Mathematical Sciences: H-Measures and the Study of Oscillations and Concentration Effects in Systems of PartialDifferential Equations of Continuum Mechanics

弱收敛的数学理论已被证明对连续介质物理中出现的问题有重要的应用，因为它在宏观水平上描述了微观变量与它们的平均值之间的关系。为了求平均非线性量，F·木拉提和首席研究员发展了一种称为补偿紧性的数学工具，并用它从相应的微观关系中获得了各种宏观性质的关系。这一工具的改进基于最先出现在同质化问题中的所谓H-测量，最近由首席调查员开发。这个项目的目标是进一步研究H-测度的数学性质，扩大它在连续介质力学的非线性偏微分方程中的应用，并定义更一般的数学对象，以弥补H-测度已知的一些局限性。特别要研究的领域包括混合物的有效性质、均匀化引起的非局部效应、复合材料中微观结构的演变以及非均质材料中的波的衰减。

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.