This project focuses on the regularity and solvability of fully nonlinear equations. The first topic is to investigate Bernstein/Liouville problems for various equations. Here the aim is to prove that the only solutions of certain problems are certain very simple or "trivial" solutions. Another topic is to prove regularity results for the solutions of fully nonlinear second order elliptic equations in 3d domains and also for the Isaacs' equation of control theory. The project will also look at deriving necessary conditions for the local isometric imbedding problem for 2-dimensional Riemannian manifolds.
The partial differential equations in the project arise not only in geometry but also in science and engineering. Results on these problems should have applications to string theory in modern physics, for nonlinear elasticity in mechanics, and for stochastic optimal control theory in engineering and economics.
