Test Data for a Second-Order Equation Operator on a Hilbert Triple and the Admissible Sets

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 $h$ be the penalty function, let $\mathrm{Sym}(H)$ carry the norm $\lVert\cdot\rVert$ of 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 $F^{+}_{\delta}$ for each real $\delta>0$. Products of sets are Cartesian products, and $|s|$ is the absolute value of a real number $s$.

1. (Test data)(i) The set of test data for $F$ is the product

$$\mathcal{W}=W\times\mathbb{R}\times H\times\mathrm{Sym}(H),$$

which is the domain of $F^{-}_{\delta}$ and of $F^{+}_{\delta}$ for every real $\delta>0$. Its elements are written $\xi=(x,r,p,X)$, and for such a $\xi$ and a real $\delta>0$ we write $F^{-}_{\delta}(\xi)=F^{-}_{\delta}(x,r,p,X)$ and $F^{+}_{\delta}(\xi)=F^{+}_{\delta}(x,r,p,X)$.

2. ($R$-bounded test data)(ii) Let $R\in\mathbb{R}$ be positive. A test datum $\xi=(x,r,p,X)\in\mathcal{W}$ is $R$-bounded if

$$h(x)<R,\qquad |r|<R,\qquad |p|_{H}<R,\qquad \lVert X\rVert<R;$$

here $h(x)$ is defined because $x\in W\subseteq D(A)\subseteq V$ by Hilbert Triples: Standing Notation and Background (iii).

3. (Admissible test data)(iii) Let $\delta,R\in\mathbb{R}$ be positive. The set $S^{-}_{\delta,R}$ consists of those $\xi\in\mathcal{W}$ that are $R$-bounded and for which there exists an $R$-bounded $\eta\in\mathcal{W}$ with

$$F^{-}_{\delta}(\xi)-F^{+}_{\delta}(\eta)<R .$$

The set $S^{+}_{\delta,R}$ consists of those $\eta\in\mathcal{W}$ that are $R$-bounded and for which there exists an $R$-bounded $\xi\in\mathcal{W}$ satisfying the same inequality. Both sets depend on $F$, on $\delta$ and on $R$; when several operators are in play they are written $S^{-}_{\delta,R}(F)$ and $S^{+}_{\delta,R}(F)$.