Fully nonlinear elliptic equations

A fully nonlinear elliptic equation is an expression of the form \[ F(D^2u, \nabla u, u, x) = 0 \text{ in } \Omega.\]

The function $F$ is supposed to satisfy the following two basic assumptions

• If $A \geq B$, $F(A,p,u,x) \geq F(B,p,u,x)$ for any values of $p \in \R^n$, $u\in \R$ and $x \in \Omega$.
• If $u \geq v$, $F(A,p,u,x) \leq F(A,p,v,x)$ for any values of $A \in \R^{n \times n}$, $p \in \R^n$ and $x \in \Omega$.