Perron's Method on a Hilbert Triple: Existence of a Viscosity Solution Between a Subsolution and a Supersolution

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), and let $F$ be a second-order equation operator on $U$ relative to $(H,V,A)$ that is degenerate elliptic. Local bounds of functions on $U$ are as fixed there.

Let $f:U\to\mathbb{R}$ be a viscosity subsolution of $F$ on $U$ that is bounded below near each point of $U$, let $g:U\to\mathbb{R}$ be a viscosity supersolution of $F$ on $U$ that is bounded above near each point of $U$, and assume that

$$f(x)\le g(x)\qquad\text{for every }x\in U .$$

Let $\mathcal{G}$ be the set of all viscosity subsolutions $v$ of $F$ on $U$ that satisfy $f(x)\le v(x)$ and $v(x)\le g(x)$ for every $x\in U$. Then $f\in\mathcal{G}$, so that $\mathcal{G}$ is nonempty; and for $x\in U$ the set $\{v(x):v\in\mathcal{G}\}$ is nonempty and bounded above by $g(x)$, so that it 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

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

Then the following hold.

1. (The supremum lies between the data)(i) For every $x\in U$, $f(x)\le u(x)$ and $u(x)\le g(x)$; consequently $u$ is bounded above near each point of $U$ and bounded below near each point of $U$.

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