Andrej V. Cherkaev and Robert V. Kohn In the past twenty years we have witnessed a renaissance
of theoretical work on the macroscopic behavior of microscopically heterogeneous mate rials.
This activity brings together a number of related themes including: ( 1) the use of weak
convergence as a rigorous yet general language for the discussion of macroscopic behavior (2)
interest in new types of questions particularly the G-closure problem motivated in large part
by applications of optimal control theory to structural optimization (3) the introduction of
new methods for bounding effective moduli including one based on com pensated compactness and
(4) the identification of deep links between the analysis of microstructures and the
multidimensional calculus of variations. This work has implications for many physical problems
involving optimal design composite materials and coherent phase transitions. As a result it
has received attention and support from numerous scientific communities including engineering
materials science and physics as well as mathematics. There is by now an extensive literature
in this area. But for various reasons certain fundamental papers were never properly published
circu lating instead as mimeographed notes or preprints. Other work appeared in poorly
distributed conference proceedings volumes. Still other work was published in standard books or
journals but written in Russian or French. The net effect is a sort of gap in the literature
which has made the subject unnecessarily difficult for newcomers to penetrate.
