On Some Nonlinear Elliptic Equations

关于一些非线性椭圆型方程摘要建议ResearchYanyan Li这个奖项是继续早期的研究紧的解决方案的Yamabe问题，并存在的解决方案，一个非常普遍的完全非线性版本的Yamabe问题。PI还提议与Luis Caffarelli合作研究Monge-Ampere方程的多值解，以了解由Monge-Ampere方程的多值解在三维空间中Y形奇点的顶点附近生成的度量的行为。黎曼几何中的许多问题可以表述为关于某些相关的非线性偏微分方程及其解的性质的问题。这些公式经常被证明是证明非常一般的几何定理的手段。结果往往依赖于流形的维数和其他几何不变量。根据这项奖励，将调查一些这类具体的知识和问题。

On some Nonlinear Elliptic Equations Abstract of Proposed ResearchYanyan LiThis award is to continue earlier studies on the compactness of solutions to the Yamabe problem, and on the existence of solutions to a very general fully nonlinear version of the Yamabe problem. The PI also proposes to study, in collaboration with Luis Caffarelli, multi-valued solutions to the Monge-Ampere equations in order to understand behavior of a metric generated by a multi-valued solutions to the Monge-Ampere equations near the vertex of a Y-shaped singularity in 3-dimensional space. Many questions in (Riemannian) geometry can be formulated as questions about the properties of certain associated nonlinear partial differential equations and their solutions. These formulations have often proved to be the means to prove very general geometrical theorems. The results often depend on the dimension of the manifolds and other geometrical invariants. Under this award, a number of specific conjectures and problems of this type will be investigated.