On Some Fully Nonlinear Elliptic Equations

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Proposal DMS-0401118PI: Yanyan Li (Rutgers University)Title: On some fully nonlinear elliptic equationsThe PI proposes to work on some conformally invariant fully nonlinearequations which arise naturally from conformal geometry. In particular he proposes to prove a very general fully nonlinear version of the Yamabeconjecture. The corresponding Liouville type theorem has been established. The main task is to obtain apriori estimates for solutions.Partial differential equations arise naturally from physics, geometry,and many other fields and form a basis for mathematical modeling ofthe physical world. The proposed research should enhance the understanding of fully nonlinear elliptic equations, an important class of equationsin modeling the physical world. The study of conformally invariant equations has found applications in geometry and topology. Part of the proposed workconcerns important analytical issues for such equations. We expect that further applications to geometry and topology can be made with conformally invariant fully nonlinear equations and related ones, and the proposed work shouldprovide analytical tools for such applications.