Viscosity Inequalities Pass to Limits of Test-Function Data

Let $n\ge1$ be a natural number, let $U\subseteq\mathbb{R}^{n}$ be an open subset of Euclidean space $\mathbb{R}^{n}$, let $\mathbb{R}$ be the real numbers with the order $\le$ of their ordered field structure and the absolute value $|\cdot|$, and let $\mathcal{S}(n)$ be the set of symmetric real $n\times n$ matrices. Write $d_{E}$ for the Euclidean distance, a metric on $\mathbb{R}^{n}$ by Euclidean Distance is a Metric on $\mathbb{R}^n$, $\lVert\,\cdot\,\rVert$ for the Euclidean norm, and $d_{\mathcal{S}(n)}$ for the distance between symmetric real matrices, a metric on $\mathcal{S}(n)$ by The Set of Symmetric Real Matrices is a Metric Space.

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 $X\in\mathcal{S}(n)$. For a function $\varphi:U\to\mathbb{R}$ of class $C^{2}$ on $U$ (via clause 3 there) and $y\in U$, write $D\varphi(y)$ for the gradient and $D^{2}\varphi(y)$ for the Hessian matrix of $\varphi$ at $y$; the latter lies in $\mathcal{S}(n)$ by claim 2 of Equality of Mixed Second Partial Derivatives and Symmetry of the Hessian, so the quadruple $\bigl(y,u(y),D\varphi(y),D^{2}\varphi(y)\bigr)$ lies in the domain of $F$. Let $u-\varphi:U\to\mathbb{R}$ be the function whose value at $y$ is $u(y)-\varphi(y)$; local extrema relative to $U$ are understood with respect to $d_{E}$, as in Viscosity Subsolution and Supersolution of a Second-Order Equation.

Assume that $F$ is continuous at $\bigl(x_{0},u(x_{0}),p,X\bigr)$, in the sense that for every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ there is $\delta\in\mathbb{R}$ with $0<\delta$ such that every $y\in U$, $s\in\mathbb{R}$, $q\in\mathbb{R}^{n}$ and $Y\in\mathcal{S}(n)$ satisfying

also satisfies

Say that the quadruple $\bigl(x_{0},u(x_{0}),p,X\bigr)$ is approximable by test data from above for $u$ if for every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ there exist $y\in U$ and $\varphi:U\to\mathbb{R}$ of class $C^{2}$ on $U$ such that $u-\varphi$ has a local maximum at $y$ relative to $U$ and

and approximable by test data from below if the same holds with local minimum at $y$ relative to $U$ in place of local maximum.

Then the following hold.

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

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