Viscosity Subsolution and Supersolution up to the Boundary

In the setting of Second-Order Equations on Euclidean Open Sets, let $n\ge1$ be a natural number, let $\Omega\subseteq\mathbb{R}^{n}$ be open, and write $\overline{\Omega}=\operatorname{cl}_{\mathbb{R}^{n}}(\Omega)$, so that $\Omega\subseteq\overline{\Omega}$ by The Closure is the Smallest Closed Superset.

Let $F$ be a second-order equation operator on $\Omega$, let $u:\overline{\Omega}\to\mathbb{R}$, and let $u|_{\Omega}:\Omega\to\mathbb{R}$ be the function whose value at $x\in\Omega$ is $u(x)$.

We say that $u$ is a viscosity subsolution of $F$ up to the boundary of $\Omega$ if $u$ is upper semicontinuous on $\overline{\Omega}$ and $u|_{\Omega}$ is a viscosity subsolution of $F$ on $\Omega$.

We say that $u$ is a viscosity supersolution of $F$ up to the boundary of $\Omega$ if $u$ is lower semicontinuous on $\overline{\Omega}$ and $u|_{\Omega}$ is a viscosity supersolution of $F$ on $\Omega$.