Second Order Nonlinear Elliptic Equations and Systems

9706887 Li这项研究是关于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组。第一种是类似于带边界流形上的Yamabe问题：对于给定的具有边界(M，g)的紧致黎曼流形，研究了具有常数量曲率和常边界平均曲率的共形于g的度量的存在性和紧性。这等价于研究一类临界指数满足一定非线性Neuman边界条件的半线性椭圆型方程正解的存在性和紧性。第二个问题是关于在n球面上规定标量曲率的问题。这个问题在维n=2，3，4中被更好地理解。这里的目标是在维n=5中寻找良好的存在准则，允许指定的曲率函数K是Morse函数。第三个是关于退化的Monge-Ampere方程的研究，特别是那些源于几何的方程。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，形成了对物理世界进行数学建模的基础。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、唯一性、光滑性和成长性等问题的回答。此外，分析经常开发出近似解的方法和对这些近似的精度的估计。

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.