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.
