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 proposed work concern nonlinear partial differential equations, which are especially important due to the nonlinear effects they are used to model. 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. The PI proposes to investigate the compactness of conformal metrics on a Riemannian manifold having constant sigma-k curvature for k larger than 1 and less than half of the dimension of the manifold. For k greater than or equal to half of the dimension of the manifold, or when the manifold is locally conformally flat, the compactness result has been proved. A success in establishing the compactness results would lead to new existence results on conformal metrics with constant sigma-k curvature. A related problem on compactness of solutions to the constant Q-curvature equations on Riemannian manifolds is also proposed. The PI has also proposed to study elliptic systems arising from composite material. The approach to the study of the compactness of solutions is to give a fine analysis of blow up solutions to the type of nonlinear elliptic equations on manifolds. Efforts will be made in advancing further and deeper understanding of solutions of conformally invariant equations.
|Effective start/end date||6/1/15 → 5/31/20|
- National Science Foundation (National Science Foundation (NSF))