Smooth Local Solutions to Fully Nonlinear Partial Differential Equations

In this dissertation we discuss the local solvability of two classes of fully nonlinear partial differential equations. In the first chapter we discuss the geometric background of our equations, state our main results and describe the methods for proof. In Chapter 2 we prove the prescribed k-curvature equations, for 2 le k le n-1, are always locally solvable, in particular the sign of the right-hand side is irrelevant. The proof is based on the observation that these equations can always be made to be elliptic, so we can use the implicit function theorem. The following three chapters are devoted to the study of degenerate hyperbolic Monge-Ampere equations. We prove they are locally solvable if the zero set of certain directional derivatives of the right-hand side has a special structure. For the proof, we carefully analyze the linearized operator and then derive the a priori estimates for the linearized equations. After these, we use Nash-Moser iteration to prove the existence of a local solution.