Viscosity Inequalities Pass to Limits of Test-Function Data

Throughout we work in the setting of Second-Order Equations on Euclidean Open Sets, whose notation is in force in a dimension $n$, a natural number with $1\le n$. That a quadruple is approximable by test data from above or from below for a function is as defined there.

Let $U\subseteq\mathbb{R}^{n}$ be open, let $F$ be a second-order equation operator on $U$, let $u:U\to\mathbb{R}$, let $x_{0}\in U$, let $p\in\mathbb{R}^{n}$ and let $X\in\mathcal{S}(n)$, and assume that $F$ is continuous at $\bigl(x_{0},u(x_{0}),p,X\bigr)$. Then the following hold.

1. (Subsolutions)¶ If $u$ is a viscosity subsolution of $F$ on $U$ and the quadruple $\bigl(x_{0},u(x_{0}),p,X\bigr)$ is approximable by test data from above for $u$, then

$$F\bigl(x_{0},u(x_{0}),p,X\bigr)\le 0 .$$

2. (Supersolutions)¶ If $u$ is a viscosity supersolution of $F$ on $U$ and the quadruple $\bigl(x_{0},u(x_{0}),p,X\bigr)$ is approximable by test data from below for $u$, then

$$0\le F\bigl(x_{0},u(x_{0}),p,X\bigr).$$