On Some Fully Nonlinear Elliptic Equations

提案DMS-0401118PI：Yanyan Li(罗格斯大学)题目：关于一些完全非线性的椭圆型方程PI建议研究一些从共形几何自然产生的共形不变的完全非线性方程。特别是，他建议证明Yamabeconess猜想的一个非常普遍的完全非线性版本。建立了相应的Liouberle型定理。它的主要任务是获得解的先验估计。偏微分方程自然而然地产生于物理学、几何学和许多其他领域，并形成了物理世界的数学模型的基础。这项拟议的研究将加强对完全非线性椭圆型方程的理解，这是模拟物理世界中的一类重要方程。共形不变方程的研究在几何学和拓扑学中都有应用。拟议工作的一部分涉及此类方程的重要分析问题。我们期望用共形不变的完全非线性方程和相关方程进一步应用于几何和拓扑学，所提出的工作将为这类应用提供分析工具。

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.