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

By clause Second-Order Equations on Euclidean Open Sets §test-functions, for every $\varphi:U\to\mathbb{R}$ of class $C^{2}$ on $U$ and every $x\in U$ the gradient $D\varphi(x)$ lies in $\mathbb{R}^{n}$ and the Hessian $D^{2}\varphi(x)$ 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 $U$,

$$F(x,u(x),D\varphi(x),D^{2}\varphi(x))\le 0 .$$

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 $U$,

$$0\le F(x,u(x),D\varphi(x),D^{2}\varphi(x)) .$$