Classical Subsolution and Supersolution 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 $u:U\to\mathbb{R}$ be of class $C^{2}$ on $U$.

By that same clause, for each $x\in U$ the gradient $Du(x)$ lies in $\mathbb{R}^{n}$ and the Hessian $D^{2}u(x)$ lies in $\mathcal{S}(n)$, so the quadruple $(x,u(x),Du(x),D^{2}u(x))$ lies in the domain of $F$.

We say that $u$ is a classical subsolution of $F$ on $U$ if

for every $x\in U$, and that $u$ is a classical supersolution of $F$ on $U$ if

for every $x\in U$.