Viscosity Subsolution and Supersolution up to the Boundary

Let $n\ge1$ be a natural number and let $\mathbb{R}$ be the real numbers. Equip Euclidean space $\mathbb{R}^n$ with the Euclidean distance $d_E$, a metric by Euclidean Distance is a Metric on $\mathbb{R}^n$, and with the topology $\mathcal{T}_{d_E}$ of its metric-open subsets, a topology by Metric Open Sets Form a Topology that agrees with Euclidean openness by Euclidean Openness Agrees with Metric Openness on $\mathbb{R}^n$. Semicontinuity refers to $d_E$.

Let $\Omega\subseteq\mathbb{R}^n$ be open and let $\overline{\Omega}=\operatorname{cl}_{\mathbb{R}^n}(\Omega)$ be its closure in $(\mathbb{R}^n,\mathcal{T}_{d_E})$, 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 $\Omega$.