Restriction of a Second-Order Equation Operator to an Open Subset

Throughout we work in the setting of Second-Order Equations on Euclidean Open Sets, whose notation, including that of Euclidean Space and Real Matrices: 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 $V\subseteq U$ be open. Let

$$F|_{V}:V\times\mathbb{R}\times\mathbb{R}^{n}\times\mathcal{S}(n)\to\mathbb{R}$$

be the function whose value at a quadruple $(x,s,p,X)$ with $x\in V$, $s\in\mathbb{R}$, $p\in\mathbb{R}^{n}$ and $X\in\mathcal{S}(n)$ is $F(x,s,p,X)$; this is well defined because $x\in V$ implies $x\in U$.

Then the following hold.

1. (The restriction is an operator)(i) $F|_{V}$ is a second-order equation operator on $V$.

2. (Degenerate ellipticity is inherited)(ii) If $F$ is degenerate elliptic, then $F|_{V}$ is degenerate elliptic.

3. (Continuity is inherited)(iii) Let $x_{0}\in V$, let $r_{0}\in\mathbb{R}$, let $p_{0}\in\mathbb{R}^{n}$ and let $X_{0}\in\mathcal{S}(n)$. If $F$ is continuous at $(x_{0},r_{0},p_{0},X_{0})$, then $F|_{V}$ is continuous at $(x_{0},r_{0},p_{0},X_{0})$. In particular, if $F$ is continuous, then $F|_{V}$ is continuous.