The Viscosity Sub- and Supersolution Properties are Local

Throughout we work in the setting of Second-Order Equations on Euclidean Open Sets, whose notation, including that of Real Matrices, Symmetric Matrices and the Semidefinite Ordering: 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, let $F$ be a second-order equation operator on $U$ and let $u:U\to\mathbb{R}$. For an open $V\subseteq U$ we write $u|_{V}:V\to\mathbb{R}$ for the function whose value at $z\in V$ is $u(z)$, and $F|_{V}$ for the restriction of $F$ to $V$, a second-order equation operator on $V$.

Then the following hold.

1. (Restriction to an open subset)¶ Assume that $F$ is continuous, and let $V\subseteq U$ be open. If $u$ is a viscosity subsolution of $F$ on $U$, then $u|_{V}$ is a viscosity subsolution of $F|_{V}$ on $V$. If $u$ is a viscosity supersolution of $F$ on $U$, then $u|_{V}$ is a viscosity supersolution of $F|_{V}$ on $V$.

2. (A local subsolution is a subsolution)¶ Suppose that for every $x\in U$ there is an open set $V$ with $x\in V$ and $V\subseteq U$ such that $u|_{V}$ is a viscosity subsolution of $F|_{V}$ on $V$. Then $u$ is a viscosity subsolution of $F$ on $U$. Likewise, if for every $x\in U$ there is such a $V$ with $u|_{V}$ a viscosity supersolution of $F|_{V}$ on $V$, then $u$ is a viscosity supersolution of $F$ on $U$.