On Some Nonlinear Elliptic Equations

点击翻译按钮获取中文摘要

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.