The Upper Semicontinuous Envelope of a Supremum of Viscosity Subsolutions is a Viscosity Subsolution

Throughout we work in the setting of Second-Order Equations on Euclidean Open Sets, whose notation, including that of Euclidean Space and Real Matrices: Standing Notation and Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation on which it rests, is in force in a dimension $n$, a natural number with $1\le n$.

Let $U\subseteq\mathbb{R}^{n}$ be open and nonempty, let $F$ be a second-order equation operator on $U$ that is continuous, and let $\mathcal{F}$ be a nonempty set whose elements are functions from $U$ to $\mathbb{R}$, each of which is a viscosity subsolution of $F$ on $U$.

Assume that $\mathcal{F}$ 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

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

Fix $x\in U$ and such a pair $c,r$. The set of values $\{v(x):v\in\mathcal{F}\}$ is nonempty, because $\mathcal{F}$ is, and is bounded above by $c$, because $d_{E}(x,x)=0\le r$ by the metric axioms; it therefore has a least upper bound by Least Upper Bound Property of the Real Numbers, unique by Uniqueness of the Supremum and of the Infimum. Let $w:U\to\mathbb{R}$ be the function given by

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

Every $y\in U$ with $d_{E}(y,x)\le r$ satisfies $w(y)\le c$, since $c$ is an upper bound of $\{v(y):v\in\mathcal{F}\}$ and $w(y)$ is its least upper bound. Hence $w$ is bounded above near each point of $U$, and its upper semicontinuous envelope $w^{*}:U\to\mathbb{R}$ is defined.

Then $w^{*}$ is a viscosity subsolution of $F$ on $U$.(ii)