Homogenization of the first initial boundary-value problem for parabolic systems: operator error estimates
aa r X i v : . [ m a t h . A P ] J a n HOMOGENIZATION OF THE FIRST INITIAL BOUNDARY-VALUEPROBLEM FOR PARABOLIC SYSTEMS: OPERATOR ERROR ESTIMATES
YU. M. MESHKOVA AND T. A. SUSLINA
Abstract.
Let
O ⊂ R d be a bounded domain of class C , . In L ( O ; C n ), we consider a self-adjoint matrix second order elliptic differential operator B D,ε , 0 < ε
1, with the Dirichletboundary condition. The principal part of the operator is given in a factorized form. The oper-ator involves first and zero order terms. The operator B D,ε is positive definite; its coefficientsare periodic and depend on x /ε . We study the behavior of the operator exponential e − B D,ε t , t >
0, as ε →
0. We obtain approximations for the exponential e − B D,ε t in the operator norm on L ( O ; C n ) and in the norm of operators acting from L ( O ; C n ) to the Sobolev space H ( O ; C n ).The results are applied to homogenization of solutions of the first initial boundary-value problemfor parabolic systems. Introduction
The paper concerns homogenization theory of periodic differential operators (DO’s). Wemention the books on homogenization [BaPa, BeLPap, ZhKO, Sa].0.1.
Statement of the problem.
Let Γ ⊂ R d be a lattice and let Ω be the elementary cell ofthe lattice Γ. For a Γ-periodic function ψ in R d , we denote ψ ε ( x ) := ψ ( x /ε ), where ε >
0, and ψ := | Ω | − R Ω ψ ( x ) d x .Let O ⊂ R d be a bounded domain of class C , . In L ( O ; C n ), we study a selfadjoint matrixstrongly elliptic second order DO B D,ε , 0 < ε
1, with the Dirichlet boundary condition. Theprincipal part of the operator B D,ε is given in a factorized form A ε = b ( D ) ∗ g ε ( x ) b ( D ), where b ( D ) is a matrix homogeneous first order DO, and g ( x ) is a Γ-periodic bounded and positivedefinite matrix-valued function in R d . (The precise assumptions on b ( D ) and g ( x ) are givenbelow in Subsection 1.3.) The operator B D,ε is given by the differential expression B ε = b ( D ) ∗ g ε ( x ) b ( D ) + d X j =1 (cid:0) a εj ( x ) D j + D j a εj ( x ) ∗ (cid:1) + Q ε ( x ) + λQ ε ( x ) (0.1)with the Dirichlet condition on ∂ O . Here a j ( x ), j = 1 , . . . , d , and Q ( x ) are Γ-periodic matrix-valued functions, in general, unbounded; a Γ-periodic matrix-valued function Q ( x ) is such that Q ( x ) > Q , Q − ∈ L ∞ . The constant λ is chosen so that the operator B D,ε is positivedefinite. (The precise assumptions on the coefficients are given below in Subsection 1.4.)The coefficients of the operator (0.1) oscillate rapidly for small ε . Let u ε ( x , t ) be the solutionof the first initial boundary-value problem: ( Q ε ( x ) ∂ t u ε ( x , t ) = − B ε u ε ( x , t ) , x ∈ O , t > u ε ( x , t ) = 0 , x ∈ ∂ O , t > Q ε ( x ) u ε ( x ,
0) = ϕ ( x ) , x ∈ O , (0.2)where ϕ ∈ L ( O ; C n ). We are interested in the behavior of the solution in the small periodlimit. Date : January 17, 2018.2000
Mathematics Subject Classification.
Primary 35B27.
Key words and phrases.
Periodic differential operators, parabolic systems, homogenization, operator errorestimates.Supported by Russian Foundation for Basic Research (grant no. 16-01-00087). The first author was supportedby “Native Towns”, a social investment program of PJSC “Gazprom Neft”, by the “Dynasty” foundation, andby the Rokhlin grant.
Main results.
It turns out that, as ε →
0, the solution u ε ( · , t ) converges in L ( O ; C n ) tothe solution u ( · , t ) of the effective problem with constant coefficients: ( Q ∂ t u ( x , t ) = − B u ( x , t ) , x ∈ O , t > u ( x , t ) = 0 , x ∈ ∂ O , t > Q u ( x ,
0) = ϕ ( x ) , x ∈ O . (0.3)Here B is the differential expression for the effective operator B D . Our first main result is theestimate k u ε ( · , t ) − u ( · , t ) k L ( O ) Cε ( t + ε ) − / e − ct k ϕ k L ( O ) , t > , (0.4)for sufficiently small ε . For fixed time t >
0, this estimate is of sharp order O ( ε ). Our secondmain result is approximation of the solution u ε ( · , t ) in the energy norm: k u ε ( · , t ) − v ε ( · , t ) k H ( O ) C ( ε / t − / + εt − ) e − ct k ϕ k L ( O ) , t > . (0.5)Here v ε ( · , t ) = u ( · , t ) + ε K D ( t ; ε ) ϕ ( · ) is the first order approximation of the solution u ε ( · , t ).The operator K D ( t ; ε ) is a corrector. It involves rapidly oscillating factors, and so depends on ε . We have k ε K D ( t ; ε ) k L → H = O (1). For fixed t , estimate (0.5) is of order O ( ε / ) due to theinfluence of the boundary layer. The presence of the boundary layer is confirmed by the factthat, in a strictly interior subdomain O ′ ⊂ O , the order of the H -estimate can be improved: k u ε ( · , t ) − v ε ( · , t ) k H ( O ′ ) Cε ( t − / δ − + t − ) e − ct k ϕ k L ( O ) , t > . Here δ = dist {O ′ ; ∂ O} .In the general case, the corrector involves a smoothing operator. We distinguish conditionsunder which it is possible to use a simpler corrector which does not include the smoothingoperator. Along with estimate (0.5), we obtain approximation of the flux g ε b ( D ) u ε ( · , t ) in the L -norm.The constants in estimates (0.4) and (0.5) are controlled in terms of the problem data; they donot depend on ϕ . Therefore, estimates (0.4) and (0.5) can be rewritten in the uniform operatortopology. In a simpler case where Q ( x ) = n , we have k e − B D,ε t − e − B D t k L ( O ) → L ( O ) Cε ( t + ε ) − / e − ct , t > , k e − B D,ε t − e − B D t − ε K D ( t ; ε ) k L ( O ) → H ( O ) C ( ε / t − / + εt − ) e − ct , t > . The resuts of such type are called operator error estimates in homogenization theory.0.3.
Operator error estimates. Survey.
Currently, the study of operator error estimatesis an actively developing area of homogenization theory. The interest in this subject arosein connection with the papers [BSu1, BSu2] by M. Sh. Birman and T. A. Suslina, where theoperator A ε of the form b ( D ) ∗ g ε ( x ) b ( D ) acting in L ( R d ; C n ) was studied. By the spectralapproach , it was proved that k ( A ε + I ) − − ( A + I ) − k L ( R d ) → L ( R d ) Cε. (0.6)Here A = b ( D ) ∗ g b ( D ) is an effective operator and g is a constant effective matrix. Approxi-mation for the operator ( A ε + I ) − in the ( L → H )-norm was obtained in [BSu4]: k ( A ε + I ) − − ( A + I ) − − εK ( ε ) k L ( R d ) → H ( R d ) Cε. (0.7)Later T. A. Suslina carried over estimates (0.6) and (0.7) to more general operator B ε of the form(0.1) acting in L ( R d ; C n ). We also mention the paper [Bo] by D. I. Borisov, where the expressionfor the effective operator B was found and approximations (0.6), (0.7) for the resolvent wereobtained. In [Bo], it was assumed that the coefficients of the operator depend not only on therapid variable, but also on the slow variable; however, the coefficients of B ε were assumed to besufficiently smooth.To parabolic systems, the spectral approach was applied in the papers [Su1, Su2] by T. A. Sus-lina, where the principal term of approximation was found, and in [Su3], where estimate withthe corrector was proved: k e − A ε t − e − A t k L ( R d ) → L ( R d ) Cε ( t + ε ) − / , t > , (0.8) k e − A ε t − e − A t − ε K ( t ; ε ) k L ( R d ) → H ( R d ) Cε ( t − / + t − ) , t > ε . (0.9) OMOGENIZATION OF THE FIRST INITIAL BOUNDARY-VALUE PROBLEM 3
In these estimates, the exponentially decreasing function of t is absent, because the bottom ofthe spectra of A ε and A is zero. The exponential of the operator B ε of the form (0.1) wasstudied in the paper [M] by Yu. M. Meshkova, where analogs of inequalities (0.8) and (0.9) wereobtained.A different approach to operator error estimates in homogenization theory was suggested byV. V. Zhikov [Zh2]. In [Zh2, ZhPas1], estimates of the form (0.6) and (0.7) for the acousticsand elasticity operators were obtained. The “modified method of the first order approximation” or the “shift method” , in the terminology of the authors, was based on analysis of the firstorder approximation to the solution and introduction of the additional parameter. Along withproblems in R d , in [Zh2, ZhPas1], homogenization problems in a bounded domain O ⊂ R d withthe Dirichlet or Neumann boundary conditions were studied. To parabolic equations, the shiftmethod was applied in [ZhPas2], where analogs of estimates (0.8) and (0.9) were proved. Furtherresults of V. V. Zhikov, S. E. Pastukhova, and their students are discussed in the recent survey[ZhPas3].Operator error estimates for the Dirichlet and Neumann problems for second order ellipticequations in a bounded domain were studied by many authors. Apparently, the first result isdue to Sh. Moskow and M. Vogelius who proved an estimate k A − D,ε − ( A D ) − k L ( O ) → L ( O ) Cε ; (0.10)see [MoV, Corollary 2.2]. Here the operator A D,ε acts in L ( O ), where O ⊂ R , and is given by − div g ε ( x ) ∇ with the Dirichlet condition on ∂ O . The matrix-valued function g ( x ) is assumedto be infinitely smooth.For arbitrary dimension, homogenization problems in a bounded domain were studied in[Zh2] and [ZhPas1]. The acoustics and elasticity operators with the Dirichlet or Neumannboundary conditions and without any smoothness assumptions on coefficients were considered.The authors obtained approximation with corrector for the inverse operator in the ( L → H )-norm with error estimate of order O ( √ ε ). The order deteriorates as compared with a similarresult in R d ; this is explained by the boundary influence. As a rough consequence, approximationof the form (0.10) with error estimate of order O ( √ ε ) was deduced. Similar results for theoperator − div g ε ( x ) ∇ in a bounded domain O ⊂ R d with the Dirichlet or Neumann boundaryconditions were obtained by G. Griso [Gr1, Gr2] with the help of the “unfolding” method. In[Gr2], for the same operator a sharp-order estimate (0.10) was proved. For elliptic systemssimilar results were independently obtained in [KeLiS] and in [PSu, Su5]. Further results and adetailed survey can be found in [Su6, Su7].For the matrix operator of the form (0.1) with the Dirichlet condition, a homogenizationproblem was studied by Q. Xu [Xu1, Xu3]. The case of the Neumann boundary condition wasstudied in [Xu2]. However, in the papers by Q. Xu, the operator is subject to a rather restrictivecondition of uniform ellipticity. Approximations of the generalized resolvent of the operator (0.1)with two-parametric error estimates were obtained in the recent paper [MSu3] by the authors(see also a brief communication [MSu4]). We focus on these results in more detail, since theyare basic for us. For ζ ∈ C \ R + , | ζ | >
1, and sufficiently small ε , we have k ( B D,ε − ζQ ε ) − − ( B D − ζQ ) − k L ( O ) → L ( O ) C ( φ ) ε | ζ | − / , (0.11) k ( B D,ε − ζQ ε ) − − ( B D − ζQ ) − − εK D ( ε ; ζ ) k L ( O ) → H ( O ) C ( φ ) (cid:0) ε / | ζ | − / + ε (cid:1) . (0.12)Note that the values C ( φ ) are controlled explicitly in terms of the problem data and the angle φ = arg ζ . Estimates (0.11) and (0.12) are uniform with respect to φ in any domain of the form { ζ = | ζ | e iφ ∈ C : | ζ | > , φ φ π − φ } with arbitrarily small φ >
0. Moreover, in [MSu3],analogs of estimates (0.11) and (0.12) in a wider domain of spectral parameter ζ were proved.We proceed to discussion of the parabolic problems in a bounded domain. In the two-dimensional case, some estimates of operator type for elliptic and parabolic equations wereobtained in [ChKonLe]. However, in [ChKonLe], the matrix g was assumed to be C ∞ -smooth,and the initial data for a parabolic equation belonged to H ( O ). In the case of arbitrary di-mension and without smoothness assumptions on coefficients, approximation for the exponentialof the operator b ( D ) ∗ g ε ( x ) b ( D ) (with the Dirichlet or Neumann conditions) was found in the YU. M. MESHKOVA AND T. A. SUSLINA paper [MSu1] by the authors: k e − A D,ε t − e − A D t k L ( O ) → L ( O ) Cε ( t + ε ) − / e − ct , t > , k e − A D,ε t − e − A D t − ε K D ( t ; ε ) k L ( O ) → H ( O ) Cε / t − / e − ct , t > ε . The method of [MSu1] was based on employing the identity e − A D,ε t = − πi Z γ e − ζt ( A D,ε − ζI ) − dζ, where γ ⊂ C is a contour enclosing the spectrum of A D,ε in positive direction. This identityallows us to deduce approximations for the operator exponential e − A D,ε t from the correspondingapproximations of the resolvent ( A D,ε − ζI ) − with two-parametric error estimates (with respectto ε and ζ ). The required approximations for the resolvent were found in [Su7].The operator with coefficients periodic in the space and time variables was studied by J. Gengand Z. Shen [GeS]. In [GeS], operator error estimates for the equation ∂ t u ε ( x , t ) = − div g ( ε − x , ε − t ) ∇ u ε ( x , t )in a bounded domain of class C , were obtained. The results of [GeS] were generalized to thecase of Lipschitz domains by Q. Xu and Sh. Zhou [XuZ].0.4. Method.
We develop the method of the paper [MSu1]. It is based upon the followingrepresentation for the solution u ε of the first initial boundary-value problem (0.2): u ε ( · , t ) = − πi R γ e − ζt ( B D,ε − ζQ ε ) − ϕ dζ , where γ ⊂ C is a suitable contour. The solution of the effectiveproblem (0.3) admits a similar representation. Hence, u ε ( · , t ) − u ( · , t ) = − πi Z γ e − ζt (cid:0) ( B D,ε − ζQ ε ) − − ( B D − ζQ ) − (cid:1) ϕ dζ. (0.13)Using the results of [MSu3] (estimate (0.11)), we obtain approximation of the resolvent for ζ ∈ γ and employ representation (0.13). This leads to (0.4). Note that the dependence of the right-hand side of (0.11) on ζ for large | ζ | is important for us. Approximation with the correctortaken into account is obtained in a similar way.0.5. Plan of the paper.
The paper consists of five sections and Appendix ( §§ §
1, wedescribe the class of operators B D,ε , introduce the effective operator B D , and formulate theneeded results about approximation of the operator ( B D,ε − ζQ ε ) − . The main results of thepaper are obtained in §
2. In §
3, these results are applied to homogenization of the solutions of thefirst initial boundary-value problem for nonhomogeneous parabolic equation. §§
4, 5 are devotedto applications of the general results. In §
4, a scalar elliptic operator with a singular potentialof order O ( ε − ) is considered. In §
5, we study an operator with a singular potential of order O ( ε − ). In Appendix ( §§ §
7; the case of a strictly interior subdomain is discussed in §
8. The needed properties of theoscillating factors in the corrector are obtained in § Notation.
Let H and H ∗ be complex separable Hilbert spaces. The symbols ( · , · ) H and k · k H stand for the inner product and the norm in H ; the symbol k · k H → H ∗ denotes the normof a linear continuous operator acting from H to H ∗ .The set of natural numbers and the set of nonnegative integers are denoted by N and Z + ,respectively. We denote R + := [0 , ∞ ). The symbols h · , · i and | · | denote the inner product andthe norm in C n ; n is the identity ( n × n )-matrix. If a is an ( m × n )-matrix, then the symbol | a | denotes the norm of a viewed as operator from C n to C m . If α = ( α , . . . , α d ) ∈ Z d + is amultiindex, | α | denotes its length: | α | = P dj =1 α j . For z ∈ C , the complex conjugate numberis denoted by z ∗ . (We use such nonstandard notation, because the upper line denotes themean value of a periodic function over the periodicity cell.) We denote x = ( x , . . . , x d ) ∈ R d , iD j = ∂ j = ∂/∂x j , j = 1 , . . . , d , D = − i ∇ = ( D , . . . , D d ). The L p -classes of C n -valuedfunctions in a domain O ⊂ R d are denoted by L p ( O ; C n ), 1 p ∞ . The Sobolev classes of C n -valued functions in a domain O ⊂ R d are denoted by H s ( O ; C n ). By H ( O ; C n ) we denote OMOGENIZATION OF THE FIRST INITIAL BOUNDARY-VALUE PROBLEM 5 the closure of C ∞ ( O ; C n ) in H ( O ; C n ). If n = 1, we write simply L p ( O ), H s ( O ), etc., butsometimes, if this does not lead to confusion, we use such simple notation for the spaces ofvector-valued or matrix-valued functions. The symbol L p ((0 , T ); H ), 1 p ∞ , denotes the L p -space of H -valued functions on the interval (0 , T ).Various constants in estimates are denoted by c, C, C , C , C (probably, with indices and marks).The main results of the present paper were announced in [MSu4].1. The results on homogenization of the Dirichlet problemfor elliptic systems
Lattices in R d . Let Γ ⊂ R d be a lattice generated by the basis a , . . . , a d ∈ R d :Γ = n a ∈ R d : a = d X j =1 ν j a j , ν j ∈ Z o , and let Ω be the elementary cell of the lattice Γ:Ω = n x ∈ R d : x = d X j =1 τ j a j , − < τ j < o . By | Ω | we denote the Lebesgue measure of the cell Ω: | Ω | = meas Ω. We put 2 r := diam Ω . Let e H (Ω) denote the subspace of functions in H (Ω), whose Γ-periodic extension to R d belongs to H ( R d ). If Φ( x ) is a Γ-periodic matrix-valued function in R d , we put Φ ε ( x ) :=Φ( x /ε ), ε >
0; Φ := | Ω | − R Ω Φ( x ) d x , Φ := (cid:0) | Ω | − R Ω Φ( x ) − d x (cid:1) − . Here, in the definition of Φit is assumed that Φ ∈ L , loc ( R d ); in the definition of Φ it is assumed that the matrix Φ is squareand nondegenerate, and Φ − ∈ L , loc ( R d ). By [Φ ε ] we denote the operator of multiplication bythe matrix-valued function Φ ε ( x ).1.2. The Steklov smoothing.
The Steklov smoothing operator S ( k ) ε , ε >
0, acts in L ( R d ; C k )(where k ∈ N ) and is given by( S ( k ) ε u )( x ) = | Ω | − Z Ω u ( x − ε z ) d z , u ∈ L ( R d ; C k ) . (1.1)We shall omit the index k in the notation and write simply S ε . Obviously, S ε D α u = D α S ε u for u ∈ H σ ( R d ; C k ) and any multiindex α such that | α | σ . Note that k S ε k H σ ( R d ) → H σ ( R d ) , σ > . (1.2)We need the following properties of the operator S ε (see [ZhPas1, Lemmas 1.1 and 1.2] or [PSu,Propositions 3.1 and 3.2]). Proposition 1.1.
For any function u ∈ H ( R d ; C k ) , we have k S ε u − u k L ( R d ) εr k Du k L ( R d ) , where r = diam Ω . Proposition 1.2.
Let Φ be a Γ -periodic function in R d such that Φ ∈ L (Ω) . Then the operator [Φ ε ] S ε is continuous in L ( R d ) and k [Φ ε ] S ε k L ( R d ) → L ( R d ) | Ω | − / k Φ k L (Ω) . The operator A D,ε . Let
O ⊂ R d be a bounded domain of class C , . In L ( O ; C n ), weconsider the operator A D,ε given formally by the differential expression A ε = b ( D ) ∗ g ε ( x ) b ( D )with the Dirichlet condition on ∂ O . Here g ( x ) is a Γ-periodic Hermitian ( m × m )-matrix-valuedfunction (in general, with complex entries). It is assumed that g ( x ) > g, g − ∈ L ∞ ( R d ).The differential operator b ( D ) is given by b ( D ) = P dj =1 b j D j , where b j , j = 1 , . . . , d , are constantmatrices of size m × n (in general, with complex entries). Assume that m > n and that thesymbol b ( ξ ) = P dj =1 b j ξ j of the operator b ( D ) has maximal rank: rank b ( ξ ) = n for 0 = ξ ∈ R d .This condition is equivalent to the estimates α n b ( θ ) ∗ b ( θ ) α n , θ ∈ S d − ; 0 < α α < ∞ , (1.3) YU. M. MESHKOVA AND T. A. SUSLINA with some positive constants α and α . From (1.3) it follows that | b j | α / , j = 1 , . . . , d. (1.4)The precise definition of the operator A D,ε is given in terms of the quadratic form a D,ε [ u , u ] = Z O h g ε ( x ) b ( D ) u , b ( D ) u i d x , u ∈ H ( O ; C n ) . (1.5)Extending u ∈ H ( O ; C n ) by zero to R d \ O and taking (1.3) into account, we find α k g − k − L ∞ k Du k L ( O ) a D,ε [ u , u ] α k g k L ∞ k Du k L ( O ) , u ∈ H ( O ; C n ) . (1.6)1.4. Lower order terms. The operator B D,ε . We study the selfadjoint operator B D,ε whoseprincipal part coincides with A ε . To define the lower order terms, we introduce Γ-periodic( n × n )-matrix-valued functions (in general, with complex entries) a j , j = 1 , . . . , d , such that a j ∈ L ρ (Ω) , ρ = 2 for d = 1 , ρ > d for d > , j = 1 , . . . , d. (1.7)Next, let Q and Q be Γ-periodic Hermitian ( n × n )-matrix-valued functions (with complexentries) such that Q ∈ L s (Ω) , s = 1 for d = 1 , s > d/ d >
2; (1.8) Q ( x ) > Q , Q − ∈ L ∞ ( R d ) . For convenience of further references, the following set of variables is called the “problem data”: d, m, n, ρ, s ; α , α , k g k L ∞ , k g − k L ∞ , k a j k L ρ (Ω) , j = 1 , . . . , d ; k Q k L s (Ω) ; k Q k L ∞ , k Q − k L ∞ ; the parameters of the lattice Γ; the domain O . (1.9)In L ( O ; C n ), we consider the operator B D,ε , 0 < ε
1, formally given by the differentialexpression B ε = b ( D ) ∗ g ε ( x ) b ( D ) + d X j =1 (cid:0) a εj ( x ) D j + D j a εj ( x ) ∗ (cid:1) + Q ε ( x ) + λQ ε ( x ) (1.10)with the Dirichlet boundary condition. Here the constant λ is chosen so that the operator B D,ε is positive definite (see (1.16) below). The precise definition of the operator B D,ε is given interms of the quadratic form b D,ε [ u , u ] = ( g ε b ( D ) u , b ( D ) u ) L ( O ) + 2Re d X j =1 ( a εj D j u , u ) L ( O ) + ( Q ε u , u ) L ( O ) + λ ( Q ε u , u ) L ( O ) , u ∈ H ( O ; C n ) . (1.11)Let us check that the form b D,ε is closed. By the H¨older inequality and the Sobolev embeddingtheorem, it can be shown that for any ν > C j ( ν ) > k a ∗ j u k L ( R d ) ν k Du k L ( R d ) + C j ( ν ) k u k L ( R d ) , u ∈ H ( R d ; C n ) ,j = 1 , . . . , d ; see [Su4, (5.11)–(5.14)]. By the change of variables y := ε − x and u ( x ) =: v ( y ),we deduce k ( a εj ) ∗ u k L ( R d ) = Z R d | a j ( ε − x ) ∗ u ( x ) | d x = ε d Z R d | a j ( y ) ∗ v ( y ) | d y ε d ν Z R d | D y v ( y ) | d y + ε d C j ( ν ) Z R d | v ( y ) | d y ν k Du k L ( R d ) + C j ( ν ) k u k L ( R d ) , u ∈ H ( R d ; C n ) , < ε . OMOGENIZATION OF THE FIRST INITIAL BOUNDARY-VALUE PROBLEM 7
Then, by (1.3), for any ν > C ( ν ) > d X j =1 k ( a εj ) ∗ u k L ( R d ) ν k ( g ε ) / b ( D ) u k L ( R d ) + C ( ν ) k u k L ( R d ) , u ∈ H ( R d ; C n ) , < ε . (1.12)If ν is fixed, then C ( ν ) depends only on d , ρ , α , the norms k g − k L ∞ , k a j k L ρ (Ω) , j = 1 , . . . , d ,and the parameters of the lattice Γ.By (1.3), for u ∈ H ( R d ; C n ) we have k Du k L ( R d ) c k ( g ε ) / b ( D ) u k L ( R d ) , (1.13)where c := α − / k g − k / L ∞ . Combining this with (1.12), we obtain2 (cid:12)(cid:12)(cid:12)(cid:12) Re d X j =1 ( D j u , ( a εj ) ∗ u ) L ( R d ) (cid:12)(cid:12)(cid:12)(cid:12) k ( g ε ) / b ( D ) u k L ( R d ) + c k u k L ( R d ) , u ∈ H ( R d ; C n ) , < ε , (1.14)where c := 8 c C ( ν ) with ν := 2 − α k g − k − L ∞ .Next, by condition (1.8) on Q , for any ν > C Q ( ν ) > | ( Q ε u , u ) L ( R d ) | ν k Du k L ( R d ) + C Q ( ν ) k u k L ( R d ) , u ∈ H ( R d ; C n ) , < ε . (1.15)For fixed ν , the constant C Q ( ν ) is controlled in terms of d , s , k Q k L s (Ω) , and the parameters ofthe lattice Γ.We fix a constant λ in (1.10) as in [MSu2, Subsection 2.8]: λ := ( C Q ( ν ∗ ) + c ) k Q − k L ∞ for ν ∗ := 2 − α k g − k − L ∞ . (1.16)We return to the form (1.11). Extending the function u ∈ H ( O ; C n ) by zero to R d \ O andusing (1.5), (1.13), (1.14), and (1.15) with ν = ν ∗ , we obtain the lower estimate for the form(1.11): b D,ε [ u , u ] > a D,ε [ u , u ] > c ∗ k Du k L ( O ) , u ∈ H ( O ; C n ); (1.17) c ∗ := 14 α k g − k − L ∞ . (1.18)Next, by (1.6), (1.14), and (1.15) with ν = 1, we have b D,ε [ u , u ] C ∗ k u k H ( R d ) , u ∈ H ( O ; C n ) , where C ∗ := max { α k g k L ∞ + 1; C Q (1) + λ k Q k L ∞ + c } . Thus, the form b D,ε is closed. Thecorresponding selfadjoint operator in L ( O ; C n ) is denoted by B D,ε .By the Friedrichs inequality, (1.17) implies that b D,ε [ u , u ] > c ∗ (diam O ) − k u k L ( O ) , u ∈ H ( O ; C n ) . (1.19)Hence, the operator B D,ε is positive definite. By (1.17) and (1.19), k u k H ( O ) c k B / D,ε u k L ( O ) , u ∈ H ( O ; C n ); (1.20) c := c − / ∗ (cid:0) O ) (cid:1) / . (1.21)We also need an auxiliary operator e B D,ε . We factorize the matrix Q ( x ): there exists aΓ-periodic matrix-valued function f ( x ) such that f , f − ∈ L ∞ ( R d ) and Q ( x ) = ( f ( x ) ∗ ) − f ( x ) − . (1.22)(For instance, one can choose f ( x ) = Q ( x ) − / .) Let e B D,ε be a selfadjoint operator in L ( O ; C n )generated by the quadratic form e b D,ε [ u , u ] := b D,ε [ f ε u , f ε u ] (1.23) YU. M. MESHKOVA AND T. A. SUSLINA on the domain Dom e b D,ε := { u ∈ L ( O ; C n ) : f ε u ∈ H ( O ; C n ) } . In other words, e B D,ε =( f ε ) ∗ B D,ε f ε . Let e B ε denote the differential expression ( f ε ) ∗ B ε f ε . Note that( B D,ε − ζQ ε ) − = f ε ( e B D,ε − ζI ) − ( f ε ) ∗ . (1.24)1.5. The effective matrix and its properties.
The effective operator for A D,ε is given bythe differential expression A = b ( D ) ∗ g b ( D ) with the Dirichlet condition on ∂ O . Here g is aconstant effective matrix of size m × m . The matrix g is expressed in terms of the solution ofan auxiliary problem on the cell. Let an ( n × m )-matrix-valued function Λ( x ) be the (weak)Γ-periodic solution of the problem b ( D ) ∗ g ( x )( b ( D )Λ( x ) + m ) = 0 , Z Ω Λ( x ) d x = 0 . (1.25)Then the effective matrix is given by g := | Ω | − Z Ω e g ( x ) d x , (1.26) e g ( x ) := g ( x )( b ( D )Λ( x ) + m ) . (1.27)It can be checked that the matrix g is positive definite.According to [BSu3, (6.28) and Subsection 7.3], the solution of problem (1.25) satisfies k Λ k H (Ω) M. (1.28)Here the constant M depends only on m , α , k g k L ∞ , k g − k L ∞ , and the parameters of the latticeΓ. The effective matrix satisfies the estimates known as the Voigt–Reuss bracketing (see, e. g.,[BSu2, Chapter 3, Theorem 1.5]). Proposition 1.3.
Let g be the effective matrix (1.26) . Then g g g. (1.29) If m = n , then g = g . From (1.29) it follows that | g | k g k L ∞ , | ( g ) − | k g − k L ∞ . (1.30)Now we distinguish the cases where one of the inequalities in (1.29) becomes an identity, see[BSu2, Chapter 3, Propositions 1.6 and 1.7]. Proposition 1.4.
The identity g = g is equivalent to the relations b ( D ) ∗ g k ( x ) = 0 , k = 1 , . . . , m, (1.31) where g k ( x ) , k = 1 , . . . , m, are the columns of the matrix g ( x ) . Proposition 1.5.
The identity g = g is equivalent to the representations l k ( x ) = l k + b ( D ) w k , l k ∈ C m , w k ∈ e H (Ω; C m ) , k = 1 , . . . , m, (1.32) where l k ( x ) , k = 1 , . . . , m, are the columns of the matrix g ( x ) − . The effective operator.
To describe how the lower order terms of the operator B D,ε arehomogenized, we consider a Γ-periodic ( n × n )-matrix-valued function e Λ( x ) which is the (weak)solution of the problem b ( D ) ∗ g ( x ) b ( D ) e Λ( x ) + d X j =1 D j a j ( x ) ∗ = 0 , Z Ω e Λ( x ) d x = 0 . (1.33)According to [Su4, (7.51) and (7.52)], we have k e Λ k H (Ω) f M , (1.34)where the constant f M depends only on n , ρ , α , k g − k L ∞ , k a j k L ρ (Ω) , j = 1 , . . . , d , and theparameters of the lattice Γ. OMOGENIZATION OF THE FIRST INITIAL BOUNDARY-VALUE PROBLEM 9
Next, we define constant matrices V and W as follows: V := | Ω | − Z Ω ( b ( D )Λ( x )) ∗ g ( x )( b ( D ) e Λ( x )) d x , (1.35) W := | Ω | − Z Ω ( b ( D ) e Λ( x )) ∗ g ( x )( b ( D ) e Λ( x )) d x . (1.36)In L ( O ; C n ), consider the quadratic form b D [ u , u ] = ( g b ( D ) u , b ( D ) u ) L ( O ) + 2Re d X j =1 ( a j D j u , u ) L ( O ) − V u , b ( D ) u ) L ( O ) − ( W u , u ) L ( O ) + ( Q u , u ) L ( O ) + λ ( Q u , u ) L ( O ) , u ∈ H ( O ; C n ) . The following estimates were proved in [MSu3, (2.22) and (2.23)]: c ∗ k Du k L ( O ) b D [ u , u ] c k u k H ( O ) , u ∈ H ( O ; C n ) , (1.37) b D [ u , u ] > c ∗ (diam O ) − k u k L ( O ) , u ∈ H ( O ; C n ) . (1.38)Here the constant c depends only on the problem data (1.9). A selfadjoint operator in L ( O ; C n )corresponding to the form b D is denoted by B D . By (1.37) and (1.38), k u k H ( O ) c k ( B D ) / u k L ( O ) , u ∈ H ( O ; C n ) , (1.39)where c is given by (1.21).Due to condition ∂ O ∈ C , , the operator B D is given by B = b ( D ) ∗ g b ( D ) − b ( D ) ∗ V − V ∗ b ( D ) + d X j =1 ( a j + a ∗ j ) D j − W + Q + λQ (1.40)on the domain H ( O ; C n ) ∩ H ( O ; C n ), and we have k ( B D ) − k L ( O ) → H ( O ) b c. (1.41)Here the constant b c depends only on the problem data (1.9). To justify this fact, we refer to thetheorems about regularity of solutions of the strongly elliptic systems (see [McL, Chapter 4]). Remark 1.6.
Instead of condition ∂ O ∈ C , , one could impose the following implicit condition:a bounded Lipschitz domain O ⊂ R d is such that estimate (1.41) holds. For such domain theresults of the paper remain true. In the case of scalar elliptic operators, wide conditions on ∂ O ensuring estimate (1.41) can be found in [KoE] and [MaSh, Chapter 7] (in particular, it sufficesto assume that ∂ O ∈ C α , α > / f := (cid:0) Q (cid:1) − / . (1.42)By (1.22), | f | k f k L ∞ = k Q − k / L ∞ , | f − | k f − k L ∞ = k Q k / L ∞ . (1.43)In what follows, we will need the operator e B D := f B D f corresponding to the quadratic form e b D [ u , u ] := b D [ f u , f u ] , u ∈ H ( O ; C n ) . (1.44)Note that ( B D − ζQ ) − = f ( e B D − ζI ) − f . Approximation of the generalized resolvent ( B D,ε − ζQ ε ) − . Now we formulate theresults of the paper [MSu3], where the behavior of the generalized resolvent ( B D,ε − ζQ ε ) − was studied. Suppose that ζ ∈ C \ R + and | ζ | >
1. The principal term of approximation ofthe generalized resolvent ( B D,ε − ζQ ε ) − was found in [MSu3, Theorem 2.5]; approximationof this resolvent in the ( L → H )-norm with the corrector taken into account was foundin [MSu3, Theorem 2.6]; an appropriate approximation of the operator g ε b ( D )( B D,ε − ζQ ε ) − (corresponding to the “flux”) was obtained in [MSu3, Proposition 10.7].We choose the numbers ε , ε ∈ (0 ,
1] according to the following condition.
Condition 1.7.
Let
O ⊂ R d be a bounded domain. Denote ( ∂ O ) ε := n x ∈ R d : dist { x ; ∂ O} < ε o . Suppose that there exists a number ε ∈ (0 , such that the strip ( ∂ O ) ε can be covered by afinite number of open sets admitting diffeomorphisms of class C , rectifying the boundary ∂ O .Denote ε := ε (1 + r ) − , where r = diam Ω . Obviously, the number ε depends only on the domain O and the lattice Γ. Note thatCondition 1.7 is ensured only by the assumption that ∂ O is Lipschitz; we imposed a morerestrictive condition ∂ O ∈ C , in order to ensure estimate (1.41). Theorem 1.8 ([MSu3]) . Let
O ⊂ R d be a bounded domain of class C , . Suppose that theassumptions of Subsections are satisfied. Let ζ = | ζ | e iφ ∈ C \ R + , | ζ | > . Denote c ( φ ) := ( | sin φ | − , φ ∈ (0 , π/ ∪ (3 π/ , π ) , , φ ∈ [ π/ , π/ . Suppose that ε is subject to Condition . Then for < ε ε we have k ( B D,ε − ζQ ε ) − − ( B D − ζQ ) − k L ( O ) → L ( O ) C c ( φ ) ε | ζ | − / . The constant C depends only on the problem data (1.9) . We fix a linear continuous extension operator P O : H σ ( O ; C n ) → H σ ( R d ; C n ) , σ > . (1.45)Such a “universal” extension operator exists for any Lipschitz bounded domain (see [R]). Wehave k P O k H σ ( O ) → H σ ( R d ) C ( σ ) O , σ > , (1.46)where the constant C ( σ ) O depends only on σ and the domain O . By R O we denote the operatorof restriction of functions in R d to the domain O . We put K D ( ε ; ζ ) := R O (cid:0) [Λ ε ] b ( D ) + [ e Λ ε ] (cid:1) S ε P O ( B D − ζQ ) − . (1.47)The corrector (1.47) is a continuous mapping of L ( O ; C n ) to H ( O ; C n ). This can be easilychecked with the help of Proposition 1.2 and relations Λ, e Λ ∈ e H (Ω). Note that k εK D ( ε ; ζ ) k L ( O ) → H ( O ) = O (1) for small ε and fixed ζ . Theorem 1.9 ([MSu3]) . Suppose that the assumptions of Theorem are satisfied. Let K D ( ε ; ζ ) be given by (1.47) . Then for ζ ∈ C \ R + , | ζ | > , and < ε ε we have k ( B D,ε − ζQ ε ) − − ( B D − ζQ ) − − εK D ( ε ; ζ ) k L ( O ) → H ( O ) C c ( φ ) ε / | ζ | − / + C c ( φ ) ε. (1.48) Let e g ( x ) be the matrix-valued function (1.27) . We put G D ( ε ; ζ ) := e g ε S ε b ( D ) P O ( B D − ζQ ) − + g ε (cid:0) b ( D ) e Λ (cid:1) ε S ε P O ( B D − ζQ ) − . (1.49) Then for ζ ∈ C \ R + , | ζ | > , and < ε ε the operator g ε b ( D )( B D,ε − ζQ ε ) − correspondingto the “flux” satisfies k g ε b ( D )( B D,ε − ζQ ε ) − − G D ( ε ; ζ ) k L ( O ) → L ( O ) e C c ( φ ) / ε / | ζ | − / . (1.50) OMOGENIZATION OF THE FIRST INITIAL BOUNDARY-VALUE PROBLEM 11
The constants C , C , and e C depend only on the problem data (1.9) . In [MSu3, Theorem 9.2], estimates in a wider domain of the spectral parameter were obtained.It was assumed that ζ ∈ C \ [ c ♭ , ∞ ), where c ♭ is a common lower bound of the operators e B D,ε and e B D . We put c ♭ := 4 − α k g − k − L ∞ k Q k − L ∞ (diam O ) − , (1.51)using relations (1.18), (1.19), (1.22), (1.23), (1.38), (1.43), and (1.44). Theorem 1.10 ([MSu3]) . Let
O ⊂ R d be a bounded domain of class C , . Suppose that theassumptions of Subsections are satisfied. Let K D ( ε ; ζ ) be the corrector (1.47) and let G D ( ε ; ζ ) be the operator (1.49) . Suppose that ζ ∈ C \ [ c ♭ , ∞ ) , where c ♭ is given by (1.51) . Denote ψ := arg ( ζ − c ♭ ) , < ψ < π , and ̺ ♭ ( ζ ) := ( c ( ψ ) | ζ − c ♭ | − , | ζ − c ♭ | < ,c ( ψ ) , | ζ − c ♭ | > . (1.52) Suppose that the number ε is subject to Condition . For < ε ε we have k ( B D,ε − ζQ ε ) − − ( B D − ζQ ) − k L ( O ) → L ( O ) C ε̺ ♭ ( ζ ) , (1.53) k ( B D,ε − ζQ ε ) − − ( B D − ζQ ) − − εK D ( ε ; ζ ) k L ( O ) → H ( O ) C (cid:0) ε / ̺ ♭ ( ζ ) / + ε | ζ | / ̺ ♭ ( ζ ) (cid:1) , (1.54) k g ε b ( D )( B D,ε − ζQ ε ) − − G D ( ε ; ζ ) k L ( O ) → L ( O ) e C (cid:0) ε / ̺ ♭ ( ζ ) / + ε | ζ | / ̺ ♭ ( ζ ) (cid:1) . (1.55) The constants C , C , and e C depend only on the problem data (1.9) . Remark 1.11.
1) In (1.52), expression c ( ψ ) | ζ − c ♭ | − is inverse to the square of the distancefrom ζ to [ c ♭ , ∞ ). 2) The number (1.51) in Theorem 1.10 can be replaced by any common lowerbound of the operators e B D,ε and e B D . Let κ > ζ = 0), B D,ε converges to B D in the norm-resolvent sense. Therefore, for sufficientlysmall ε one can take c ♭ = λ k Q k − L ∞ − κ , where λ is the first eigenvalue of the operator B D .Under such choice of c ♭ , the constants in estimates become dependent on κ . 3) It makes senseto use estimates (1.53)–(1.55) for bounded values of | ζ | and small ε̺ ♭ ( ζ ). In this case, the value ε / ̺ ♭ ( ζ ) / + ε | ζ | / ̺ ♭ ( ζ ) is controlled in terms of Cε / ̺ ♭ ( ζ ) / . For large | ζ | and for φ separated from the points 0 and 2 π , it is preferable to use Theorems 1.8 and 1.9.1.8. Removal of the smoothing operator in the corrector.
It turns out that the smoothingoperator in the corrector can be removed under some additional assumptions on the matrix-valued functions Λ( x ) and e Λ( x ). Condition 1.12.
Suppose that the Γ -periodic solution Λ( x ) of problem (1.25) is bounded, i. e., Λ ∈ L ∞ ( R d ) . Some cases ensuring that Condition 1.12 is satisfied were distinguished in [BSu4, Lemma 8.7].
Proposition 1.13 ([BSu4]) . Suppose that at least one of the following assumptions is satisfied :1 ◦ ) d ◦ ) dimension d > is arbitrary, and the differential expression A ε is given by A ε = D ∗ g ε ( x ) D ,where g ( x ) is a symmetric matrix with real entries ;3 ◦ ) dimension d is arbitrary, and g = g , i. e., relations (1.32) are satisfied.Then Condition holds. In order to remove S ε in the term of the corrector involving e Λ ε , it suffices to impose thefollowing condition. Condition 1.14.
Suppose that the Γ -periodic solution e Λ( x ) of problem (1.33) is such that e Λ ∈ L p (Ω) , p = 2 for d = 1 , p > for d = 2 , p = d for d > . The following result was checked in [Su4, Proposition 8.11].
Proposition 1.15 ([Su4]) . Suppose that at least one of the following assumptions is satisfied :1 ◦ ) d ◦ ) dimension d is arbitrary, and A ε is given by A ε = D ∗ g ε ( x ) D , where g ( x ) is a symmetricmatrix with real entries.Then Condition is satisfied. Remark 1.16. If A ε = D ∗ g ε ( x ) D , where g ( x ) is a symmetric matrix with real entries, thenfrom [LaU, Chapter III, Theorem 13.1] it follows that Λ , e Λ ∈ L ∞ and the norm k Λ k L ∞ does notexceed a constant depending on d , k g k L ∞ , k g − k L ∞ , and Ω, while the norm k e Λ k L ∞ is controlledin terms of d , ρ , k g k L ∞ , k g − k L ∞ , k a j k L ρ (Ω) , j = 1 , . . . , d , and Ω. In this case, Conditions 1.12and 1.14 hold.In [MSu3, Theorem 7.6] the following result was obtained. Theorem 1.17 ([MSu3]) . Suppose that the assumptions of Theorem are satisfied. Supposethat Λ( x ) is subject to Condition and e Λ( x ) satisfies Condition . We put K D ( ε ; ζ ) := (Λ ε b ( D ) + e Λ ε )( B D − ζQ ) − , (1.56) G D ( ε ; ζ ) := e g ε b ( D )( B D − ζQ ) − + g ε (cid:0) b ( D ) e Λ (cid:1) ε ( B D − ζQ ) − . (1.57) Then for ζ ∈ C \ R + , | ζ | > , and < ε ε we have k ( B D,ε − ζQ ε ) − − ( B D − ζQ ) − − εK D ( ε ; ζ ) k L ( O ) → H ( O ) C c ( φ ) ε / | ζ | − / + C c ( φ ) ε, k g ε b ( D )( B D,ε − ζQ ε ) − − G D ( ε ; ζ ) k L ( O ) → L ( O ) e C c ( φ ) ε / | ζ | − / + e C c ( φ ) ε. Here the constants C , e C are as in (1.48) and (1.50) . The constants C and e C depend onlyon the problem data (1.9) , p , and the norms k Λ k L ∞ , k e Λ k L p (Ω) . Approximations in a wider domain of the spectral parameter were found in [MSu3, Theo-rem 9.8].
Theorem 1.18 ([MSu3]) . Suppose that the assumptions of Theorem and Conditions , are satisfied. Let K D ( ε ; ζ ) be the corrector (1.56) . Let G D ( ε ; ζ ) be given by (1.57) . Thenfor < ε ε and ζ ∈ C \ [ c ♭ , ∞ ) we have k ( B D,ε − ζQ ε ) − − ( B D − ζQ ) − − εK D ( ε ; ζ ) k L ( O ) → H ( O ) C (cid:0) ε / ̺ ♭ ( ζ ) / + ε | ζ | / ̺ ♭ ( ζ ) (cid:1) , k g ε b ( D )( B D,ε − ζQ ε ) − − G D ( ε ; ζ ) k L ( O ) → L ( O ) e C (cid:0) ε / ̺ ♭ ( ζ ) / + ε | ζ | / ̺ ♭ ( ζ ) (cid:1) . Here the constants C and e C depend only on the problem data (1.9) , p , and the norms k Λ k L ∞ , k e Λ k L p (Ω) . According to [MSu3, Remarks 7.9 and 9.9], we observe the following.
Remark 1.19.
If only Condition 1.12 (respectively, Condition 1.14) is satisfied, then thesmoothing operator S ε can be removed in the term of the corrector involving Λ ε (respectively,in the term containing e Λ ε ).1.9. The case where the corrector is equal to zero.
Suppose that g = g , i. e., relations(1.31) hold. Then the Γ-periodic solution of problem (1.25) is equal to zero: Λ( x ) = 0. Supposein addition that d X j =1 D j a j ( x ) ∗ = 0 . (1.58)Then the Γ-periodic solution of problem (1.33) is also equal to zero: e Λ( x ) = 0. According to[MSu3, Propositions 7.10 and 9.12], in this case the ( L → H )-estimate of sharp order O ( ε )holds. Proposition 1.20 ([MSu3]) . Suppose that relations (1.31) and (1.58) are satisfied. ◦ . Under the assumptions of Theorem , for ζ ∈ C \ R + , | ζ | > , and < ε we have k ( B D,ε − ζQ ε ) − − ( B D − ζQ ) − k L ( O ) → H ( O ) C c ( φ ) ε. (1.59) OMOGENIZATION OF THE FIRST INITIAL BOUNDARY-VALUE PROBLEM 13 ◦ . Under the assumptions of Theorem , for ζ ∈ C \ [ c ♭ , ∞ ) and < ε we have k ( B D,ε − ζQ ε ) − − ( B D − ζQ ) − k L ( O ) → H ( O ) ( C + C | ζ | / ) ε̺ ♭ ( ζ ) . (1.60) The constants C , C , and C depend only on the problem data (1.9) . Estimates in a strictly interior subdomain.
It is possible to improve the H -estimatesin a strictly interior subdomain O ′ of the domain O . In Theorems 8.1 and 9.14 of [MSu3], thefollowing result was obtained. Theorem 1.21 ([MSu3]) . Let O ′ be a strictly interior subdomain of the domain O . Denote δ := min (cid:8)
1; dist {O ′ ; ∂ O} (cid:9) . (1.61)1 ◦ . Under the assumptions of Theorem , for ζ ∈ C \ R + , | ζ | > , and < ε ε we have k ( B D,ε − ζQ ε ) − − ( B D − ζQ ) − − εK D ( ε ; ζ ) k L ( O ) → H ( O ′ ) c ( φ ) ε ( C ′ | ζ | − / δ − + C ′′ ) , k g ε b ( D )( B D,ε − ζQ ε ) − − G D ( ε ; ζ ) k L ( O ) → L ( O ′ ) c ( φ ) ε (cid:0) e C ′ | ζ | − / δ − + e C ′′ (cid:1) . The constants C ′ , C ′′ , e C ′ , and e C ′′ depend only on the problem data (1.9) . ◦ . Under the assumptions of Theorem , for ζ ∈ C \ [ c ♭ , ∞ ) and < ε ε we have k ( B D,ε − ζQ ε ) − − ( B D − ζQ ) − − εK D ( ε ; ζ ) k L ( O ) → H ( O ′ ) ε (cid:0) C ′ δ − ̺ ♭ ( ζ ) / + C ′′ | ζ | / ̺ ♭ ( ζ ) (cid:1) , (1.62) k g ε b ( D )( B D,ε − ζQ ε ) − − G D ( ε ; ζ ) k L ( O ) → L ( O ′ ) ε (cid:0) e C ′ δ − ̺ ♭ ( ζ ) / + e C ′′ | ζ | / ̺ ♭ ( ζ ) (cid:1) . (1.63) The constants C ′ , C ′′ , and e C ′ , e C ′′ depend only on the problem data (1.9) . If the matrix-valued functions Λ( x ) and e Λ( x ) satisfy some additional assumptions, this resultremains true with a simpler corrector. Approximations for ζ ∈ C \ R + , | ζ | >
1, were foundin [MSu3, Theorem 8.2].
Theorem 1.22 ([MSu3]) . Suppose that the assumptions of Theorem ◦ ) are satisfied. Sup-pose that the matrix-valued functions Λ( x ) and e Λ( x ) satisfy Conditions and , respec-tively. Let K D ( ε ; ζ ) and G D ( ε ; ζ ) be the operators defined by (1.56) and (1.57) . Then for < ε ε and ζ ∈ C \ R + , | ζ | > , we have k ( B D,ε − ζQ ε ) − − ( B D − ζQ ) − − εK D ( ε ; ζ ) k L ( O ) → H ( O ′ ) c ( φ ) ε ( C ′ | ζ | − / δ − + C ) , k g ε b ( D )( B D,ε − ζQ ε ) − − G D ( ε ; ζ ) k L ( O ) → L ( O ′ ) c ( φ ) ε ( e C ′ | ζ | − / δ − + e C ) . The constants C ′ and e C ′ are as in Theorem . The constants C and e C depend on theproblem data (1.9) , p , and the norms k Λ k L ∞ , k e Λ k L p (Ω) . Approximations in a wider domain of the parameter ζ are obtained in [MSu3, Theorem 9.15]. Theorem 1.23 ([MSu3]) . Suppose that the assumptions of Theorem ◦ ) are satisfied. Sup-pose that the matrix-valued functions Λ( x ) and e Λ( x ) are subject to Conditions and ,respectively. Let K D ( ε ; ζ ) be the corrector (1.56) , and let G D ( ε ; ζ ) be the operator (1.57) . Thenfor ζ ∈ C \ [ c ♭ , ∞ ) and < ε ε we have k ( B D,ε − ζQ ε ) − − ( B D − ζQ ) − − εK D ( ε ; ζ ) k L ( O ) → H ( O ′ ) ε (cid:0) C ′ δ − ̺ ♭ ( ζ ) / + C | ζ | / ̺ ♭ ( ζ ) (cid:1) , k g ε b ( D )( B D,ε − ζQ ε ) − − G D ( ε ; ζ ) k L ( O ) → L ( O ′ ) ε (cid:0) e C ′ δ − ̺ ♭ ( ζ ) / + e C | ζ | / ̺ ♭ ( ζ ) (cid:1) . Here the constants C ′ and e C ′ are as in (1.62) and (1.63) . The constants C and e C dependon the problem data (1.9) , p , and the norms k Λ k L ∞ , k e Λ k L p (Ω) . Statement of the problem. Main results
Statement of the problem.
We study the behavior of the solution of the first initialboundary-value problem Q ε ( x ) ∂ u ε ∂t ( x , t ) = − B ε u ε ( x , t ) , x ∈ O , t > u ε ( · , t ) | ∂ O = 0 , t > Q ε ( x ) u ε ( x ,
0) = ϕ ( x ) , x ∈ O . (2.1)Here ϕ ∈ L ( O ; C n ). (The solution is understood in the weak sense.) Let us find relationbetween u ε ( · , t ) and ϕ . According to (1.22), the function s ε ( x , t ) := ( f ε ( x )) − u ε ( x , t ) is thesolution of the problem ∂ s ε ∂t ( x , t ) = − e B ε s ε ( x , t ) x ∈ O , t > s ε ( · , t ) | ∂ O = 0 , t > s ε ( x ,
0) = ( f ε ( x )) ∗ ϕ ( x ) , x ∈ O . Then s ε ( · , t ) = e − e B D,ε t ( f ε ) ∗ ϕ and u ε ( · , t ) = f ε s ε ( · , t ) = f ε e − e B D,ε t ( f ε ) ∗ ϕ . Our goal is to study the behavior of the generalized solution u ε of the first initial boundary-value problem (2.1) in the small period limit. In other words, we are interested in approximationsof the sandwiched operator exponential f ε e − e B D,ε t ( f ε ) ∗ for small ε .The corresponding effective problem is given by Q ∂ u ∂t ( x , t ) = − B u ( x , t ) , x ∈ O , t > u ( · , t ) | ∂ O = 0 , t > Q u ( x ,
0) = ϕ ( x ) , x ∈ O . (2.2)By (1.42), the solution of the effective problem is given by u ( · , t ) = f e − e B D t f ϕ ( · ) . (2.3)2.2. The properties of the operator exponential.
We prove the following simple statementabout estimates for the operator exponentials e − e B D,ε t and e − e B D t . Lemma 2.1.
For < ε we have k e − e B D,ε t k L ( O ) → L ( O ) e − c ♭ t , t > , (2.4) k f ε e − e B D,ε t k L ( O ) → H ( O ) c t − / e − c ♭ t/ , t > , (2.5) k e − e B D t k L ( O ) → L ( O ) e − c ♭ t , t > , (2.6) k f e − e B D t k L ( O ) → H ( O ) c t − / e − c ♭ t/ , t > , (2.7) k f e − e B D t k L ( O ) → H ( O ) e ct − e − c ♭ t/ , t > . (2.8) Here the constants c and c ♭ are given by (1.21) and (1.51) . The constant e c depends only on theproblem data (1.9) .Proof. Since the number c ♭ defined by (1.51) is a common lower bound of the operators e B D,ε and e B D , estimates (2.4) and (2.6) are obvious.By (1.20) and (1.23), k f ε e − e B D,ε t k L ( O ) → H ( O ) c k B / D,ε f ε e − e B D,ε t k L ( O ) → L ( O ) = c k e B / D,ε e − e B D,ε t k L ( O ) → L ( O ) . (2.9)Since e B D,ε > c ♭ I , then k e B / D,ε e − e B D,ε t k L ( O ) → L ( O ) sup x > c ♭ x / e − xt e − c ♭ t/ sup x > c ♭ x / e − xt/ t − / e − c ♭ t/ . (2.10)Combining this with (2.9), we obtain inequality (2.5). Similarly, (1.39) and (1.44) imply esti-mate (2.7). OMOGENIZATION OF THE FIRST INITIAL BOUNDARY-VALUE PROBLEM 15
From (1.41), (1.43), and the identity e B D = f B D f it follows that k f e − e B D t k L ( O ) → H ( O ) b c k B D f e − e B D t k L ( O ) → L ( O ) b c k f − k L ∞ k e B D e − e B D t k L ( O ) → L ( O ) . Hence, k f e − e B D t k L ( O ) → H ( O ) b c k f − k L ∞ sup x > c ♭ xe − xt b c k f − k L ∞ t − e − c ♭ t/ . This proves estimate (2.8) with the constant e c = b c k f − k L ∞ . (cid:3) Approximation of the solution in L ( O ; C n ) .Theorem 2.2. Let
O ⊂ R d be a bounded domain of class C , . Suppose that the assumptionsof Subsections are satisfied. Let B D,ε be the operator in L ( O ; C n ) corresponding tothe quadratic form (1.11) . Let B D be the operator in L ( O ; C n ) given by expression (1.40) on H ( O ; C n ) ∩ H ( O ; C n ) . We put e B D,ε = ( f ε ) ∗ B D,ε f ε and e B D = f B D f , where the matrix-valued function f is defined by (1.22) , and the matrix f is given by (1.42) . Let u ε be the solutionof problem (2.1) , and let u be the solution of the corresponding effective problem (2.2) . Supposethat the number ε is subject to Condition . Then for < ε ε we have k u ε ( · , t ) − u ( · , t ) k L ( O ) C ε ( t + ε ) − / e − c ♭ t/ k ϕ k L ( O ) , t > . In the operator terms, k f ε e − e B D,ε t ( f ε ) ∗ − f e − e B D t f k L ( O ) → L ( O ) C ε ( t + ε ) − / e − c ♭ t/ , t > . (2.11) Here the constant c ♭ is given by (1.51) . The constant C depends only on the problem data (1.9) .Proof. The proof is based on the results of Theorems 1.8, 1.10, and representations for theexponentials of the operators e B D,ε , e B D in terms of the contour integrals of the correspondingresolvents.We have (see, e. g., [Ka, Chapter IX, Section 1.6]) e − e B D,ε t = − πi Z γ e − ζt ( e B D,ε − ζI ) − dζ, t > . (2.12)Here γ ⊂ C is a contour enclosing the spectrum of the operator e B D,ε in positive direction. Theexponential of the operator e B D satisfies a similar representation. Since the constant (1.51) is acommon lower bound of the operators e B D,ε and e B D , it is convenient to choose the contour ofintegration as follows: γ = { ζ ∈ C : Im ζ > , Re ζ = Im ζ + c ♭ / } ∪ { ζ ∈ C : Im ζ , Re ζ = − Im ζ + c ♭ / } . Multiplying (2.12) by f ε from the left and by ( f ε ) ∗ from the right and using identity (1.24),we obtain f ε e − e B D,ε t ( f ε ) ∗ = − πi Z γ e − ζt ( B D,ε − ζQ ε ) − dζ, t > . Similarly, f e − e B D t f = − πi Z γ e − ζt ( B D − ζQ ) − dζ, t > . Hence, f ε e − e B D,ε t ( f ε ) ∗ − f e − e B D t f = − πi Z γ e − ζt (cid:0) ( B D,ε − ζQ ε ) − − ( B D − ζQ ) − (cid:1) dζ. (2.13)By Theorems 1.8 and 1.10, we estimate the difference of the generalized resolvents for ζ ∈ γ uniformly in arg ζ . Recall the notation ψ = arg ( ζ − c ♭ ). Note that for ζ ∈ γ and ψ = π/ ψ = 3 π/ | ζ | = √ c ♭ /
2. We apply Theorem 1.10 for ζ ∈ γ such that | ζ | ˇ c , whereˇ c := max { √ c ♭ / } . (2.14) Obviously, ψ ∈ ( π/ , π/
4) on the contour γ and ρ ♭ ( ζ ) {
1; 8 c − ♭ } =: C , ζ ∈ γ. (2.15)Therefore, (1.53) implies that k ( B D,ε − ζQ ε ) − − ( B D − ζQ ) − k L ( O ) → L ( O ) C C ε C ′ | ζ | − / ε,ζ ∈ γ, | ζ | ˇ c, < ε ε ; C ′ := C C ˇ c / . (2.16)For other ζ ∈ γ , we have | sin φ | > − / , ζ ∈ γ, | ζ | > ˇ c, (2.17)and, by Theorem 1.8, k ( B D,ε − ζQ ε ) − − ( B D − ζQ ) − k L ( O ) → L ( O ) C ′′ | ζ | − / ε,ζ ∈ γ, | ζ | > ˇ c, < ε ε , (2.18)where C ′′ := 5 / C . As a result, combining (2.16) and (2.18), for 0 < ε ε we have k ( B D,ε − ζQ ε ) − − ( B D − ζQ ) − k L ( O ) → L ( O ) b C | ζ | − / ε, ζ ∈ γ, (2.19)where b C := max { C ′ ; C ′′ } .From (2.13) and (2.19) it follows that k f ε e − e B D,ε t ( f ε ) ∗ − f e − e B D t f k L ( O ) → L ( O ) π − b C εt − / Γ(1 / e − c ♭ t/ . Taking into account that Γ(1 /
2) = π / , we find k f ε e − e B D,ε t ( f ε ) ∗ − f e − e B D t f k L ( O ) → L ( O ) π − / b C εt − / e − c ♭ t/ ˇ C ε ( t + ε ) − / e − c ♭ t/ , t > ε , (2.20)where ˇ C := 2 √ π − / b C . For t ε we use a rough estimate k f ε e − e B D,ε t ( f ε ) ∗ − f e − e B D t f k L ( O ) → L ( O ) k f k L ∞ e − c ♭ t √ k f k L ∞ ε ( t + ε ) − / e − c ♭ t/ , t ε . (2.21)Relations (2.20) and (2.21) imply the required inequality (2.11) with the constant C :=max { ˇ C ; 2 √ k f k L ∞ } . (cid:3) Approximation of the solution in H ( O ; C n ) . We introduce a corrector K D ( t ; ε ) := R O (cid:16) [Λ ε ] S ε b ( D ) + [ e Λ ε ] S ε (cid:17) P O f e − e B D t f . (2.22)For t > L ( O ; C n ) to H ( O ; C n ). Indeed,according to (2.8), for t > f e − e B D t f is continuous from L ( O ; C n ) to H ( O ; C n ).Hence, the operator b ( D ) P O f e − e B D t f is continuous from L ( O ; C n ) to H ( R d ; C m ). Obviously,the operator P O f e − e B D t f is also continuous from L ( O ; C n ) to H ( R d ; C n ). It remains to usethe continuity of the operators [Λ ε ] S ε : H ( R d ; C m ) → H ( R d ; C n ) and [ e Λ ε ] S ε : H ( R d ; C n ) → H ( R d ; C n ) which follows from Proposition 1.2 and relations Λ , e Λ ∈ e H (Ω).We put e u ( · , t ) := P O u ( · , t ). By v ε we denote the first order approximation of the solution u ε of problem (2.1): e v ε ( · , t ) = e u ( · , t ) + ε Λ ε S ε b ( D ) e u ( · , t ) + ε e Λ ε S ε e u ( · , t ) , v ε ( · , t ) := e v ε ( · , t ) | O . (2.23)So, v ε ( · , t ) = f e − e B D t f ϕ ( · ) + ε K D ( t ; ε ) ϕ ( · ). Theorem 2.3.
Suppose that the assumptions of Theorem are satisfied. Suppose that thematrix-valued functions Λ( x ) and e Λ( x ) are Γ -periodic solutions of the problems (1.25) and (1.33) , OMOGENIZATION OF THE FIRST INITIAL BOUNDARY-VALUE PROBLEM 17 respectively. Let S ε be the Steklov smoothing operator (1.1) , and let P O be the extension oper-ator (1.45) . We put e u ( · , t ) = P O u ( · , t ) . Suppose that v ε is defined by (2.23) . Then for < ε ε and t > we have k u ε ( · , t ) − v ε ( · , t ) k H ( O ) C ( ε / t − / + εt − ) e − c ♭ t/ k ϕ k L ( O ) . In the operator terms, k f ε e − e B D,ε t ( f ε ) ∗ − f e − e B D t f − ε K D ( t ; ε ) k L ( O ) → H ( O ) C ( ε / t − / + εt − ) e − c ♭ t/ , (2.24) where K D ( t ; ε ) is the corrector (2.22) . Suppose that the matrix-valued function e g ( x ) is definedby (1.27) . For < ε ε and t > the flux p ε := g ε b ( D ) u ε satisfies (cid:13)(cid:13) p ε ( · , t ) − e g ε S ε b ( D ) e u ( · , t ) − g ε (cid:0) b ( D ) e Λ (cid:1) ε S ε e u ( · , t ) (cid:13)(cid:13) L ( O ) e C ε / t − / e − c ♭ t/ k ϕ k L ( O ) . In the operator terms, k g ε b ( D ) f ε e − e B D,ε t ( f ε ) ∗ − G D ( t ; ε ) k L ( O ) → L ( O ) e C ε / t − / e − c ♭ t/ . (2.25) Here G D ( t ; ε ) := e g ε S ε b ( D ) P O f e − e B D t f + g ε (cid:0) b ( D ) e Λ (cid:1) ε S ε P O f e − e B D t f . The constants C and e C depend only on the problem data (1.9) .Proof. As in the proof of Theorem 2.2, we use representations for the sandwiched operatorexponentials in terms of the contour integrals of the corresponding generalized resolvents. Wehave f ε e − e B D,ε t ( f ε ) ∗ − f e − e B D t f − ε K D ( t ; ε )= − πi Z γ e − ζt (cid:0) ( B D,ε − ζQ ε ) − − ( B D − ζQ ) − − εK D ( ε ; ζ ) (cid:1) dζ. (2.26)Here K D ( ε ; ζ ) is the operator (1.47).Similarly to (2.16)–(2.19), by Theorems 1.9 and 1.10, k ( B D,ε − ζQ ε ) − − ( B D − ζQ ) − − εK D ( ε ; ζ ) k L ( O ) → H ( O ) b C (cid:16) ε / | ζ | − / + ε (cid:17) , ζ ∈ γ, < ε ε , (2.27)with the constant b C := max { C ′ ; C ′′ } , where C ′ := (1+ˇ c ) / C C and C ′′ := max { C ; 25 C } .Relations (2.26) and (2.27) imply the required estimate (2.24) with the constant C := 2 π − Γ(3 / b C .Similarly, the identity g ε b ( D ) f ε e − e B D,ε t ( f ε ) ∗ − G D ( t ; ε ) = − πi Z γ e − ζt (cid:0) g ε b ( D )( B D,ε − ζQ ε ) − − G D ( ε ; ζ ) (cid:1) dζ (2.28)and estimates (1.50), (1.55) yield the inequality (2.25) with the constant e C := 2 π − Γ(3 /
4) max n / e C ; 2ˇ c / (1 + ˇ c ) / e C C o . (cid:3) By Remark 1.11(2), we observe the following.
Remark 2.4.
Let λ be the first eigenvalue of the operator B D , and let κ > ε ◦ thenumber λ k Q k − L ∞ − κ/ e B D,ε for all 0 < ε ε ◦ .Therefore, we can shift the integration contour so that it will intersect the real axis at thepoint c := λ k Q k − L ∞ − κ instead of c ♭ /
2. By this way, we obtain estimates (2.11), (2.24), and(2.25) with e − c ♭ t/ replaced by e − c t in the right-hand sides. The constants in estimates becomedependent on κ . Estimates for small time.
Note that for 0 < t < ε it makes no sense to apply estimates(2.24) and (2.25), since it is better to use the following simple statement (which is valid, however,for all t > Proposition 2.5.
Suppose that the assumptions of Theorem are satisfied. Then for t > and < ε we have k f ε e − e B D,ε t ( f ε ) ∗ − f e − e B D t f k L ( O ) → H ( O ) C t − / e − c ♭ t/ , (2.29) k g ε b ( D ) f ε e − e B D,ε t ( f ε ) ∗ k L ( O ) → L ( O ) e C t − / e − c ♭ t/ , (2.30) k g b ( D ) f e − e B D t f k L ( O ) → L ( O ) e C t − / e − c ♭ t/ , (2.31) where the constants C := 2 c k f k L ∞ and e C := k g k / L ∞ k f k L ∞ depend only on the problem data (1.9) .Proof. Inequality (2.29) follows from (1.43), (2.5), and (2.7).Next, by (1.23), k g ε b ( D ) f ε e − e B D,ε t ( f ε ) ∗ k L ( O ) → L ( O ) k g k / L ∞ k f k L ∞ k e B / D,ε e − e B D,ε t k L ( O ) → L ( O ) . Together with (2.10), this yields (2.30). By (1.43) and (1.44), estimate (2.31) is checked similarly. (cid:3)
Removal of the smoothing operator S ε in the corrector. It is possible to removethe smoothing operator in the corrector if the matrix-valued functions Λ( x ) and e Λ( x ) satisfyConditions 1.12 and 1.14, respectively. The following result is checked similarly to Theorem 2.3by using Theorems 1.17 and 1.18. Theorem 2.6.
Suppose that the assumptions of Theorem are satisfied. Suppose that thematrix-valued functions Λ( x ) and e Λ( x ) satisfy Conditions and , respectively. We put K D ( t ; ε ) := (Λ ε b ( D ) + e Λ ε ) f e − e B D t f , (2.32) G D ( t ; ε ) := e g ε b ( D ) f e − e B D t f + g ε (cid:0) b ( D ) e Λ (cid:1) ε f e − e B D t f . (2.33) Then for t > and < ε ε we have k f ε e − e B D,ε t ( f ε ) ∗ − f e − e B D t f − ε K D ( t ; ε ) k L ( O ) → H ( O ) C (cid:16) ε / t − / + εt − (cid:17) e − c ♭ t/ , k g ε b ( D ) f ε e − e B D,ε t ( f ε ) ∗ − G D ( t ; ε ) k L ( O ) → L ( O ) e C (cid:16) ε / t − / + εt − (cid:17) e − c ♭ t/ . The constants C and e C depend on the problem data (1.9) , p , and the norms k Λ k L ∞ and k e Λ k L p (Ω) . By Remark 1.19, we observe the following.
Remark 2.7.
If only Condition 1.12 (Condition 1.14, respectively) is satisfied, then the smooth-ing operator S ε can be removed in the term of the corrector containing Λ ε ( e Λ ε , respectively).2.7. The case of smooth boundary.
It is also possible to remove the smoothing operator S ε in the corrector by increasing smoothness of the boundary. In this subsection, we consider thecase where d >
3, because for d Lemma 2.8.
Let k > be an integer. Let O ⊂ R d be a bounded domain with the boundary ∂ O of class C k − , . Then for t > the operator e − e B D t is a continuous mapping of L ( O ; C n ) to H q ( O ; C n ) , q k , and k e − e B D t k L ( O ) → H q ( O ) b C q t − q/ e − c ♭ t/ , t > . (2.34) The constant b C q depends only on q and the problem data (1.9) . OMOGENIZATION OF THE FIRST INITIAL BOUNDARY-VALUE PROBLEM 19
Proof.
It suffices to check estimate (2.34) for integer q ∈ [0 , k ]; then the result for non-integer q follows by interpolation. For q = 0 , , q be an integer such that 2 q k . By theorems about regularity of solutionsof strongly elliptic systems (see, e. g., [McL, Chapter 4]), the operator ( e B D ) − is continuousfrom H σ ( O ; C n ) to H σ +2 ( O ; C n ) under the assumption ∂ O ∈ C σ +1 , , where σ ∈ Z + . We alsotake into account that the operator ( e B D ) − / is continuous from L ( O ; C n ) to H ( O ; C n ). Itfollows that, under the assumptions of lemma, for integer q ∈ [2 , k ] the operator ( e B D ) − q/ is acontinuous mapping of L ( O ; C n ) to H q ( O ; C n ). We have k ( e B D ) − q/ k L ( O ) → H q ( O ) ˇC q , (2.35)where the constant ˇC q depends on q and the problem data (1.9). From (2.35) it follows that k e − e B D t k L ( O ) → H q ( O ) ˇC q k ( e B D ) q/ e − e B D t k L ( O ) → L ( O ) ˇC q sup x > c ♭ x q/ e − xt ˇC q t − q/ e − c ♭ t/ sup x > x q/ e − x/ b C q t − q/ e − c ♭ t/ , where b C q := ˇC q ( q/e ) q/ . (cid:3) Using Lemma 2.8, the properties of the matrix-valued functions Λ( x ) and e Λ( x ), and theproperties of the operator S ε , we can estimate the difference of the correctors (2.22) and (2.32). Lemma 2.9.
Let d > . Let O ⊂ R d be a bounded domain of class C d/ , if d is even and of class C ( d +1) / , if d is odd. Let K D ( t ; ε ) be the operator (2.22) , and let K D ( t ; ε ) be the operator (2.32) .Then for < ε and t > we have kK D ( t ; ε ) − K D ( t ; ε ) k L ( O ) → H ( O ) b C d ( t − + t − d/ − / ) e − c ♭ t/ . (2.36) The constant b C d depends only on the problem data (1.9) . Lemma 2.9 and Theorem 2.3 imply the following result.
Theorem 2.10.
Suppose that the assumptions of Theorem are satisfied, and d > . Supposethat the domain O satisfies the assumptions of Lemma . Let K D ( t ; ε ) be the corrector (2.32) .Let G D ( t ; ε ) be the operator (2.33) . Then for t > and < ε ε we have k f ε e − e B D,ε t ( f ε ) ∗ − f e − e B D t f − ε K D ( t ; ε ) k L ( O ) → H ( O ) C d ( ε / t − / + εt − d/ − / ) e − c ♭ t/ , (2.37) k g ε b ( D ) f ε e − e B D,ε t ( f ε ) ∗ − G D ( t ; ε ) k L ( O ) → L ( O ) e C d ( ε / t − / + εt − d/ − / ) e − c ♭ t/ . (2.38) The constants C d and e C d depend only on the problem data (1.9) . The proofs of Lemma 2.9 and Theorem 2.10 are presented in Appendix (see §
7) in order notto clutter the main presentation. Clearly, it is convenient to apply Theorem 2.10 if t is separatedfrom zero. For small t the order of the factor ( ε / t − / + εt − d/ − / ) grows with dimension.This is a “charge” for the removal of the smoothing operator. Remark 2.11.
Instead of the smoothness assumption on ∂ O from Lemma 2.9, we could imposethe following implicit condition: a bounded domain O with Lipschitz boundary is such thatestimate (2.34) holds for q = d/ The case of zero corrector.
Suppose that g = g , i. e., relations (1.31) are satisfied.Suppose also that condition (1.58) is satisfied. Then the Γ-periodic solutions of problems (1.25)and (1.33) are equal to zero: Λ( x ) = 0 and e Λ( x ) = 0. Using Proposition 1.20, we obtain thefollowing result. Proposition 2.12.
Suppose that relations (1.31) and (1.58) are satisfied. Then, under theassumptions of Theorem , for < ε we have k f ε e − e B D,ε t ( f ε ) ∗ − f e − e B D t f k L ( O ) → H ( O ) C εt − e − c ♭ t/ , t > , (2.39) where the constant C depends only on the problem data (1.9) .Proof. We rely on identity (2.13). For | ζ | ˇ c , where ˇ c is the constant (2.14), we use (1.60)and (2.15). For | ζ | > ˇ c we apply (1.59) and (2.17). As a result, we see that for 0 < ε k ( B D,ε − ζQ ε ) − − ( B D − ζQ ) − k L ( O ) → H ( O ) b C ε, ζ ∈ γ ; b C := max (cid:8)(cid:0) C + C (1 + ˇ c ) / (cid:1) C ; 25 C (cid:9) . Together with (2.13), this yields (2.39) with the constant C := 2 π − b C . (cid:3) Special case.
Now, we assume that g = g , i. e., relations (1.32) are satisfied. Then,by Proposition 1.13(3 ◦ ), Condition 1.12 is satisfied. By [BSu3, Remark 3.5], the matrix-valuedfunction (1.27) is constant and coincides with g , i. e., e g ( x ) = g = g . Thus, e g ε b ( D ) f e − e B D t f = g b ( D ) f e − e B D t f .Suppose in addition that relation (1.58) is satisfied. Then e Λ( x ) = 0. The following result canbe deduced from Theorem 2.3 and Proposition 1.1. Proposition 2.13.
Suppose that the relations (1.32) and (1.58) are satisfied. Then, under theassumptions of Theorem , for < ε ε and t > we have k g ε b ( D ) f ε e − e B D,ε t ( f ε ) ∗ − g b ( D ) f e − e B D t f k L ( O ) → L ( O ) e C ′ ε / t − / e − c ♭ t/ . (2.40) The constant e C ′ depends only on the problem data (1.9) .Proof. From Theorem 2.3 it follows that k g ε b ( D ) f ε e − e B D,ε t ( f ε ) ∗ − g S ε b ( D ) P O f e − e B D t f k L ( O ) → L ( O ) e C ε / t − / e − c ♭ t/ . (2.41)On the one hand, Proposition 1.1 and relations (1.3), (1.30), (1.43), (1.46), (2.8) imply that k g ( S ε − I ) b ( D ) P O f e − e B D t f k L ( O ) → L ( R d ) ε k g k L ∞ r α / k P O f e − e B D t f k L ( O ) → H ( R d ) ε k g k L ∞ k f k L ∞ r α / C (2) O e ct − e − c ♭ t/ . (2.42)On the other hand, from (1.2), (1.3), (1.30), (1.43), (1.46), and (2.7) it follows that k g ( S ε − I ) b ( D ) P O f e − e B D t f k L ( O ) → L ( R d ) k g k L ∞ α / k P O f e − e B D t f k L ( O ) → H ( R d ) k g k L ∞ k f k L ∞ α / C (1) O c t − / e − c ♭ t/ . (2.43)By (2.42) and (2.43), k g ( S ε − I ) b ( D ) P O f e − e B D t f k L ( O ) → L ( R d ) ˇ C ε / t − / e − c ♭ t/ , where ˇ C := k g k L ∞ k f k L ∞ α / (cid:0) r C (1) O C (2) O e cc (cid:1) / . Combining this with (2.41), we obtainestimate (2.40) with the constant e C ′ := e C + ˇ C . (cid:3) Estimates in a strictly interior subdomain.
Using Theorem 1.21, we improve errorestimates in a strictly interior subdomain.
Theorem 2.14.
Suppose that the assumptions of Theorem are satisfied. Let O ′ be a strictlyinterior subdomain of the domain O , and let δ be defined by (1.61) . Then for < ε ε and t > we have k f ε e − e B D,ε t ( f ε ) ∗ − f e − e B D t f − ε K D ( t ; ε ) k L ( O ) → H ( O ′ ) ε ( C t − / δ − + C t − ) e − c ♭ t/ , (2.44) k g ε b ( D ) f ε e − e B D,ε t ( f ε ) ∗ − G D ( t ; ε ) k L ( O ) → L ( O ′ ) ε ( e C t − / δ − + e C t − ) e − c ♭ t/ . The constants C , C , e C , and e C depend only on the problem data (1.9) .Proof. The proof is based on application of Theorem 1.21 and relations (2.26), (2.28). Also,estimates (2.15) and (2.17) are used. We omit the details. (cid:3)
OMOGENIZATION OF THE FIRST INITIAL BOUNDARY-VALUE PROBLEM 21
The following result is checked similarly with the help of Theorems 1.22 and 1.23.
Theorem 2.15.
Suppose that the assumptions of Theorem are satisfied. Suppose thatthe matrix-valued functions Λ( x ) and e Λ( x ) satisfy Conditions and , respectively. Let K D ( t ; ε ) be the corrector (2.32) , and let G D ( t ; ε ) be the operator (2.33) . Then for t > and < ε ε we have k f ε e − e B D,ε t ( f ε ) ∗ − f e − e B D t f − ε K D ( t ; ε ) k L ( O ) → H ( O ′ ) ε ( C t − / δ − + C t − ) e − c ♭ t/ , k g ε b ( D ) f ε e − e B D,ε t ( f ε ) ∗ − G D ( t ; ε ) k L ( O ) → L ( O ′ ) ε ( e C t − / δ − + e C t − ) e − c ♭ t/ . The constants C and e C are the same as in Theorem . The constants C and e C dependon the problem data (1.9) , p , and the norms k Λ k L ∞ , k e Λ k L p (Ω) . Note that it is possible to remove the smoothing operator S ε in the corrector in estimatesof Theorem 2.14 without any additional assumptions on the matrix-valued functions Λ( x ) and e Λ( x ). For this, the additional smoothness of the boundary is not required. We consider the casewhere d > t > e − e B D t is continuous from L ( O ; C n ) to H ( O ; C n ) and estimate (2.8)holds. Moreover, the following property of “regularity improvement” inside the domain is valid:for t > e − e B D t is continuous from L ( O ; C n ) to H σ ( O ′ ; C n ) for any integer σ > k e − e B D t k L ( O ) → H σ ( O ′ ) C ′ σ t − / ( δ − + t − ) ( σ − / e − c ♭ t/ ,t > , σ ∈ N , σ > . (2.45)The constant C ′ σ depends on σ and the problem data (1.9). For the scalar parabolic equations,the property of “regularity improvement” inside the domain was obtained in [LaSoU, Chapter 3, § e B D . It is easy to deduce thequalified estimates (2.45), noticing that the derivatives D α u (where u is the function (2.3)with ϕ ∈ L ( O ; C n )) are solutions of a parabolic equation Q ∂ t D α u = − B D α u . We multiplythis equation by χ D α u and integrate over the cylinder O × (0 , t ). Here χ is a smooth cut-offfunction equal to zero near the lateral surface and the bottom of the cylinder. The standardanalysis of the corresponding integral identity together with the already known inequalities ofLemma 2.1 leads to estimates (2.45).Using the properties of the matrix-valued functions Λ( x ) and e Λ( x ), and also the propertiesof the operator S ε , we can deduce the following statement from relation (2.45). Lemma 2.16.
Suppose that the assumptions of Theorem are satisfied and that d > . Let K D ( t ; ε ) be the operator (2.32) . Denote h d ( δ ; t ) := t − + t − / ( δ − + t − ) d/ . (2.46) Let r = diam Ω . Then for < ε (4 r ) − δ and t > we have kK D ( t ; ε ) − K D ( t ; ε ) k L ( O ) → H ( O ′ ) C ′′ d h d ( δ ; t ) e − c ♭ t/ . (2.47) The constant C ′′ d depends only on the problem data (1.9) . From Lemma 2.16 and Theorem 2.14 we deduce the following result.
Theorem 2.17.
Suppose that the assumptions of Theorem are satisfied, and d > . Let K D ( t ; ε ) be the corrector (2.32) , and let G D ( t ; ε ) be the operator (2.33) . Let r = diam Ω . Thenfor < ε min { ε ; (4 r ) − δ } and t > we have k f ε e − e B D,ε t ( f ε ) ∗ − f e − e B D t f − ε K D ( t ; ε ) k L ( O ) → H ( O ′ ) ε C d h d ( δ ; t ) e − c ♭ t/ , (2.48) k g ε b ( D ) f ε e − e B D,ε t ( f ε ) ∗ − G D ( t ; ε ) k L ( O ) → L ( O ′ ) ε e C d h d ( δ ; t ) e − c ♭ t/ . (2.49) Here h d ( δ ; t ) is given by (2.46) , the constants C d and e C d depend only on the problem data (1.9) . The proofs of Lemma 2.16 and Theorem 2.17 are presented in Appendix (see §
8) in order notto clutter the main presentation. Clearly, it is convenient to apply Theorem 2.17 if t is separatedfrom zero. For small t the order of the factor h d ( δ ; t ) grows with dimension. This is a “charge”for removal of the smoothing operator. Homogenization of the first initial boundary value problemfor nonhomogeneous equation
The principal term of approximation.
In this section, we study the behavior of thesolution of the first initial boundary value problem for a nonhomogeneous parabolic equation: Q ε ( x ) ∂ u ε ∂t ( x , t ) = − B ε u ε ( x , t ) + F ( x , t ) , x ∈ O , t > u ε ( · , t ) | ∂ O = 0 , t > Q ε ( x ) u ε ( x ,
0) = ϕ ( x ) , x ∈ O . (3.1)Here F ∈ H r ( T ) := L r ((0 , T ); L ( O ; C n )), 0 < T ∞ , with some 1 r ∞ . Then u ε ( · , t ) = f ε e − e B D,ε t ( f ε ) ∗ ϕ ( · ) + t Z f ε e − e B D,ε ( t − e t ) ( f ε ) ∗ F ( · , e t ) d e t. (3.2)The corresponding effective problem takes the form Q ∂ u ∂t ( x , t ) = − B u ( x , t ) + F ( x , t ) , x ∈ O , t > u ( · , t ) | ∂ O = 0 , t > Q u ( x ,
0) = ϕ ( x ) , x ∈ O . (3.3)The solution of this problem is given by u ( · , t ) = f e − e B D t f ϕ ( · ) + t Z f e − e B D ( t − e t ) f F ( · , e t ) d e t. (3.4)Subtracting (3.4) from (3.2) and using Theorem 2.2 (see (2.11)), we conclude that for 0 < ε ε and t > k u ε ( · , t ) − u ( · , t ) k L ( O ) C ε ( t + ε ) − / e − c ♭ t/ k ϕ k L ( O ) + C ε L ( ε ; t ; F ) , where L ( ε ; t ; F ) := t Z e − c ♭ ( t − e t ) / ( ε + t − e t ) − / k F ( · , e t ) k L ( O ) d e t. Estimating the term L ( ε ; t ; F ), for the case 1 < r ∞ we obtain the following result. Its proofis completely analogous to the proof of Theorem 5.1 from [MSu1]. Theorem 3.1.
Suppose that
O ⊂ R d is a bounded domain of class C , . Suppose that theassumptions of Subsections are satisfied. Let u ε be the solution of problem (3.1) , and let u be the solution of the effective problem (3.3) with ϕ ∈ L ( O ; C n ) and F ∈ H r ( T ) , < T ∞ ,with some < r ∞ . Then for < ε ε and < t < T we have k u ε ( · , t ) − u ( · , t ) k L ( O ) C ε ( t + ε ) − / e − c ♭ t/ k ϕ k L ( O ) + c r θ ( ε, r ) k F k H r ( T ) . Here θ ( ε, r ) is given by θ ( ε, r ) = ε − /r , < r < ,ε ( | ln ε | + 1) / , r = 2 ,ε, < r ∞ . (3.5) The constant c r depends only on r and the problem data (1.9) . By analogy with the proof of Theorem 5.2 from [MSu1], we can deduce approximation of thesolution of problem (3.1) in H r ( T ) from Theorem 2.2. Theorem 3.2.
Suppose that the assumptions of Theorem are satisfied. Let u ε and u bethe solutions of problems (3.1) and (3.3) , respectively, with ϕ ∈ L ( O ; C n ) and F ∈ H r ( T ) , < T ∞ , for some r < ∞ . Then for < ε ε we have k u ε − u k H r ( T ) c r ′ θ ( ε, r ′ ) k ϕ k L ( O ) + C ε k F k H r ( T ) . Here θ ( ε, · ) is given by (3.5) , r − + ( r ′ ) − = 1 . The constant C depends only on the problemdata (1.9) , the constant c r ′ depends on the same parameters and r . OMOGENIZATION OF THE FIRST INITIAL BOUNDARY-VALUE PROBLEM 23
Remark 3.3.
For the case where ϕ = 0 and F ∈ H ∞ ( T ), Theorem 3.1 implies that k u ε − u k H ∞ ( T ) c ∞ ε k F k H ∞ ( T ) , < ε ε . Approximation of the solution in H ( O ; C n ) . Now, we obtain approximation of thesolution of problem (3.1) in the H ( O ; C n )-norm with the help of Theorem 2.3. The difficultiesarise in consideration of the integral term in (3.2), because estimate (2.24) “deteriorates” forsmall t . Assuming that t > ε , we divide the integration interval in (3.2) into two parts:(0 , t − ε ) and ( t − ε , t ). On the interval (0 , t − ε ) we apply (2.24), and on ( t − ε , t ) we use(2.29).Denote w ε ( · , t ) := f e − e B D ε f − u ( · , t − ε ) , (3.6)where u is the solution of problem (3.3). By (3.4), w ε ( · , t ) = f e − e B D t f ϕ ( · ) + t − ε Z f e − e B D ( t − e t ) f F ( · , e t ) d e t. The following statement can be checked similarly to Theorem 5.4 from [MSu1].
Theorem 3.4.
Suppose that the assumptions of Theorem are satisfied. Suppose that u ε and u are the solutions of problems (3.1) and (3.3) , respectively, with ϕ ∈ L ( O ; C n ) and F ∈ H r ( T ) , < T ∞ , for some < r ∞ . Let w ε ( · , t ) be given by (3.6) . Let Λ( x ) and e Λ( x ) be the Γ -periodic matrix solutions of problems (1.25) and (1.33) , respectively. Suppose that P O is a linear continuous extension operator (1.45) . Let S ε be the Steklov smoothing operator (1.1) .We put e w ε ( · , t ) := P O w ε ( · , t ) and denote v ε ( · , t ) := u ( · , t ) + ε Λ ε S ε b ( D ) e w ε ( · , t ) + ε e Λ ε S ε e w ε ( · , t ) . Let p ε ( · , t ) := g ε b ( D ) u ε ( · , t ) , and let e g ( x ) be the matrix-valued function (1.27) . We put q ε ( · , t ) := e g ε S ε b ( D ) e w ε ( · , t ) + g ε (cid:0) b ( D ) e Λ (cid:1) ε S ε e w ε ( · , t ) . Then for < ε ε and ε t < T we have k u ε ( · , t ) − v ε ( · , t ) k H ( O ) C ε / t − / e − c ♭ t/ k ϕ k L ( O ) + ˇ c r ω ( ε, r ) k F k H r ( T ) , k p ε ( · , t ) − q ε ( · , t ) k L ( O ) e C ε / t − / e − c ♭ t/ k ϕ k L ( O ) + e c r ω ( ε, r ) k F k H r ( T ) . Here constants ˇ c r and e c r depend only on the problem data (1.9) and r , and ω ( ε, r ) := ε − /r , < r < ,ε / ( | ln ε | + 1) / , r = 4 ,ε / , < r ∞ . (3.7)Since the right-hand side of estimate (2.25) grows slowly than the right-hand side in estimate(2.24), as t →
0, for r > p ε in terms of h ε ( · , t ) := e g ε S ε b ( D ) e u ( · , t ) + g ε (cid:0) b ( D ) e Λ (cid:1) ε S ε e u ( · , t ) . (3.8) Proposition 3.5.
Suppose that the assumptions of Theorem are satisfied. Suppose that u ε and u are the solutions of problems (3.1) and (3.3) , respectively, with ϕ ∈ L ( O ; C n ) and F ∈ H r ( T ) , < T ∞ , for some < r ∞ . Let p ε ( · , t ) = g ε b ( D ) u ε ( · , t ) and let h ε ( · , t ) begiven by (3.8) . Then for < t < T and < ε ε we have k p ε ( · , t ) − h ε ( · , t ) k L ( O ) e C ε / t − / e − c ♭ t/ k ϕ k L ( O ) + C ( r )24 ε / k F k H p ( t ) . (3.9) The constant C ( r )24 depends only on the problem data (1.9) and r .Proof. To check estimate (3.9), we use inequality (2.25) and identities (3.2), (3.4). If r = ∞ , wededuce (3.9) with C ( ∞ )24 := (2 /c ♭ ) / Γ(1 / e C . If 4 < r < ∞ , we apply the H¨older inequality: k p ε ( · , t ) − h ε ( · , t ) k L ( O ) e C ε / t − / e − c ♭ t/ k ϕ k L ( O ) + e C ε / k F k H r ( t ) I r ( ε, t ) /r ′ , r − + ( r ′ ) − = 1 . Here I r ( ε, t ) := t Z τ − r ′ / e − c ♭ r ′ τ/ dτ ( c ♭ r ′ / r ′ / − Γ(1 − r ′ / . This implies (3.9) with the constant C ( r )24 := ( c ♭ r ′ / / − /r ′ Γ(1 − r ′ / /r ′ e C . (cid:3) Combining Proposition 2.5 and Theorem 2.6, we deduce the following result.
Theorem 3.6.
Suppose that the assumptions of Theorem are satisfied. Suppose that thematrix-valued functions Λ( x ) and e Λ( x ) satisfy Conditions and , respectively. Denote ˇ v ε ( · , t ) := u ( · , t ) + ε Λ ε b ( D ) w ε ( · , t ) + ε e Λ ε w ε ( · , t ) , (3.10)ˇ q ε ( · , t ) := e g ε b ( D ) w ε ( · , t ) + g ε (cid:0) b ( D ) e Λ (cid:1) ε w ε ( · , t ) . (3.11) Then for < ε ε and ε t < T we have k u ε ( · , t ) − ˇ v ε ( · , t ) k H ( O ) C ε / t − / e − c ♭ t/ k ϕ k L ( O ) + c ′ r ω ( ε, r ) k F k H r ( t ) , k p ε ( · , t ) − ˇ q ε ( · , t ) k L ( O ) e C ε / t − / e − c ♭ t/ k ϕ k L ( O ) + c ′′ r ω ( ε, r ) k F k H r ( t ) . The constants c ′ r and c ′′ r depend only on the initial data (1.9) , r , p , and the norms k Λ k L ∞ , k e Λ k L p (Ω) . For the case of sufficiently smooth boundary, we could apply Theorem 2.10. However, becauseof the strong growth of the right-hand side in estimates (2.37), (2.38) for small t , we obtain ameaningful result only in the three-dimensional case and only for r > Proposition 3.7.
Suppose that the assumptions of Theorem are satisfied with d = 3 and r > . Suppose that ∂ O ∈ C , . Let ˇ v ε and ˇ q ε be given by (3.10) and (3.11) . Then for < ε ε and ε t < T we have k u ε ( · , t ) − ˇ v ε ( · , t ) k H ( O ) C ( ε / t − / + εt − / ) e − c ♭ t/ k ϕ k L ( O ) + e c ′ r ε / − /r k F k H r ( t ) , k p ε ( · , t ) − ˇ q ε ( · , t ) k L ( O ) e C ( ε / t − / + εt − / ) e − c ♭ t/ k ϕ k L ( O ) + e c ′′ r ε / − /r k F k H r ( t ) . The constants e c ′ r and e c ′′ r depend only on the problem data (1.9) and r . Approximation of the solution in a strictly interior subdomain.
From Theo-rem 2.14 and Proposition 2.5 we deduce the following result.
Theorem 3.8.
Suppose that the assumptions of Theorem are satisfied. Let O ′ be a strictlyinterior subdomain of the domain O . Let δ be given by (1.61) . Then for < ε ε and ε t < T we have k u ε ( · , t ) − v ε ( · , t ) k H ( O ′ ) ε ( C t − / δ − + C t − ) e − c ♭ t/ k ϕ k L ( O ) + k r ϑ ( ε, δ, r ) k F k H r ( t ) , k p ε ( · , t ) − q ε ( · , t ) k L ( O ′ ) ε ( e C t − / δ − + e C t − ) e − c ♭ t/ k ϕ k L ( O ) + e k r ϑ ( ε, δ, r ) k F k H r ( t ) . Here the constants k r and e k r depend only on the problem data (1.9) and r , and ϑ ( ε, δ, r ) := ( εδ − + ε − /r , < r < ∞ ,εδ − + ε ( | ln ε | + 1) , r = ∞ . Finally, under Conditions 1.12 and 1.14, Theorem 2.15 implies the following result.
Theorem 3.9.
Suppose that the assumtions of Theorem are satisfied. Suppose that thematrix-valued functions Λ( x ) and e Λ( x ) satisfy Conditions and , respectively. Supposethat ˇ v ε and ˇ q ε are given by (3.10) and (3.11) . Then for < ε ε and ε t < T we have k u ε ( · , t ) − ˇ v ε ( · , t ) k H ( O ′ ) ε ( C t − / δ − + C t − ) e − c ♭ t/ k ϕ k L ( O ) + ˇ k r ϑ ( ε, δ, r ) k F k H r ( t ) , k p ε ( · , t ) − ˇ q ε ( · , t ) k L ( O ′ ) ε ( e C t − / δ − + e C t − ) e − c ♭ t/ k ϕ k L ( O ) + b k r ϑ ( ε, δ, r ) k F k H r ( t ) . The constants ˇ k r and b k r depend only on the problem data (1.9) , r , p , and the norms k Λ k L ∞ , k e Λ k L p (Ω) . OMOGENIZATION OF THE FIRST INITIAL BOUNDARY-VALUE PROBLEM 25
Applications
For elliptic systems in the whole space R d , the examples considered below were studied in [Su4,MSu2]. For elliptic systems in a bounded domain, these examples were considered in [MSu3].4. Scalar elliptic operator with a singular potential
Description of the operator.
We consider the case where n = 1, m = d , b ( D ) = D , and g ( x ) is a Γ-periodic symmetric ( d × d )-matrix-valued function with real entries such that g, g − ∈ L ∞ and g ( x ) >
0. Obviously (see (1.3)), α = α = 1 and b ( D ) ∗ g ε ( x ) b ( D ) = − div g ε ( x ) ∇ .Next, let A ( x ) = col { A ( x ) , . . . , A d ( x ) } , where A j ( x ), j = 1 , . . . , d , are Γ-periodic real-valuedfunctions such that A j ∈ L ρ (Ω) , ρ = 2 for d = 1 , ρ > d for d > j = 1 , . . . , d. (4.1)Let v ( x ) and V ( x ) be real-valued Γ-periodic functions such that v, V ∈ L s (Ω) , s = 1 for d = 1 , s > d/ d > Z Ω v ( x ) d x = 0 . (4.2)In L ( O ), we consider the operator B D,ε given formally by the differential expression B ε = ( D − A ε ( x )) ∗ g ε ( x )( D − A ε ( x )) + ε − v ε ( x ) + V ε ( x ) (4.3)with the Dirichlet condition on ∂ O . The precise definition of the operator B D,ε is given in termsof the quadratic form b D,ε [ u, u ] = Z O (cid:0) h g ε ( D − A ε ) u, ( D − A ε ) u i + ( ε − v ε + V ε ) | u | (cid:1) d x , u ∈ H ( O ) . It is easily seen (cf. [Su4, Subsection 13.1]) that expression (4.3) can be written as B ε = D ∗ g ε ( x ) D + d X j =1 (cid:0) a εj ( x ) D j + D j ( a εj ( x )) ∗ (cid:1) + Q ε ( x ) . (4.4)Here Q ( x ) is a real-valued function defined by Q ( x ) = V ( x ) + h g ( x ) A ( x ) , A ( x ) i . (4.5)The complex-valued functions a j ( x ) are given by a j ( x ) = − η j ( x ) + iξ j ( x ) , j = 1 , . . . , d. (4.6)Here η j ( x ) are the components of the vector-valued function η ( x ) = g ( x ) A ( x ), and the functions ξ j ( x ) are defined by ξ j ( x ) = − ∂ j Φ( x ), where Φ( x ) is the Γ-periodic solution of the problem∆Φ( x ) = v ( x ), R Ω Φ( x ) d x = 0. We have v ( x ) = − d X j =1 ∂ j ξ j ( x ) . (4.7)It is easy to check that the functions (4.6) satisfy condition (1.7) with a suitable ρ ′ dependingon ρ and s , and the norms k a j k L ρ ′ (Ω) are controlled in terms of k g k L ∞ , k A k L ρ (Ω) , k v k L s (Ω) ,and the parameters of the lattice Γ. (See [Su4, Subsection 13.1].) The function (4.5) satisfiescondition (1.8) with a suitable s ′ = min { s ; ρ/ } .Let Q ( x ) be a positive definite and bounded Γ-periodic function. According to (1.10), weintroduce a positive definite operator B D,ε := B D,ε + λQ ε . Here the constant λ is chosenaccording to condition (1.16) for the operator B D,ε with the coefficients g , a j , j = 1 , . . . , d , Q ,and Q defined above. The operator B D,ε is given by B ε = ( D − A ε ( x )) ∗ g ε ( x )( D − A ε ( x )) + ε − v ε ( x ) + V ε ( x ) + λQ ε ( x ) . (4.8)We are interested in the behavior of the exponential of the operator e B D,ε := f ε B D,ε f ε , where f ( x ) := Q ( x ) − / . For the scalar elliptic operator (4.8), the problem data (1.9) are reduced to the following setof parameters: d, ρ, s ; k g k L ∞ , k g − k L ∞ , k A k L ρ (Ω) , k v k L s (Ω) , kVk L s (Ω) , k Q k L ∞ , k Q − k L ∞ ; the parameters of the lattice Γ; the domain O . (4.9)4.2. The effective operator.
Let us write down the effective operator. In the case underconsideration, the Γ-periodic solution of problem (1.25) is a row: Λ( x ) = i Ψ( x ), Ψ( x ) =( ψ ( x ) , . . . , ψ d ( x )), where ψ j ∈ e H (Ω) is the solution of the problemdiv g ( x )( ∇ ψ j ( x ) + e j ) = 0 , Z Ω ψ j ( x ) d x = 0 . Here e j , j = 1 , . . . , d , is the standard orthonormal basis in R d . Clearly, the functions ψ j ( x ) arereal-valued, and the entries of Λ( x ) are purely imaginary. By (1.27), the columns of the ( d × d )-matrix-valued function e g ( x ) are the vector-valued functions g ( x )( ∇ ψ j ( x ) + e j ), j = 1 , . . . , d .The effective matrix is defined according to (1.26): g = | Ω | − R Ω e g ( x ) d x . Clearly, e g ( x ) and g have real entries.According to (4.6) and (4.7), the periodic solution of problem (1.33) is represented as e Λ( x ) = e Λ ( x ) + i e Λ ( x ), where the real-valued Γ-periodic functions e Λ ( x ) and e Λ ( x ) are the solutions ofthe problems − div g ( x ) ∇ e Λ ( x ) + v ( x ) = 0 , Z Ω e Λ ( x ) d x = 0; − div g ( x ) ∇ e Λ ( x ) + div g ( x ) A ( x ) = 0 , Z Ω e Λ ( x ) d x = 0 . The column V (see (1.35)) has the form V = V + iV , where V , V are the columns with realentries defined by V = | Ω | − Z Ω ( ∇ Ψ( x )) t g ( x ) ∇ e Λ ( x ) d x , V = −| Ω | − Z Ω ( ∇ Ψ( x )) t g ( x ) ∇ e Λ ( x ) d x . According to (1.36), the constant W is given by W = | Ω | − Z Ω (cid:16) h g ( x ) ∇ e Λ ( x ) , ∇ e Λ ( x ) i + h g ( x ) ∇ e Λ ( x ) , ∇ e Λ ( x ) i (cid:17) d x . The effective operator for B D,ε acts as follows B D u = − div g ∇ u + 2 i h∇ u, V + η i + ( − W + Q + λQ ) u, u ∈ H ( O ) ∩ H ( O ) . The corresponding differential expression can be written as B = ( D − A ) ∗ g ( D − A ) + V + λQ , (4.10)where A = ( g ) − ( V + g A ) , V = V + h g A , A i − h g A , A i − W. Let f := ( Q ) − / . Denote e B D := f B D f .4.3. Approximation of the sandwiched operator exponential.
According to Remark 1.16,in the case under consideration, Conditions 1.12 and 1.14 are satisfied, and the norms k Λ k L ∞ and k e Λ k L ∞ are estimated in terms of the problem data (4.9). Therefore, we can use the correctorwhich does not involve the smoothing operator: K D ( t ; ε ) := (cid:16) [Λ ε ] D + [ e Λ ε ] (cid:17) f e − e B D t f = (cid:16) [Ψ ε ] ∇ + [ e Λ ε ] (cid:17) f e − e B D t f . (4.11)The operator (2.33) takes the form G D ( t ; ε ) = − i G D ( t ; ε ), where G D ( t ; ε ) = e g ε ∇ f e − e B D t f + g ε ( ∇ e Λ) ε f e − e B D t f . (4.12)Theorems 2.2 and 2.6 imply the following result. OMOGENIZATION OF THE FIRST INITIAL BOUNDARY-VALUE PROBLEM 27
Proposition 4.1.
Suppose that the assumptions of Subsections and are satisfied. Sup-pose that the operators K D ( t ; ε ) and G D ( t ; ε ) are given by (4.11) and (4.12) , respectively. Sup-pose that the number ε is subject to Condition . Then for < ε ε we have k f ε e − e B D,ε t f ε − f e − e B D t f k L ( O ) → L ( O ) C ε ( t + ε ) − / e − c ♭ t/ , t > k f ε e − e B D,ε t f ε − f e − e B D t f − ε K D ( t ; ε ) k L ( O ) → H ( O ) C ( ε / t − / + εt − ) e − c ♭ t/ , t > k g ε ∇ f ε e − e B D,ε t f ε − G D ( t ; ε ) k L ( O ) → L ( O ) e C ( ε / t − / + εt − ) e − c ♭ t/ , t > . The constants C , C , and e C depend only on the problem data (4.9) . Homogenization of the first initial boundary-value problem for parabolic equa-tion with a singular potential.
Consider the first initial boundary-value problem for nonho-mogeneous parabolic equation with a singular potential: Q ε ( x ) ∂u ε ∂t ( x , t ) = − ( D − A ε ( x )) ∗ g ε ( x )( D − A ε ( x )) u ε ( x , t ) − (cid:0) ε − v ε ( x ) + V ε ( x ) + λQ ε ( x ) (cid:1) u ε ( x , t ) + F ( x , t ) , x ∈ O , t > u ε ( · , t ) | ∂ O = 0 , t > Q ε ( x ) u ε ( x ,
0) = ϕ ( x ) , x ∈ O . Here ϕ ∈ L ( O ) and F ∈ H r ( T ) := L r ((0 , T ); L ( O )), 0 < T ∞ , for some 1 r ∞ .According to (3.3) and (4.10), the effective problem takes the form Q ∂u ∂t ( x , t ) = − ( D − A ) ∗ g ( D − A ) u ( x , t ) − (cid:0) V + λQ (cid:1) u ( x , t )+ F ( x , t ) , x ∈ O , t > u ( · , t ) | ∂ O = 0 , t > Q u ( x ,
0) = ϕ ( x ) , x ∈ O . Applying Theorems 3.1 and 3.6, we obtain the following result.
Proposition 4.2.
Suppose that the number ε is subject to Condition . Suppose that theassumptions of Subsection are satisfied, and < r ∞ . Then for < ε ε and < t < T we have k u ε ( · , t ) − u ( · , t ) k L ( O ) C ε ( t + ε ) − / e − c ♭ t/ k ϕ k L ( O ) + c r θ ( ε, r ) k F k H r ( T ) . Here θ ( ε, r ) is given by (3.5) .Assuming that t > ε , we put w ε ( · , t ) := f e − e B D ε f − u ( · , t − ε ) . Denote ˇ v ε ( · , t ) := u ( · , t )+ ε Ψ ε ∇ w ε ( · , t )+ ε e Λ ε w ε ( · , t ) and ˇ q ε ( · , t ) := e g ε ∇ w ε ( · , t )+ g ε (cid:0) ∇ e Λ (cid:1) ε w ε ( · , t ) . In addition,assume that < r ∞ . Then for < ε ε and ε t < T we have k u ε ( · , t ) − ˇ v ε ( · , t ) k H ( O ) C ε / t − / e − c ♭ t/ k ϕ k L ( O ) + c ′ r ω ( ε, r ) k F k H r ( t ) , k g ε ∇ u ε ( · , t ) − ˇ q ε ( · , t ) k L ( O ) e C ε / t − / e − c ♭ t/ k ϕ k L ( O ) + c ′′ r ω ( ε, r ) k F k H r ( t ) . Here ω ( ε, r ) is given by (3.7) . The constants C , C , and e C depend only on the problem data (4.9) . The constants c r , c ′ r , and c ′′ r depend on the same parameters and on r . The scalar operator with a strongly singularpotential of order ε − Homogenization of the first initial boundary-value problem for parabolic equation with astrongly singular potential was studied in [AlCPiSiVa]. Some motivations can be found in[AlCPiSiVa, § Description of the operator.
Let ˇ g ( x ) be a Γ-periodic symmetric ( d × d )-matrix-valuedfunction in R d with real entries such that ˇ g, ˇ g − ∈ L ∞ and ˇ g ( x ) >
0. Let ˇ v ( x ) be a real-valuedΓ-periodic function such thatˇ v ∈ L s (Ω) , s = 1 for d = 1 , s > d/ d > . By ˇ A we denote the operator in L ( R d ) corresponding to the quadratic form Z R d (cid:0) h ˇ g ( x ) D u, D u i + ˇ v ( x ) | u | (cid:1) d x , u ∈ H ( R d ) . Adding a constant to the potential ˇ v ( x ), we assume that the bottom of the spectrum of ˇ A is thepoint zero. Then the operator ˇ A admits a factorization with the help of the eigenfunction ofthe operator D ∗ ˇ g ( x ) D + ˇ v ( x ) on the cell Ω (with periodic boundary conditions) correspondingto the eigenvalue λ = 0 (see [BSu2, Chapter 6, Subsection 1.1]). Apparently, such factorizationtrick was first used in homogenization problems in [Zh1, K].In L ( O ), we consider the operator ˇ A D given by the expression D ∗ ˇ g ( x ) D + ˇ v ( x ) with theDirichlet condition on ∂ O . The precise definition of the operator ˇ A D is given in terms of thequadratic form ˇ a D [ u, u ] = Z O (cid:0) h ˇ g ( x ) D u, D u i + ˇ v ( x ) | u | (cid:1) d x , u ∈ H ( O ) . (5.1)The operator ˇ A D inherits factorization of the operator ˇ A . To describe this factorization, weconsider the equation D ∗ ˇ g ( x ) D ω ( x ) + ˇ v ( x ) ω ( x ) = 0 . (5.2)There exists a Γ-periodic solution ω ∈ e H (Ω) of this equation defined up to a constant factor.We can fix this factor so that ω ( x ) > Z Ω ω ( x ) d x = | Ω | . (5.3)Moreover, the solution is positive definite and bounded: 0 < ω ω ( x ) ω < ∞ . The norms k ω k L ∞ and k ω − k L ∞ are controlled in terms of k ˇ g k L ∞ , k ˇ g − k L ∞ , and k ˇ v k L s (Ω) . Note that ω and ω − are multipliers in H ( O ).Substituting u = ωz and taking (5.2) into account, we represent the form (5.1) asˇ a D [ u, u ] = Z O ω ( x ) h ˇ g ( x ) D z, D z i d x , u = ωz, z ∈ H ( O ) . Hence, the differential expression for the operator ˇ A D admits a factorizationˇ A = ω − D ∗ g D ω − , g = ω ˇ g. (5.4)Now, we consider the operator ˇ A D,ε with rapidly oscillating coefficients acting in L ( O ) andgiven by ˇ A ε = ( ω ε ) − D ∗ g ε D ( ω ε ) − , g = ω ˇ g, (5.5)with the Dirichlet boundary condition. In the initial terms, expression (5.5) takes the formˇ A ε = D ∗ ˇ g ε D + ε − ˇ v ε . (5.6)Next, let A ( x ) = col { A ( x ) , . . . , A d ( x ) } , where A j ( x ) are Γ-periodic real-valued functionssatisfying (4.1). Let b v ( x ) and ˇ V ( x ) be Γ-periodic real-valued functions such that b v, ˇ V ∈ L s (Ω) , s = 1 for d = 1 , s > d/ d > Z Ω b v ( x ) ω ( x ) d x = 0 . (5.7)In L ( O ), we consider the operator e B D,ε given formally by the differential expression e B ε = ( D − A ε ) ∗ ˇ g ε ( D − A ε ) + ε − ˇ v ε + ε − b v ε + ˇ V ε with the Dirichlet condition on ∂ O . The precise definition is given in terms of the quadraticform. OMOGENIZATION OF THE FIRST INITIAL BOUNDARY-VALUE PROBLEM 29
We put v ( x ) := b v ( x ) ω ( x ) , V ( x ) := ˇ V ( x ) ω ( x ) . (5.8)By (5.5) and (5.6), we have e B D,ε = ( ω ε ) − B D,ε ( ω ε ) − , where the operator B D,ε is given bythe expression (4.3) with the Dirichlet condition on ∂ O ; g is defined by (5.4), and v , V aregiven by (5.8). By (5.7) and the properties of ω , the coefficients v and V satisfy (4.2). Thenthe operator B D,ε can be represented in the form (4.4), where a j , j = 1 , . . . , d , and Q areconstructed in terms of g , A , v , and V according to (4.5), (4.6).The constant λ is chosen according to condition (1.16) for the operator with the same coeffi-cients g , a j , j = 1 , . . . , d , and Q , as the coefficients of B D,ε , and the coefficient Q ( x ) := ω ( x ).Then the operators e B D,ε := e B D,ε + λI and B D,ε := B D,ε + λQ ε are related by e B D,ε =( ω ε ) − B D,ε ( ω ε ) − .The following set of parameters is called the “problem data”: d, ρ, s ; k ˇ g k L ∞ , k ˇ g − k L ∞ , k A k L ρ (Ω) , k ˇ v k L s (Ω) , k b v k L s (Ω) , k ˇ V k L s (Ω) ;the parameters of the lattice Γ; the domain O . (5.9)5.2. Homogenization of the first initial boundary-value problem for the parabolicequation with strongly singular potential.
We apply Proposition 4.1 to the operator e B D,ε described in Subsection 5.1. We have f ( x ) = ω ( x ) − , whence, by (5.3), f = 1 and e B D = B D .The coefficients g , A , and V of the effective operator are constructed in terms of g , A , v , and V (see (5.5) and (5.8)), as described in Subsection 4.2. We apply the results to homogenizationof the solution of the first initial boundary-value problem ∂u ε ∂t ( x , t ) = − ( D − A ε ( x )) ∗ ˇ g ε ( x )( D − A ε ( x )) u ε ( x , t ) − (cid:0) ε − ˇ v ε + ε − b v ε ( x ) + ˇ V ε ( x ) + λI (cid:1) u ε ( x , t ) , x ∈ O , t > u ε ( · , t ) | ∂ O = 0 , t > u ε ( x ,
0) = ω ε ( x ) − ϕ ( x ) , x ∈ O . Here ϕ ∈ L ( O ). (For simplicity, we consider a homogeneous equation.) Then u ε ( · , t ) = e − e B D,ε t ( ω ε ) − ϕ .Let u be the solution of the homogenized problem ∂u ∂t ( x , t ) = − ( D − A ) ∗ g ( D − A ) u ( x , t ) − (cid:0) V + λ (cid:1) u ( x , t ) , x ∈ O , t > u ( · , t ) | ∂ O = 0 , t > u ( x ,
0) = ϕ ( x ) , x ∈ O . Proposition 4.1 implies the following result.
Proposition 5.1.
Suppose that the assumptions of Subsection are satisfied. Denote ˇ v ε ( · , t ) := u ( · , t ) + ε Ψ ε ∇ u ( · , t ) + ε e Λ ε u ( · , t ) , ˇ q ε ( · , t ) := e g ε ∇ u ( · , t ) + g ε ( ∇ e Λ) ε u ( · , t ) . Then for < ε ε we have k ( ω ε ) − u ε ( · , t ) − u ( · , t ) k L ( O ) C ε ( t + ε ) − / e − c ♭ t/ k ϕ k L ( O ) , t > k ( ω ε ) − u ε ( · , t ) − ˇ v ε ( · , t ) k H ( O ) C ( ε / t − / + εt − ) e − c ♭ t/ k ϕ k L ( O ) , k g ε ∇ ( ω ε ) − u ε ( · , t ) − ˇ q ε ( · , t ) k L ( O ) e C ( ε / t − / + εt − ) e − c ♭ t/ k ϕ k L ( O ) ,t > . The constants C , C , and e C depend on the problem data (5.9) . Note that, in the presence of a strongly singular potential in the equation, not the solution u ε , but the product ( ω ε ) − u ε admits a “good approximation”. This shows that the nature ofthe results of § § Appendix
In Appendix, we consider the case where d > S ε in the case of sufficiently smooth boundary (Lemma 2.9 andTheorem 2.10) and in the case of a strictly interior subdomain (Lemma 2.16 and Theorem 2.17).6. The properties of the matrix-valued functions Λ and e ΛWe need the following results; see [PSu, Lemma 2.3] and [MSu2, Lemma 3.4].
Lemma 6.1.
Let Λ be the Γ -periodic solution of problem (1.25) . Then for any function u ∈ C ∞ ( R d ) and ε > we have Z R d | ( D Λ) ε ( x ) | | u ( x ) | d x β k u k L ( R d ) + β ε Z R d | Λ ε ( x ) | | D u ( x ) | d x . The constants β and β depend on m , d , α , α , k g k L ∞ , and k g − k L ∞ . Lemma 6.2.
Let e Λ be the Γ -periodic solution of problem (1.33) . Then for any function u ∈ C ∞ ( R d ) and < ε we have Z R d | ( D e Λ) ε ( x ) | | u ( x ) | d x e β k u k H ( R d ) + e β ε Z R d | e Λ ε ( x ) | | D u ( x ) | d x . The constants e β and e β depend only on n , d , α , α , ρ , k g k L ∞ , k g − k L ∞ , the norms k a j k L ρ (Ω) , j = 1 , . . . , d , and the parameters of the lattice Γ . Below in § x ) and e Λ( x ). Lemma 6.3.
Suppose that a matrix-valued function Λ( x ) is the Γ -periodic solution of prob-lem (1.25) . Let d > and l = d/ . ◦ . For < ε and u ∈ H l − ( R d ; C m ) we have Λ ε u ∈ L ( R d ; C n ) and k Λ ε u k L ( R d ) C (0) k u k H l − ( R d ) . (6.1)2 ◦ . For < ε and u ∈ H l ( R d ; C m ) we have Λ ε u ∈ H ( R d ; C n ) and k Λ ε u k H ( R d ) C (1) ε − k u k L ( R d ) + C (2) k u k H l ( R d ) . (6.2) The constants C (0) , C (1) , and C (2) depend on m , d , α , α , k g k L ∞ , k g − k L ∞ , and the parametersof the lattice Γ .Proof. It suffices to check (6.1) and (6.2) for u ∈ C ∞ ( R d ; C m ). Substituting x = ε y , ε d/ u ( x ) = U ( y ), we obtain k Λ ε u k L ( R d ) Z R d | Λ( ε − x ) | | u ( x ) | d x = Z R d | Λ( y ) | | U ( y ) | d y = X a ∈ Γ Z Ω+ a | Λ( y ) | | U ( y ) | d y X a ∈ Γ k Λ k L ν (Ω) k U k L ν ′ (Ω+ a ) , (6.3)where ν − + ( ν ′ ) − = 1. We choose ν so that the embedding H (Ω) ֒ → L ν (Ω) is continuous,i. e., ν = d ( d − − . Then k Λ k L ν (Ω) c Ω k Λ k H (Ω) , (6.4)where the constant c Ω depends only on the dimension d and the lattice Γ. We have 2 ν ′ = d .Since the embedding H l − (Ω) ֒ → L d (Ω) is continuous, we have k U k L d (Ω+ a ) c ′ Ω k U k H l − (Ω+ a ) , (6.5)where the constant c ′ Ω depends only on the dimension d and the lattice Γ. Now, from (6.3)–(6.5)it follows that Z R d | Λ ε ( x ) | | u ( x ) | d x c Ω c ′ Ω k Λ k H (Ω) k U k H l − ( R d ) . (6.6) OMOGENIZATION OF THE FIRST INITIAL BOUNDARY-VALUE PROBLEM 31
Obviously, for 0 < ε k U k H l − ( R d ) k u k H l − ( R d ) . Combining this with (6.6) and(1.28), we see that Z R d | Λ ε ( x ) | | u ( x ) | d x c Ω c ′ Ω M k u k H l − ( R d ) , u ∈ C ∞ ( R d ; C m ) , (6.7)which proves estimate (6.1) with the constant C (0) := ( c Ω c ′ Ω ) / M .Next, by Lemma 6.1, k D (Λ ε u ) k L ( R d ) ε − Z R d | ( D Λ) ε ( x ) u ( x ) | d x + 2 Z R d | Λ ε ( x ) | | Du ( x ) | d x β ε − Z R d | u ( x ) | d x + 2(1 + β ) Z R d | Λ ε ( x ) | | Du ( x ) | d x . (6.8)From (6.7) (with u replaced by the derivatives ∂ j u ) it follows that Z R d | Λ ε ( x ) | | Du ( x ) | d x c Ω c ′ Ω M k u k H l ( R d ) , u ∈ C ∞ ( R d ; C m ) . (6.9)As a result, relations (6.7)–(6.9) imply inequality (6.2) with the constants C (1) := (2 β ) / and C (2) := M (3 + 2 β ) / ( c Ω c ′ Ω ) / . (cid:3) Using the extension operator P O satisfying estimates (1.46), we deduce the following statementfrom Lemma 6.3(1 ◦ ). Corollary 6.4.
Suppose that the assumptions of Lemma are satisfied. Then the operator [Λ ε ] is continuous from H l − ( O ; C m ) to L ( O ; C n ) and k [Λ ε ] k H l − ( O ) → L ( O ) C (0) C ( l − O . The following statement can be checked similarly to Lemma 6.3, by using Lemma 6.2 andestimate (1.34).
Lemma 6.5.
Suppose that a matrix-valued function e Λ( x ) is the Γ -periodic solution of prob-lem (1.33) . Let d > and l = d/ . ◦ . For < ε and u ∈ H l − ( R d ; C n ) we have e Λ ε u ∈ L ( R d ; C n ) and k e Λ ε u k L ( R d ) e C (0) k u k H l − ( R d ) . ◦ . For < ε and u ∈ H l ( R d ; C n ) we have e Λ ε u ∈ H ( R d ; C n ) and k e Λ ε u k H ( R d ) e C (1) ε − k u k H ( R d ) + e C (2) k u k H l ( R d ) . The constants e C (0) := ( c Ω c ′ Ω ) / f M , e C (1) := (2 e β ) / , e C (2) := √ e β + 1) / ( c Ω c ′ Ω ) / f M depend only on the problem data (1.9) . Using the extension operator P O , we deduce the following corollary from Lemma 6.5(1 ◦ ). Corollary 6.6.
Under the assumptions of Lemma , the operator [ e Λ ε ] is continuous from H l − ( O ; C n ) to L ( O ; C n ) and k [ e Λ ε ] k H l − ( O ) → L ( O ) e C (0) C ( l − O . Removal of the smoothing operator in the correctorin the case of sufficiently smooth boundary
Proof of Lemma 2.9.
Suppose that the assumptions of Lemma 2.9 are satisfied. Let u be given by (2.3), where ϕ ∈ L ( O ; C n ). We put e u ( · , t ) = P O u ( · , t ) . According to (2.22) and (2.32), we have K D ( t ; ε ) ϕ = (cid:0) Λ ε S ε b ( D ) + e Λ ε S ε (cid:1)e u ( · , t ) , (7.1) K D ( t ; ε ) ϕ = (cid:0) Λ ε b ( D ) + e Λ ε (cid:1) u ( · , t ) . (7.2)We need to estimate the following value kK D ( t ; ε ) ϕ − K D ( t ; ε ) ϕ k H ( O ) k Λ ε (cid:0) ( S ε − I ) b ( D ) e u (cid:1) ( · , t ) k H ( R d ) + k e Λ ε (cid:0) ( S ε − I ) e u (cid:1) ( · , t ) k H ( R d ) . (7.3)Under the above assumptions, by Lemma 2.8, we have u ∈ H l +1 ( O ; C n ), whence e u ∈ H l +1 ( R d ; C n ). This gives us possibility to apply Lemma 6.3(2 ◦ ) to estimate the first summandin the right-hand side of (7.3): k Λ ε (cid:0) ( S ε − I ) b ( D ) e u (cid:1) ( · , t ) k H ( R d ) C (1) ε − k (cid:0) ( S ε − I ) b ( D ) e u (cid:1) ( · , t ) k L ( R d ) + C (2) k (cid:0) ( S ε − I ) b ( D ) e u (cid:1) ( · , t ) k H l ( R d ) , (7.4)where l = d/
2. The first term in the right-hand side of (7.4) is estimated with the help ofProposition 1.1 and relations (1.3), (1.43), (1.46), (2.3), and (2.8): ε − k (cid:0) ( S ε − I ) b ( D ) e u (cid:1) ( · , t ) k L ( R d ) r k D b ( D ) e u ( · , t ) k L ( R d ) r α / C (2) O k u ( · , t ) k H ( O ) C (3) t − e − c ♭ t/ k ϕ k L ( O ) , (7.5)where C (3) := r α / C (2) O e c k f k L ∞ . To estimate the second term in the right-hand side of (7.4),we apply (1.2) and (1.3): k (cid:0) ( S ε − I ) b ( D ) e u (cid:1) ( · , t ) k H l ( R d ) α / k e u ( · , t ) k H l +1 ( R d ) . (7.6)By (1.43), (1.46), (2.3), and Lemma 2.8, k e u ( · , t ) k H l +1 ( R d ) C ( l +1) O b C l +1 k f k L ∞ t − ( l +1) / e − c ♭ t/ k ϕ k L ( O ) . (7.7)From (7.6) and (7.7) it follows that k (cid:0) ( S ε − I ) b ( D ) e u (cid:1) ( · , t ) k H l ( R d ) C (4) t − ( l +1) / e − c ♭ t/ k ϕ k L ( O ) , (7.8)where C (4) := 2 α / C ( l +1) O b C l +1 k f k L ∞ .Now we estimate the second term in the right-hand side of (7.3). By Lemma 6.5(2 ◦ ), k e Λ ε (cid:0) ( S ε − I ) e u (cid:1) ( · , t ) k H ( R d ) e C (1) ε − k ( S ε − I ) e u ( · , t ) k H ( R d ) + e C (2) k ( S ε − I ) e u ( · , t ) k H l ( R d ) , l = d/ . (7.9)The first summand in the right-hand side of (7.9) is estimated by using Proposition 1.1 andrelations (1.43), (1.46), (2.3), (2.8): ε − k ( S ε − I ) e u ( · , t ) k H ( R d ) r C (2) O k u ( · , t ) k H ( O ) C (5) t − e − c ♭ t/ k ϕ k L ( O ) ; C (5) := r C (2) O e c k f k L ∞ . (7.10)The second summand in (7.9) is estimated with the help of (1.2) and (7.7): k ( S ε − I ) e u ( · , t ) k H l ( R d ) k e u ( · , t ) k H l ( R d ) k e u ( · , t ) k H l +1 ( R d ) C (6) t − ( l +1) / e − c ♭ t/ k ϕ k L ( O ) ; C (6) := 2 C ( l +1) O b C l +1 k f k L ∞ . (7.11)As a result, relations (7.3)–(7.5) and (7.8)–(7.11) imply that kK D ( t ; ε ) ϕ − K D ( t ; ε ) ϕ k H ( O ) ( C (7) t − + C (8) t − ( l +1) / ) e − c ♭ t/ k ϕ k L ( O ) , OMOGENIZATION OF THE FIRST INITIAL BOUNDARY-VALUE PROBLEM 33 where l = d/ C (7) := C (1) C (3) + e C (1) C (5) , and C (8) := C (2) C (4) + e C (2) C (6) . This provesestimate (2.36) with the constant b C d := max { C (7) ; C (8) } . (cid:3) Proof of Theorem 2.10.
Inequality (2.37) directly follows from (2.24) and (2.36). Here-with, C d := 2( b C d + C ). Above, we took into account that for t > εt − does notexceed ε / t − / , and for t εt − d/ − / since d > (cid:13)(cid:13)(cid:13) g ε b ( D ) (cid:16) f ε e − e B D,ε t ( f ε ) ∗ − f e − e B D t f − ε (cid:0) Λ ε b ( D ) + e Λ ε (cid:1) f e − e B D t f (cid:17)(cid:13)(cid:13)(cid:13) L → L k g k L ∞ ( dα ) / C d ( ε / t − / + εt − d/ − / ) e − c ♭ t/ . (7.12)We have εg ε b ( D ) (cid:0) Λ ε b ( D ) + e Λ ε (cid:1) f e − e B D t f = g ε (cid:16) ( b ( D )Λ) ε + (cid:0) b ( D ) e Λ (cid:1) ε (cid:17) f e − e B D t f + ε d X k,j =1 g ε b k Λ ε b j D k D j f e − e B D t f + ε d X j =1 g ε b j e Λ ε D j f e − e B D t f . (7.13)The norm of the second summand in the right-hand side of (7.13) is estimated with the help of(1.4), (1.43), Lemma 2.8, and Corollary 6.4: ε (cid:13)(cid:13)(cid:13) d X k,j =1 g ε b k Λ ε b j D k D j f e − e B D t f (cid:13)(cid:13)(cid:13) L ( O ) → L ( O ) C (9) εt − ( l +1) / e − c ♭ t/ , (7.14) l = d/ C (9) := α dC (0) C ( l − O b C l +1 k g k L ∞ k f k L ∞ . The third summand in the right-hand sideof (7.13) is estimated by using (1.4), (1.43), Lemma 2.8, and Corollary 6.6: ε (cid:13)(cid:13)(cid:13) d X j =1 g ε b j e Λ ε D j f e − e B D t f (cid:13)(cid:13)(cid:13) L ( O ) → L ( O ) C (10) εt − ( l +1) / e − c ♭ t/ , (7.15)where l = d/ C (10) := ( dα ) / e C (0) C ( l − O b C l +1 k g k L ∞ k f k L ∞ . As a result, relations (7.12)–(7.15) imply inequality (2.38) with the constant e C d := k g k L ∞ ( dα ) / C d + C (9) + C (10) . (cid:3) Removal of the smoothing operator in the correctorin a strictly interior subdomain
One property of the operator S ε . Now we proceed to estimates in a strictly interiorsubdomain. We start with one simple property of the operator S ε .Let O ′ be a strictly interior subdomain of the domain O , and let δ be given by (1.61). Denote O ′′ := { x ∈ O : dist { x ; ∂ O} > δ/ } , O ′′′ := { x ∈ O : dist { x ; ∂ O} > δ/ } . Lemma 8.1.
Let S ε be the operator (1.1) . Let r = diam Ω . Suppose that v ∈ L ( R d ; C m ) and v ∈ H σ ( O ′′′ ; C m ) with some σ ∈ Z + . Then for < ε (4 r ) − δ we have S ε v ∈ H σ ( O ′′ ; C m ) and k S ε v k H σ ( O ′′ ) k v k H σ ( O ′′′ ) . Proof.
According to (1.1), k S ε v k H σ ( O ′′ ) = | Ω | − X | α | σ Z O ′′ d x (cid:12)(cid:12)(cid:12)(cid:12) Z Ω D α v ( x − ε z ) d z (cid:12)(cid:12)(cid:12)(cid:12) | Ω | − X | α | σ Z O ′′ d x Z Ω | D α v ( x − ε z ) | d z . (8.1)Since 0 < εr δ/
4, for x ∈ O ′′ and z ∈ Ω we have x − ε z ∈ O ′′′ . Hence, changing the order ofintegration in (8.1), we obtain the required estimate. (cid:3) A cut-off function χ ( x ) . We fix a smooth cut-off function χ ( x ) such that χ ∈ C ∞ ( R d ) , χ ( x ) χ ( x ) = 1 , x ∈ O ′ ;supp χ ⊂ O ′′ ; | D α χ ( x ) | κ σ δ − σ , | α | = σ, σ ∈ N . (8.2)The constants κ σ depend only on d , σ , and the domain O . Lemma 8.2.
Suppose that χ ( x ) is a cut-off function satisfying conditions (8.2) . Let k ∈ Z + . ◦ . For any function v ∈ H k ( R d ; C m ) we have k χ v k H k ( R d ) C (11) k k X j =0 δ − ( k − j ) k v k H j ( O ′′ ) . (8.3)2 ◦ . For any function v ∈ H k +1 ( R d ; C m ) we have k χ v k H k +1 / ( R d ) C (11) k +1 / (cid:18) k +1 X j =0 δ − ( k +1 − j ) k v k H j ( O ′′ ) (cid:19) / (cid:18) k X i =0 δ − ( k − i ) k v k H i ( O ′′ ) (cid:19) / . (8.4) The constants C (11) k and C (11) k +1 / depend on d , k , and the domain O .Proof. Inequality (8.3) follows from the Leibniz formula for the derivatives of the product χ v and from the estimates for the derivatives of χ (see (8.2)). To check (8.4), we should also takeinto account that k w k H k +1 / ( R d ) k w k H k +1 ( R d ) k w k H k ( R d ) , w ∈ H k +1 ( R d ; C m ) . (cid:3) Proof of Lemma 2.16.
Suppose that the assumptions of Lemma 2.16 are satisfied. Let u be given by (2.3) with ϕ ∈ L ( O ; C n ). According to (1.43) and (2.7), (2.8), we have k Du ( · , t ) k L ( O ) k u ( · , t ) k H ( O ) c k f k L ∞ t − / e − c ♭ t/ k ϕ k L ( O ) , (8.5) k Du ( · , t ) k H ( O ) k u ( · , t ) k H ( O ) e c k f k L ∞ t − e − c ♭ t/ k ϕ k L ( O ) . (8.6)Let e u = P O u . Relations (7.1) and (7.2) remain valid. We need to estimate the following value: kK D ( t ; ε ) ϕ − K D ( t ; ε ) ϕ k H ( O ′ ) k Λ ε χ (cid:0) ( S ε − I ) b ( D ) e u (cid:1) ( · , t ) k H ( R d ) + k e Λ ε χ (cid:0) ( S ε − I ) e u (cid:1) ( · , t ) k H ( R d ) . (8.7)Recall (cf. discussion in Subsection 2.10) that u ( · , t ) ∈ H σ ( O ′′′ ; C n ) for any σ ∈ Z + . Then thefunction e u ( · , t ) satisfies the assumptions of Lemma 8.1 for any σ ∈ Z + . Hence, ( S ε e u )( · , t ) ∈ H σ ( O ′′ ; C n ) for 0 < ε (4 r ) − δ . Then we can apply Lemma 6.3(2 ◦ ) to estimate the firstsummand in the right-hand side of (8.7): k Λ ε χ (cid:0) ( S ε − I ) b ( D ) e u (cid:1) ( · , t ) k H ( R d ) C (1) ε − k χ (cid:0) ( S ε − I ) b ( D ) e u (cid:1) ( · , t ) k L ( R d ) + C (2) k χ (cid:0) ( S ε − I ) b ( D ) e u (cid:1) ( · , t ) k H l ( R d ) , (8.8) l = d/
2. The first term in the right-hand side of (8.8) is estimated by using inequality (7.5)(which holds without additional smoothness assumption on ∂ O ): ε − k χ (cid:0) ( S ε − I ) b ( D ) e u (cid:1) ( · , t ) k L ( R d ) C (3) t − e − c ♭ t/ k ϕ k L ( O ) . (8.9)Now, we consider the second summand in the right-hand side of (8.8). Obviously, k χ (cid:0) ( S ε − I ) b ( D ) e u (cid:1) ( · , t ) k H l ( R d ) k χ ( S ε b ( D ) e u )( · , t ) k H l ( R d ) + k χb ( D ) e u ( · , t ) k H l ( R d ) . (8.10)To estimate the second term in the right-hand side of (8.10), we apply Lemma 8.2 and (1.4). If l = d/ d is even), we have k χb ( D ) e u ( · , t ) k H l ( R d ) C (11) l ( dα ) / l X j =0 δ − ( l − j ) k Du ( · , t ) k H j ( O ′′ ) . (8.11) OMOGENIZATION OF THE FIRST INITIAL BOUNDARY-VALUE PROBLEM 35 If l = d/ k + 1 /
2, then k χb ( D ) e u ( · , t ) k H l ( R d ) C (11) l ( dα ) / (cid:16) k +1 X j =0 δ − ( k +1 − j ) k Du ( · , t ) k H j ( O ′′ ) (cid:17) / × (cid:16) k X σ =0 δ − ( k − σ ) k Du ( · , t ) k H σ ( O ′′ ) (cid:17) / . (8.12)The norms of Du ( · , t ) in L ( O ; C n ) and in H ( O ; C n ) are estimated in (8.5) and (8.6). By(1.43), (2.3), and (2.45) (with O ′ replaced by O ′′ ), k Du ( · , t ) k H σ ( O ′′ ) C ′ σ +1 k f k L ∞ σ t − / ( δ − + t − ) σ/ e − c ♭ t/ k ϕ k L ( O ) , (8.13) σ >
2. Using (8.5), (8.6), and (8.11)–(8.13), we arrive at the inequality k χb ( D ) e u ( · , t ) k H l ( R d ) C (12) t − / ( δ − + t − ) d/ e − c ♭ t/ k ϕ k L ( O ) . (8.14)The constant C (12) depends only on the problem data (1.9).To estimate the first term in the right-hand side of (8.10), we apply Lemmas 8.1 and 8.2.Assume that 0 < ε (4 r ) − δ . By (1.4), in the case of integer l , we have k χ ( S ε b ( D ) e u )( · , t ) k H l ( R d ) C (11) l ( dα ) / l X σ =0 δ − ( l − σ ) k Du ( · , t ) k H σ ( O ′′′ ) . (8.15)The norms of Du ( · , t ) in L ( O ; C n ) and in H ( O ; C n ) are estimated in (8.5) and (8.6). By(1.43), (2.3) and (2.45) (with O ′ replaced by O ′′′ ) k Du ( · , t ) k H σ ( O ′′′ ) C ′ σ +1 k f k L ∞ σ t − / ( δ − + t − ) σ/ e − c ♭ t/ k ϕ k L ( O ) , (8.16) σ >
2. From (8.5), (8.6), (8.15), and (8.16) it follows that k χ ( S ε b ( D ) e u )( · , t ) k H l ( R d ) C (13) t − / ( δ − + t − ) d/ e − c ♭ t/ k ϕ k L ( O ) . (8.17)The constant C (13) depends only on the problem data (1.9). Estimate (8.17) in the case ofhalf-integer l is checked similarly. Combining (8.8)–(8.10), (8.14), and (8.17), we estimate thefirst summand in the right-hand side of (8.7): k Λ ε χ (cid:0) ( S ε − I ) b ( D ) e u (cid:1) ( · , t ) k H ( R d ) C (14) (cid:0) t − + t − / ( δ − + t − ) d/ (cid:1) e − c ♭ t/ k ϕ k L ( O ) . (8.18)Here C (14) := max { C (1) C (3) ; C (2) ( C (12) + C (13) ) } .The second summand in the right-hand side of (8.7) is estimated by Lemma 6.5(2 ◦ ): k e Λ ε χ (cid:0) ( S ε − I ) e u (cid:1) ( · , t ) k H ( R d ) e C (1) ε − k χ (cid:0) ( S ε − I ) e u (cid:1) ( · , t ) k H ( R d ) + e C (2) k χ (cid:0) ( S ε − I ) e u (cid:1) ( · , t ) k H l ( R d ) , (8.19)where l = d/
2. To estimate the first summand in the right-hand side of (8.19), we use (8.2) andinequality (7.10) (which holds without extra smoothness assumption on the boundary): ε − k χ (cid:0) ( S ε − I ) e u (cid:1) ( · , t ) k H ( R d ) ε − k (cid:0) ( S ε − I ) e u (cid:1) ( · , t ) k H ( R d ) + ε − k ( D χ ) (cid:0) ( S ε − I ) e u (cid:1) ( · , t ) k L ( R d ) C (5) t − e − c ♭ t/ k ϕ k L ( O ) + ε − κ δ − k ( S ε − I ) e u ( · , t ) k L ( R d ) . Combining this with Proposition 1.1 and relations (1.43), (1.46), (2.3), and (2.7), we obtain ε − k χ (cid:0) ( S ε − I ) e u (cid:1) ( · , t ) k H ( R d ) C (15) ( δ − t − / + t − ) e − c ♭ t/ k ϕ k L ( O ) , (8.20)where C (15) := max { C (5) ; κ r C (1) O c k f k L ∞ } .If l = d/ k χ (cid:0) ( S ε − I ) e u (cid:1) ( · , t ) k H l ( R d ) C (11) l l X σ =0 δ − ( l − σ ) k u ( · , t ) k H σ ( O ′′′ ) , (8.21) < ε (4 r ) − δ . The norms of u in L ( O ; C n ), H ( O ; C n ), and H ( O ; C n ) are estimated byLemma 2.1 and relations (1.43), (2.3). For σ > k u ( · , t ) k H σ ( O ′′′ ) is estimated byusing (2.45) (with O ′ replaced by O ′′′ ): k u ( · , t ) k H σ ( O ′′′ ) C ′ σ +1 k f k L ∞ σ t − / ( δ − + t − ) σ/ e − c ♭ t/ k ϕ k L ( O ) . Combining these arguments with (8.21), we deduce that k χ (cid:0) ( S ε − I ) e u (cid:1) ( · , t ) k H l ( R d ) C (16) t − / ( δ − + t − ) d/ e − c ♭ t/ k ϕ k L ( O ) , (8.22)with the constant C (16) depending only on the problem data (1.9). For the case of half-integer l , estimate (8.22) is checked similarly. As a result, relations (8.19), (8.20), and (8.22) imply thefollowing estimate for the second summand in the right-hand side of (8.7): k e Λ ε χ (cid:0) ( S ε − I ) e u (cid:1) ( · , t ) k H ( R d ) e C (1) C (15) ( δ − t − / + t − ) e − c ♭ t/ k ϕ k L ( O ) + e C (2) C (16) t − / ( δ − + t − ) d/ e − c ♭ t/ k ϕ k L ( O ) . Together with (8.7) and (8.18), this implies inequality (2.47) with the constant C ′′ d := C (14) + e C (1) C (15) + e C (2) C (16) . We have taken into account that the term δ − t − / does not exceed t − / ( δ − + t − ) d/ . (cid:3) Proof of Theorem 2.17.
Inequality (2.48) follows directly from (2.44) and (2.47). Here-with, C d := max { C ; C } + C ′′ d .Let us check (2.49). From (2.48), (1.4), and (2.32) it follows that k g ε b ( D ) (cid:0) f ε e − e B D,ε t ( f ε ) ∗ − ( I + ε Λ ε b ( D ) + ε e Λ ε ) f e − e B D t f (cid:1) k L ( O ) → L ( O ′ ) k g k L ∞ ( dα ) / C d εh d ( δ ; t ) e − c ♭ t/ . (8.23)Let us apply identity (7.13). The norm of the second summand in the right-hand side of(7.13) is estimated with the help of (1.4), (8.2), and Lemma 6.3(1 ◦ ): ε (cid:13)(cid:13)(cid:13) d X k,j =1 g ε b k Λ ε b j D k D j f e − e B D t f (cid:13)(cid:13)(cid:13) L ( O ) → L ( O ′ ) εα k g k L ∞ C (0) d X k,j =1 k χD k D j f e − e B D t f k L ( O ) → H l − ( R d ) , l = d/ . (8.24)Next, we apply Lemma 8.2. If l is integer, (1.43) yields d X k,j =1 k χD k D j f e − e B D t f k L ( O ) → H l − ( R d ) dC (11) l − k f k L ∞ l − X i =0 δ − ( l − − i ) k f e − e B D t k L ( O ) → H i +2 ( O ′′ ) . (8.25)The norm k f e − e B D t k L ( O ) → H ( O ) satisfies (2.8). If i >
1, relations (1.43) and (2.45) (with O replaced by O ′′ ) imply that k f e − e B D t k L ( O ) → H i +2 ( O ′′ ) C ′ i +2 k f k L ∞ i +1 t − / ( δ − + t − ) ( i +1) / e − c ♭ t/ . Combining this with (2.8), (8.24), and (8.25), we obtain ε (cid:13)(cid:13)(cid:13) d X k,j =1 g ε b k Λ ε b j D k D j f e − e B D t f (cid:13)(cid:13)(cid:13) L ( O ) → L ( O ′ ) C (17) εt − / ( δ − + t − ) d/ e − c ♭ t/ , (8.26)where the constant C (17) depends only on the problem data (1.9). If l is half-integer, inequality(8.26) is checked by using Lemma 8.2(2 ◦ ). OMOGENIZATION OF THE FIRST INITIAL BOUNDARY-VALUE PROBLEM 37
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Chebyshev Laboratory, St. Petersburg State University, 14 Liniya V.O., 29b, 199178, St. Pe-tersburg, Russia
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