Stability of Viscosity Sub- and Supersolutions on a Hilbert Triple: Limits of Uniformly Bounded, Equicontinuous Sequences under Convergence of the Operators on Bounded Test Data

In the setting of Hilbert Triples: Standing Notation and Background, the set $H$ is open in $H$ by Hilbert Triples: Standing Notation and Background §open-sets, with $W=D(A)$ and $V\cap H=V$ there. Assume that every sequence in $V$ that is bounded in $V$ has a subsequence that converges in $H$.

Let $F$ and, for each $N\in\mathbb{N}$, $F_{N}$ be second-order equation operators on $H$ relative to $(H,V,A)$ such that $(F_{N})_{N\in\mathbb{N}}$ converges to $F$ on bounded test data. Let $C\in\mathbb{R}$ be nonnegative, let $\omega$ be a modulus of continuity, and let $u_{N}:H\to\mathbb{R}$, for $N\in\mathbb{N}$, satisfy

$$|u_{N}(x)|\le C\quad\text{and}\quad|u_{N}(x)-u_{N}(y)|\le\omega\bigl(|x-y|_{H}\bigr)\qquad\text{for all }N\in\mathbb{N}\text{ and }x,y\in H .$$

For $x\in H$ the sequence $(u_{N}(x))_{N\in\mathbb{N}}$ is a bounded sequence of real numbers, since $|u_{N}(x)|\le C$; let $\bar{u}(x)$ be its limit superior and $\underline{u}(x)$ its limit inferior. Then the following hold.

1. (Subsolutions)¶ If $u_{N}$ is a viscosity subsolution of $F_{N}$ on $H$ for every $N\in\mathbb{N}$, then $\bar{u}$ is a viscosity subsolution of $F$ on $H$.

2. (Supersolutions)¶ If $u_{N}$ is a viscosity supersolution of $F_{N}$ on $H$ for every $N\in\mathbb{N}$, then $\underline{u}$ is a viscosity supersolution of $F$ on $H$.