Second-Order Equation Operator Convex in the Value, Gradient and Matrix Variables

In the setting of Second-Order Equations on Euclidean Open Sets, let $n\ge1$ be a natural number, let $U\subseteq\mathbb{R}^{n}$ be open and let $F$ be a second-order equation operator on $U$. For $X,Y\in\mathcal{S}(n)$ and $t\in\mathbb{R}$ the matrix $(1-t)X+tY$ lies in $\mathcal{S}(n)$ by Second-Order Equations on Euclidean Open Sets §matrices.

We say that $F$ is convex in $(r,p,X)$ if for every $x\in U$, all $r,s\in\mathbb{R}$, all $p,q\in\mathbb{R}^{n}$, all $X,Y\in\mathcal{S}(n)$ and every $t\in\mathbb{R}$ with $0\le t\le1$,