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 §open-sets, 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 §inner-product-space; 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 §bound, 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

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

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

and

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 §operator.

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

such that

The conclusion is that of The Test-Datum Estimate at a Maximum Point of the Doubled Function §estimate, and 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 $8\alpha$.