Viscosity Solution of a Second-Order Equation

In the setting of Second-Order Equations on Euclidean Open Sets, let $n\ge1$ be a natural number, let $U\subseteq\mathbb{R}^{n}$ be open, let $F$ be a second-order equation operator on $U$, and let $u:U\to\mathbb{R}$.

We say that $u$ is a viscosity solution of $F$ on $U$ if $u$ is both a viscosity subsolution of $F$ on $U$ and a viscosity supersolution of $F$ on $U$.