Proof of The Test-Datum Estimate at a Maximum Point of the Doubled Function

Each result cited is universally quantified over the data in its own statement. Write $\Theta:V\times V\to\mathbb{R}$ for the function in the hypothesis, so that $\Theta$ attains a maximum at $(\hat{x},\hat{y})$.

Step 1 (test functions). Let $\varphi:H\to\mathbb{R}$ be given by $\varphi(x)=\tfrac{\alpha}{2}|x-\hat{y}|_{H}^{2}+\langle p,x\rangle_{H}$. By claims 3 and 1 of Affine and Quadratic Functions on a Real Hilbert Space are of Class $C^2$ its two summands belong to $C^{2}(H)$, with gradients $\alpha(x-\hat{y})$ and $p$ and Hessians $\alpha I_{H}$ and $0_{\mathrm{Sym}}$ at every $x\in H$, so by Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space (ii) $\varphi\in C^{2}(H)$ with

$$D\varphi(\hat{x})=\alpha(\hat{x}-\hat{y})+p,\qquad D^{2}\varphi(\hat{x})=\alpha I_{H}.$$

Let $\psi:H\to\mathbb{R}$ be given by $\psi(y)=-\tfrac{\alpha}{2}|y-\hat{x}|_{H}^{2}-\langle q,y\rangle_{H}$; by the same two lemmas together with Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space (iii), $\psi\in C^{2}(H)$ with

$$D\psi(\hat{y})=-\alpha(\hat{y}-\hat{x})-q=\alpha(\hat{x}-\hat{y})-q,\qquad D^{2}\psi(\hat{y})=-\alpha I_{H}.$$

For $x\in V$ one has $\Theta(x,\hat{y})=\bigl(u^{-}_{\delta}(x)-\varphi(x)\bigr)-v^{+}_{\delta}(\hat{y})-\langle q,\hat{y}\rangle_{H}$, and $\Theta(x,\hat{y})\le\Theta(\hat{x},\hat{y})$; hence the function on $V$ with value $u^{-}_{\delta}(x)-\varphi(x)$ at $x$ satisfies

$$u^{-}_{\delta}(x)-\varphi(x)\le u^{-}_{\delta}(\hat{x})-\varphi(\hat{x})\qquad\text{for every }x\in V,$$

and so has a local maximum relative to $V$ at $\hat{x}$, the condition of Local Maximum of a Function Relative to a Subset of a Metric Space holding with any positive radius. Likewise, for $y\in V$ one has $\Theta(\hat{x},y)=u^{-}_{\delta}(\hat{x})-\langle p,\hat{x}\rangle_{H}-\bigl(v^{+}_{\delta}(y)-\psi(y)\bigr)$, so the function with value $v^{+}_{\delta}(y)-\psi(y)$ at $y$ has a local minimum relative to $V$ at $\hat{y}$.

Step 2 (test data). Applying Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple (i) with $\delta$, the test function $\varphi$, the point $\hat{x}$ and the number $\varepsilon$ yields $x_{1}\in D(A)$, $s_{1}\in\mathbb{R}$, $p_{1}\in H$ and $X_{1}\in\mathrm{Sym}(H)$ with

$$|x_{1}-\hat{x}|_{H}<\varepsilon,\quad |u^{-}_{\delta}(x_{1})-u^{-}_{\delta}(\hat{x})|<\varepsilon,\quad |s_{1}-u^{-}_{\delta}(\hat{x})|<\varepsilon,\quad |p_{1}-D\varphi(\hat{x})|_{H}<\varepsilon,\quad \lVert X_{1}-\alpha I_{H}\rVert<\varepsilon,$$

and $F^{-}_{\delta}(x_{1},s_{1},p_{1},X_{1})\le\varepsilon$. Applying Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple (ii) with $\delta$, $\psi$, $\hat{y}$ and $\varepsilon$ yields $y_{1}\in D(A)$, $t_{1}\in\mathbb{R}$, $q_{1}\in H$ and $Y_{1}\in\mathrm{Sym}(H)$ with

$$|y_{1}-\hat{y}|_{H}<\varepsilon,\quad |v^{+}_{\delta}(y_{1})-v^{+}_{\delta}(\hat{y})|<\varepsilon,\quad |t_{1}-v^{+}_{\delta}(\hat{y})|<\varepsilon,\quad |q_{1}-D\psi(\hat{y})|_{H}<\varepsilon,\quad \lVert Y_{1}+\alpha I_{H}\rVert<\varepsilon,$$

and $-\varepsilon\le F^{+}_{\delta}(y_{1},t_{1},q_{1},Y_{1})$.

Step 3 (the data are admissible). By Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity (iv), $u^{-}_{\delta}(x_{1})\le C-\delta h(x_{1})$, so, using $C\le B$, $-u^{-}_{\delta}(\hat{x})\le B$ and $\varepsilon\le1$,

$$\delta h(x_{1})\le C-u^{-}_{\delta}(x_{1})\le C-u^{-}_{\delta}(\hat{x})+\varepsilon\le 2B+1 ,$$

whence $h(x_{1})\le\frac{2B+1}{\delta}<R$. By the same claim $-C+\delta h(y_{1})\le v^{+}_{\delta}(y_{1})$, so $\delta h(y_{1})\le v^{+}_{\delta}(y_{1})+C\le v^{+}_{\delta}(\hat{y})+\varepsilon+C\le2B+1$ and $h(y_{1})<R$. Next $|s_{1}|\le|u^{-}_{\delta}(\hat{x})|+\varepsilon\le B+1<R$ and $|t_{1}|\le B+1<R$, because $B+1\le3B+2<R$ and $0\le B$.

Since $|I_{H}(x,y)|=|\langle x,y\rangle_{H}|\le|x|_{H}|y|_{H}$ for $x,y\in H$ by The Cauchy–Schwarz Inequality in a Real Inner Product Space, claim 2 of Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity gives $\lVert I_{H}\rVert\le1$, and hence $\lVert\alpha I_{H}\rVert\le\alpha$ and $\lVert-\alpha I_{H}\rVert\le\alpha$ by the homogeneity of the norm in Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity (iii). Therefore, by the triangle inequality of that same claim,

$$\lVert X_{1}\rVert\le\lVert\alpha I_{H}\rVert+\lVert X_{1}-\alpha I_{H}\rVert\le\alpha+1<R,\qquad \lVert Y_{1}\rVert\le\alpha+1<R .$$

Finally, by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity (i),

$$|p_{1}|_{H}\le|D\varphi(\hat{x})|_{H}+\varepsilon\le\alpha|\hat{x}-\hat{y}|_{H}+|p|_{H}+\varepsilon\le G+\sigma+\varepsilon\le G+2<G+2\alpha<R,$$

using $1<\alpha$, and likewise $|q_{1}|_{H}\le G+2<R$. Thus $\xi_{1}=(x_{1},s_{1},p_{1},X_{1})$ and $\eta_{1}=(y_{1},t_{1},q_{1},Y_{1})$ are $R$-bounded test data, and

$$F^{-}_{\delta}(\xi_{1})-F^{+}_{\delta}(\eta_{1})\le\varepsilon+\varepsilon\le2<\alpha+1<R .$$

Hence $\xi_{1}\in S^{-}_{\delta,R}$ and $\eta_{1}\in S^{+}_{\delta,R}$ by Test Data for a Second-Order Equation Operator on a Hilbert Triple and the Admissible Sets (iii), each datum serving as the witness required for the other.

Step 4 (moving the gradient arguments). Put $P=\alpha(x_{1}-y_{1})$, $\tau_{1}=|P-p_{1}|_{H}$ and $\tau_{2}=|P-q_{1}|_{H}$; these are nonnegative. By the triangle inequality,

$$\tau_{1}\le\alpha\bigl|(x_{1}-y_{1})-(\hat{x}-\hat{y})\bigr|_{H}+\bigl|\alpha(\hat{x}-\hat{y})+p-p_{1}\bigr|_{H}+|p|_{H}\le\alpha(\varepsilon+\varepsilon)+\varepsilon+\sigma=(2\alpha+1)\varepsilon+\sigma ,$$

and the same bound holds for $\tau_{2}$, since $D\psi(\hat{y})=\alpha(\hat{x}-\hat{y})-q$ and $|q|_{H}\le\sigma$.

Apply the first condition of The Shift-Continuity Condition on Admissible Test Data (i) to $\xi_{1}$ with the perturbations $P-p_{1}\in H$ and $0_{\mathrm{Sym}}\in\mathrm{Sym}(H)$: since $p_{1}+(P-p_{1})=P$, $X_{1}+0_{\mathrm{Sym}}=X_{1}$ and $\lVert 0_{\mathrm{Sym}}\rVert=0$,

$$F^{-}_{\delta}(x_{1},s_{1},P,X_{1})\le F^{-}_{\delta}(x_{1},s_{1},p_{1},X_{1})+\omega(\tau_{1})\le\varepsilon+\omega(\tau_{1}).$$

Apply the second condition to $\eta_{1}$ with the perturbations $P-q_{1}$ and $0_{\mathrm{Sym}}$:

$$-\varepsilon-\omega(\tau_{2})\le F^{+}_{\delta}(y_{1},t_{1},q_{1},Y_{1})-\omega(\tau_{2})\le F^{+}_{\delta}(y_{1},t_{1},P,Y_{1}).$$

Step 5 (the structure condition). We have $x_{1},y_{1}\in D(A)=W$, $|t_{1}|\le B+1\le3B+2$, $1<\alpha$, $0<\delta<1$ and $P=\alpha(x_{1}-y_{1})$, so The First-Order Structure Condition for a Second-Order Equation Operator on a Hilbert Triple (i), applied to the structure pair $(\omega_{1},\omega_{2})$ at $3B+2$ with the value argument $t_{1}$ and the form arguments $X_{1}$ and $Y_{1}$, gives

$$F^{+}_{\delta}(y_{1},t_{1},P,Y_{1})-\omega_{1}\Bigl(\alpha|x_{1}-y_{1}|_{H}^{2}+\tfrac{1}{\alpha}\Bigr)-\omega_{2}\bigl(\delta\,(h(x_{1})+h(y_{1})+1),\,\alpha\bigr)\ \le\ F^{-}_{\delta}(x_{1},t_{1},P,X_{1}).$$

Write $\Omega$ for the sum $\omega_{1}\bigl(\alpha|x_{1}-y_{1}|_{H}^{2}+\tfrac{1}{\alpha}\bigr)+\omega_{2}\bigl(\delta\,(h(x_{1})+h(y_{1})+1),\,\alpha\bigr)$; it is nonnegative, because $\omega_{1}$ and the function $t\mapsto\omega_{2}(t,\alpha)$ are moduli of continuity, whose values are nonnegative by clause 1 of Modulus of Continuity.

Step 6 (properness and conclusion). Since $|s_{1}-u^{-}_{\delta}(\hat{x})|<\varepsilon$ and $|t_{1}-v^{+}_{\delta}(\hat{y})|<\varepsilon$,

$$u^{-}_{\delta}(\hat{x})-v^{+}_{\delta}(\hat{y})\le(s_{1}+\varepsilon)-(t_{1}-\varepsilon)=s_{1}-t_{1}+2\varepsilon . \tag{$*$}$$

Suppose first that $t_{1}\le s_{1}$. By Step 3 we have $0\le\delta h(x_{1})\le2B+1$, $|s_{1}|\le B+1$ and $|t_{1}|\le B+1$, so both $s_{1}+\delta h(x_{1})$ and $t_{1}+\delta h(x_{1})$ lie between $-(3B+2)$ and $3B+2$, and $t_{1}+\delta h(x_{1})\le s_{1}+\delta h(x_{1})$. Applying Locally Strictly Proper Second-Order Equation Operator on a Hilbert Triple (i) at the level $3B+2$, with the point $x_{1}$, the gradient argument $P+\delta Ax_{1}\in H$ and the form argument $X_{1}|_{V}+\delta I_{V}\in\mathrm{Sym}(V)$, and reading the two resulting values of $F$ through Second-Order Equation Operator on an Open Subset of a Hilbert Triple and Its $\delta$-Shifts (ii), we obtain

$$\lambda(s_{1}-t_{1})=\lambda\bigl((s_{1}+\delta h(x_{1}))-(t_{1}+\delta h(x_{1}))\bigr)\le F^{-}_{\delta}(x_{1},s_{1},P,X_{1})-F^{-}_{\delta}(x_{1},t_{1},P,X_{1}).$$

Combining this with Steps 4 and 5,

$$\lambda(s_{1}-t_{1})\le\varepsilon+\omega(\tau_{1})-F^{+}_{\delta}(y_{1},t_{1},P,Y_{1})+\Omega\ \le\ 2\varepsilon+\omega(\tau_{1})+\omega(\tau_{2})+\Omega .$$

Multiplying $(*)$ by the positive number $\lambda$ and adding $2\lambda\varepsilon$ to the previous display gives the asserted estimate.

Suppose now that $s_{1}<t_{1}$. Then $(*)$ gives $u^{-}_{\delta}(\hat{x})-v^{+}_{\delta}(\hat{y})<2\varepsilon$, so $\lambda\bigl(u^{-}_{\delta}(\hat{x})-v^{+}_{\delta}(\hat{y})\bigr)<2\lambda\varepsilon$, and the asserted estimate holds because its remaining right-hand terms are nonnegative: the values of the modulus of continuity $\omega$ are nonnegative by clause 1 of Modulus of Continuity, $\Omega$ is nonnegative by Step 5, and $0<2\varepsilon$.

In both cases $x_{1},y_{1}\in D(A)$ satisfy $|x_{1}-\hat{x}|_{H}<\varepsilon$, $|y_{1}-\hat{y}|_{H}<\varepsilon$, $|u^{-}_{\delta}(x_{1})-u^{-}_{\delta}(\hat{x})|<\varepsilon$ and $|v^{+}_{\delta}(y_{1})-v^{+}_{\delta}(\hat{y})|<\varepsilon$ by Step 2, and $\tau_{1},\tau_{2}$ satisfy the stated bounds by Step 4.