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

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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.