Tatiana Aleksandrovna Suslina

Second order periodic differential operators. Threshold properties and homogenization

The vector periodic differential operators (DO’s) A admitting a factorization A = X ∗X , where X is a first order homogeneous DO, are considered in L2(R). Many operators of mathematical physics have this form. The effects that depend only on a rough behavior of the spectral expansion of A in a small neighborhood of zero are called threshold effects at the point λ = 0. An example of a threshold effect is the behavior of a DO in the small period limit (the homogenization effect). Another example is related to the negative discrete spectrum of the operator A−αV , α > 0, where V (x) ≥ 0 and V (x) → 0 as |x| → ∞. “Effective characteristics”, such as the homogenized medium, effective mass, effective Hamiltonian, etc., arise in these problems. The general approach to these problems proposed in this paper is based on the spectral perturbation theory for operator-valued functions admitting analytic factorization. Most of the arguments are carried out in abstract terms. As to applications, the main attention is paid to homogenization of DO’s.

Homogenization with corrector term for periodic elliptic differential operators

We continue to study the class of matrix periodic elliptic differential operators Aε in Rd with coefficients oscillating rapidly (i.e., depending on x/ε). This class was introduced in the authors’ earlier work of 2001 and 2003. The problem of homogenization in the small period limit is considered. Approximation for the resolvent (Aε + I)−1 is obtained in the operator norm in L2(R) with error term of order ε2. The so-called corrector is taken into account. We develop the approach of our paper of 2003, where approximation with no corrector term but with remainder term of order ε was found. The paper is based on the operator-theoretic material obtained in our paper in the previous issue of this journal. Though the present paper is a continuation of the earlier work, it can be read independently.

Homogenization with corrector for periodic differential operators. Approximation of solutions in the Sobolev class ¹(ℝ^{})

Investigation of a class of matrix periodic elliptic second-order differential operators Aε in Rd with rapidly oscillating coefficients (depending on x/ε) is continued. The homogenization problem in the small period limit is studied. Approximation for the resolvent (Aε + I)−1 in the operator norm from L2(R) to H1(Rd) is obtained with an error of order ε. In this approximation, a corrector is taken into account. Moreover, the (L2 → L2)-approximations of the so-called fluxes are obtained.

Threshold approximations with corrector for the resolvent of a factorized selfadjoint operator family

In a Hilbert space we consider the family of operators admitting a factorization A(t) = X(t)∗X(t), where X(t) = X0 + tX1, t ∈ R. We suppose that the subspace N = KerA(0) is finite-dimensional. For the resolvent (A(t) + e2I)−1, we obtain an approximation in the operator norm on a fixed interval |t| ≤ t0 for small values of e. This approximation contains the so called 1⁄2corrector ; the remainder term is of order O(1). The results are aimed on applications to homogenization of periodic differential operators in the small period limit. The paper develops and strengthens the results of [BSu, Chapter 1].

Homogenization of a stationary periodic Maxwell system

The homogenization problem is considered for a stationary periodic Maxwell system in R3 in the small period limit. The behavior of four fields is studied, namely, of the strength of the electric field, the strength of the magnetic field, the electric displacement vector, and the magnetic displacement vector. Each field is represented as a sum of two terms. For some terms uniform approximations in the L2(R)-norm are obtained, together with a precise order estimate for the remainder term. §0. Introduction 0.1. In the present paper, we consider the homogenization problem for a stationary periodic Maxwell system in the small period limit. This problem was studied intensively; in particular, it was discussed in the books [BaPa, BeLP, ZhKO, Sa]. However, the known results give only the weak convergence of solutions to the solution of the “homogenized” system with constant coefficients. We rely on the abstract approach developed in [BSu1, BSu2]. This approach makes it possible to establish the convergence of resolvents in the operator L2-norm to the resolvent of the homogenized problem, and simultaneously gives a remainder estimate of precise order. At the same time, the Maxwell operator can be included in the class of differential operators studied in [BSu1, BSu2] only in the case where one of two periodic characteristics of the medium is constant. In [BSu2, Chapter 7], the homogenization problem for the Maxwell operator was considered in the case where μ = 1. Here we study the much more difficult general case, which requires essential modification of the technique. The detailed comparison of the methods and results of the present paper and those of [BSu2] is given below in Subsection 0.8. 0.2. Setting of the problem. We denote G = L2(R; C) and J = {f ∈ G : div f = 0}. Let Γ be a lattice of periods in R, and let Ω ⊂ R be the elementary cell of Γ. Assume that the dielectric permittivity η(x) and the magnetic permeability μ(x) are Γ-periodic matrix-valued functions and that η and μ are bounded and uniformly positive. We denote by u the strength of the electric field and by v the strength of the magnetic field; w = ηu is the electric displacement vector, and z = μv is the magnetic displacement vector. We write the Maxwell operator M in terms of the displacement vectors, assuming that w and z are solenoidal. Then M = M(η, μ) acts in the space J ⊕ J and is given by the formula M(η, μ) = ( 0 i curl μ−1 −i curl η−1 0 ) on the natural domain. The point λ = i is a regular point for M. 2000 Mathematics Subject Classification. Primary 35P20, 35Q60.

Operator error estimates for homogenization of the elliptic Dirichlet problem in a bounded domain

Let

Homogenization in the Sobolev class ¹(ℝ^{}) for second order periodic elliptic operators with the inclusion of first order terms

\mathcal{O} \subset \mathbb{R}^d

Operator error estimates in the homogenization problem for nonstationary periodic equations

be a bounded domain of class

On homogenization for a periodic elliptic operator in a strip

C^2

Homogenization with corrector for a stationary periodic Maxwell system

. In the Hilbert space