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

证明了弱收敛的数学理论 对连续统中出现的问题有重要的应用 因为它描述了微观和微观之间的关系 变量及其在宏观水平上的平均值。 为了 平均非线性量，一个数学工具， 补偿紧性是由F.穆拉特和 首席研究员，并已被用来获得各种 宏观与微观的关系 关系 该工具的改进，基于所谓的H- 首先出现在同质化问题上的措施， 最近由首席研究员开发。 本项目的目标是进一步研究 H-测度的数学性质，扩大了其 非线性偏微分方程的应用 连续介质力学，并定义更一般的数学对象 这可能会弥补一些已知的限制H- 措施 特别要研究的领域包括有效特性 混合物，由均匀化引起的非局部效应， 复合材料微观结构的演变与阻尼 非均匀材料中的波。

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.