The Shift-Continuity Condition on Admissible Test Data

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 $\mathrm{Sym}(H)$ with its norm $\lVert\cdot\rVert$ and its sums of forms be as in Hilbert Triples: Standing Notation and Background (vi), and let $F$ be a second-order equation operator on $U$ relative to $(H,V,A)$, with $\delta$-shifts $F^{-}_{\delta}$ and $F^{+}_{\delta}$ and with the admissible sets $S^{-}_{\delta,R}$ and $S^{+}_{\delta,R}$ of test data.

1. (Shift modulus at a level)(i) Let $\delta,R\in\mathbb{R}$ satisfy $0<\delta<1$ and $0<R$, and let $\omega$ be a modulus of continuity. We say that $\omega$ is a shift modulus for $F$ at $(\delta,R)$ if the following two conditions hold for every $q\in H$ and every $Y\in\mathrm{Sym}(H)$. First, every $(x,r,p,X)\in S^{-}_{\delta,R}$ satisfies

$$F^{-}_{\delta}(x,r,p+q,X+Y)\ \le\ F^{-}_{\delta}(x,r,p,X)+\omega\bigl(|q|_{H}+\lVert Y\rVert\bigr).$$

Secondly, every $(y,s,p',X')\in S^{+}_{\delta,R}$ satisfies

$$F^{+}_{\delta}(y,s,p',X')-\omega\bigl(|q|_{H}+\lVert Y\rVert\bigr)\ \le\ F^{+}_{\delta}(y,s,p'+q,X'+Y).$$

The arguments of $\omega$ are nonnegative, being sums of norms.

2. (The shift-continuity condition)(ii) The operator $F$ satisfies the shift-continuity condition if for all $\delta,R\in\mathbb{R}$ with $0<\delta<1$ and $0<R$ there is a shift modulus for $F$ at $(\delta,R)$.