Second-Order Equation Operator on a Euclidean Open Set

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 set of real numbers, and let $\mathcal{S}(n)$ be the set of symmetric real $n\times n$ matrices.

We write $U\times\mathbb{R}\times\mathbb{R}^n\times\mathcal{S}(n)$ for the set whose elements are the ordered quadruples $(x,r,p,X)$ with $x\in U$, $r\in\mathbb{R}$, $p\in\mathbb{R}^n$ and $X\in\mathcal{S}(n)$; this is the Cartesian product iterated from the left, the quadruple $(x,r,p,X)$ being written for the nested pair $(((x,r),p),X)$.

A second-order equation operator on $U$ is a function

its value at the quadruple $(x,r,p,X)$ being written $F(x,r,p,X)$.