Second Order Nonlinear Elliptic Equations and Systems

9706887李本研究涉及三个相关领域的几何问题，都是基于求解各种类型的非线性二阶偏微分方程。第一部分是关于有边界流形上的Yamabe问题的类比：研究对于给定的边界为（M,g）的紧致黎曼流形，具有恒定的标量曲率和恒定的边界平均曲率的与g共形的度量的存在性和紧性。这相当于研究了一类满足非线性诺依曼边界条件的临界指数半线性椭圆方程正解的存在性和紧性。第二个是关于n球上标量曲率的规定问题。这个问题在n=2,3,4维中更容易理解。这里的目标是在n = 5维中寻找一个好的存在准则，使得规定的曲率函数K是莫尔斯函数。第三是关于退化的蒙日-安培方程的研究，特别是那些由几何引起的。偏微分方程自然地产生于物理学、几何学和许多其他领域，并构成了物理世界数学建模的基础。数学分析的作用与其说是创建方程，不如说是创建关于解的定性和定量信息。这可能包括对存在、独特性、平滑性和成长等问题的回答。此外，分析常常发展出解的近似方法和对这些近似精度的估计。

9706887 Li This research is on geometric problems in three related areas, all based on solving nonlinear second order partial differential equations of various types. The first concerns an analogue of the Yamabe problem on manifolds with boundaries: To study, for a given compact Riemannian manifold with boundary (M,g), the existence and compactness of metrics conformal to g that have constant scalar curvature and constant boundary mean curvature. This is equivalent to the study of the existence and compactness of positive solutions to a certain semilinear elliptic equation with critical exponent satisfying certain nonlinear Neuman boundary conditions. The second concerns the problem of prescribing scalar curvature on the n-sphere. The problem is better understood in dimension n=2,3,4. The goal here is to search for good existence criterion in dimension n = 5 that allow the prescribed curvature functions K to be Morse functions. The third concerns the study of degenerate Monge-Ampere equations, especially those arising from geometry. Partial differential equations arise naturally from physics, geometry, and many other fields and form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to create qualitative and quantitative information about the solutions. This may include answers to questions about existence, uniqueness, smoothness, and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations.