The Second-Order Structure Condition for an 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 $h$ be the penalty function, let $\mathrm{Sym}(H)$ with its norm $\lVert\cdot\rVert$ and its order $\preceq$ 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 $F^{+}_{\delta}$. For a real $\alpha$ and $x,y\in H$, $\alpha(x-y)$ denotes the scalar multiple by $\alpha$ of the difference $x-y$ in $H$, and $\tfrac{1}{\alpha}$ is the quotient for $\alpha\ne0$. We write $3=2+1$ and $6=3+3$, and $|z|_{H}^{2}=|z|_{H}|z|_{H}$. For $x\in W$ one has $x\in D(A)\subseteq V$ by Hilbert Triples: Standing Notation and Background (iii), so $h(x)$ is defined, and $0\le h(x)$ by The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds (i).

1. (Pairs admitted at a penalty strength)(i) Let $\alpha\in\mathbb{R}$ be positive. A pair $(X,Y)$ of members of $\mathrm{Sym}(H)$ is admitted at $\alpha$ if $X\preceq Y$, $\lVert X\rVert\le6\alpha$, $\lVert Y\rVert\le6\alpha$, and

$$-3\alpha\bigl(|z|_{H}^{2}+|w|_{H}^{2}\bigr)\ \le\ X(z,z)-Y(w,w)\ \le\ 3\alpha\,|z-w|_{H}^{2}\qquad\text{for all }z,w\in H .$$

2. (Second-order structure pair at a level)(ii) Let $R\in\mathbb{R}$ be positive, let $\omega_{1}$ be a modulus of continuity, and let $\omega_{2}$ be a function with values in $\mathbb{R}$ on the set of all pairs $(t,\alpha)$ of real numbers with $0\le t$ and $1<\alpha$, such that for every real $\alpha>1$ the function on the set of nonnegative reals with value $\omega_{2}(t,\alpha)$ at $t$ is a modulus of continuity. We say that $(\omega_{1},\omega_{2})$ is a second-order structure pair for $F$ at $R$ if

$$-\omega_{1}\Bigl(\alpha|x-y|_{H}^{2}+\tfrac{1}{\alpha}\Bigr)-\omega_{2}\bigl(\delta\,(h(x)+h(y)+1),\,\alpha\bigr)\ \le\ F^{-}_{\delta}\bigl(x,r,\alpha(x-y),X\bigr)-F^{+}_{\delta}\bigl(y,r,\alpha(x-y),Y\bigr)$$

for all $x,y\in W$, every $r\in\mathbb{R}$ with $-R\le r\le R$, all $\alpha,\delta\in\mathbb{R}$ with $1<\alpha$ and $0<\delta<1$, and every pair $(X,Y)$ of members of $\mathrm{Sym}(H)$ admitted at $\alpha$ in the sense of clause 1. The arguments of the two moduli are nonnegative: $\alpha|x-y|_{H}^{2}+\tfrac{1}{\alpha}$ is a sum of a product of nonnegative reals and a positive quotient, and $\delta\,(h(x)+h(y)+1)$ is the product of the positive real $\delta$ and the positive real $h(x)+h(y)+1$.

3. (The second-order structure condition)(iii) The operator $F$ satisfies the second-order structure condition if for every positive $R\in\mathbb{R}$ there is a second-order structure pair for $F$ at $R$.

The pairs admitted at $\alpha$ are exactly those carrying the three conclusions Lions' Lemma on a Hilbert Space: Test Data and Form Bounds at a Sequentially Strict Maximum of a Quadratically Penalised Difference (ii), Lions' Lemma on a Hilbert Space: Test Data and Form Bounds at a Sequentially Strict Maximum of a Quadratically Penalised Difference (iii) and Lions' Lemma on a Hilbert Space: Test Data and Form Bounds at a Sequentially Strict Maximum of a Quadratically Penalised Difference (iv) of the doubling lemma at a quadratic penalty of strength $\alpha$. Requiring the displayed inequality only for those pairs is what distinguishes this condition from The First-Order Structure Condition for a Second-Order Equation Operator on a Hilbert Triple (ii), which imposes it for all $X,Y\in\mathrm{Sym}(H)$; the second-order condition is therefore the weaker of the two.