Classical Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple

In the setting of Hilbert Triples: Standing Notation and Background, let $U\subseteq H$ be open in $H$, with $W=D(A)\cap U$ and the class $C^{2}(U)$ as in Hilbert Triples: Standing Notation and Background (v), let $Y\mapsto Y|_{V}$ be the restriction of forms to $V$, let $F$ be a second-order equation operator on $U$ relative to $(H,V,A)$, and let $u\in C^{2}(U)$. For $x\in W$ the gradient $Du(x)$ lies in $H$ and the restricted Hessian $D^{2}u(x)|_{V}$ lies in $\mathrm{Sym}(V)$, so $(x,u(x),Du(x),D^{2}u(x)|_{V})$ lies in the domain of $F$. The inequalities below are required only at points of $W$, because $F$ is defined only there.

1. (Classical subsolution)(i) The function $u$ is a classical subsolution of $F$ on $U$ if

$$F\bigl(x,u(x),Du(x),D^{2}u(x)|_{V}\bigr)\le0\qquad\text{for every }x\in W.$$

2. (Classical supersolution)(ii) The function $u$ is a classical supersolution of $F$ on $U$ if

$$0\le F\bigl(x,u(x),Du(x),D^{2}u(x)|_{V}\bigr)\qquad\text{for every }x\in W.$$

3. (Classical solution)(iii) The function $u$ is a classical solution of $F$ on $U$ if it is both a classical subsolution and a classical supersolution of $F$ on $U$, that is, if $F(x,u(x),Du(x),D^{2}u(x)|_{V})=0$ for every $x\in W$.