The mathematical theory of weak convergence has been shown to have important applications to problems arising in continuum physics because it describes the relations between microscopic variables and their averages at a macroscopic level. In order to average nonlinear quantities, a mathematical tool called compensated compactness has been developed by F. Murat and the principal investigator and it has been used to obtain various relations of a macroscopic nature from corresponding microscopic relations. An improvement of this tool, based on so called H- measures that first appeared in questions of homogenization, has been developed recently by the principal investigator. The goal of this project is to further study the mathematical properties of H-measures, enlarge the class of its applications to nonlinear partial differential equations of continuum mechanics, and define more general mathematical objects that could remedy some of the already known limitations of H- measures. Particular areas to be studied include effective properties of mixtures, nonlocal effects induced by homogenization, evolution of microstructures in composite materials and damping of waves in heterogeneous materials.
