The Dirichlet Boundary Value Problem in Strongly Perforated Domains with Fine-Grained Boundary -- The Dirichlet Boundary Value Problem in Strongly Perforated Domains with Complex Boundary -- Strongly Connected Domains -- The Neumann Boundary Value Problems in Strongly Perforated Domains -- Nonstationary Problems and Spectral Problems -- Differential Equations with Rapidly Oscillating Coefficients -- Homogenized Conjugation Conditions.

Homogenization is a method for modeling processes in microinhomogeneous media, which are encountered in radiophysics, filtration theory, rheology, elasticity theory, and other domains of mechanics, physics, and technology. These processes are described by PDEs with rapidly oscillating coefficients or boundary value problems in domains with complex microstructure. From the technical point of view, given the complexity of these processes, the best techniques to solve a wide variety of problems involve constructing appropriate macroscopic (homogenized) models. The present monograph is a comprehensive study of homogenized problems, based on the asymptotic analysis of boundary value problems as the characteristic scales of the microstructure decrease to zero. The work focuses on the construction of nonstandard models: non-local models, multicomponent models, and models with memory. Along with complete proofs of all main results, numerous examples of typical structures of microinhomogeneous media with their corresponding homogenized models are provided. Graduate students, applied mathematicians, physicists, and engineers will benefit from this monograph, which may be used in the classroom or as a comprehensive reference text.