − (cid:0) τ ε [ A, T ε ]( D u + D U ) , D ( u + ε − u +0 − εU + ε ) (cid:1) R d × Q − ε (cid:0) τ ε A T ε D U, D ( u + ε − u +0 − εU + ε ) (cid:1) R d × Q − ( A ε ( I − S ε ) u , u + ε ) R d + εµ ( U ε , u + ε ) R d . Using the inequalities (7.10), (7.11) and (7.12) with u + ε − u +0 − εU + ε in place of u + ε and then applying the estimates (4.6), (5.2), (5.3) and (6.2) + , we see that the normsof the operators associated with the first three forms on the right are of order ε .As for the last two forms, we write ( A ε ( I − S ε ) u , u + ε ) R d = (cid:0) ( I − S ε ) u , g + ¯ µu + ε (cid:1) R d and ε ( U ε , u + ε ) R d = ε ( S ε U ε , u + ε ) R d + ε ( U ε , ( I − S ε ) u + ε ) R d . OMOGENIZATION FOR LOCALLY PERIODIC ELLIPTIC OPERATORS 19
Then, by Lemma 5.4 and the estimates (4.6) and (3.5) + , | ( A ε ( I − S ε ) u , u + ε ) R d | ≤ (cid:107) ( I − S ε ) u (cid:107) , R d (cid:0) (cid:107) g (cid:107) , R d + | µ |(cid:107) u + ε (cid:107) , R d (cid:1) (cid:46) ε (cid:107) f (cid:107) , R d (cid:107) g (cid:107) , R d , while, by Lemmas 5.4 and 5.7 and the estimates (5.10) and (3.5) + , ε | ( U ε , u + ε ) R d | ≤ ε (cid:107)S ε U ε (cid:107) , R d (cid:107) u + ε (cid:107) , R d + ε (cid:107) U ε (cid:107) , R d (cid:107) ( I − S ε ) u + ε (cid:107) , R d (cid:46) ε (cid:107) f (cid:107) , R d (cid:107) g (cid:107) , R d . The proof is complete. (cid:3)
Acknowledgment
The author is grateful to T. A. Suslina for helpful discussions.
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Saint Petersburg State University, Universitetskaya nab. 7/9, Saint Petersburg199034, Russia