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密切联系的完全非线性椭圆方程。它集中于三个主题：实或复流形上的全非线性椭圆方程的估计；双曲空间中的渐近平台问题；复蒙日-安培方程的正则性和弱解问题。这些主题都是密切相关的，具有强烈的共同特征，即大多数主要问题都集中在或归结为对某些完全非线性偏微分方程的解建立某些先验估计。然而，这些问题中的每一个都提出了独特的技术挑战。全非线性椭圆方程在理解数学难题方面发挥着关键作用，在图像处理、光学反射器设计、最优质量输运和数学物理等方面有着重要的应用。为了解决一个完全非线性的二阶方程，一个中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（但不是假设一致椭圆）方程接近最优条件下得出这种估计的技术。这一方向的突破将对整个非线性偏微分方程领域及其应用产生广泛的影响。

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.