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A fully nonlinear partial differential equation on a domain $\\Omega \\subset {\\bf R}^n$ is one of the form

$$ f(x,u, Du, D^2u) = 0 \\eqno{(*)} $$

for a continuous function $f: \\Omega \\ times {\\bf R}^n \\ times {\\bf R}^n \\ times {\\ rm Sym}^2( {\\bf R}^n ) \\ to {\\bf R}^n$. Substantial insight into solving such equations can be gained by considering $(*)$ to be the boundary of the set

$$ F \\equiv \\{ f \\geq 0 \\}. $$

The geometry of this set encodes much of importance for questions of existence and uniqueness of viscosity solutions. It also elucidates when the Maximum Principle and the Strong Maximum Principle hold, it dictates necessary boundary conditions for solving the Dirichlet Problem, and it brings forward the useful notion of a monotonicity cone for the equation.

These methods lead to rather complete existence and uniqueness results for the Dirichlet Problem for highly degenerate elliptic equations. A number of examples will be discussed including all branches of the homogeneous Monge- Ampere equation over ${\\bf R}^n$ , ${\\bf C}$ and ${\\bf H}$. The methods also lead to removable singularity results for solutions and subsolutions.

This is all joint work with Reese Harvey.