Nonlinear elliptic equations

University Of Washington, Seattle WA

Investigators

Abstract

The research activity into this proposed project will deepen our understanding of two intimately connected mathematical fields, partial differential equations and differential geometry, which may be viewed as extensions of advanced calculus. Simultaneously, the project will also have impact on the areas on which the equations studied in the project rest: some equations provide the mathematical foundation for mirror symmetry in the string theory of modern physics, which is a unified way to describe our physical universe; another equation is an effective model in material science; solutions to the so called Isaacs equations lead to the optimal strategy for certain random processes, for example, in engineering and finance; Also Hessian equations are related to nonlinear elasticity theory in mechanics, which studies the mechanisms whereby a material that is stretched returns to its original size and shape. The objectives for special Lagrangian equations are to derive Schauder and Calderon-Zygmund estimates for the equations with critical and supercritical phases, to answer the question whether any homogeneous order two solution in dimension five or higher is trivial or not, and to study low regularity of continuous viscosity solutions to the equations with subcritical phases. The purposes for self similar solutions to mean curvature flows are to classify Lagrangian translating solutions and study uniqueness of embedded sphere shrinker in 3-d Euclidean space. The aim for symmetric Hessian equations is to investigate Hessian estimates for quadratic Hessian equations in dimension four and higher and also scalar curvature equations, to obtain Schauder and Calderon-Zygmund estimates for 3-d quadratic Hessian equations, and to study the Liouville problem for k-symmetric Hessian equations. The attempt for fully nonlinear elliptic equations such as Isaacs equations in 3-d is to study the regularity for general fully nonlinear elliptic equations in 3-d, in particular for equations in the form of linear combinations of k-symmetric Hessians and finitely piecewise linear Isaacs equations. The plan for complex Monge-Ampere equations is to show the triviality of any global solution to complex Monge-Ampere equations including self-shrinking equations for the Kahler Ricci flow with certain necessary restrictions.