Department of Mathematics and Materials Research Institute, Penn State University, University Park, PA 16802
Vladimir State Pedagogical University, 11 Stroitelei Avenue, Vladimir, 600024, Russian Federation
Homogenization problems were first studied in the late nineteenth century (Poisson, Maxwell, Rayleigh) and early twentieth century (Einstein). These studies were based on deep physical intuition allowing these outstanding physicists to solve several specific important problems such as calculating the effective conductivity of a two-phase conductor and the effective viscosity of suspensions. It was not until the early 1960s that homogenization began to gain a rigorous mathematical footing which enabled it to be applied to a wide variety of problem in physics and mechanics. A number of mathematical tools such as the asymptotic analysis of PDEs, variational bounds, heterogeneous multiscale method, and the probabilistic techniques of averaging were developed. Although this theory is a well-established area of mathematics, many fascinating problems remain open. Interesting examples of such problems can be found in the papers of this issue.
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