Mathematical Sciences: Microlocal Measures and the Study of Oscillations, Concentration Effects and Comparison Mechanics in Nonlinear Partial Differential Equations

9401310鞑靼在过去的二十年里，首席研究人员开创了大多数新的数学工具，这些工具现在被用于研究出现在连续介质力学或物理学中的微观结构(通常是连续介质力学或物理学的非线性偏微分方程式：均匀化、补偿紧致性、H措施)。本提案涉及理论性质的研究，目的是统一许多零碎的成果，并创造一个更强大的工具。它一方面将经典分析的问题与在现有材料中观察到的微观结构的描述联系起来，另一方面，它的一个重要应用是设计新材料，其微观结构创造了一些比现有材料更好的性能。在过去的二十年里，首席研究人员率先创造了大多数新的数学工具，这些工具现在被用于研究材料科学和工程中重要情况下出现的微观结构。这项提议涉及的理论问题有助于描述如何设计新材料，其精细的结构有助于改善现有材料的性能。

9401310 Tartar In the last twenty years, the Principal Investigator has pioneered the creation of most of the new mathematical tools which are now used for studying microstructures appearing in the (usually nonlinear partial differential equations of Continuum Mechanics or Physics: Homogenization, Compensated Compactness, H-measures. The present Proposal deals with research of a theoretical nature, with the purposes of unifying many fragmentary results and of creating a still more powerful tool. It links questions of Classical Analysis on one side to the description of microstructures observed in existing materials on the other side, one important application being to devise new materials whose microstructure creates some improved properties over existing materials. In the last twenty years, the Principal Investigator has pioneered the creation of most of the new mathematical tools which are now used for studying the microstructures appearing in important situations in Material Sciences and Engineering. This Proposal deals with theoretical questions which can help describing how to devise new materials whose structure at a fine level is responsible for improved properties over existing materials.