Fully Nonlinear Equations

全非线性方程项目摘要于元本项目主要研究全非线性方程的正则性和可解性。第一个课题是研究各种方程的Bernstein/Liouville问题。这里的目的是证明某些问题的唯一解是某些非常简单或“微不足道”的解。另一个课题是在三维域上证明完全非线性二阶椭圆方程解的正则性结果，以及控制理论中的Isaacs方程的正则性结果。该项目还将着眼于推导二维黎曼流形局部等距嵌入问题的必要条件。项目中的偏微分方程不仅出现在几何中，而且出现在科学和工程中。这些问题的结果在现代物理学中的弦理论、力学中的非线性弹性理论以及工程和经济学中的随机最优控制理论中都有应用。

Fully Nonlinear EquationsProject AbstractYu YuanThis project focuses on the regularity and solvability of fully nonlinear equations. The first topic is to investigate Bernstein/Liouville problems for various equations. Here the aim is to prove that the only solutions of certain problems are certain very simple or "trivial" solutions. Another topic is to prove regularity results for the solutions of fully nonlinear second order elliptic equations in 3d domains and also for the Isaacs' equation of control theory. The project will also look at deriving necessary conditions for the local isometric imbedding problem for 2-dimensional Riemannian manifolds. The partial differential equations in the project arise not only in geometry but also in science and engineering. Results on these problems should have applications to stringtheory in modern physics, for nonlinear elasticity in mechanics, and for stochastic optimal control theory in engineering and economics.