Nonlinear Elliptic Equations and Systems

提出的工作的第一部分涉及蒙日-安培方程，包括Jorgens， Calabi和pogorlov的经典结果的扩展，这些结果表明这些方程的整个解是二次多项式。研究欧几里德空间外域上蒙日-安培方程的狄利克雷问题。本文将研究仿射伯恩斯坦问题的相关问题。第二部分研究黎曼流形上的最佳Sobolev不等式和流形上的Yamabe问题。这包括在黎曼流形上建立一种新形式的最佳Sobolev不等式，以及关于有边界流形上的Yamabe问题的存在性和紧性结果。在这类研究中，精确的点对爆破解的估计发挥着重要作用。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。

The first part of the proposed work concerns Monge-Ampere equations, including extensions of classical results of Jorgens, Calabi and Pogorelov which state that entire solutions to such equations are quadratic polynomials. Dirichlet problem for Monge-Ampere equations in exterior domains of Euclidean space will also be investigated. Related problems concerning the affine Bernstein problem will be studied. The second part of the proposed work concerns best Sobolev inequality on Riemannian manifolds and the Yamabe problem on manifolds with boundary.This includes efforts in establishing a new form of best Sobolev inequalities on Riemannian manifolds as well as existence and compactness results concerning the Yamabe problem on manifolds with boundary. Sharp pointwise estimates to blow up solutions play important roles in such studies.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.