The Test-Datum Estimate at a Maximum Point 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 $u,v:H\to\mathbb{R}$ 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 $\delta,\alpha,\sigma,\varepsilon,B,G,R\in\mathbb{R}$ 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 maximum 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

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

Finally, let $\lambda$ be a properness constant for $F$ at $3B+2$, let $(\omega_{1},\omega_{2})$ be a 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

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

such that

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