Classical Subsolution and Supersolution of a Second-Order Equation

Let $n$ be a natural number, let $U\subseteq\mathbb{R}^n$ be an open subset of Euclidean space $\mathbb{R}^n$, let $\mathbb{R}$ be the set of real numbers with the order $\le$ of its ordered field structure, let $\mathcal{S}(n)$ be the set of symmetric real $n\times n$ matrices, let $F$ be a second-order equation operator on $U$, and let $u:U\to\mathbb{R}$ be of class $C^2$ on $U$.

For each $x\in U$ write $Du(x)$ for the gradient of $u$ at $x$ and $D^2u(x)$ for the Hessian matrix of $u$ at $x$; by Equality of Mixed Second Partial Derivatives and Symmetry of the Hessian the latter lies in $\mathcal{S}(n)$, so the quadruple $(x,u(x),Du(x),D^2u(x))$ 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 $x\in U$.