The Scope of the First-Order Structure Condition

In the setting of Hilbert Triples: Standing Notation and Background, the set $H$ is open in $H$ by Hilbert Triples: Standing Notation and Background §open-sets, so that $W=D(A)$ there. Let $\mathrm{Sym}(H)$, $\mathrm{Sym}(V)$, the restriction $Y\mapsto Y|_{V}$ of a form and the identity form $I_{V}$ be as in Hilbert Triples: Standing Notation and Background §restriction, let $I_{H}$ be the identity form of $H$, let $h$ be the penalty function, and let $F$ be a second-order equation operator on $H$ relative to $(H,V,A)$ with $\delta$-shifts $F^{-}_{\delta}$ and $F^{+}_{\delta}$.

The inequality required by The First-Order Structure Condition for a Second-Order Equation Operator on a Hilbert Triple §pair is quantified over all $X,Y\in\mathrm{Sym}(H)$ independently, whereas the two moduli bounding it from below are evaluated only at $\alpha|x-y|_{H}^{2}+\tfrac{1}{\alpha}$ and at $\delta(h(x)+h(y)+1)$ and $\alpha$, and so do not see the form arguments at all. Fix therefore a positive $R$, points $x,y\in D(A)$, a real $r$ with $|r|\le R$, and $\alpha>1$ and $0<\delta<1$. Holding $Y$ fixed and letting $X$ vary, the inequality says that $F^{-}_{\delta}(x,r,\alpha(x-y),X)$ is bounded below by a real that does not depend on $X$; holding $X$ fixed and letting $Y$ vary, it says that $F^{+}_{\delta}(y,r,\alpha(x-y),Y)$ is bounded above by a real that does not depend on $Y$.

The condition is thus not only a compatibility between the first-order terms of $F$ and the penalty $h$: it also demands that $F$ depend on its form argument boundedly, in the sense just described. For an operator that does not depend on that argument at all, such as one that is first order, the demand is empty, which is what allows A Comparison Principle on a Hilbert Triple under the First-Order Structure Condition to require no relation whatever between $X$ and $Y$. At the other extreme, consider an operator carrying a summand $-X(e,e)$, where $X$ denotes its form argument as in Second-Order Equation Operator on an Open Subset of a Hilbert Triple and Its $\delta$-Shifts §operator and $e\in V$ is a fixed nonzero vector, so that $0<|e|_{H}$ by Elementary Identities in a Real Inner Product Space §vanishing. Let the first form argument of the structure inequality be the multiple $t\,I_{H}$, with $t$ positive. The form that $F^{-}_{\delta}$ then feeds to $F$ has at $(e,e)$ the value $t|e|_{H}^{2}+\delta|e|_{V}^{2}$, which grows without bound as $t$ does, so that $F^{-}_{\delta}(x,r,\alpha(x-y),t\,I_{H})$ is not bounded below as $t$ grows. Such an operator therefore violates the lower bound above and cannot satisfy the first-order structure condition, however its first-order terms are arranged.

A second-order theory consequently cannot ask for its structure inequality at arbitrary pairs of forms. It asks for it only at the pairs a Crandall-Ishii-type lemma actually produces at a near-maximum point of a doubled function, which come coupled by a one-sided inequality between them and by a bound on their norms. That is a different hypothesis, not a weakening of this one, and the comparison argument that uses it must construct such a pair before the inequality can be applied.