Nonlinear Elliptic Equations and Applications

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This project will continue earlier studies by the principal investigator on the existence and compactness of solutions to a fully nonlinear version of the Yamabe problem. Such equations have been studied extensively. The project will have a special focus on compactness issues related to the aforementioned problem. The clarification of these issues is needed for further progress to be made on it. The research should lead to a better understanding of nonlinear elliptic, but not necessarily uniformly elliptic, partial differential equations, as well as of nonlinear degenerate elliptic equations. The principal investigator will also study elliptic systems that arise in the modeling of composite materials.Partial differential equations arise naturally in physics, engineering, geometry, and many other fields, and they form the basis for modeling many phenomena in the physical world. The particular class of "fully nonlinear elliptic" equations is especially important from this perspective. For instance, such equations turn up in the study of composite materials. This project will contribute to a basic understanding of fully nonlinear elliptic equations, thereby providing scientists and engineers with sharpened insight into various physical processes and ultimately enhancing the quality of, say, consumer products manufactured from composites. As part of the project, the principal investigator will train Ph.D. students, many of whom are expected to continue their careers as educators. They, in turn, will convey to even younger generations both their mathematical knowledge and the long-term value of mathematical research not only to science and engineering but also, in the end, to society.