Leonid Berlyand is a Professor of Mathematics, and a member of the Materials Research Institute and  Huck Institutes for Life Sciences  at the  Pennsylvania State University (USA) as well as an Honorary Professor (Professoris Honoris Cavza) at the Moscow State University. Leonid Berlyand is an internationally recognized expert in applied mathematics. He has made fundamental contributions to homogenization theory, mathematical Ginzburg-Landau theory, and mathematical biology.Volodymyr Rybalko works as a senior research fellow in Mathematical Division, B.Verkin Institute for Low Temperature Physics and Engineering of National Academy of Sciences of Ukraine. He is an expert in the area of PDEs, he is known, in particular, for his contributions in homogenization theory.

• Chapter 1- Preliminaries.- Chapter 2- What is Homogenization and Multiscale? First Examples.- Chapter 3- Brief History and Surprising Examples in Homogenization.- Chapter 4- Formal Two-scale Asymptotic Expansions and the Corrector Problem.- Chapter 5- Compensated Compactness and Oscillating Test-functions.- Chapter 6- Two-scale Convergence.- Chapter 7- Examples of Explicit Effective Coefficients: Laminated Structures and 2D Checkerboards.- Chapter 8- Introduction to Stochastic Homogenization.- Chapter 9- G-Convergence in Nonlinear Homogenization Problems.- Chapter 10- An Example of a Nonlinear Problem: Homogenization of Plasticity and Limit Loads.- Chapter 11- Continuum Limits for Discrete Problems with Fine Scales.- References.- Appendix: Regular and Singular Perturbations and Boundary Layers.- Index.

“This book is an invitation to the young researcher to start using modern asymptotic methods of the applied mathematics of multiscale systems.” (Adrian Muntean, zbMATH 1441.35001, 2020)