Berlyand L., Rybalko V. Getting Acquainted with Homogenization and Multiscale

Springer, 2018. — 187 p. — (Compact Textbooks in Mathematics). — ISBN: 978-3-030-01776-7.The objective of this book is to navigate beginning graduate students in mathematics and engineering through a mature field of multiscale problems in homogenization theory and to provide an idea of its broad scope. An overview of a wide spectrum of homogenization techniques ranging from classical two-scale asymptotic expansions to Gamma convergence and the rapidly developing field of stochastic homogenization is presented. The mathematical proofs and definitions are supplemented with intuitive explanations and figures to make them easier to follow. A blend of mathematics and examples from materials science and engineering is designed to teach a mixed audience of mathematical and non-mathematical students.Preliminaries
What Is Homogenization and Multiscale? First Examples
Brief History and Surprising Examples in Homogenization
Formal Two-Scale Asymptotic Expansions and the Corrector Problem
Compensated Compactness and Oscillating Test Functions
Two-Scale Convergence
Examples of Explicit Effective Coefficients: Laminated Structures and 2D Checkerboards
Introduction to Stochastic Homogenization
Γ-Convergence in Nonlinear Homogenization Problems
An Example of a Nonlinear Problem: Homogenization of Plasticity and Limit Loads
Continuum Limits for Discrete Problems with Fine Scales