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

Conventions. From the setting we use only the real numbers, Euclidean space with its distance $d_{E}$ and notion of openness, and the set $\mathcal{S}(n)$ with its ordering $\preceq$ and distance $d_{\mathcal{S}(n)}$. Throughout, $x\in V$ implies $x\in U$, and $F|_{V}(x,s,p,X)=F(x,s,p,X)$ for every such $x$ and all $s\in\mathbb{R}$, $p\in\mathbb{R}^{n}$, $X\in\mathcal{S}(n)$.

Proof of claim 1. By the definition of a second-order equation operator, an operator on $V$ is a function from $V\times\mathbb{R}\times\mathbb{R}^{n}\times\mathcal{S}(n)$ to $\mathbb{R}$, where $V$ is an open subset of $\mathbb{R}^{n}$. The set $V$ is open by hypothesis, and $F|_{V}$ is by construction a function on that set of quadruples with values in $\mathbb{R}$, so it is a second-order equation operator on $V$.

Proof of claim 2. Assume $F$ is degenerate elliptic, and let $x\in V$, $r\in\mathbb{R}$, $p\in\mathbb{R}^{n}$ and $X,Y\in\mathcal{S}(n)$ satisfy $X\preceq Y$. Since $x\in U$, degenerate ellipticity of $F$ gives $F(x,r,p,Y)\le F(x,r,p,X)$, that is

$$F|_{V}(x,r,p,Y)\le F|_{V}(x,r,p,X).$$

As $x$, $r$, $p$, $X$ and $Y$ were arbitrary, $F|_{V}$ is degenerate elliptic.

Proof of claim 3. Assume $F$ is continuous at $(x_{0},r_{0},p_{0},X_{0})$ and let $\varepsilon\in\mathbb{R}$ be positive. By clause Continuity of a Second-Order Equation Operator (i) there is a positive $\delta\in\mathbb{R}$ such that all $y\in U$, $s\in\mathbb{R}$, $q\in\mathbb{R}^{n}$ and $Y\in\mathcal{S}(n)$ satisfying

$$d_{E}(y,x_{0})<\delta,\qquad|s-r_{0}|<\delta,\qquad\lVert q-p_{0}\rVert<\delta,\qquad d_{\mathcal{S}(n)}(Y,X_{0})<\delta$$

also satisfy $\bigl|F(y,s,q,Y)-F(x_{0},r_{0},p_{0},X_{0})\bigr|<\varepsilon$. Every $y\in V$ lies in $U$, and $F|_{V}$ agrees with $F$ at every quadruple with first entry in $V$; hence the same $\delta$ witnesses the required condition for $F|_{V}$, and $F|_{V}$ is continuous at $(x_{0},r_{0},p_{0},X_{0})$.

Finally, if $F$ is continuous, that is continuous at every quadruple in the sense of clause Continuity of a Second-Order Equation Operator (ii), then in particular it is continuous at every quadruple whose first entry lies in $V$, so $F|_{V}$ is continuous at every such quadruple; these are exactly the quadruples at which continuity of $F|_{V}$ is required. $\blacksquare$