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.
The project focused on several areas which we list below. 1. The study of various types of wave motion on two dimensional spheres and flat space. In the main we obtained results for non-linear wave equations whose elliptic parts were conformally invariant, that is part of the wave equation remained invariant under conformal transformations of the two dimensional sphere or conformal transformations of two dimensional Euclidean space. The investigationalso focused on studying when solutions to such wave equations exist, and if they do exist, does the existence persist for all time or if there is a breakdown in the solution for somefinite time, commonly called a blow-up phenomena. We showed that such blow-up phenomena is related to the fact that the initial data has an energy threshold above a certain value determined by a typical solution to the elliptic problem called a bubble in the literature. 2. The second part of our work was devoted to investigate Cauchy-Riemann structures or CR structures on compact, three dimensional manifolds. We obtained results where we could find invariant conditions when such manifolds could be embedded in Complex n-dimensional space. 3. Our last set of investigations focused on the properties of composite materials and analysing properties of the interface that arises in constructing objects made out of composites where the shape and mass is specified and the objective is to create an object with maximum rigidity, that is as strong as possible.
