Let O ⊂ R d be a bounded domain of class C 2 . In the Hilbert space L2(O; C n ), we consider a matrix elliptic second order differential operator AD,ε with the Dirichlet boundary condition. Here ε > 0 is the small parameter. The coefficients of the operator are periodic and depend on x/ε. A sharp order operator error estimateHere A 0 D is the effective operator with constant coefficients and with the Dirichlet boundary condition.
Abstract. Let O ⊂ R d be a bounded domain with the boundary of class C 1,1 . In L2(O; C n ), a matrix elliptic second order differential operator AN,ε with the Neumann boundary condition is considered. Here ε > 0 is a small parameter, the coefficients of AN,ε are periodic and depend on x/ε. There are no regularity assumptions on the coefficients. It is shown that the resolvent (AN,ε + λI) −1 converges in the L2(O; C n )-operator norm to the resolvent of the effective operator A 0 N with constant coefficients, as ε → 0. A sharp order error estimate (AN,ε + λI)Cε is obtained. Approximation for the resolvent (AN,ε + λI) −1 in the norm of operators acting from L2(O; C n ) to the Sobolev spaceApproximation is given by the sum of the operator (A 0 N + λI) −1 and the first order corrector. In a strictly interior subdomain O ′ a similar approximation with an error O(ε) is obtained.
We study homogenization in the small period limit for a periodic parabolic Cauchy problem in R d and prove that the solutions converge in L 2 (R d ) to the solution of the homogenized problem for each t > 0. For the L 2 (R d )-norm of the difference, we obtain an order-sharp estimate uniform with respect to the L 2 (R d )-norm of the initial value. 1.We study homogenization of the Cauchy problem for the parabolic systemin the small period limit ε → 0. Here u ε (x, t) is a C n -valued function of x ∈ R d and t 0, A ε is a matrix elliptic operator with periodic coefficients depending on ε −1 x, and ρ is a periodic positive matrix function. For the precise definition of the operator A ε , see Secs. 2 and 4 below. The homogenization problem for parabolic equations has been intensively studied by traditional methods of homogenization theory (e.g., see the books [1-3]). The corresponding results give convergence (in an appropriate function class) of the solutions u ε to the solution u 0 of the homogenized system with constant effective coefficients. We use the abstract operator-theoretic approach developed for elliptic systems in the papers [4,5], which allows us not only to prove the convergence of u ε to u 0 in the L 2 (R d )-norm (for a given t) but also to obtain an order-sharp estimate, uniform with respect to the L 2 (R d )-norm of the initial value φ, for the norm of the difference u ε − u 0 . More precisely, we obtain an estimate of the order of εt −1/2 for the operator norm of the difference of the resolving operators for problem (1) and the "homogenized" Cauchy problem (see Theorems 3 and 4 below).Let H be a separable Hilbert space. The symbol · H stands for the norm in H, and by · H→H we denote the norm of a bounded operator in H. Next, | · | is the norm of a vector in C n , and the identity (n × n) matrix is denoted by 1 n . We use the notation x = (x 1 , . . . , x d ) ∈ R d , iD j = ∂/∂x j , j = 1, . . . , d, and D = −i∇ = (D 1 , . . . , D d ). The L p -spaces of C n -valued functions in a domain O ⊂ R d are denoted by L p (O; C n ), 1 p ∞. The Sobolev classes of C n -valued functions are denoted by H s (O; C n ), s ∈ R.2. Factorized second-order operators. We consider self-adjoint matrix second-order differential operators in R d admitting a factorization of the form X * X , where X is a homogeneous first-order differential operator. This class of operators was introduced and studied in detail in [4,5]. Let n, m ∈ N, and let m ≥ n. We set G := L 2 (R d ; C n ) and G * := L 2 (R d ; C m ). Let b(D) : G → G * be a homogeneous first-order differential operator with constant coefficients. It has a symbol b(ξ), which is a linear homogeneous (m × n) matrix function of ξ ∈ R d . Suppose that rank b(ξ) = n, 0 = ξ ∈ R d . ThenConsider an (n × n) matrix function f (x) and an (m × m) matrix function h(x) (in general, with complex entries). Suppose that they are bounded and the inverses are also bounded:(3) *
Abstract. Matrix periodic elliptic second order differential operators B ε in R d with rapidly oscillating coefficients (depending on x/ε) are studied. The principal part of the operator is given in a factorized form b(D) * g(ε −1 x)b(D), where g is a periodic, bounded and positive definite matrix-valued function and b(D) is a matrix first order operator whose symbol is a matrix of maximal rank. The operator also has zero and first order terms with unbounded coefficients. The problem of homogenization in the small period limit is considered. Approximation for the generalized resolvent of the operator B ε is obtained in the operator norm in L 2 (R d ; C n ) with error term O(ε). Also, approximation for this resolvent is obtained in the norm of operators acting from L 2 (R d ; C n ) to H 1 (R d ; C n ) with error term of order ε and with the corrector taken into account. The general results are applied to homogenization problems for the Schrödinger operator and the two-dimensional Pauli operator with potentials involving singular terms. §0. Introduction Main results.We study the problem of homogenization in the small period limit for matrix differential operators (DO's) B ε , ε > 0, acting in the space L 2 (R d ; C n ). Let Γ be a lattice in R d , and let Ω be the cell of Γ. We use the notation ϕ ε (x) = ϕ(ε −1 x) for any measurable Γ-periodic function ϕ(x) in R d . The principal part A ε of the operator B ε is given in a factorized formwhere b(D) is a matrix homogeneous first order DO and g(x) is a bounded and positive definite matrix-valued function in R d , periodic with respect to the lattice Γ. (The precise assumptions on b(D) and g(x) are given below in §5.) Homogenization problems for the operator A ε have been studied in detail in a series of papers [BSu1,BSu2,BSu3,BSu4,BSu5]. Now we study more general operators B ε : we add zero and first order terms to A ε :
Abstract. Matrix periodic differential operators (DO's)are considered. The operators are assumed to admit a factorization of the form A = X * X , where X is a homogeneous first order DO. Let A ε = A(ε −1 x, D), ε > 0. The behavior of the solutions u ε (x, τ) of the Cauchy problem for the Schrödinger equation i∂ τ u ε = A ε u ε , and also the behavior of those for the hyperbolic equation ∂ 2 τ u ε = −A ε u ε , is studied as ε → 0. Let u 0 be the solution of the corresponding homogenized problem. Estimates of order ε are obtained for theThe estimates are uniform with respect to the norm of initial data in the Sobolev space H s (R d ; C n ), where s = 3 in the case of the Schrödinger equation and s = 2 in the case of the hyperbolic equation. The dependence of the constants in estimates on the time τ is traced, which makes it possible to obtain qualified error estimates for small ε and large |τ | = O(ε −α ) with appropriate α < 1. §0. Introduction 0.1. The class of operators. The present paper is a continuation of the authors' investigations [BSu1, BSu2, BSu3, BSu4, BSu5, Su1, Su2] in homogenization theory for a class of (matrix) periodic differential operators (DO's) acting in the space L 2 (R d ; C n ). This class is rather wide and includes many operators of mathematical physics.We consider matrix elliptic positive second order DO's in L 2 (R d ; C n ) that admit a factorization of the formHere b(D) is a homogeneous matrix first order DO with constant coefficients. Its symbol b(ξ) is an (m × n)-matrix of rank n (we assume that m ≥ n). It is assumed that the matrix-valued functions f (x) (of size n × n) and g(x) (of size m × m) are periodic with respect to some lattice Γ in R d and thatFor a more precise description of the operators (0.1), see Subsection 4.1. It is convenient to start with a narrower class of operators of the formand accordingly, to accept the "two-level"order of exposition.
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