Restriction of Viscosity Sub- and Supersolutions to an Open Subset

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 $U$, let $V\subseteq U$ be open, and let $F|_{V}$ be the restriction of $F$, a second-order equation operator on $V$ by Restriction of a Second-Order Equation Operator to an Open Subset §operator. Let $u:U\to\mathbb{R}$ and let $u|_{V}:V\to\mathbb{R}$ be its restriction, with value $u(x)$ at $x\in V$.

Then the following hold.

1. (Subsolutions)¶ If $u$ is a viscosity subsolution of $F$ on $U$, then $u|_{V}$ is a viscosity subsolution of $F|_{V}$ on $V$.

2. (Supersolutions)¶ If $u$ is a viscosity supersolution of $F$ on $U$, then $u|_{V}$ is a viscosity supersolution of $F|_{V}$ on $V$.