Fully Nonlinear Elliptic and Parabolic Equations

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The research activity in this project will deepen our understanding of two intimately connected mathematical fields: partial differential equations and differential geometry, which are just super calculus. Simultaneously, the project will also have impact on the areas where the equations to be investigated rest: special Lagrangian equations and complex Monge-Ampere 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; maximal surface equations are directly from the fascinating general relativity, which fundamentally changed our understanding of the world; mean curvature flow is an effective model in material science; Hessian equations are also 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 of research on special Lagrangian equations are to derive Schauder and Calderon-Zygmund estimates for equations with critical and supercritical phases, to answer whether any homogeneous order two solution in dimension five or higher is trivial, to study low regularity of continuous viscosity solutions to the equations with subcritical phases, and to resolve exterior Liouville problems with constraints as well as (entire) Liouville problem for the complex version of the special Lagrangian equation. The aim of research on symmetric sigma-k equations is to investigate Hessian estimates for sigma-2 equations in dimension four and higher and also sigma-2 principle curvature equations, to obtain Schauder and Calderon-Zygmund estimates for 3-d sigma-2 equations, and to study the Liouville problem for sigma-k equations. The plan for complex and real Monge-Ampere equations is to demonstrate the triviality of any global solution to complex Monge-Ampere equations including self-shrinking equations for the Kahler Ricci flow with certain necessary restrictions and to derive regularity of solutions to the real Monge-Ampere equations under a necessary noncollapsing condition. The attempt for maximal surface equations is to study the Bernstein problems for exterior solutions and regularity for solutions under a noncollapsing condition. The purposes for the study on fully nonlinear parabolic equations are to show uniqueness and existence for viscosity solutions to parabolic Monge-Ampere equations under certain necessary conditions and to derive Hessian estimates for Lagrangian mean curvature flow under certain convexity condition.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.