On operator error estimates for homogenization of hyperbolic systems with periodic coefficients

In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider a selfadjoint matrix strongly elliptic second order differential operator $\mathcal{A}_\varepsilon$, $\varepsilon>0$. The coefficients of the operator $\mathcal{A}_\varepsilon$ are periodic and depend on $\mathbf{x}/\varepsilon$. We study the behavior of the operator $\mathcal{A}_\varepsilon ^{-1/2}\sin (\tau \mathcal{A}_\varepsilon ^{1/2})$, $\tau\in\mathbb{R}$, in the small period limit. The principal term of approximation in the $(H^1\rightarrow L_2)$-norm for this operator is found. Approximation in the $(H^2\rightarrow H^1)$-operator norm with the correction term taken into account is also established. The results are applied to homogenization for the solutions of the nonhomogeneous hyperbolic equation $\partial ^2_\tau \mathbf{u}_\varepsilon =-\mathcal{A}_\varepsilon \mathbf{u}_\varepsilon +\mathbf{F}$.


Introduction
The paper is devoted to homogenization of periodic differential operators (DO's). A broad literature is devoted to homogenization theory, see, e. g., the books [BaPa, BeLPap, Sa, ZhKO]. We use the spectral approach to homogenization problems based on the Floquet-Bloch theory and the analytic perturbation theory. 0.1. The class of operators. In L 2 (R d ; C n ), we consider a matrix elliptic second order DO A ε admitting a factorization A ε = b(D) * g(x/ε)b(D), ε > 0. Here b(D) = 0.2. Operator error estimates for elliptic and parabolic problems. In a series of papers [BSu1,BSu2,BSu3,BSu4] by M. Sh. Birman and T. A. Suslina, an abstract operator-theoretic (spectral) approach to homogenization problems in R d was developed. This approach is based on the scaling transformation, the Floquet-Bloch theory, and the analytic perturbation theory.
A typical homogenization problem is to study the behavior of the solution u ε of the equation A ε u ε + u ε = F, where F ∈ L 2 (R d ; C n ), as ε → 0. It turns out that the solutions u ε converge in some sense to the solution u 0 of the homogenized equation A 0 u 0 + u 0 = F. Here is the effective operator and g 0 is the constant effective matrix. The way to construct g 0 is well known in homogenization theory.
In [BSu1], it was shown that This estimate is order-sharp. The constant C is controlled explicitly in terms of the problem data. Inequality (0.1) means that the resolvent (A ε + I) −1 converges to the resolvent of the effective operator in the L 2 (R d ; C n )-operator norm, as ε → 0. Moreover, Cε.
Results of this type are called operator error estimates in homogenization theory.
In [BSu4], approximation of the resolvent (A ε + I) −1 in the (L 2 → H 1 )-operator norm was found: Here the correction term K(ε) is taken into account. It contains a rapidly oscillating factor and so depends on ε. Herewith, εK(ε) L 2 →H 1 = O(1). In contrast to the traditional corrector of homogenization theory, the operator K(ε) contains an auxiliary smoothing operator Π ε (see (9.6) below).
(0.5) 0.4. Approximation for the solutions of hyperbolic systems with the correction term taken into account. Operator error estimates with the correction term for nonstationary equations of Schrödinger type and hyperbolic type previously have not been established. So, we discuss the known "classical" homogenization results that cannot be written in the uniform operator topology. These results concern the operators in a bounded domain O ⊂ R d . Approximation for the solution of the hyperbolic equation with the zero initial data and a non-zero right-hand side was obtained in [BeLPap,Chapter 2,Subsec. 3.6]. In [BeLPap], it was shown that the difference of the solution and the first order approximation strongly converges to zero in L 2 ((0, T ); H 1 (O)). The error estimate was not established. The case of zero initial data and non-zero right-hand side was also considered in [BaPa,Chapter 4,Section 5]. In [BaPa], the complete asymptotic expansion of the solution was constructed and the estimate of order O(ε 1/2 ) for the difference of the solution and the first order approximation in the H 1 -norm on the cylinder O × (0, T ) was obtained. Herewith, the right-hand side was assumed to be C ∞ -smooth. It is natural to be interested in the approximation with the correction term for the solutions of hyperbolic systems with non-zero initial data, i. e., in approximation of the operator cosine cos(τ A 1/2 ε ) in some suitable sense. One could expect the correction term in this case to be of similar structure as for elliptic and parabolic problems. However, in [BrOtFMu] it was observed that this is true only for very special class of initial data. In the general case, approximation with the corrector was found in [BraLe, CaDiCoCalMaMarG], but the correction term was non-local because of the dispersion of waves in inhomogeneous media. Dispersion effects for homogenization of the wave equation were discussed in [ABriV, ConOrV, ConSaMaBalV] via the Floquet-Bloch theory and the analytic perturbation theory. Operator error estimates have not been obtained. 0.5. Main results. Our goal is to refine estimate (0.4) with respect to the type of the operator norm without any additional assumptions and to find an approximation for the operator A −1/2 ε sin(τ A 1/2 ε ) in the (H 2 → H 1 )-norm. We wish to apply the results to problem (0.5) with ϕ = 0 and non-zero F and ψ.
Our first main result is the estimate (0.6) (Under additional assumptions on the operator, improvement of estimate (0.6) with respect to the type of the norm was obtained by M. A. Dorodnyi and T. A. Suslina in the forthcoming paper [DSu3] that is, actually, major revision of [DSu2].) Our second main result is the approximation Cε(1 + |τ |), (0.7) ε > 0, τ ∈ R. In the general case, the corrector contains the smoothing operator. We distinguish the cases when the smoothing operator can be removed. Also we show that the smoothing operator naturally arising from our method can be replaced by the Steklov smoothing. The latter is more convenient for homogenization problems in a bounded domain. Using of the Steklov smoothing is borrowed from [ZhPas1].
The results are applied to homogenization of the system (0.5) with ϕ = 0. A more general equation Q(x/ε)∂ 2 τ u ε (x, τ ) = −A ε u ε (x, τ ) + Q(x/ε)F(x, τ ) is also considered. Here Q(x) is a Γperiodic (n × n)-matrix-valued function such that Q(x) > 0 and Q, Q −1 ∈ L ∞ . In Introduction, we discuss only the case Q = 1 n for simplicity. 0.6. Method. We apply the method of [BSu5,DSu2] carrying out all the constructions for the operator A −1/2 ε sin(τ A 1/2 ε ). To obtain the result with the correction term, we borrow some technical tools from [Su3]. By the scaling transformation, inequality (0.6) is equivalent to Because of the presence of differentiation in the definition of H 1 -norm, by the scaling transformation, inequality (0.7) reduces to the estimate of order O(ε): For this reason, in estimate (0.9), we use the ,,smoothing operator" ε 2 (−∆ + ε 2 I) −1 instead of the operator ε(−∆+ε 2 I) −1/2 which was used in estimate (0.8) of order O(1). Thus, the principal term of approximation of the operator A −1/2 ε sin(τ A 1/2 ε ) is obtained in the (H 1 → L 2 )-norm, but approximation in the energy class is given in the (H 2 → H 1 )-norm.
To obtain estimates (0.8) and (0.9), using the unitary Gelfand transformation (see Section 4.2 below), we decompose the operator A into the direct integral of operators A(k) acting in the space L 2 on the cell of periodicity and depending on the parameter k ∈ R d called the quasimomentum. We study the family A(k) by means of the analytic perturbation theory with respect to the onedimensional parameter |k|. Then we should make our constructions and estimates uniform in the additional parameter θ := k/|k|. Herewith, a good deal of considerations can be done in the framework of an abstract operator-theoretic scheme. 0.7. Plan of the paper. The paper consists of three chapters. Chapter I (Sec. 1-3) contains necessary operator-theoretic material. Chapter II (Sec. 4-8) is devoted to periodic DO's. In Sec. 4-6, the class of operators under consideration is introduced, the direct integral decomposition is described, and the effective characteristics are found. In Sec. 7 and 8, the approximations for the operator-valued function A −1/2 sin(ε −1 τ A 1/2 ) are obtained and estimates (0.8) and (0.9) are proven. In Chapter III (Sec. 9 and 10), homogenization for hyperbolic systems is considered. In Sec. 9, the main results of the paper in operator terms (estimates (0.6) and (0.7)) are obtained. Afterwards, in Sec. 10, these results are applied to homogenization for solutions of the nonhomogeneous hyperbolic systems. Section 11 is devoted to applications of the general results to the acoustics equation, the operator of elasticity theory and the model equation of electrodynamics. 0.8. Acknowledgement. The author is grateful to T. A. Suslina for attention to work and numerous comments that helped to improve the quality of presentation. 0.9. Notation. Let H and H * be separable Hilbert spaces. The symbols (·, ·) H and · H mean the inner product and the norm in H, respectively; the symbol · H→H * denotes the norm of a bounded linear operator acting from H to H * . Sometimes we omit the indices if this does not lead to confusion. By I = I H we denote the identity operator in H. If A : H → H * is a linear operator, then Dom A denotes the domain of A. If N is a subspace of H, then N ⊥ := H ⊖ N.
The symbol ·, · denotes the inner product in C n , | · | means the norm of a vector in C n ; 1 n is the unit matrix of size n × n. If a is an (m × n)-matrix, then |a| denotes its norm as a linear operator from C n to C m ; a * means the Hermitian conjugate (n × m)-matrix.
The classes L p of C n -valued functions on a domain O ⊂ R d are denoted by L p (O; C n ), 1 p ∞. The Sobolev spaces of order s of C n -valued functions on a domain O ⊂ R d are denoted by H s (O; C n ). By S(R d ; C n ) we denote the Schwartz class of C n -valued functions in R d . If n = 1, then we simply write L p (O), H s (O) and so on, but sometimes we use such simplified notation also for the spaces of vector-valued or matrix-valued functions. The symbol L p ((0, T ); H), 1 p ∞, stands for L p -space of H-valued functions on the interval (0, T ). Next, The Laplace operator is denoted by ∆ = ∂ 2 /∂x 2 1 + · · · + ∂ 2 /∂x 2 d . By C, C, C, c, c (probably, with indices and marks) we denote various constants in estimates. The absolute constants are denoted by β with various indices.
Chapter I. Abstract scheme 1. Preliminaries 1.1. Quadratic operator pencils. Let H and H * be separable complex Hilbert spaces. Suppose that X 0 : H → H * is a densely defined and closed operator, and that X 1 : H → H * is a bounded operator. On the domain Dom X(t) = Dom X 0 , consider the operator X(t) := X 0 + tX 1 , t ∈ R. Our main object is a family of operators (1.1) that are selfadjoint in H and non-negative. The operator A(t) acting in H is generated by the closed quadratic form We assume that the point λ 0 = 0 is isolated in the spectrum of A 0 and 0 < n := dim N < ∞, n n * := dim N * ∞. By d 0 we denote the distance from the point zero to the rest of the spectrum of A 0 and by F (t, s) we denote the spectral projection of the operator A(t) for the interval [0, s]. Fix δ > 0 such that 8δ < d 0 . Next, we choose a number t 0 > 0 such that Then (see [BSu1,Chapter 1,(1.3)]) F (t, δ) = F (t, 3δ) and rank F (t, δ) = n for |t| t 0 . We often write F (t) instead of F (t, δ). Let P and P * be the orthogonal projections of H onto N and of H * onto N * , respectively.
1.2. Operators Z and R. Let D := Dom X 0 ∩ N ⊥ , and let u ∈ H * . Consider the following equation for the element ψ ∈ D (cf. [BSu1, Chapter 1, (1.7)]): The equation is understood in the weak sense. In other words, ψ ∈ D satisfies the identity Equation (1.3) has a unique solution ψ, and X 0 ψ H * u H * . Now, let ω ∈ N and u = −X 1 ω. The corresponding solution of equation (1.3) is denoted by ψ(ω). We define the bounded operator Z : H → H by the identities Now, we introduce an operator R : N → N * (see [BSu1,Chapter 1,Subsec. 1.2]) as follows: Rω = X 0 ψ(ω) + X 1 ω ∈ N * . Another description of R is given by the formula R = P * X 1 | N .
1.3. The spectral germ. The selfadjoint operator S := R * R : N → N is called the spectral germ of the operator family (1.1) at t = 0 (see [BSu1,Chapter 1,Subsec. 1.3]). This operator also can be written as S = P X * 1 P * X 1 | N . So, S X 1 2 . (1.5) The spectral germ S is called nondegenerate, if Ker S = {0} or, equivalently, rank R = n.
In [BSu1,Chapter 1,Subsec. 1.6] it was shown that the numbers γ l and the elements ω l , l = 1, . . . , n, are eigenvalues and eigenvectors of the operator S: Sω l = γ l ω l , l = 1, . . . , n. (1.7) The numbers γ l and the vectors ω l , l = 1, . . . , n, are called threshold characteristics at the bottom of the spectrum of the operator family A(t).
Lemma 1.1. For |t| t 0 and t = 0 we have (1.20) The constant C 6 is defined below in (1.23) and depends only on δ, X 1 , and c * .
Proof. For t = 0 the operator under the norm sign in (2.1) is understood as a limit for t → 0. Using the Taylor series expansion for the sine function, we see that this limit is equal to zero. Now, let t = 0. We put (2.5) By (1.8) and (1.16), the operator-valued function (2.5) satisfies the following estimate: (2.8) (Cf. the proof of Theorem 2.5 from [BSu5].) So, (2.9) By virtue of (1.8) and (1.10), from (2.9) we derive the inequality (2.10) 2.2. Approximation in the "energy" norm. Now, we obtain another approximation for the operator A(t) −1/2 sin(τ A(t) 1/2 ) (in the "energy" norm).
Theorem 2.4. For τ ∈ R, ε > 0, and |t| t 0 we have Let H be yet another separable Hilbert space. Let X(t) = X 0 + t X 1 : H → H * be a family of operators of the same form as X(t), and suppose that X(t) satisfies the assumptions of Subsection 1.1.
Let M : H → H be an isomorphism. Suppose that M Dom X 0 = Dom X 0 ; X 0 = X 0 M ; X 1 = X 1 M . Then X(t) = X(t)M . Consider the family of operators (3.1) Obviously, 2) In what follows, all the objects corresponding to the family (3.1) are supplied with the upper mark " ". Note that N = M N, n = n, N * = N * , n * = n * , and P * = P * .
We denote Let Q N be the block of Q in the subspace N: As was shown in [Su2, Proposition 1.2], the orthogonal projection P of the space H onto N and the orthogonal projection P of the space H onto N satisfy the following relation: According to [BSu1,Chapter 1,Subsec. 1.5], the spectral germs S and S satisfy For the operator family (3.1) we introduce the operator Z Q acting in H and taking an element u ∈ H to the solution ϕ Q of the problem where ω := P u. Equation (3.5) is understood in the weak sense. As was shown in [BSu2,Lemma 6.1], the operator Z for A(t) and the operator Z Q satisfy 3.2. The principal term of approximation for the sandwiched operator A(t) −1/2 sin(τ A(t) 1/2 ). In this subsection, we find an approximation for the operator A(t) −1/2 sin(τ A(t) 1/2 ), where A(t) is given by (3.2), in terms of the germ S of A(t) and the isomorphism M . It is convenient to border the operator A(t) −1/2 sin(τ A(t) 1/2 ) by appropriate factors.
Proposition 3.1. Suppose that the assumptions of Subsec. 3.1 are satisfied. Then for τ ∈ R and |t| t 0 we have (3.7) Here t 0 is defined according to (1.2), and C 7 is the constant from (2.10) depending only on δ, X 1 , and c * .

Approximation with the corrector.
Proposition 3.2. Under the assumptions of Subsec. 3.1, for τ ∈ R and |t| t 0 we have (3.14) The constant C 9 is defined in (2.18) and depends only on δ, X 1 , and c * .
Chapter II. Periodic differential operators in L 2 (R d ; C n ) In the present chapter, we describe the class of matrix second order differential operators admitting a factorization of the form A = X * X , where X is a homogeneous first order DO. This class was introduced and studied in [BSu1, Chapter 2].

Factorized second order operators
4.1. Lattices Γ and Γ. Let Γ be a lattice in R d generated by the basis a 1 , . . . , a d : and let Ω be the elementary cell of the lattice Γ: The basis b 1 , . . . , b d dual to a 1 , . . . , a d is defined by the relations b l , a j = 2πδ lj . This basis generates the lattice Γ dual to Γ: Let Ω be the first Brillouin zone of the lattice Γ: (4.1) Let |Ω| be the Lebesgue measure of the cell Ω: |Ω| = meas Ω, and let | Ω| = meas Ω. We put 2r 1 := diam Ω. The maximal radius of the ball containing in clos Ω is denoted by r 0 . Note that With the lattice Γ, we associate the discrete Fourier transformation which is a unitary mapping of l 2 ( Γ) onto L 2 (Ω): Below by H 1 (Ω; C n ) we denote the subspace of functions from H 1 (Ω; C n ) whose Γ-periodic extension to R d belongs to H 1 loc (R d ; C n ). We have and convergence of the series in the right-hand side of (4.5) is equivalent to the relation u ∈ H 1 (Ω; C n ). From (4.1), (4.4), and (4.5) it follows that and in the definition of ψ it is assumed that the matrix ψ(x) is square and nondegenerate, and 4.2. The Gelfand transformation. Initially, the Gelfand transformation U is defined on the functions of the Schwartz class by the formula Since Under the Gelfand transformation, the operator of multiplication by a bounded periodic function in L 2 (R d ; C n ) turns into multiplication by the same function on the fibers of the direct integral.
The operator D applied to v ∈ H 1 (R d ; C n ) turns into the operator D + k applied to v(k, ·) ∈ H 1 (Ω; C n ).

4.3.
Factorized second order operators. Let b(D) be a matrix first order DO of the form d j=1 b j D j , where b j , j = 1, . . . , d, are constant matrices of size m × n (in general, with complex entries). We always assume that m n. Suppose that the symbol b(ξ) = d j=1 b j ξ j , ξ ∈ R d , of the operator b(D) has maximal rank: rank b(ξ) = n for 0 = ξ ∈ R d . This condition is equivalent to the existence of constants α 0 , α 1 > 0 such that (4.7) From (4.7) it follows that be positive definite and bounded together with the inverse matrix: (4.9) In L 2 (R d ; C n ), consider DO A formally given by the differential expression 10) The precise definition of the operator A is given in terms of the quadratic form Using the Fourier transformation and assumptions (4.7), (4.9), it is easily seen that (4.11) Thus, the form a[·, ·] is closed and non-negative.
The operator A admits a factorization of the form A = X * X , where

Direct integral decomposition for the operator A
5.1. The forms a(k) and the operators A(k). We put and consider the closed operator X (k) : in L 2 (Ω; C n ) is formally given by the differential expression with the periodic boundary conditions. The precise definition of the operator A(k) is given in terms of the closed quadratic form a(k) [u, u] Using the discrete Fourier transformation (4.3) and assumptions (4.7), (4.9), it is easily seen that So, by the compactness of the embedding H 1 (Ω; C n ) ֒→ L 2 (Ω; C n ), the spectrum of A(k) is discrete and the resolvent is compact. By (4.6) and the lower estimate (5.3), From (4.2) and (4.5) with k = 0 it follows that Combining this with the lower estimate (5.3) for k = 0, we see that the distance d 0 from the point zero to the rest of the spectrum of A(0) satisfies 5.2. Direct integral decomposition for A. Using the Gelfand transformation, we decompose the operator A into the direct integral of the operators A(k): This means the following. If v ∈ Dom a, then v(k, ·) = (U v)(k, ·) ∈ d for a. e. k ∈ Ω, (5.9) Conversely, if v ∈ Ω ⊕L 2 (Ω; C n ) dk satisfies (5.9) and the integral in (5.10) is finite, then v ∈ Dom a and (5.10) holds.

5.3.
Incorporation of the operators A(k) into the abstract scheme. For d > 1 the operators A(k) depend on the multidimensional parameter k. According to [BSu1, Chapter 2], we consider the onedimensional parameter t := |k|. We will apply the scheme of Chapter I. Herewith, all our considerations will depend on the additional parameter θ = k/|k| ∈ S d−1 , and we need to make our estimates uniform with respect to θ. The spaces H and H * are defined by (5.1). Let Then A(t, θ) = X(t, θ) * X(t, θ). According to (5.5) and (5.6), N = Ker X 0 = Ker A(0), dim N = n. The distance d 0 from the point zero to the rest of the spectrum of A(0) satisfied estimate (5.7). As was shown in [BSu1,Chapter 2,Sec. 3], the condition n n * = dim Ker X * 0 is also fulfilled. Thus, all the assumptions of Section 1 are valid.
In Subsection 1.1, it was required to choose the number δ < d 0 /8. Taking (5.4) and (5.7) into account, we put This allows us to take the following number in the role of t 0 (see (1.2)). Obviously, t 0 r 0 /2, and the ball |k| t 0 lies in Ω. It is important that c * , δ, and t 0 (see (5.4), (5.11), (5.13)) do not depend on the parameter θ. From (5.4) it follows that the spectral germ S(θ) (which now depends on θ) is nondegenerate: (5.14) It is important that the spectral germ is nondegenerate uniformly in θ.
6. The operator A. The effective matrix. The effective operator 6.1. The operator A. In the case where f = 1 n , we agree to mark all the objects by the upper hat ,, ". We have H = H = L 2 (Ω; C n ). For the operator is denoted by A(t; θ). If f = 1 n , the kernel (5.6) takes the form Let P be the orthogonal projection of H onto the subspace N. Then P is the operator of averaging over the cell: 6.2. The effective matrix. In accordance with [BSu1, Chapter 3, Sec. 1], the spectral germ S(θ) of the operator family A(t, θ) acting in N can be represented as where b(θ) is the symbol of the operator b(D) and g 0 is the so-called effective matrix. The constant positive (m × m)-matrix g 0 is defined as follows. Assume that a Γ-periodic (n × m)matrix-valued function Λ ∈ H 1 (Ω) is the weak solution of the problem Then the effective matrix g 0 is given by It turns out that the matrix g 0 is positive definite. In the case where f = 1 n , estimate (5.14) takes the form S(θ) c * I N . (6.10) We also need the following inequalities obtained in [BSu3,(6.28) and Subsec. 7.3]: 6.3. The effective operator A 0 . By (6.6) and the homogeneity of the symbol b(k), we have 14) The matrix S(k) is the symbol of the differential operator and called the effective operator for the operator A.
Let A 0 (k) be the operator family in L 2 (Ω; C n ) corresponding to the effective operator A 0 .
Proposition 6.1. Let g 0 be the effective matrix (6.9). Then From inequalities (6.18) it follows that Now, we distinguish the cases where one of the inequalities in (6.18) becomes an identity. See [BSu1, Chapter 3, Propositions 1.6 and 1.7].
Theorem 8.1. Under the assumptions of Subsection 8.1, for ε > 0 and τ ∈ R we have The constants C 12 and C 13 depend only on m, α 0 , α 1 , g L∞ , g −1 L∞ , f L∞ , f −1 L∞ , and the parameters of the lattice Γ.
Proposition 8.3. Let l = 1 for d = 1, l > 1 for d = 2, and l = d/2 for d 3. Then the operator A 1/2 [Λ] is a continuous mapping of H l (R d ; C m ) to L 2 (R d ; C n ), and (8.5) Here the constant C d depends only on m, n, d, α 0 , α 1 , g L∞ , g −1 L∞ , and the parameters of the lattice Γ; for d = 2 it depends also on l.
Proof of Theorem 8.2. Taking into account that the matrix-valued function (7.4) is the symbol of the operator A 0 and the function χ Ω (ξ) is the symbol of Π, using (4.7), (7.3), and (7.7) we For d 4, we can take l 2 in Proposition 8.3. So, combining (8.5) and (8.6), we have Combining this with (8.3), we arrive at estimate (8.4) with C 14 = C 13 + C ′ 14 . 8.3. On the possibility of removal of the operator Π from the corrector. Sufficient conditions on Λ. It is possible to eliminate the operator Π for d 5 by imposing the following assumption on the matrix-valued function Λ.

Condition 8.4. The operator [Λ] is continuous from
Actually, it is sufficient to impose the following condition to remove Π for d 5.
Condition 8.5. Assume that the periodic solution Λ of problem (6.7) belongs to L d (Ω).
To prove Proposition 8.6 we need the following statement.
Lemma 8.7. Let d 3. Assume that Condition 8.5 is satisfied. Then the operator g 1/2 b(D)[Λ] is a continuous mapping of H 2 (R d ; C m ) to L 2 (R d ; C m ) and (8.7) The constant C Λ depends only on d, α 0 , α 1 , g L∞ , g −1 L∞ , Λ L d (Ω) , and the parameters of the lattice Γ.
Proof. The proof is quite similar to the proof of Proposition 8.8 from [Su4].
For d 5, removal of the operator Π in the corrector also can be achieved by increasing the degree of the operator R(ε). In the application to homogenization of the hyperbolic Cauchy problem, this corresponds to more restrictive assumptions on the regularity of the initial data.
The proof of the following result is quite similar to that of Theorem 8.2.
Chapter III. Homogenization problem for hyperbolic systems

Approximation of the sandwiched operator
be the operator of multiplication by the function ψ ε (x). Our main object is the operator A ε , ε > 0, acting in L 2 (R d ; C n ) and formally given by the differential expression 2) The precise definitions of these operators are given in terms of the corresponding quadratic forms. The coefficients of the operators (9.1) and (9.2) oscillate rapidly as ε → 0.
Our goal is to approximate the sandwiched operator A −1/2 ε sin(τ A 1/2 ε ). The results are applied to homogenization of the solutions of the Cauchy problem for hyperbolic systems. 9.1. The principal term of approximation. Let T ε be the unitary scaling transformation in The operator A 0 satisfies a similar identity. Next, Note that for any s the operator (H 0 + I) s/2 is an isometric isomorphism of the Sobolev space Using these arguments, from (8.2) we deduce the following result.
9.3. The case where d 4. Now we apply Theorem 8.2. By the scaling transformation, (8.4) implies that Combining this with (9.9), we obtain Let us estimate the (L 2 → L 2 )-norm of the corrector. By the scaling transformation, (9.23) The (H s → L 2 )-norm of the operator [Λ] was estimated in [Su3,Proposition 11.3].
Proposition 9.7. Let s = 0 for d = 1, s > 0 for d = 2, s = d/2 − 1 for d 3. Then the operator [Λ] is a continuous mapping of H s (R d ; C m ) to L 2 (R d ; C m ), and there the constant C d depends only on d, m, n, α 0 , g L∞ , g −1 L∞ , and the parameters of the lattice Γ; in the case d = 2 it depends also on s. Now we consider only the case d 4. So, by Proposition 9.7, (9.24) Thus, we need to estimate the operator b(D)f 0 ( in the (L 2 → H 1 )-norm. By (6.19), (7.3), and (7.5), for any d we have (9.25) The following result is a direct consequence of (9.4) and (9.22)-(9.25).
Under Condition 9.11, the operator [(DΛ) ε ] is bounded from H 1 to L 2 due to the following result obtained in [PSu, Corollary 2.4].
Lemma 9.12. Under Condition 9.11, for any function u ∈ H 1 (R d ) and ε > 0 we have The constants c 1 and c 2 depend on m, d, α 0 , α 1 , g L∞ , and g −1 L∞ . Some cases where Condition 9.11 is fulfilled automatically were distinguished in [BSu4,Lemma 8.7].
Proposition 9.13. Suppose that at least one of the following assumptions is satisfied: 1 • ) d 2; 2 • ) the dimension d 1 is arbitrary and the operator A ε has the form A ε = D * g ε (x)D, where g(x) is symmetric matrix with real entries; 3 • ) the dimension d is arbitrary and g 0 = g, i. e., relations (6.21) are true.
Then Condition 9.11 is fulfilled.
Surely, if Λ ∈ L ∞ , then Condition 8.5 holds automatically. Then, by Proposition 8.6, for d 5, the assumptions of Theorem 9.9 are satisfied.
Proposition 9.15. Suppose that relations (6.20) hold. Then under the assumptions of Theorem 9.1, for 0 s 1 and τ ∈ R, 0 < ε 1 we have C 2 (s)(1 + |τ |)ε s . 9.7. The Steklov smoothing operator. Another approximation with the corrector. Let us show that the result of Theorem 9.5 remains true with the operator Π ε replaced by another smoothing operator. The operator S ε , ε > 0, acting in L 2 (R d ; C n ) and defined by the relation is called the Steklov smoothing operator.
The following properties of the operator S ε were checked in [ZhPas1, Lemmata 1.1 and 1.2] (see also [PSu, Propositions 3.1 and 3.2]).
Proposition 9.16. For any function u ∈ H 1 (R d ; C n ) we have Proposition 9.17. Let Φ(x) be a Γ-periodic function in R d such that Φ ∈ L 2 (Ω). Then the operator [Φ ε ]S ε is bounded in L 2 (R d ; C n ), and We also need the following statement obtained in [PSu,Lemma 3.5].
Remark 9.20. Similarly to the proof of Theorem 9.6, using the properties of the Steklov smoothing, one can check that the estimate of the form (9.13) remains true with Π ε replaced by S ε .
Theorem 10.1. Let u ε be the solution of problem (10.1) and let u 0 be the solution of the effective problem (10.4). 1 • . Let ψ ∈ H 1 (R d ; C n ) and let F ∈ L 1,loc (R; H 1 (R d ; C n )). Then for τ ∈ R and ε > 0 we have 2 • . Let ψ ∈ H 2 (R d ; C n ) and let F ∈ L 1,loc (R; H 2 (R d ; C n )). Let Λ(x) be the Γ-periodic solution of problem (6.7). Let Π ε be the smoothing operator (9.6). By v ε we denote the first order approximation: v Then for τ ∈ R and ε > 0 we have Let S ε be the Steklov smoothing operator (9.33). We puť Then for τ ∈ R and ε > 0 we have Remark 10.2. If d 4 (or d 5 and Condition 8.4 is satisfied), then we can use Theorem 9.8 (respectively, Theorem 9.9), i. e., the estimate of the form (10.7) is valid with v ε replaced by Theorem 9.10 implies the following statement.
Let ψ ∈ H d/2 (R d ; C n ) and let F ∈ L 1,loc (R; H d/2 (R d ; C n )). Let u ε and u 0 be the solutions of problems (10.1) and (10.4) respectively. Let v (0) ε be given by (10.8). Then for τ ∈ R and 0 < ε 1 we have Applying Theorems 9.2 and 9.6, we arrive at the following result.
Theorem 10.5. Let u ε be the solution of problem (10.1), and let u 0 be the solution of the effective problem (10.4). 1 • . Let ψ ∈ L 2 (R d ; C n ) and F ∈ L 1,loc (R; L 2 (R d ; C n )). Then Let v ε be given by (10.6). Then for τ ∈ R we have Remark 10.6. Taking Remark 9.20 into account, we see that the results of Theorems 10.4(2 • ) and 10.5(2 • ) remain true with the operator Π ε replaced by the Steklov smoothing S ε , i. e., with v ε replaced byv ε . This only changes the constants in estimates.
Applying Theorem 9.14, we make the following observation.
Remark 10.7. For 0 < ε 1, under Condition 9.11, the analogs of Theorems 10.1, 10.4, and 10.5 are valid with the operators Π ε and S ε replaced by the identity operator.
The assertion 2 • follows from (10.26) by the Banach-Steinhaus theorem. The result 3 • is a consequence of (10.27) and the Banach-Steinhaus theorem.
Remark 10.11. Using Proposition 9.17, it is easily seen that the results of Lemma 10.9 are valid with the operator Π ε replaced by the operator S ε . Hence, by using (10.11) and interpolation, we deduce the analog of Theorem 10.10 with Π ε replaced by S ε . This only changes the constants in estimates.

10.3.
On the possibility to remove Π ε from approximation of the flux.
Theorem 10.12. Under the assumptions of Theorem 10.8, let d 4. Then for τ ∈ R and 0 < ε 1 we have The constant C 30 depends only on m, n, d, α 0 , α 1 , g L∞ , g −1 L∞ , f L∞ , f −1 L∞ , and the parameters of the lattice Γ.
The following statement can be checked by analogy with the proof of Theorem 10.12.
To obtain interpolational results without any smoothing operator, we need to prove the analog of Lemma 10.9 without Π ε . I. e., we want to prove (L 2 → L 2 )-boundedness of the operator The following property of g was obtained in [Su3,Proposition 9.6]. (The one dimensional case will be considered in Subsection 10.4 below.) Proposition 10.15. Let l > 1 for d = 2, and l = d/2 for d 3. The operator [ g] is a continuous mapping of H l (R d ; C m ) to L 2 (R d ; C m ), and The constant C ′ d depends only d, m, n, α 0 , α 1 , g L∞ , g −1 L∞ , and the parameters of the lattice Γ; for d = 2 it depends also on l.
So, for d 2, we can not expect the (L 2 → L 2 )-boundedness of the operator (10.35). The (H 2 → L 2 )-continuity of the operator (10.35) was used in Theorem 10.12 and, under Condition 8.4, in Theorem 10.13. (The (H 2 → L 2 )-boundedness of [ g] follows from [MaSh, Subsection 1.3.2, Lemma 1].) So, without any additional conditions on Λ, using Proposition 10.15, we can obtain some interpolational results only for d 3.

Applications of the general results
The following examples were previously considered in [BSu1,BSu5,DSu3,DSu4].
11.1. The acoustics equation. In L 2 (R d ), we consider the operator where g(x) is a periodic symmetric matrix with real entries. Assume that g(x) > 0, g, g −1 ∈ L ∞ . The operator A describes a periodic acoustical medium. The operator (11.1) is a particular case of the operator (6.1). Now we have n = 1, m = d, b(D) = D, α 0 = α 1 = 1. Consider the operator A ε = D * g ε (x)D, whose coefficients oscillate rapidly for small ε. Let us write down the effective operator. In the case under consideration, the Γ-periodic solution of problem (6.7) is a row: Here e j , j = 1, . . . , d, is the standard orthonormal basis in R d . Clearly, the functions Φ j (x) are real-valued, and the entries of Λ(x) are purely imaginary. By (6.8), the columns of the (d × d)matrix-valued function g(x) are the vector-valued functions g(x) (∇Φ j (x) + e j ), j = 1, . . . , d. The effective matrix is defined according to (6.9): g 0 = |Ω| −1 Ω g(x) dx. Clearly, g(x) and g 0 have real entries. If d = 1, then m = n = 1, whence g 0 = g.
Let Q(x) be a Γ-periodic function on R d such that Q(x) > 0, Q, Q −1 ∈ L ∞ . The function Q(x) describes the density of the medium.
The constants C 6 (s), C 7 (s), and C 8 (s) depend only on s, d, g L∞ , g −1 L∞ , and parameters of the lattice Γ.
11.2. The operator of elasticity theory. Let d 2. We represent the operator of elasticity theory in the form used in [BSu1,Chapter 5,§2]. Let ζ be an orthogonal second rank tensor in R d ; in the standard orthonormal basis in R d , it can be represented by a matrix ζ = {ζ jl } d j,l=1 . We shall consider symmetric tensors ζ, which will be identified with vectors ζ * ∈ C m , 2m = d(d + 1), by the following rule. The vector ζ * is formed by all components ζ jl , j l, and the pairs (j, l) are put in order in some fixed way. Let χ be an (m × m)-matrix, χ = diag {χ (j,l) }, where χ (j,l) = 1 for j = l and χ (j,l) = 2 for j < l. Then | | |ζ| | | 2 = χζ * , ζ * C m .
Let u ∈ H 1 (R d ; C d ) be the displacement vector. Then the deformation tensor is given by e(u) = 1 2 ∂u j ∂x l + ∂u l ∂x j . The corresponding vector is denoted by e * (u). The relation b(D)u = −ie * (u) determines an (m × d)-matrix homogeneous DO b(D) uniquely; the symbol of this DO is a matrix with real entries. For instance, with an appropriate ordering, we have Let σ(u) be the stress tensor, and let σ * (u) be the corresponding vector. The Hooke law can be represented by the relation σ * (u) = g(x)e * (u), where g(x) is an (m × m) matrix (which gives a ,,concise" description of the Hooke tensor). This matrix characterizes the parameters of the elastic (in general, anisotropic) medium. We assume that g(x) is Γ-periodic and such that g(x) > 0, and g, g −1 ∈ L ∞ .
The energy of elastic deformations is given by the quadratic form (11.4) The operator W generated by this form is the operator of elasticity theory. Thus, the operator 2W = b(D) * g(x)b(D) = A is of the form (6.1) with n = d and m = d(d + 1)/2.
In the case of an isotropic medium, the expression for the form (11.4) simplifies significantly and depends only on two functional Lamé parameters λ(x), µ(x): The parameter µ is the shear modulus. The modulus λ(x) may be negative. Often, another parameter κ(x) = λ(x) + 2µ(x)/d is introduced instead of λ(x); κ is called the modulus of volume compression. In the isotropic case, the conditions that ensure the positive definiteness of the matrix g(x) are µ(x) µ 0 > 0, κ(x) κ 0 > 0. We write down the ,,isotropic" matrices g for d = 2 and d = 3: Consider the operator W ε = 1 2 A ε with rapidly oscillating coefficients. The effective matrix g 0 and the effective operator W 0 = 1 2 A 0 are defined by the general rules (see (6.8), (6.9), and (6.15)).
Let Q(x) be a Γ-periodic (d × d)-matrix-valued function such that Q(x) > 0, Q, Q −1 ∈ L ∞ . Usually, Q(x) is a scalar-valued function describing the density of the medium. We assume that Q(x) is a matrix-valued function in order to take possible anisotropy into account.
Theorems 10.4 and 10.10 can be applied to problem (11.5). If d = 2, then Condition 9.11 is satisfied according to Proposition 9.13. So, we can use Theorem 9.14. If d = 3, then Theorem 9.8 is applicable.
The application of Theorems 9.1 and 9.8 gives the following result.
To approximate the flux, we apply Theorem 10.12. The matrix g = g(1 + b(D)Λ) has a blockdiagonal structure, see [BSu3,Subsection 14.3]): the upper left (3 × 3) block is represented by the matrix with the columns ∇ Φ j (x) + c j , j = 1, 2, 3. We denote this block by a(x). The element at the right lower corner is equal to ν. The other elements are zero. Then, by (11.6) and (11.10), We arrive at the following statement.