Homogenization of Partial Differential Equations

Preface (S. 6-7)

This book is devoted to homogenization problems for partial differential equations describing various physical phenomena in microinhomogeneous media. This direction in the theory of partial differential equations has been intensively developed for the last forty years, it finds numerous applications in radiophysics, filtration theory, rheology, elasticity theory, and many other areas of physics, mechanics, and engineering sciences.

A medium is called microinhomogeneous if its local parameters can be described by functions rapidly varying with respect to the space variables. We will always assume that the length scale of oscillations is much less than the linear sizes of the domain in which a physical process is considered but much greater than the sizes of molecules, so that the process can be described using the differential equations of the mechanics of solids. These differential equations either have rapidly oscillating coefficients (with respect to the space variables) or are considered in domains with complex microstructure, such as domains with fine-grained boundary [112] (called later by the better-known term strongly perforated domains). The microstructure is understood as the local structure of a domain or the coefficients of equations in the scale of microinhomogeneities.

Obviously, it is practically impossible to solve the corresponding boundary (initial boundary) value problems by either analytical or numerical methods. However, if the microscale is much less than the characteristic scale of the process under investigation (e.g., the wavelength), then it is possible to give a macroscopic description of the process. If it is the case, the medium usually has stable characteristics (heat conductivity, dielectric permeability, etc.), which, in general, may differ substantially from the local characteristics. Such stable characteristics are referred to as homogenized, or effective, characteristics, because they are usually determined by methods of the homogenization theory for differential equations or the relevant mean field methods, effective medium methods, etc.

The term homogenization is associated, first of all, with methods of nonlinear mechanics and ordinary differential equations developed by Poincare, Krylov, Bogolyubov, and Mitropolskii (see, e.g., [21, 123]). For partial differential equations, homogenization problems have been studied by physicists from Maxwell`s times, but they remained for a long time outside the interests of mathematicians. However, since the mid I 960s, homogenization theory for partial differential equations began to be intensively developed by mathematicians as well, which was motivated not only by numerous applications (first of all, in the theory of composite media [142]) but also by the emergence of new deep ideas and concepts important for mathematics itself.

Currently, there is a great number of publications devoted to mathematical aspects of homogenization such as asymptotic analysis, two-scale convergence, G-convergence, and r -convergence. Making no claim to cite all of the available monographs on the subject, we would like to mention the books by Allaire [3], Bakhvalov and Panasenko [9], Bensoussan, Lions, and Papanicolaou [13], Braides and Defranceschi [26], Cioranescu and Donato [42].