The Second-Order Test-Datum Estimate at a Sequentially Strict Maximum of the Doubled Function

In the setting of Hilbert Triples: Standing Notation and Background, the set $H$ is open in $H$, since every open ball of $(H,d_{H})$ is a subset of $H$; accordingly $W=D(A)\cap H=D(A)$ and $V\cap H=V$ in the notation of Hilbert Triples: Standing Notation and Background (v), and $h$ is the penalty function. Let $F$ be a second-order equation operator on $H$ relative to $(H,V,A)$, with $\delta$-shifts $F^{-}_{\delta}$ and $F^{+}_{\delta}$. Let $H\times H$ be the product of $H$ with itself, a real inner product space by Properties of the Product of Two Real Inner Product Spaces (i); sequentially strict maxima of a function on a subset of $H\times H$ are taken with respect to its distance.

Let $u,v:H\to\mathbb{R}$ and let $C\in\mathbb{R}$ satisfy $0\le C$, $u(x)\le C$ and $-C\le v(x)$ for every $x\in H$; then $u$ is bounded above near each point of $H$ and $v$ is bounded below near each point of $H$ by Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity (iv), so that for each real $\delta>0$ the $\delta$-envelopes $u^{-}_{\delta}$ and $v^{+}_{\delta}$ are defined on $V$. Assume that $u$ is a viscosity subsolution of $F$ on $H$ and that $v$ is a viscosity supersolution of $F$ on $H$.

Let $(e_{k})_{k\in\mathbb{N}}$ be an orthonormal basis of $H$ with $e_{k}\in V$ for every $k\in\mathbb{N}$, and assume that $F$ is tail-insensitive along $(e_{k})_{k\in\mathbb{N}}$.

Let $\delta,\alpha,\sigma,\varepsilon,B,G,R\in\mathbb{R}$ satisfy

$$0<\delta<1,\qquad 1<\alpha,\qquad 0<\sigma\le1,\qquad 0<\varepsilon\le1,\qquad 0\le G,\qquad C\le B,$$

let $p,q\in H$ satisfy $|p|_{H}\le\sigma$ and $|q|_{H}\le\sigma$, and let $\hat{x},\hat{y}\in V$ be such that the function $V\times V\to\mathbb{R}$ whose value at $(x,y)$ is

$$u^{-}_{\delta}(x)-v^{+}_{\delta}(y)-\tfrac{\alpha}{2}|x-y|_{H}^{2}-\langle p,x\rangle_{H}-\langle q,y\rangle_{H}$$

attains a sequentially strict maximum on $V\times V$ at $(\hat{x},\hat{y})$. Assume

$$\delta\,h(\hat{x})\le B,\qquad \delta\,h(\hat{y})\le B,\qquad |u^{-}_{\delta}(\hat{x})|\le B,\qquad |v^{+}_{\delta}(\hat{y})|\le B,\qquad \alpha\,|\hat{x}-\hat{y}|_{H}\le G,$$

and

$$\frac{2B+1}{\delta}<R,\qquad 3B+2<R,\qquad 8\alpha+2<R,\qquad G+2\alpha<R .$$

Finally, let $\lambda$ be a properness constant for $F$ at $3B+2$, let $(\omega_{1},\omega_{2})$ be a second-order structure pair for $F$ at $3B+2$, and let $\omega$ be a shift modulus for $F$ at $(\delta,R)$. In the estimate below, $\tfrac{1}{\alpha}$ is the quotient, and $h(x_{1})$ and $h(y_{1})$ are defined because $D(A)\subseteq V$ by Hilbert Triples: Standing Notation and Background (iii).

(i)Then there exist $x_{1},y_{1}\in D(A)$ and $\tau_{1},\tau_{2}\in\mathbb{R}$ with

$$|x_{1}-\hat{x}|_{H}<\varepsilon,\qquad |y_{1}-\hat{y}|_{H}<\varepsilon,\qquad |u^{-}_{\delta}(x_{1})-u^{-}_{\delta}(\hat{x})|<\varepsilon,\qquad |v^{+}_{\delta}(y_{1})-v^{+}_{\delta}(\hat{y})|<\varepsilon,$$ $$0\le\tau_{1}\le(2\alpha+1)\varepsilon+\sigma,\qquad 0\le\tau_{2}\le(2\alpha+1)\varepsilon+\sigma,$$

for which the following two clauses hold.

1. (Test data at the nearby points)(ii) There are $s,t\in\mathbb{R}$ and a pair $(X,Y)$ of members of $\mathrm{Sym}(H)$ admitted at $\alpha$ with $|s-u^{-}_{\delta}(\hat{x})|<\varepsilon$ and $|t-v^{+}_{\delta}(\hat{y})|<\varepsilon$ such that

$$F^{-}_{\delta}\bigl(x_{1},s,\alpha(x_{1}-y_{1}),X\bigr)\le\varepsilon+\omega(\tau_{1})\qquad\text{and}\qquad-\varepsilon-\omega(\tau_{2})\le F^{+}_{\delta}\bigl(y_{1},t,\alpha(x_{1}-y_{1}),Y\bigr).$$

2. (The estimate)(iii) The points $x_{1},y_{1}$ and the numbers $\tau_{1},\tau_{2}$ above satisfy

$$\lambda\bigl(u^{-}_{\delta}(\hat{x})-v^{+}_{\delta}(\hat{y})\bigr)\ \le\ 2\lambda\varepsilon+2\varepsilon+\omega(\tau_{1})+\omega(\tau_{2})+\omega_{1}\Bigl(\alpha|x_{1}-y_{1}|_{H}^{2}+\tfrac{1}{\alpha}\Bigr)+\omega_{2}\bigl(\delta\,(h(x_{1})+h(y_{1})+1),\,\alpha\bigr).$$

Clause 2 is the inequality of The Test-Datum Estimate at a Maximum Point of the Doubled Function (i), and clause 1 records the test data from which it is derived. The hypotheses differ from those of that lemma in three respects: the maximum of the doubled function is required to be sequentially strict, the structure pair is required only at second order, and a basis along which $F$ is tail-insensitive is supplied. The bound $8\alpha+2<R$ replaces $\alpha+1<R$ because the second-order data produced here carry forms of norm at most $8\alpha$.