Nonlinear Elliptic Equations and Systems, and Applications

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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 turn up in the theoretical study of composite materials. This project contributes to a better 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 trains 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.At a technical level, the PI has made valuable contributions in the areas of the project and the work supported by this award is a natural continuation of his earlier work. One part of the project concerns a long-standing 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. Closely related work includes a fully nonlinear Nirenberg problem and a fully nonlinear Loewner-Nirenberg problem. There has not been enough understanding for such type of equations, especially comparing to that available for fully nonlinear uniformly elliptic equations of second order where the theory is much more mature. The study of the open problem should lead to 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 will provide new tools in the study of this and other important nonlinear partial differential equations arising from geometry and physics. Another part of the project concerns elliptic equations and systems arising in the study of fluids and composite materials. In particular, new tools are developed to study a long-standing open problem on the existence of smooth solutions to the incompressible stationary Navier-Stokes equations on a flat torus of dimension sixteen.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.