The principle investigator will continue his study of special Lagrangian equations, Isaacs equations, symmetric Hessian equations, and complex Monge-Ampere equations. The theory of a priori estimates and solvability for fully nonlinear uniformly elliptic equations (with the convexity condition in arbitrary 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. Only preliminary attempts have been made to deal with such issues, mainly in the saddle case. 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. This project seeks to remedy that state of affairs.
Investigations into the aforementioned concrete equations will further our knowledge of two 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 closely related to nonlinear elasticity theory in mechanics, which studies the mechanisms whereby a material that is stretched returns to its original size and shape.
