Proof of Stability of Viscosity Sub- and Supersolutions on a Hilbert Triple: Limits of Uniformly Bounded, Equicontinuous Sequences under Convergence of the Operators on Bounded Test Data

Each result cited below is universally quantified over the data in its own statement. Elementary arithmetic and order facts about real numbers, the limit laws and passage to the limit in non-strict inequalities for real sequences, the principle that $a\le b+\eta$ for every real $\eta>0$ implies $a\le b$, the approximation of suprema and infima, the monotonicity of squares of nonnegative numbers, and the growth $k\le n_{k}$ of the indices of a subsequence are used without further mention; they are carried by The Real Numbers: Standing Notation and Background.

Conventions. We take $U=H$ in the clause on open sets of the setting: $H$ is nonempty, as $0_{H}\in H$, and open in $H$; $V\cap H=V$ carries the metric $d_{H}$, and $W=D(A)$. Accordingly local maxima relative to $V$, continuity and semicontinuity of functions on $H$ or on $V$ refer to the metric space $(H,d_{H})$, where $d_{H}(x,y)=|x-y|_{H}$, and continuity is equivalent to sequential continuity, by the topological vocabulary of the Hilbert-space setting. The norm of $H$ obeys the triangle inequality and $|\lambda x|_{H}=|\lambda|\,|x|_{H}$ by the background results on inner product spaces. The set $\mathrm{Sym}(H)$ is a vector space by Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity §vector-space, and its norm $\lVert\cdot\rVert$ obeys the triangle inequality and $\lVert\lambda b\rVert=|\lambda|\,\lVert b\rVert$ by Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity §norm-axioms. The penalty function is $h(x)=\tfrac12|x|_{V}^{2}\ge 0$ for $x\in V$, by Hilbert Triples: Standing Notation and Background §penalty and The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds §nonneg. To avoid a clash with the penalty parameter $\delta$, radii supplied by the definition of a modulus of continuity are called $\tau$. We record two consequences of that definition.

(M1) If $(t_{k})_{k\in\mathbb{N}}$ is a sequence of nonnegative reals converging to $0$, then $\omega(t_{k})\to 0$. Indeed, given $\eta>0$, condition 2 of Modulus of Continuity gives $\tau>0$ with $\omega(t)\le\eta$ whenever $0\le t\le\tau$; for all large $k$ we have $t_{k}\le\tau$, hence $0\le\omega(t_{k})\le\eta$, using condition 1.

(M2) If $v:H\to\mathbb{R}$ satisfies $|v(x)-v(y)|\le\omega(|x-y|_{H})$ for all $x,y\in H$, then $v$ is continuous on $H$. Indeed, given $x\in H$ and $\eta>0$, take $\tau>0$ with $\omega(t)\le\eta/2$ for $0\le t\le\tau$; then every $y\in H$ with $d_{H}(x,y)<\tau$ satisfies $|v(y)-v(x)|\le\eta/2<\eta$, which is continuity at $x$ relative to $H$.

Step 0 (a core claim). Parts 1 and 2 will both be derived (Steps 8 and 9) from the following claim, proved in Steps 1 to 7, in which the hypothesis of the theorem that every sequence in $V$ bounded in $V$ has a subsequence converging in $H$ remains in force.

Core claim. Let $G$ and $G_{N}$, for $N\in\mathbb{N}$, be second-order equation operators on $H$ relative to $(H,V,A)$, with $\delta$-shifts $G^{\pm}_{\delta}$ and $G^{\pm}_{N,\delta}$, such that $(G_{N})_{N\in\mathbb{N}}$ converges to $G$ on bounded test data. Let $w_{N}:H\to\mathbb{R}$, for $N\in\mathbb{N}$, and $w:H\to\mathbb{R}$ satisfy the following three conditions. (a) For all $N\in\mathbb{N}$ and $x,y\in H$,

$$|w_{N}(x)|\le C,\qquad |w_{N}(x)-w_{N}(y)|\le\omega(|x-y|_{H}),\qquad |w(x)-w(y)|\le\omega(|x-y|_{H}).$$

(b) For every $x\in H$ there are natural numbers $N_{1}<N_{2}<\cdots$ such that $w_{N_{j}}(x)\to w(x)$ as $j\to\infty$. (c) For every $z\in H$ and every real $\eta>0$ there is $k\in\mathbb{N}$ with $w_{N}(z)<w(z)+\eta$ for every $N\ge k$. If $w_{N}$ is a viscosity subsolution of $G_{N}$ on $H$ for every $N\in\mathbb{N}$, then $w$ is a viscosity subsolution of $G$ on $H$.

Step 1 (the envelopes). By (a) and (M2), every $w_{N}$ and $w$ is continuous on $H$. Let $\delta>0$ be real. By Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity §semicontinuous-case, applied with $U=H$, each $w_{N}$ and $w$ is bounded above near each point of $H$, and the envelopes, which are the upper semicontinuous envelopes on $V$ of $w_{N}-\delta h$ and $w-\delta h$, satisfy

$$(w_{N})^{-}_{\delta}(x)=w_{N}(x)-\delta h(x),\qquad w^{-}_{\delta}(x)=w(x)-\delta h(x)\qquad\text{for every }x\in V .$$

In particular the definition of a viscosity subsolution applies to $w$ and $G$, and to each $w_{N}$ and $G_{N}$.

Step 2 (fixing the data). Fix, in this order, a real $\delta>0$, a function $\varphi\in C^{2}(H)$, a point $\hat{x}\in V$ at which the function $V\to\mathbb{R}$, $x\mapsto w^{-}_{\delta}(x)-\varphi(x)$, has a local maximum relative to $V$, and a real $\varepsilon>0$. We must produce $y\in D(A)$, $s\in\mathbb{R}$, $q\in H$ and $Y\in\mathrm{Sym}(H)$ satisfying the six requirements of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple §subsolution for $w$, $G$, $\delta$, $\varphi$, $\hat{x}$ and $\varepsilon$. By the definition of a local maximum and Step 1 there is a real $r_{0}>0$ such that, with

$$\hat{M}=w(\hat{x})-\delta h(\hat{x})-\varphi(\hat{x}),$$ $$w(x)-\delta h(x)-\varphi(x)\le\hat{M}\qquad\text{for every }x\in V\text{ with }|x-\hat{x}|_{H}<r_{0}. \tag{2.1}$$

Since $\varphi$ is continuous on $H$ by Basic Properties of Differentiability on an Open Subset of a Real Inner Product Space §continuous, there is a real $\rho>0$ with $|\varphi(x)-\varphi(\hat{x})|<1$ whenever $|x-\hat{x}|_{H}<\rho$. Put $r=\tfrac12\min\{\rho,r_{0}\}$ and $\bar{B}=\{x\in H:|x-\hat{x}|_{H}\le r\}$, so that $r<\rho$ and $r<r_{0}$. Next put

$$\alpha=\frac{\varepsilon}{8\,(1+\lVert I_{H}\rVert)},$$

so that $\alpha>0$ and $\alpha\lVert I_{H}\rVert<\varepsilon/8$, and define $\psi,\varphi_{1}:H\to\mathbb{R}$ by $\psi(x)=\tfrac{\alpha}{2}|x-\hat{x}|_{H}^{2}$ and $\varphi_{1}(x)=\varphi(x)+\psi(x)$. By Affine and Quadratic Functions on a Real Hilbert Space are of Class $C^2$ §quadratic, applied in $H$ with $\alpha$ and $y_{0}=\hat{x}$, we have $\psi\in C^{2}(H)$ with $D\psi(x)=\alpha(x-\hat{x})$ and $D^{2}\psi(x)=\alpha I_{H}$ for every $x\in H$; hence by Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space §sum we have $\varphi_{1}\in C^{2}(H)$ and

$$D\varphi_{1}(x)=D\varphi(x)+\alpha(x-\hat{x}),\qquad D^{2}\varphi_{1}(x)=D^{2}\varphi(x)+\alpha I_{H}\qquad\text{for every }x\in H. \tag{2.2}$$

Moreover $\varphi_{1}(\hat{x})=\varphi(\hat{x})$ and $\varphi\le\varphi_{1}$. Writing $\Phi(x)=w(x)-\delta h(x)-\varphi_{1}(x)$ for $x\in V$, (2.1) gives

$$\Phi(x)\le\hat{M}-\tfrac{\alpha}{2}|x-\hat{x}|_{H}^{2}\quad\text{for every }x\in V\cap\bar{B},\qquad \Phi(\hat{x})=\hat{M}. \tag{2.3}$$

Claim A. Let $(z_{k})_{k\in\mathbb{N}}$ be a sequence in $V$ converging in $H$ to a point $z\in H$, and let $a\in\mathbb{R}$ be such that for every real $\eta>0$ there is $k_{0}\in\mathbb{N}$ with $\delta h(z_{k})\le a+\eta$ for every $k\ge k_{0}$. Then $z\in V$ and $\delta h(z)\le a$. Proof. Given $\eta>0$ and such a $k_{0}$, the sequence $(z_{k+k_{0}})_{k\in\mathbb{N}}$ lies in $V$, converges in $H$ to $z$, and satisfies $h(z_{k+k_{0}})\le(a+\eta)/\delta$ for every $k$; by The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds §closed-sublevel we get $z\in V$ and $\delta h(z)\le a+\eta$. As $\eta>0$ was arbitrary, $\delta h(z)\le a$.

Step 3 (localised maximisers). By (b) at the point $\hat{x}$, fix natural numbers $N_{1}<N_{2}<\cdots$ with $w_{N_{j}}(\hat{x})\to w(\hat{x})$. For $j\in\mathbb{N}$ write $v_{j}=w_{N_{j}}$ and

$$\Phi_{j}(x)=v_{j}(x)-\delta h(x)-\varphi_{1}(x)=(v_{j})^{-}_{\delta}(x)-\varphi_{1}(x)\qquad(x\in V),$$

the second equality by Step 1. Put $c=h(\hat{x})+(2C+2)/\delta$ and $K=\{x\in V\cap\bar{B}:h(x)\le c\}$. Since $C\ge 0$, $\hat{x}\in K$; and $c\ge 0$.

(3.1) For every $j$ and every $x\in V\cap\bar{B}$ with $h(x)>c$, $\Phi_{j}(x)<\Phi_{j}(\hat{x})$. For $x\in V\cap\bar{B}$ we have $|x-\hat{x}|_{H}\le r<\rho$, so $\varphi_{1}(x)\ge\varphi(x)>\varphi(\hat{x})-1$, and $v_{j}(x)\le C$ by (a); thus $\Phi_{j}(x)<C-\delta h(x)-\varphi(\hat{x})+1$. If $h(x)>c$, then $\delta h(x)>\delta h(\hat{x})+2C+2$, so $\Phi_{j}(x)<-C-\delta h(\hat{x})-\varphi(\hat{x})-1<v_{j}(\hat{x})-\delta h(\hat{x})-\varphi_{1}(\hat{x})=\Phi_{j}(\hat{x})$, using $v_{j}(\hat{x})\ge -C$ and $\varphi_{1}(\hat{x})=\varphi(\hat{x})$.

(3.2) Every sequence in $K$ has a subsequence converging in $H$ to a point of $K$. Let $(z_{m})_{m\in\mathbb{N}}$ lie in $K$. From $\tfrac12|z_{m}|_{V}^{2}=h(z_{m})\le c$ we get $|z_{m}|_{V}\le 1+2c$ (if $|z_{m}|_{V}>1$ then $|z_{m}|_{V}<|z_{m}|_{V}^{2}\le 2c$), so $(z_{m})$ is bounded in $V$, and by the compactness hypothesis of the theorem it has a subsequence $(z_{m_{k}})_{k\in\mathbb{N}}$ converging in $H$ to some $z\in H$. By The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds §closed-sublevel, applied to this subsequence with the constant $c$, we get $z\in V$ and $h(z)\le c$. Finally $|z-\hat{x}|_{H}\le|z-z_{m_{k}}|_{H}+r$ for every $k$, and $|z-z_{m_{k}}|_{H}\to0$, so $|z-\hat{x}|_{H}\le r$. Hence $z\in K$.

(3.3) For every $j$ there is $x_{j}\in K$ with $\Phi_{j}(x)\le\Phi_{j}(x_{j})$ for every $x\in V\cap\bar{B}$. Fix $j$. For $x\in K$, the bound in the proof of (3.1) and $h(x)\ge0$ give $\Phi_{j}(x)<C-\varphi(\hat{x})+1$; so $\{\Phi_{j}(x):x\in K\}$ is nonempty and bounded above, with supremum $M_{j}$. Choose $z_{m}\in K$ with $\Phi_{j}(z_{m})>M_{j}-\tfrac1m$ for $m\in\mathbb{N}$, and by (3.2) a subsequence $(z_{m_{k}})$ converging in $H$ to some $z\in K$. For every $k$,

$$\delta h(z_{m_{k}})=v_{j}(z_{m_{k}})-\varphi_{1}(z_{m_{k}})-\Phi_{j}(z_{m_{k}})<v_{j}(z_{m_{k}})-\varphi_{1}(z_{m_{k}})-M_{j}+\tfrac{1}{m_{k}} ,$$

and the right-hand side converges to $v_{j}(z)-\varphi_{1}(z)-M_{j}$, because $v_{j}$ is continuous (Step 1), $\varphi_{1}$ is continuous by Basic Properties of Differentiability on an Open Subset of a Real Inner Product Space §continuous, and $m_{k}\ge k$. Claim A therefore gives $\delta h(z)\le v_{j}(z)-\varphi_{1}(z)-M_{j}$, that is, $\Phi_{j}(z)\ge M_{j}$. Put $x_{j}=z$. Then $\Phi_{j}(x)\le M_{j}\le\Phi_{j}(x_{j})$ for $x\in K$, while for $x\in(V\cap\bar{B})\setminus K$ we have $h(x)>c$ and so $\Phi_{j}(x)<\Phi_{j}(\hat{x})\le\Phi_{j}(x_{j})$ by (3.1) and $\hat{x}\in K$.

Step 4 (convergence of the maximisers). Since $\hat{x}\in V\cap\bar{B}$, (3.3) gives $\Phi_{j}(x_{j})\ge\Phi_{j}(\hat{x})=v_{j}(\hat{x})-\delta h(\hat{x})-\varphi(\hat{x})$, that is,

$$\delta h(x_{j})\le v_{j}(x_{j})-\varphi_{1}(x_{j})-v_{j}(\hat{x})+\delta h(\hat{x})+\varphi(\hat{x})\qquad\text{for every }j. \tag{4.0}$$

We show: (4.1) $x_{j}\to\hat{x}$ in $H$; (4.2) $v_{j}(x_{j})\to w(\hat{x})$; (4.3) $h(x_{j})\to h(\hat{x})$.

Proof of (4.1). Suppose not. Then there is a real $\eta_{0}>0$ with $|x_{j}-\hat{x}|_{H}\ge\eta_{0}$ for infinitely many $j$, so there are indices $j_{1}<j_{2}<\cdots$ with $|x_{j_{k}}-\hat{x}|_{H}\ge\eta_{0}$ for all $k$. By (3.2) applied to $(x_{j_{k}})_{k}$ there are indices $i_{1}<i_{2}<\cdots$, each of the form $j_{k}$, and a point $z\in K$ with $x_{i_{l}}\to z$ in $H$. Since $|z-\hat{x}|_{H}\ge\eta_{0}-|x_{i_{l}}-z|_{H}$ for every $l$, we get $|z-\hat{x}|_{H}\ge\eta_{0}$. Let $\eta>0$. By (a), $v_{i_{l}}(x_{i_{l}})\le v_{i_{l}}(z)+\omega(|x_{i_{l}}-z|_{H})$. By (c) at $z$ there is $k$ with $w_{N}(z)<w(z)+\eta$ for $N\ge k$; since $N_{i_{l}}\ge i_{l}\ge l$, we get $v_{i_{l}}(z)<w(z)+\eta$ for $l\ge k$. By (M1), $\omega(|x_{i_{l}}-z|_{H})\le\eta$ for all large $l$. By continuity of $\varphi_{1}$, $\varphi_{1}(x_{i_{l}})>\varphi_{1}(z)-\eta$ for all large $l$. Since $v_{j}(\hat{x})\to w(\hat{x})$ and $i_{l}\ge l$, $v_{i_{l}}(\hat{x})>w(\hat{x})-\eta$ for all large $l$. Inserting these four bounds in (4.0) with $j=i_{l}$ gives, for all large $l$,

$$\delta h(x_{i_{l}})\le w(z)-\varphi_{1}(z)-w(\hat{x})+\delta h(\hat{x})+\varphi(\hat{x})+4\eta=w(z)-\varphi_{1}(z)-\hat{M}+4\eta .$$

As $\eta>0$ was arbitrary, Claim A (with $a=w(z)-\varphi_{1}(z)-\hat{M}$) gives $\delta h(z)\le w(z)-\varphi_{1}(z)-\hat{M}$, that is, $\Phi(z)\ge\hat{M}$. Since $z\in K\subseteq V\cap\bar{B}$, (2.3) gives $\Phi(z)\le\hat{M}-\tfrac{\alpha}{2}|z-\hat{x}|_{H}^{2}$. Hence $\tfrac{\alpha}{2}|z-\hat{x}|_{H}^{2}\le0$, so $|z-\hat{x}|_{H}=0$ as $\alpha>0$, contradicting $|z-\hat{x}|_{H}\ge\eta_{0}>0$.

Proof of (4.2). By (a), $|v_{j}(x_{j})-v_{j}(\hat{x})|\le\omega(|x_{j}-\hat{x}|_{H})$, which tends to $0$ by (4.1) and (M1); and $v_{j}(\hat{x})\to w(\hat{x})$.

Proof of (4.3). By (4.2), (4.1), the continuity of $\varphi_{1}$ and $\varphi_{1}(\hat{x})=\varphi(\hat{x})$, the right-hand side of (4.0) converges to $w(\hat{x})-\varphi(\hat{x})-w(\hat{x})+\delta h(\hat{x})+\varphi(\hat{x})=\delta h(\hat{x})$; so for every $\eta>0$, $\delta h(x_{j})<\delta h(\hat{x})+\eta$ for all large $j$. Conversely, suppose that for some $\eta>0$ the inequality $\delta h(x_{j})\le\delta h(\hat{x})-\eta$ held for infinitely many $j$. These $x_{j}$ form a subsequence converging in $H$ to $\hat{x}$ by (4.1), and Claim A with $a=\delta h(\hat{x})-\eta$ would give $\delta h(\hat{x})\le\delta h(\hat{x})-\eta$, which is absurd. Hence for every $\eta>0$, $\delta h(x_{j})>\delta h(\hat{x})-\eta$ for all large $j$. Thus $h(x_{j})\to h(\hat{x})$.

Step 5 (applying the subsolution property of $v_{j}$). By (4.1) fix $j_{0}\in\mathbb{N}$ with $|x_{j}-\hat{x}|_{H}<r/2$ for every $j\ge j_{0}$. Let $j\ge j_{0}$. Every $x\in V$ with $d_{H}(x_{j},x)<r/2$ satisfies $|x-\hat{x}|_{H}\le|x-x_{j}|_{H}+|x_{j}-\hat{x}|_{H}<r$, so $x\in V\cap\bar{B}$, and by (3.3) and Step 3, $(v_{j})^{-}_{\delta}(x)-\varphi_{1}(x)=\Phi_{j}(x)\le\Phi_{j}(x_{j})=(v_{j})^{-}_{\delta}(x_{j})-\varphi_{1}(x_{j})$. Thus the function $x\mapsto(v_{j})^{-}_{\delta}(x)-\varphi_{1}(x)$ on $V$ has a local maximum at $x_{j}\in V$ relative to $V$. Since $v_{j}=w_{N_{j}}$ is a viscosity subsolution of $G_{N_{j}}$ on $H$ and $\varphi_{1}\in C^{2}(H)$, Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple §subsolution, applied with $\delta$, $\varphi_{1}$, the point $x_{j}$ and the tolerance $1/j$, provides $y_{j}\in D(A)$, $s_{j}\in\mathbb{R}$, $q_{j}\in H$ and $Y_{j}\in\mathrm{Sym}(H)$ with

$$|y_{j}-x_{j}|_{H}<\tfrac1j,\qquad|(v_{j})^{-}_{\delta}(y_{j})-(v_{j})^{-}_{\delta}(x_{j})|<\tfrac1j,\qquad|s_{j}-(v_{j})^{-}_{\delta}(x_{j})|<\tfrac1j, \tag{5.1}$$ $$|q_{j}-D\varphi_{1}(x_{j})|_{H}<\tfrac1j,\qquad\lVert Y_{j}-D^{2}\varphi_{1}(x_{j})\rVert<\tfrac1j,\qquad G^{-}_{N_{j},\delta}(y_{j},s_{j},q_{j},Y_{j})\le\tfrac1j, \tag{5.2}$$

where $G^{-}_{N_{j},\delta}$ is the $\delta$-shift of $G_{N_{j}}$. These are chosen for every $j\ge j_{0}$.

Step 6 (limits of the data). As $j\to\infty$ through $j\ge j_{0}$:

(6.1) $y_{j}\to\hat{x}$ in $H$, since $|y_{j}-\hat{x}|_{H}\le|y_{j}-x_{j}|_{H}+|x_{j}-\hat{x}|_{H}<\tfrac1j+|x_{j}-\hat{x}|_{H}$ and (4.1).

(6.2) $h(y_{j})\to h(\hat{x})$. By Step 1 the second inequality of (5.1) reads $|v_{j}(y_{j})-\delta h(y_{j})-v_{j}(x_{j})+\delta h(x_{j})|<\tfrac1j$; with (a) this gives $|\delta h(y_{j})-\delta h(x_{j})|<\tfrac1j+\omega(|y_{j}-x_{j}|_{H})$, which tends to $0$ by (5.1) and (M1). Now use (4.3).

(6.3) $w^{-}_{\delta}(y_{j})\to w^{-}_{\delta}(\hat{x})$. By Step 1, $w^{-}_{\delta}(y_{j})=w(y_{j})-\delta h(y_{j})$; by (a), $|w(y_{j})-w(\hat{x})|\le\omega(|y_{j}-\hat{x}|_{H})\to0$ by (6.1) and (M1); with (6.2), $w^{-}_{\delta}(y_{j})\to w(\hat{x})-\delta h(\hat{x})=w^{-}_{\delta}(\hat{x})$.

(6.4) $s_{j}\to w^{-}_{\delta}(\hat{x})$. By Step 1 the third inequality of (5.1) reads $|s_{j}-v_{j}(x_{j})+\delta h(x_{j})|<\tfrac1j$; use (4.2) and (4.3).

(6.5) $q_{j}\to D\varphi(\hat{x})$ in $H$. By (2.2), $|q_{j}-D\varphi(\hat{x})|_{H}\le|q_{j}-D\varphi_{1}(x_{j})|_{H}+|D\varphi(x_{j})-D\varphi(\hat{x})|_{H}+\alpha|x_{j}-\hat{x}|_{H}$. The first term is below $\tfrac1j$; the second tends to $0$ because the gradient map of $\varphi$ is continuous on $H$ (The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space §c2 and The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space §c1) and (4.1); the third tends to $0$ by (4.1).

(6.6) By (2.2), in the vector space $\mathrm{Sym}(H)$,

$$Y_{j}-D^{2}\varphi(\hat{x})=\bigl(Y_{j}-D^{2}\varphi_{1}(x_{j})\bigr)+\bigl(D^{2}\varphi(x_{j})-D^{2}\varphi(\hat{x})\bigr)+\alpha I_{H},$$

so by the triangle inequality and homogeneity of $\lVert\cdot\rVert$, (5.2) and $\alpha\lVert I_{H}\rVert<\varepsilon/8$,

$$\lVert Y_{j}-D^{2}\varphi(\hat{x})\rVert<\tfrac1j+\lVert D^{2}\varphi(x_{j})-D^{2}\varphi(\hat{x})\rVert+\tfrac{\varepsilon}{8},$$

where the middle term tends to $0$ because the Hessian map of $\varphi$ is continuous on $H$ into $(\mathrm{Sym}(H),d_{\mathrm{Sym}})$ by The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space §c2, and (4.1).

Step 7 (the operator inequality and the choice of $j$). Put

$$R=\max\bigl\{1,\ 2h(\hat{x})+2,\ |w^{-}_{\delta}(\hat{x})|+1,\ |D\varphi(\hat{x})|_{H}+1,\ \lVert D^{2}\varphi(\hat{x})\rVert+\varepsilon+1\bigr\}>0 .$$

By (6.2), (6.4), (6.5) and (6.6) fix $j_{1}\ge j_{0}$ such that for every $j\ge j_{1}$: $h(y_{j})\le h(\hat{x})+1$, $|s_{j}-w^{-}_{\delta}(\hat{x})|<1$, $|q_{j}-D\varphi(\hat{x})|_{H}<1$ and $\tfrac1j+\lVert D^{2}\varphi(x_{j})-D^{2}\varphi(\hat{x})\rVert<1$. For $j\ge j_{1}$ we then have $|y_{j}|_{V}^{2}=2h(y_{j})\le 2h(\hat{x})+2\le R\le R^{2}$, so $|y_{j}|_{V}\le R$; $|s_{j}|\le R$; $|q_{j}|_{H}\le R$; and $\lVert Y_{j}\rVert\le\lVert Y_{j}-D^{2}\varphi(\hat{x})\rVert+\lVert D^{2}\varphi(\hat{x})\rVert<1+\tfrac{\varepsilon}{8}+\lVert D^{2}\varphi(\hat{x})\rVert\le R$. Since $y_{j}\in D(A)$ and $Y_{j}\in\mathrm{Sym}(H)$, the quadruple $(y_{j},s_{j},q_{j},Y_{j})$ is a test datum bounded by $R$. By Convergence of Second-Order Equation Operators on a Hilbert Triple on Bounded Test Data §convergence, applied to $(G_{N})$ and $G$ with $\delta$, $R$ and the tolerance $\varepsilon/2$, fix $N_{0}\in\mathbb{N}$ such that $|G^{-}_{N,\delta}(\xi)-G^{-}_{\delta}(\xi)|\le\varepsilon/2$ for every $N\ge N_{0}$ and every test datum $\xi$ bounded by $R$. For $j\ge\max\{j_{1},N_{0}\}$ we have $N_{j}\ge j\ge N_{0}$, so by (5.2)

$$G^{-}_{\delta}(y_{j},s_{j},q_{j},Y_{j})\le G^{-}_{N_{j},\delta}(y_{j},s_{j},q_{j},Y_{j})+\tfrac{\varepsilon}{2}\le\tfrac1j+\tfrac{\varepsilon}{2}.$$

Finally, by (6.1), (6.3), (6.4), (6.5) and (6.6), choose $j\ge\max\{j_{1},N_{0}\}$ with $\tfrac1j<\tfrac{\varepsilon}{4}$, $|y_{j}-\hat{x}|_{H}<\varepsilon$, $|w^{-}_{\delta}(y_{j})-w^{-}_{\delta}(\hat{x})|<\varepsilon$, $|s_{j}-w^{-}_{\delta}(\hat{x})|<\varepsilon$, $|q_{j}-D\varphi(\hat{x})|_{H}<\varepsilon$ and $\lVert D^{2}\varphi(x_{j})-D^{2}\varphi(\hat{x})\rVert<\tfrac{\varepsilon}{4}$. Then $y=y_{j}\in D(A)=W$, $s=s_{j}$, $q=q_{j}$ and $Y=Y_{j}$ satisfy

$$|y-\hat{x}|_{H}<\varepsilon,\qquad|w^{-}_{\delta}(y)-w^{-}_{\delta}(\hat{x})|<\varepsilon,\qquad|s-w^{-}_{\delta}(\hat{x})|<\varepsilon,$$ $$|q-D\varphi(\hat{x})|_{H}<\varepsilon,\qquad\lVert Y-D^{2}\varphi(\hat{x})\rVert<\tfrac{\varepsilon}{4}+\tfrac{\varepsilon}{4}+\tfrac{\varepsilon}{8}<\varepsilon,\qquad G^{-}_{\delta}(y,s,q,Y)\le\tfrac{\varepsilon}{4}+\tfrac{\varepsilon}{2}<\varepsilon .$$

Since $\delta$, $\varphi$, $\hat{x}$ and $\varepsilon$ were arbitrary and $w$ is bounded above near each point of $H$ (Step 1), $w$ is a viscosity subsolution of $G$ on $H$ by Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple §subsolution. This proves the core claim.

Step 8 (Part 1). Assume every $u_{N}$ is a viscosity subsolution of $F_{N}$ on $H$, and apply the core claim with $G=F$, $G_{N}=F_{N}$, $w_{N}=u_{N}$ and $w=\bar{u}$. The space $(H,d_{H})$ is a metric space by Real Hilbert Spaces: Standing Notation and Background §space, and the displayed hypothesis of the theorem is the hypothesis of Limits of a Uniformly Bounded Sequence of Real Functions with a Common Modulus of Continuity: Bounds, Extraction and Uniform Convergence on Sequentially Compact Sets for it, with the same $C$, $\omega$, $u_{N}$, $\bar{u}$ and $\underline{u}$. Condition (a) holds by that hypothesis and Limits of a Uniformly Bounded Sequence of Real Functions with a Common Modulus of Continuity: Bounds, Extraction and Uniform Convergence on Sequentially Compact Sets §modulus; condition (b) holds by Limits of a Uniformly Bounded Sequence of Real Functions with a Common Modulus of Continuity: Bounds, Extraction and Uniform Convergence on Sequentially Compact Sets §extraction. For (c), let $z\in H$ and $\eta>0$, and let $A_{k}=\{u_{m}(z):m\ge k\}$. By Limit Superior of a Bounded Sequence of Real Numbers, $\bar{u}(z)=\inf\{\sup A_{k}:k\in\mathbb{N}\}$, so $\bar{u}(z)+\eta$ is not a lower bound of this set and there is $k$ with $\sup A_{k}<\bar{u}(z)+\eta$; then $u_{N}(z)\le\sup A_{k}<\bar{u}(z)+\eta$ for every $N\ge k$. The core claim shows that $\bar{u}$ is a viscosity subsolution of $F$ on $H$.

Step 9 (Part 2). Assume every $u_{N}$ is a viscosity supersolution of $F_{N}$ on $H$. Let $\tilde{F}$ and $\tilde{F}_{N}$ be the operators $\tilde{F}(x,r,p,X)=-F(x,-r,-p,-X)$ and $\tilde{F}_{N}(x,r,p,X)=-F_{N}(x,-r,-p,-X)$ of Sign Reversal for Second-Order Equations on a Hilbert Triple: Subsolutions of $F$ are Supersolutions of $\tilde F$, applied with $U=H$; they are second-order equation operators on $H$ relative to $(H,V,A)$ by Sign Reversal for Second-Order Equations on a Hilbert Triple: Subsolutions of $F$ are Supersolutions of $\tilde F$ §operator.

$(\tilde{F}_{N})$ converges to $\tilde{F}$ on bounded test data. Let $\delta,R,\varepsilon>0$ and take $N_{0}$ from Convergence of Second-Order Equation Operators on a Hilbert Triple on Bounded Test Data §convergence for $(F_{N})$ and $F$. Let $N\ge N_{0}$ and let $(x,r,p,Y)$ be a test datum bounded by $R$. Then $(x,-r,-p,-Y)$ is also a test datum bounded by $R$, since $-Y\in\mathrm{Sym}(H)$, $|-r|=|r|$, $|-p|_{H}=|p|_{H}$ and $\lVert -Y\rVert=\lVert Y\rVert$, the last by Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity §norm-axioms with the scalar $-1$. By Sign Reversal for Second-Order Equations on a Hilbert Triple: Subsolutions of $F$ are Supersolutions of $\tilde F$ §shifted, applied to $F_{N}$ and to $F$,

$$\bigl|\tilde{F}^{-}_{N,\delta}(x,r,p,Y)-\tilde{F}^{-}_{\delta}(x,r,p,Y)\bigr|=\bigl|F^{+}_{N,\delta}(x,-r,-p,-Y)-F^{+}_{\delta}(x,-r,-p,-Y)\bigr|\le\varepsilon,$$ $$\bigl|\tilde{F}^{+}_{N,\delta}(x,r,p,Y)-\tilde{F}^{+}_{\delta}(x,r,p,Y)\bigr|=\bigl|F^{-}_{N,\delta}(x,-r,-p,-Y)-F^{-}_{\delta}(x,-r,-p,-Y)\bigr|\le\varepsilon,$$

where $\tilde{F}^{\pm}_{N,\delta}$ are the $\delta$-shifts of $\tilde{F}_{N}$. This is the required convergence.

By Sign Reversal for Second-Order Equations on a Hilbert Triple: Subsolutions of $F$ are Supersolutions of $\tilde F$ §viscosity (its second equivalence, applied to $u_{N}$ and $F_{N}$), each $-u_{N}$ is a viscosity subsolution of $\tilde{F}_{N}$ on $H$. Apply the core claim with $G=\tilde{F}$, $G_{N}=\tilde{F}_{N}$, $w_{N}=-u_{N}$ and $w=-\underline{u}$. Condition (a): $|-u_{N}(x)|=|u_{N}(x)|\le C$, $|(-u_{N}(x))-(-u_{N}(y))|=|u_{N}(x)-u_{N}(y)|\le\omega(|x-y|_{H})$, and $|(-\underline{u}(x))-(-\underline{u}(y))|=|\underline{u}(x)-\underline{u}(y)|\le\omega(|x-y|_{H})$ by Limits of a Uniformly Bounded Sequence of Real Functions with a Common Modulus of Continuity: Bounds, Extraction and Uniform Convergence on Sequentially Compact Sets §modulus, applied as in Step 8. Condition (b): by Limits of a Uniformly Bounded Sequence of Real Functions with a Common Modulus of Continuity: Bounds, Extraction and Uniform Convergence on Sequentially Compact Sets §extraction there are $N'_{1}<N'_{2}<\cdots$ with $u_{N'_{j}}(x)\to\underline{u}(x)$, hence $-u_{N'_{j}}(x)\to-\underline{u}(x)$. Condition (c): let $z\in H$, $\eta>0$ and $A_{k}=\{u_{m}(z):m\ge k\}$; by Limit Inferior of a Bounded Sequence of Real Numbers, $\underline{u}(z)=\sup\{\inf A_{k}:k\in\mathbb{N}\}$, so $\underline{u}(z)-\eta$ is not an upper bound of this set and there is $k$ with $\inf A_{k}>\underline{u}(z)-\eta$; then $u_{N}(z)>\underline{u}(z)-\eta$, that is, $-u_{N}(z)<-\underline{u}(z)+\eta$, for every $N\ge k$. The core claim shows that $-\underline{u}$ is a viscosity subsolution of $\tilde{F}$ on $H$. By Sign Reversal for Second-Order Equations on a Hilbert Triple: Subsolutions of $F$ are Supersolutions of $\tilde F$ §viscosity (its second equivalence, applied to $\underline{u}$ and $F$, together with its final sentence on local boundedness), $\underline{u}$ is a viscosity supersolution of $F$ on $H$. This proves Part 2.