Harmonic Analysis and PDE

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The project considers various problems in the area of partial differential equations, including free boundary problems, variational problems, and geometric problems. These problems have been chosen because they are not technical and confined to narrow areas but are interdisciplinary in spirit. For example, the problems on composite materials are derived from engineering questions. As the principal investigator has done in the past and expects to continue to do now, he will rely for the resolution of these problems on techniques and tools from several areas of mathematics (e.g., partial differential equations, geometry, the calculus of variations, mathematical physics). Some of the problems under investigation are also amenable to numerical simulation. This, the principal investigator hopes, will provide additional insight.The chief impact of this project on areas outside of mathematics lies in the set of problems that focus on the design of composite materials. Lightweight and strong composites are playing increasingly important roles in fields such as aerospace engineering, medicine, etc. Theoretical studies like those to be undertaken in this project, in which one tries to maximize rigidity, should furnish valuable insight into the fabrication of such smart materials. There could, of course, be other areas of application. For instance, a survey of the literature reveals that the principal investigator's earlier work on composites has found use in ecological studies.