Homogenization of Partial Differential Equations

Homogenization is a method for modelling processes in complex structures. These processes are far too complex for analytic and numerical methods and are best described by PDEs with rapidly oscillating coefficients - a technique that has become increasingly important in the last three decades due to its multiple applications in the areas of optimization, radiophysics, filtration theory, rheology, elasticity theory, and other domains of mechanics, physics, and technology. The present monograph is a comprehensive study of homogenization problems describing various physical processes in micro-inhomogeneous media. From the technical viewpoint the work focuses on the construction of nonstandard models for media characterized by several small-scale parameters (multiscale models). A variety of techniques are used -- specifically functional analysis, the spectral theory for differential operators, the Laplace transform, and, most importantly, a new variational PDE method for studying the asymptotic behavior of solutions of stationary boundary value problems. This new method can be applied to a wide variety of problems. Key topics in this systematic exposition include asymptotic analysis, Dirichlet- and Neumann-type boundary value problems, differential equations with rapidly oscillating coefficients, homogenization, homogenized and non-local models. Along with complete proofs of all main results, numerous examples of typical structures of micro-inhomogeneous media with their corresponding homogenized models are provided. Applied mathematicians, advanced-level graduate students, physicists, engineers, and specialists in mechanics will benefit from this monograph, which may be used in the classroom or as a comprehensive reference text. TOC:Introduction - The Dirichlet boundary value problem in strongly perforated domains - The Dirichlet boundary value problem in domains with complex boundary - Strongly connected domains - The Neumann boundary value problems in strongly connected domains - Non-stationary problems and spectral problems - Differential equations with rapidly oscillating coefficients - Homogenized conjugation conditions - Bibliography