The Viscosity Property Depends Only on the Values on the Trace of $V$

In the setting of Hilbert Triples: Standing Notation and Background, let $U\subseteq H$ be nonempty and open in $H$, and let $F$ be a second-order equation operator on $U$ relative to $(H,V,A)$. Local bounds of functions on $U$, the $\delta$-envelopes $u^{-}_{\delta}$ and $u^{+}_{\delta}$, and the viscosity notions of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple are as fixed there. Let $u,\tilde{u}:U\to\mathbb{R}$ satisfy

$$u(x)=\tilde{u}(x)\qquad\text{for every }x\in V\cap U .$$

Then the following hold.

1. (The $\delta$-envelopes agree)(i) If $u$ and $\tilde{u}$ are bounded above near each point of $U$, then $u^{-}_{\delta}(x)=\tilde{u}^{-}_{\delta}(x)$ for every real $\delta>0$ and every $x\in V\cap U$. If $u$ and $\tilde{u}$ are bounded below near each point of $U$, then $u^{+}_{\delta}(x)=\tilde{u}^{+}_{\delta}(x)$ for every real $\delta>0$ and every $x\in V\cap U$.

2. (Subsolutions)(ii) If $u$ and $\tilde{u}$ are bounded above near each point of $U$ and $u$ is a viscosity subsolution of $F$ on $U$, then $\tilde{u}$ is a viscosity subsolution of $F$ on $U$.

3. (Supersolutions)(iii) If $u$ and $\tilde{u}$ are bounded below near each point of $U$ and $u$ is a viscosity supersolution of $F$ on $U$, then $\tilde{u}$ is a viscosity supersolution of $F$ on $U$.

4. (Solutions)(iv) If $u$ and $\tilde{u}$ are bounded above near each point of $U$ and bounded below near each point of $U$, and $u$ is a viscosity solution of $F$ on $U$, then $\tilde{u}$ is a viscosity solution of $F$ on $U$.