Viscosity 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, 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 $u:U\to\mathbb{R}$.

Regard $\mathbb{R}^n$ as a metric space through the Euclidean distance $d_E$, which is a metric by Euclidean Distance is a Metric on $\mathbb{R}^n$; semicontinuity on $U$ and local extrema relative to $U$ are understood with respect to this metric.

For a function $\varphi:U\to\mathbb{R}$ that is of class $C^2$ on $U$, let $u-\varphi:U\to\mathbb{R}$ be the function whose value at $y\in U$ is $u(y)-\varphi(y)$, and for $x\in U$ write $D\varphi(x)$ for the gradient of $\varphi$ at $x$ and $D^2\varphi(x)$ for the Hessian matrix of $\varphi$ at $x$; by Equality of Mixed Second Partial Derivatives and Symmetry of the Hessian the latter lies in $\mathcal{S}(n)$, so the quadruple $(x,u(x),D\varphi(x),D^2\varphi(x))$ lies in the domain of $F$.

We say that $u$ is a viscosity subsolution of $F$ on $U$ if $u$ is upper semicontinuous on $U$ and, for every $\varphi:U\to\mathbb{R}$ of class $C^2$ on $U$ and every $x\in U$ at which $u-\varphi$ has a local maximum relative to $U$,

We say that $u$ is a viscosity supersolution of $F$ on $U$ if $u$ is lower semicontinuous on $U$ and, for every $\varphi:U\to\mathbb{R}$ of class $C^2$ on $U$ and every $x\in U$ at which $u-\varphi$ has a local minimum relative to $U$,