Fully nonlinear elliptic and parabolic equations

University Of Washington, Seattle WA

Investigators

Abstract

This project concentrates on the study of special Lagrangian equations, symmetric Hessian equations, Isaacs equations, complex Monge-Ampere equations, and their parabolic versions (e.g., Lagrangian mean curvature flows). The theory of regularity and solvability for fully nonlinear uniformly elliptic and parabolic equations (with the convexity condition in general dimensions and without the convexity hypothesis in dimension two) is well developed. The concrete equations just listed either do not satisfy the convexity condition or do not exhibit uniform ellipticity or parabolicity. Only preliminary attempts have been made in the general saddle cases. Substantial advances have been achieved for the symmetric Hessian equations and the complex Monge-Ampere equations, yet there is still no Schauder or Calderon-Zygmund theory for these equations; and surprisingly the regularity problem for the quadratic symmetric Hessian equations in general dimension still remains open. This project seeks to address these fundamental issues. Investigations into the aforementioned equations will further our knowledge of two closely related mathematical fields, partial differential equations and differential geometry. Moreover, the project will also have impact on the areas where these equations arise. 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. Solutions to Isaacs equations lead to the optimal strategy for certain random processes, for example, in engineering and finance. 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. Part of the research also involves participation of graduate students.