Fully Nonlinear Elliptic Equations and Related Geometric Problems

这一建议涉及由几何问题引起的或与几何问题有很强联系的完全非线性的椭圆型方程。它集中在三个主题上：实或复流形上完全非线性椭圆型方程的估计；双曲空间中的渐近平台问题；以及复Monge-Ampere方程的正则性和弱解问题。这些主题都是密切相关的，有一个很强的共同特征，即大多数主要问题都集中在或归结为建立某些完全非线性偏微分方程解的某种先验估计。完全非线性椭圆型方程是理解数学难题的关键，在图像处理、光学反射镜设计、最优传质、数学物理等领域有着重要的应用。求解完全非线性二阶方程的一个核心问题是建立透视解的先验估计。我们的目标是寻找在实流形或复流形上的一般椭圆型(但不假定为一致椭圆型)方程在接近最优的条件下获得此类估计的技巧。这方面的突破将对整个完全非线性偏微分方程组及其应用领域产生广泛的影响。

This proposal concerns fully nonlinear elliptic equations that arise from or have strong connections with problems in geometry. It focuses on three topics: estimates for fully nonlinear elliptic equations on real or complex manifolds; asymptotic Plateau problems in hyperbolic space; and questions on regularity and weak solutions of the complex Monge-Ampere equation. These topics are all closely related, with the strong common feature that most of the major questions are centred at or reduce to establishing certain a priori estimates for solutions of some fully nonlinear partial differential equations. Yet each of these questions presents unique technical challenges.Fully nonlinear elliptic equations play key roles in understanding difficult problems in mathematics, and have important applications such as in image processing, optical reflector designs, optimal mass transport, and mathematical physics. A central issue in order to solve a fully nonlinear second order equation is to establish a priori estimates for perspective solutions. Our goal is to search for techniques to derive such estimates under conditions which are close to optimal for general elliptic (but not assumed uniformly elliptic) equations on real or complex manifolds. Breakthroughs in this direction would have broad impacts to the whole field of fully nonlinear PDEs and their applications.