Degenerate Elliptic Second-Order Equation Operator on a Hilbert Triple

In the setting of Hilbert Triples: Standing Notation and Background, let $U\subseteq H$ be open in $H$, with $W=D(A)\cap U$ as in Hilbert Triples: Standing Notation and Background (v), let $\preceq$ be the order on $\mathrm{Sym}(V)$, and let $F$ be a second-order equation operator on $U$ relative to $(H,V,A)$.

(i) We say that $F$ is degenerate elliptic if for every $x\in W$, every $r\in\mathbb{R}$, every $p\in H$, and all $X,Y\in\mathrm{Sym}(V)$ with $X\preceq Y$,

$$F(x,r,p,Y)\le F(x,r,p,X).$$