The Pointwise Supremum of a Locally Uniformly Bounded Family of Viscosity Subsolutions on a Hilbert Triple is a Viscosity Subsolution

In the setting of Hilbert Triples: Standing Notation and Background, let $U\subseteq H$ be nonempty and open in $H$, with $W=D(A)\cap U$ and the class $C^{2}(U)$ as in Hilbert Triples: Standing Notation and Background (v), let $F$ be a second-order equation operator on $U$ relative to $(H,V,A)$, and let $\mathcal{S}$ be a nonempty set whose elements are functions from $U$ to $\mathbb{R}$, each of which is a viscosity subsolution of $F$ on $U$. For a real $\delta>0$, $w^{-}_{\delta}$ denotes the $\delta$-envelope of a function $w$ on $U$ that is bounded above near each point of $U$, and local bounds are as fixed there.

Assume that $\mathcal{S}$ is locally uniformly bounded above:(i) for every $x\in U$ there are $c\in\mathbb{R}$ and a positive $r\in\mathbb{R}$ such that

$$v(y)\le c\qquad\text{for every }v\in\mathcal{S}\text{ and every }y\in U\text{ with }d_{H}(y,x)\le r .$$

Fix $x_{0}\in U$ and such a pair $c,r$. The set $\{v(x_{0}):v\in\mathcal{S}\}$ is nonempty, because $\mathcal{S}$ is, and is bounded above by $c$, because $d_{H}(x_{0},x_{0})=0\le r$ by the metric axioms; it therefore has a least upper bound, as recorded in The Real Numbers: Standing Notation and Background (iv). Let $u:U\to\mathbb{R}$ be the function given by

$$u(x)=\sup\{v(x):v\in\mathcal{S}\}\qquad\text{for }x\in U .$$

Then the following hold.

1. (The supremum is locally bounded and dominates the family)(ii) The function $u$ is bounded above near each point of $U$, and $v(x)\le u(x)$ for every $v\in\mathcal{S}$ and every $x\in U$. Consequently, for every real $\delta>0$ and every $v\in\mathcal{S}$,

$$v^{-}_{\delta}(x)\le u^{-}_{\delta}(x)\qquad\text{for every }x\in V\cap U .$$

2. (The supremum is a viscosity subsolution)(iii) The function $u$ is a viscosity subsolution of $F$ on $U$.