main-content

## Über dieses Buch

It was mainly during the last two decades that the theory of homogenization or averaging of partial differential equations took shape as a distinct mathe­ matical discipline. This theory has a lot of important applications in mechanics of composite and perforated materials, filtration, disperse media, and in many other branches of physics, mechanics and modern technology. There is a vast literature on the subject. The term averaging has been usually associated with the methods of non­ linear mechanics and ordinary differential equations developed in the works of Poincare, Van Der Pol, Krylov, Bogoliubov, etc. For a long time, after the works of Maxwell and Rayleigh, homogeniza­ tion problems for· partial differential equations were being mostly considered by specialists in physics and mechanics, and were staying beyond the scope of mathematicians. A great deal of attention was given to the so called disperse media, which, in the simplest case, are two-phase media formed by the main homogeneous material containing small foreign particles (grains, inclusions). Such two-phase bodies, whose size is considerably larger than that of each sep­ arate inclusion, have been discovered to possess stable physical properties (such as heat transfer, electric conductivity, etc.) which differ from those of the con­ stituent phases. For this reason, the word homogenized, or effective, is used in relation to these characteristics. An enormous number of results, approximation formulas, and estimates have been obtained in connection with such problems as electromagnetic wave scattering on small particles, effective heat transfer in two-phase media, etc.

## Inhaltsverzeichnis

### 1. Homogenization of Second Order Elliptic Operators with Periodic Coefficients

Abstract
This chapter is intended to give a thorough description of the model homogenization problem. The fundamental homogenization theorem is proved here by two methods, viz., the method of compensated compactness and that of asymptotic expansions. Numerous examples are given to illustrate the computation of the homogenized matrix. Apart from the standard results of the homogenization theory, usually recorded in monographic literature, we consider here such questions as the derivation of explicit formulas in two-dimensional problems, residual diffusion, estimates for the homogenized matrix.
V. V. Jikov, S. M. Kozlov, O. A. Oleinik

### 2. An Introduction to the Problems of Diffusion

Abstract
Consider the following Cauchy problem for the equation describing diffusion in a stationary periodic medium:
$$2\frac{{\partial u}}{{\partial t}} - div\left( {A\left( x \right)\nabla u} \right) = 0, u\left| {_{t = 0} = f\left( x \right).} \right.$$
V. V. Jikov, S. M. Kozlov, O. A. Oleinik

### 3. Elementary Soft and Stiff Problems

Abstract
For a large class of homogenization problems the given periodic matrix A(x) satisfies the usual inequality v1 IA(x) ≤ v2 I (v1, v2 > 0) at all points of ℝ m outside certain subsets, called inclusions, where the matrix A(x) is degenerate. Two main types of degeneration are usually considered: if A(x) = 0, then we speak of soft inclusions, or a soft problem; if, on the other hand, the inverse matrix B = 0, then we are dealing with stiff inclusions, or a stiff problem. As a rule, the matrix A(x) is constant outside the inclusions, and therefore we have a kind of two-phase periodic medium.
V. V. Jikov, S. M. Kozlov, O. A. Oleinik

### 4. Homogenization of Maxwell Equations

Abstract
If the space ℝ3 is regarded as a nonconducting medium, the electromagnetic field can be described by the following system of equations
$$\begin{gathered}{\rho _1}\frac{{\partial {u_1}}}{{\partial t}} = {\operatorname{curlu} _2}, \hfill \\ {\rho _2}\frac{{\partial {u_2}}}{{\partial t}} = - {\operatorname{curlu} _1},\forall x \in {\mathbb{R}^3},t \geqslant 0. \hfill \\ \end{gathered}$$
(4.1)
V. V. Jikov, S. M. Kozlov, O. A. Oleinik

### 5. G-Convergence of Differential Operators

Abstract
Consider a class of measurable symmetric matrices A(x) ={a ij (x)} which satisfy the inequality
$$\begin{gathered}{v_1}\left| \xi \right.\left| {^2} \right. \leqslant \xi \cdot A\xi \leqslant {v_2}\left| \xi \right.\left| {^2} \right.,\forall x \in Q,\xi \in {\mathbb{R}^m}, \hfill \\ \hfill \\ \end{gathered}$$
(5.1)
where Q is a bounded domain, and v1, v2 are positive constants. This class of matrices will be denoted by ε (v1, v2, Q).
V. V. Jikov, S. M. Kozlov, O. A. Oleinik

### 6. Estimates for the Homogenized Matrix

Abstract
The bounds specified by the Voigt — Reiss inequality established in Section 1.6 are usually too wide and give little information about the homogenized matrix. The problem of tighter bounds has been the subject of intensive research in physics and continuum mechanics, especially in the theory of dispersion of electromagnetic waves on small particles and the theory of elasticity for microscopically non-homogeneous media. After the classical works of Maxwell [1] and Rayleigh [1], an enormous amount of facts accumulated in this direction. For a long time, preference has been given to the potential theory methods, and it was only the case of two-phase media that the analysts were concerned with; an important role in the previous studies belongs to the geometric properties of the inclusions.
V. V. Jikov, S. M. Kozlov, O. A. Oleinik

### 7. Homogenization of Elliptic Operators with Random Coefficients

Abstract
In this chapter we consider the problem of homogenization for elliptic operators
$$\frac{\partial }{{\partial {x_i}}}({a_{ij}}(y)\frac{\partial }{{\partial {x_j}}}), {\kern 1pt} y = {\varepsilon ^{ - 1}}x,$$
assuming that the matrix A(y) = {a ij (y)} is statistically stationary with respect to the spatial variable y ∈ ℝ m , or equivalently, that A(y) is a typical realization of a stationary random field.
V. V. Jikov, S. M. Kozlov, O. A. Oleinik

### 8. Homogenization in Perforated Random Domains

Abstract
Soft and stiff problems with periodic inclusions were considered in Chapter 3. Here we consider an arrangement of random inclusions.
V. V. Jikov, S. M. Kozlov, O. A. Oleinik

### 9. Homogenization and Percolation

Abstract
The phenomenon of percolation can be conveniently modeled by a random structure of chess-board type, or random checkered tessellation. A structure of this kind is obtained if we split the plane into squares, painting each square, independently, black or white with probability p or 1 − p, respectively, where 0 ≤ p ≤ 1. Then the union of all black squares forms a random set F. Any two black squares are thought of as neighboring, or linked, if they have a common side or a vertex. In accordance with this arrangement, a finite number of black squares is said to form a path if these squares can be enumerated in such a way that any two consecutive numbers correspond to neighboring or linked squares. A set K consisting of black squares is connected if for any pair of squares belonging to K there exists a path containing these squares and belonging to K. Obviously, the set F is a union of separate connected components, which are called clusters.
V. V. Jikov, S. M. Kozlov, O. A. Oleinik

### 10. Some Asymptotic Problems for a Non-Divergent Parabolic Equation with Random Stationary Coefficients

Abstract
An important class of equations with random stationary coefficients consists of equations with periodic coefficients. Therefore, it would be useful to recall the formulation of some asymptotic problems for such equations and indicate a method for their solution.
V. V. Jikov, S. M. Kozlov, O. A. Oleinik

### 11. Spectral Problems in Homogenization Theory

Abstract
The behavior of eigenvalues and eigenfunctions of the boundary value problems considered in the preceding chapters is studied here in the context of the homogenization theory. Our analysis is primarily based on the theorems (proved in Section 11.1) about spectral properties of a sequence of abstract operators. Direct application of these theorems allows us to describe asymptotic properties of eigenvalues and eigenfunctions for a wide class of boundary value problems with a small parameter which arise in the homogenization theory of differential operators.
V. V. Jikov, S. M. Kozlov, O. A. Oleinik

### 12. Homogenization in Linear Elasticity

Abstract
The state of an elastic body is usually characterized by the displacement vector, the strain tensor, and the stress tensor.
V. V. Jikov, S. M. Kozlov, O. A. Oleinik

### 13. Estimates for the Homogenized Elasticity Tensor

Abstract
Estimates similar to those established in Chapter 6 for the homogenized matrix are proved here for the homogenized elasticity tensor. The problem of the attainability of these estimates in the case of two-phase media will be discussed in detail.
V. V. Jikov, S. M. Kozlov, O. A. Oleinik

### 14. Elements of the Duality Theory

Abstract
This chapter is aimed at setting the variational boundary value problems, as well as the justification of the duality formulas.
V. V. Jikov, S. M. Kozlov, O. A. Oleinik

### 15. Homogenization of Nonlinear Variational Problems

Abstract
Let (Ω, μ) be a probability space with an ergodic dynamical system T(t), t ∈ ℝ m . A function f (ω, ξ) defined in Ω × ℝ m is called a random Lagrangian, if f is convex in ξ ∈ ℝ m and measurable in ωΩ for every ξ ∈ ℝ m . The simplest special case of a random Lagrangian is f (x, ξ) periodic in x ∈ ℝ m .
V. V. Jikov, S. M. Kozlov, O. A. Oleinik

### 16. Passing to the Limit in Nonlinear Variational Problems

Abstract
It is often necessary to have a reasonable definition of the “limit problem” for a given sequence of nonlinear variational problems. For instance, the theory of homogenization deals with a special sequence of Lagrangians
$${f_\varepsilon }(x,\xi ) = f({\varepsilon ^{ - 1}}x,\xi ), f(y,\xi ) is periodic in y$$
and its object consists in passing to the limit, as ε→ 0, in the variational problem
$${E^\varepsilon } = \mathop {u \in W_0^\alpha }\limits^{\inf } (Q) \int_Q {\{ {f_\varepsilon }(x, \nabla u) - p* \cdot \nabla u\} } dx$$
or some other variational problems involving the Lagrangian f ε . The expression “passing to the limit” means that we have to find a kind of “limit Lagrangian” f such that
$$\varepsilon \xrightarrow{{\lim }}0{E^\varepsilon } = \mathop {u \in W_0^\alpha }\limits^{\inf } (Q) \int_Q {\{ f(x, \nabla u) = p* \cdot \nabla u\} } dx$$
and f is independent of p*.
V. V. Jikov, S. M. Kozlov, O. A. Oleinik

### 17. Basic Properties of Abstract Γ-Convergence

Abstract
We start with a topological aspect of Γ-convergence, namely, with the notion of the topological limit of a sequence of sets; then Γ-convergence of functions will be defined in terms of the convergence of their epigraphs.
V. V. Jikov, S. M. Kozlov, O. A. Oleinik

Many problems arising in the theory of plasticity and geometry involve Lagrangians of the form |ξ|, $$\sqrt {1 + {{\left| \xi \right|}^2}}$$. The associated functionals are defined on the Sobolev space W1(Q) which is not reflexive, and therefore the variational problem may happen to possess no solution. For this reason it is natural to find a wider space containing W1(Q) and to extend the integral functional to that space, so that the minimum could be attained on the new space. This problem has been the subject of intensive research in the theory of plasticity.