Regularity for Fully Nonlinear Equations

University Of Washington, Seattle WA

Investigators

Abstract

PI: Yu Yuan, University of Washington DMS-0200784 Abstract: The theory of a priori estimates and solvability for fully nonlinear equations with the convexity condition are well developed. While other concrete equations like Isaacs equations from the stochastic optimal control theory and special Lagrangian equations from calibration geometry do not have the usual convexity condition. Only preliminary attempts were made toward this direction in recent years. Though the complex Monge-Ampere equations have the usual convexity condition, the holomorphic invariance is too large. There is no theory of a priori estimates for the complex Monge-Ampere equations with non-smooth right hand side. This project concentrates on the following four parts. In part one, the objective is to derive Holder a priori estimates for finitely piecewise linear Isaacs equations by further employment of the ideas in recent work. In part two, the attempt is to study the regularity for general fully nonlinear elliptic equations in 3-d in the light of recent work that any homogeneous order two solution to fully nonlinear elliptic equation in 3-d must be linear. In part three, the purpose is to answer whether any homogeneous order two solution to special Lagrangian equation of dimension four or higher is trivial. In part four, the aim is to study the Bernstein problem for complex Monge-Ampere equations. Differential equations and differential geometry are further applications of Newton's calculus to the investigation of laws and shapes of nature, and even some phenomena of our real world. This project deals with some particular equations, like Isaacs equations from optimal stochastic control theory, special Lagrangian equations and complex Monge-Ampere equations from differential geometry. Understanding those equations would have impacts on not only the related mathematical fields, but also fields outside mathematics like physics.