Fully Nonlinear Elliptic Equations and Related Geometric Problems

这个建议涉及完全非线性椭圆方程，这些方程产生于几何问题或与几何问题有很强的联系。它侧重于三个主题：估计完全非线性椭圆方程的真实的或复杂的流形;渐近高原问题在双曲空间;和问题的正则性和弱解的复杂蒙赫安培方程。这些主题都是密切相关的，具有强大的共同特征，即大多数主要问题都集中在或减少到建立某些先验估计的解决方案，一些完全非线性偏微分方程。完全非线性椭圆方程在理解数学难题方面发挥着关键作用，并且在图像处理、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二阶方程的中心问题是建立一个先验估计的前景的解决方案。我们的目标是寻找技术，以获得这样的估计条件下，接近最佳的一般椭圆（但不假设一致椭圆）方程的真实的或复杂的流形。这一方向的突破将对全非线性偏微分方程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.