Theory of Nonlinear Elliptic Equations and Systems

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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 and systems is especially important from this perspective. For instance, such equations and systems play a role in the theoretical study of composite materials. This project will contribute to a basic understanding of fully nonlinear elliptic equations and systems, thereby providing scientists and engineers with sharpened insight into various physical processes and ultimately enhancing the quality of 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 and help convey the long-term value of mathematical research not only to science and engineering but also, in the end, to society. At a technical level, this project is a natural continuation of the principal investigator's earlier work, which includes valuable contributions to this research area. One part of the project concerns a longstanding open problem on the existence and compactness of solutions to a fully nonlinear Yamabe problem. This is equivalent to solving, on a Riemannian manifold, a fully nonlinear elliptic, but not uniformly elliptic, partial differential equation of second order. The current understanding of this type of equations is insufficient, especially compared to what is known for fully nonlinear uniformly elliptic second order equations, where theories are much more mature. The study of this open problem should lead to a better understanding of these elliptic, but not uniformly elliptic, equations. It will also lead to a better understanding of degenerate elliptic fully nonlinear equations of second order. This project will provide new analytical tools for the study of these and other important nonlinear partial differential equations arising from geometry and physics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.