Proof of The Pointwise Supremum of a Locally Uniformly Bounded Family of Viscosity Subsolutions on a Hilbert Triple is a Viscosity Subsolution

Each result cited below is universally quantified over the data appearing in its own statement, and is applied here to the data named at the point of use. We write $|\cdot|_{H}$ and $d_{H}$ for the norm and distance of $H$, so that $d_{H}(x,y)=|x-y|_{H}$ by Real Inner Product Spaces (iii), and we use the triangle inequality of The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity (i) and the symmetry of $d_{H}$ from the metric axioms without further mention. Every member of $\mathcal{S}$, being a viscosity subsolution of $F$ on $U$, is bounded above near each point of $U$, so its $\delta$-envelopes are defined for every real $\delta>0$.

Claim 1. Let $x\in U$ and let $c\in\mathbb{R}$ and the positive $r\in\mathbb{R}$ be as supplied for $x$ by the hypothesis The Pointwise Supremum of a Locally Uniformly Bounded Family of Viscosity Subsolutions on a Hilbert Triple is a Viscosity Subsolution (i). For $y\in U$ with $d_{H}(y,x)\le r$, the number $c$ is an upper bound of $\{v(y):v\in\mathcal{S}\}$, whose least upper bound is $u(y)$; hence $u(y)\le c$. Thus $c$ belongs to the set $A_{u}(x)$ of Upper and Lower Semicontinuous Envelopes of a Real-Valued Function, and as $x$ was arbitrary, $u$ is bounded above near each point of $U$ by Upper and Lower Semicontinuous Envelopes of a Real-Valued Function (i). For $v\in\mathcal{S}$ and $x\in U$, $v(x)$ belongs to $\{w(x):w\in\mathcal{S}\}$ and $u(x)$ is an upper bound of that set, so $v(x)\le u(x)$ by Upper Bound and Least Upper Bound. The last assertion is then claim 5 of Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity, applied to the pair $v$, $u$ with $\delta'=\delta$.

Claim 2. Let $\delta>0$, let $\varphi\in C^{2}(U)$, let $\hat{x}\in V\cap U$ be a point at which the function $V\cap U\to\mathbb{R}$ with value $u^{-}_{\delta}(x)-\varphi(x)$ at $x$ has a local maximum relative to $V\cap U$, and let $\varepsilon>0$. We must produce witnesses as in Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple (i).

Step 1: a closed ball on which the maximum is global. Since $U$ is open in $H$ there is a positive $\rho_{1}$ with $B_{d_{H}}(\hat{x},\rho_{1})\subseteq U$, by Open Subset of a Metric Space, and by Local Maximum of a Function Relative to a Subset of a Metric Space there is a positive $\delta_{1}$ such that every $x\in V\cap U$ with $d_{H}(\hat{x},x)<\delta_{1}$ satisfies $u^{-}_{\delta}(x)-\varphi(x)\le u^{-}_{\delta}(\hat{x})-\varphi(\hat{x})$. Put $r_{0}=\tfrac{1}{2}\min\{\rho_{1},\delta_{1}\}$, positive by claim 2 of Elementary Properties of the Minimum of Two Elements and claim 8 of Elementary Order Arithmetic in an Ordered Field, and satisfying $r_{0}<\rho_{1}$ and $r_{0}<\delta_{1}$ by claim 1 of the former and claim 2 of the latter. Let

$$K=\bar{B}_{d_{H}}(\hat{x},r_{0})=\{x'\in H:d_{H}(\hat{x},x')\le r_{0}\},\qquad A=V\cap K,\qquad m=u^{-}_{\delta}(\hat{x})-\varphi(\hat{x}).$$

Then $K\subseteq U$, because $d_{H}(\hat{x},x')\le r_{0}<\rho_{1}$ puts $x'$ in $B_{d_{H}}(\hat{x},\rho_{1})$; $K$ is closed in $H$ by claim 3 of Elementary Properties of the Closed Ball in a Metric Space; $\hat{x}\in K$ by claim 1 of that lemma, so $\hat{x}\in A$ and $A$ is nonempty; and

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

since $A\subseteq V\cap U$ and $d_{H}(\hat{x},x)\le r_{0}<\delta_{1}$ for $x\in A$.

Step 2: a radius on which $\varphi$ and its derivatives are nearly constant. The function $\varphi$ is continuous on $U$ by claim 2 of Basic Properties of Differentiability on an Open Subset of a Real Inner Product Space, and its gradient map and Hessian map are continuous on $U$, as maps into $(H,d_{H})$ and 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 (i) and The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space (ii). Applying Continuous Map Between Metric Spaces three times at $\hat{x}$, with the positive numbers $\tfrac{\varepsilon}{16}$, $\tfrac{\varepsilon}{8}$ and $\tfrac{\varepsilon}{8}$ respectively, and taking for $\sigma$ half the minimum of the three resulting radii together with $r_{0}$ and $\tfrac{\varepsilon}{2}$, we obtain a positive $\sigma$ with $\sigma\le r_{0}$ and $\sigma\le\tfrac{\varepsilon}{2}$ such that every $z\in U$ with $d_{H}(\hat{x},z)\le\sigma$ satisfies

$$|\varphi(z)-\varphi(\hat{x})|<\tfrac{\varepsilon}{16},\qquad |D\varphi(z)-D\varphi(\hat{x})|_{H}<\tfrac{\varepsilon}{8},\qquad \lVert D^{2}\varphi(z)-D^{2}\varphi(\hat{x})\rVert<\tfrac{\varepsilon}{8}.$$

Step 3: the parameters. Put

$$\mu=\tfrac{\varepsilon}{16},\qquad \lambda=\min\bigl\{\tfrac{1}{8},\tfrac{\sigma}{8}\bigr\},\qquad \beta=\min\bigl\{\tfrac{\mu\lambda^{2}}{4},\tfrac{\varepsilon}{32}\bigr\},\qquad \varepsilon'=\min\bigl\{\tfrac{\varepsilon}{8},\tfrac{\sigma}{4}\bigr\},$$

all positive by claim 2 of Elementary Properties of the Minimum of Two Elements. By claim 1 of that lemma and claim 5 of Elementary Arithmetic in an Ordered Field we record, for later use,

$$16\mu\lambda=\varepsilon\lambda\le\tfrac{\varepsilon}{8},\qquad 2\mu\lambda^{2}=\tfrac{\varepsilon}{8}\lambda^{2}\le\tfrac{\varepsilon}{8},\qquad 4\lambda\le\tfrac{\sigma}{2},\qquad 3\beta\le\mu\lambda^{2},\qquad 3\beta\le\tfrac{\varepsilon}{8},\qquad 2\mu=\tfrac{\varepsilon}{8},$$

using $\lambda\le\tfrac{1}{8}\le 1$ for the first two. Finally, by continuity of $\varphi$ at $\hat{x}$ again, choose a positive $\rho_{2}\le\tfrac{\sigma}{4}$ such that every $z\in U$ with $d_{H}(\hat{x},z)\le\rho_{2}$ satisfies $|\varphi(z)-\varphi(\hat{x})|<\beta$.

Step 4: an almost optimal member of the family. The function $V\cap U\to\mathbb{R}$ with value $u(x)-\delta h(x)$ at $x$ is bounded above near each point of $V\cap U$ by The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds (vi), and $u^{-}_{\delta}$ is its upper semicontinuous envelope by The $\delta$-Envelopes $u^-_\delta$ and $u^+_\delta$ of a Function on an Open Subset of a Hilbert Triple (i). By claim 5 of Properties of the Upper Semicontinuous Envelope, applied at $\hat{x}$ with the positive number $\min\{\rho_{2},\beta\}$, there is $z\in V\cap U$ with

$$d_{H}(z,\hat{x})\le\rho_{2}\qquad\text{and}\qquad \bigl|\bigl(u(z)-\delta h(z)\bigr)-u^{-}_{\delta}(\hat{x})\bigr|<\beta .$$

Since $u(z)$ is the least upper bound of $\{v(z):v\in\mathcal{S}\}$, claim 3 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$, applied with the positive number $\beta$, gives $v\in\mathcal{S}$ with $u(z)-\beta<v(z)$. By claim 1 of Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity, $v(z)-\delta h(z)\le v^{-}_{\delta}(z)$, and by claim 9 of Properties of the Absolute Value in an Ordered Field, $u^{-}_{\delta}(\hat{x})-\beta<u(z)-\delta h(z)$; hence

$$u^{-}_{\delta}(\hat{x})-2\beta<v^{-}_{\delta}(z).$$

Also $d_{H}(z,\hat{x})\le\rho_{2}\le\tfrac{\sigma}{4}\le r_{0}$, so $z\in A$, and $|\varphi(z)-\varphi(\hat{x})|<\beta$ by the choice of $\rho_{2}$.

Step 5: the perturbed maximum. Let $\Phi:A\to\mathbb{R}$ be given by $\Phi(x)=v^{-}_{\delta}(x)-\varphi(x)$; this is defined because $A\subseteq V\cap U$. By claim 1 of the present proposition and Step 1, $\Phi(x)\le u^{-}_{\delta}(x)-\varphi(x)\le m$ for every $x\in A$, so $\Phi$ is bounded above and $\sup_{x\in A}\Phi(x)\le m$. By claim 3 of The $\delta$-Envelopes on an Open Subset, under Penalisation of a Continuous Function, and on a Closed Subset, applied to $v$ with the closed set $K$ and the continuous function $\varphi$, $\Phi$ has closed superlevel sets in $H$. Moreover, by Step 4,

$$\Phi(z)=v^{-}_{\delta}(z)-\varphi(z)>\bigl(u^{-}_{\delta}(\hat{x})-2\beta\bigr)-\bigl(\varphi(\hat{x})+\beta\bigr)=m-3\beta ,$$

so that $\sup_{x\in A}\Phi(x)\le m<\Phi(z)+3\beta\le\Phi(z)+\mu\lambda^{2}$.

Applying A Perturbed Maximum Principle of Borwein-Preiss Type in a Real Hilbert Space to the real Hilbert space $H$, the nonempty set $A$, the function $\Phi$, the positive numbers $\mu$ and $\lambda$ and the point $x_{0}=z$, we obtain $\bar{y}\in H$ and $\bar{x}\in A$ such that, with $\Psi(x)=\Phi(x)-\mu\,|x-\bar{y}|_{H}^{2}$,

$$|\bar{x}-z|_{H}\le 4\lambda,\qquad |\bar{x}-\bar{y}|_{H}\le 8\lambda,\qquad \sup_{x\in A}\Phi(x)\le\Phi(\bar{x})+2\mu\lambda^{2},$$

and $\Psi$ attains a sequentially strict maximum on $A$ at $\bar{x}$. In particular $\Psi(x)\le\Psi(\bar{x})$ for every $x\in A$, by Sequentially Strict Maxima and Minima on a Subset of a Metric Space (i). Two consequences are recorded. First,

$$d_{H}(\hat{x},\bar{x})\le d_{H}(\hat{x},z)+d_{H}(z,\bar{x})\le\tfrac{\sigma}{4}+4\lambda\le\tfrac{\sigma}{4}+\tfrac{\sigma}{2}=\tfrac{3\sigma}{4}\le\sigma ,$$

so the three estimates of Step 2 apply at $\bar{x}$. Secondly, $\Phi(\bar{x})\ge\Phi(z)-2\mu\lambda^{2}>m-3\beta-2\mu\lambda^{2}$, whence

$$v^{-}_{\delta}(\bar{x})=\Phi(\bar{x})+\varphi(\bar{x})>u^{-}_{\delta}(\hat{x})+\bigl(\varphi(\bar{x})-\varphi(\hat{x})\bigr)-3\beta-2\mu\lambda^{2}>u^{-}_{\delta}(\hat{x})-\tfrac{\varepsilon}{16}-\tfrac{\varepsilon}{8}-\tfrac{\varepsilon}{8},$$

using $\varphi(\bar{x})-\varphi(\hat{x})>-\tfrac{\varepsilon}{16}$ from Step 2 and claim 3 of Properties of the Absolute Value in an Ordered Field. In the other direction $\bar{x}\in A$, so $v^{-}_{\delta}(\bar{x})\le u^{-}_{\delta}(\bar{x})\le m+\varphi(\bar{x})=u^{-}_{\delta}(\hat{x})+\bigl(\varphi(\bar{x})-\varphi(\hat{x})\bigr)<u^{-}_{\delta}(\hat{x})+\tfrac{\varepsilon}{16}$. Writing $\Delta=v^{-}_{\delta}(\bar{x})-u^{-}_{\delta}(\hat{x})$, we have shown

$$-\tfrac{5\varepsilon}{16}<\Delta<\tfrac{\varepsilon}{16}.$$

Step 6: testing $v$. Let $q_{0}:H\to\mathbb{R}$ be given by $q_{0}(x)=\tfrac{2\mu}{2}|x-\bar{y}|_{H}^{2}$. By claim 3 of Affine and Quadratic Functions on a Real Hilbert Space are of Class $C^2$, applied with $\alpha=2\mu$ and $y_{0}=\bar{y}$, and by claim 4 of that lemma, the restriction of $q_{0}$ to $U$ belongs to $C^{2}(U)$ with $Dq_{0}(x)=2\mu\,(x-\bar{y})$ and $D^{2}q_{0}(x)=2\mu I_{H}$. Hence, by claim 2 of Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space, the function $\psi=\varphi+q_{0}$ on $U$ belongs to $C^{2}(U)$, with

$$D\psi(x)=D\varphi(x)+2\mu\,(x-\bar{y}),\qquad D^{2}\psi(x)=D^{2}\varphi(x)+2\mu I_{H}\qquad (x\in U).$$

For $x\in A$ we have $\Psi(x)=v^{-}_{\delta}(x)-\psi(x)$. We claim that the function $V\cap U\to\mathbb{R}$ with value $v^{-}_{\delta}(x)-\psi(x)$ at $x$ has a local maximum at $\bar{x}$ relative to $V\cap U$. Indeed, let $x\in V\cap U$ satisfy $d_{H}(\bar{x},x)<\tfrac{r_{0}}{4}$. By Step 5 and $\sigma\le r_{0}$ we have $d_{H}(\hat{x},\bar{x})\le\tfrac{3\sigma}{4}\le\tfrac{3r_{0}}{4}$, so

$$d_{H}(\hat{x},x)\le d_{H}(\hat{x},\bar{x})+d_{H}(\bar{x},x)<\tfrac{3r_{0}}{4}+\tfrac{r_{0}}{4}=r_{0};$$

hence $x\in K$ and so $x\in A$, and therefore $v^{-}_{\delta}(x)-\psi(x)=\Psi(x)\le\Psi(\bar{x})=v^{-}_{\delta}(\bar{x})-\psi(\bar{x})$. Thus $\tfrac{r_{0}}{4}$ is a witnessing radius in Local Maximum of a Function Relative to a Subset of a Metric Space.

Since $v$ is a viscosity subsolution of $F$ on $U$, Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple (i), applied with $\delta$, the test function $\psi$, the point $\bar{x}$ and the tolerance $\varepsilon'$, yields $y\in W$, $s\in\mathbb{R}$, $q\in H$ and $Y\in\mathrm{Sym}(H)$ with

$$|y-\bar{x}|_{H}<\varepsilon',\quad |v^{-}_{\delta}(y)-v^{-}_{\delta}(\bar{x})|<\varepsilon',\quad |s-v^{-}_{\delta}(\bar{x})|<\varepsilon',$$ $$|q-D\psi(\bar{x})|_{H}<\varepsilon',\quad \lVert Y-D^{2}\psi(\bar{x})\rVert<\varepsilon',\quad F^{-}_{\delta}(y,s,q,Y)\le\varepsilon' .$$

Step 7: these are witnesses for $u$. We verify the six conditions of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple (i) for $u$, $\delta$, $\varphi$, $\hat{x}$ and $\varepsilon$, with the same $y,s,q,Y$. Note first that

$$d_{H}(\hat{x},y)\le d_{H}(\hat{x},\bar{x})+d_{H}(\bar{x},y)\le\tfrac{3\sigma}{4}+\varepsilon'\le\tfrac{3\sigma}{4}+\tfrac{\sigma}{4}=\sigma ,$$

so the estimates of Step 2 apply at $y$ as well; moreover $y\in W\subseteq V\cap U$ and $d_{H}(\hat{x},y)\le\sigma\le r_{0}$, so $y\in A$.

(i) $|y-\hat{x}|_{H}=d_{H}(\hat{x},y)\le\sigma\le\tfrac{\varepsilon}{2}<\varepsilon$ by claim 8 of Elementary Order Arithmetic in an Ordered Field.

(ii) Since $y\in A$, Step 1 gives $u^{-}_{\delta}(y)-u^{-}_{\delta}(\hat{x})\le\varphi(y)-\varphi(\hat{x})<\tfrac{\varepsilon}{16}$. In the other direction, claim 1 of the present proposition and Step 6 give

$$u^{-}_{\delta}(y)\ge v^{-}_{\delta}(y)>v^{-}_{\delta}(\bar{x})-\varepsilon'=u^{-}_{\delta}(\hat{x})+\Delta-\varepsilon'>u^{-}_{\delta}(\hat{x})-\tfrac{5\varepsilon}{16}-\tfrac{\varepsilon}{8}.$$

Since $\tfrac{5\varepsilon}{16}+\tfrac{\varepsilon}{8}=\tfrac{7\varepsilon}{16}<\varepsilon$ and $\tfrac{\varepsilon}{16}<\varepsilon$, claim 9 of Properties of the Absolute Value in an Ordered Field gives $|u^{-}_{\delta}(y)-u^{-}_{\delta}(\hat{x})|<\varepsilon$.

(iii) $|s-u^{-}_{\delta}(\hat{x})|\le|s-v^{-}_{\delta}(\bar{x})|+|\Delta|<\varepsilon'+\tfrac{5\varepsilon}{16}\le\tfrac{\varepsilon}{8}+\tfrac{5\varepsilon}{16}=\tfrac{7\varepsilon}{16}<\varepsilon$, by claim 5 of Properties of the Absolute Value in an Ordered Field and the bounds on $\Delta$ from Step 5.

(iv) By the triangle inequality, Step 6 and Step 2,

$$|q-D\varphi(\hat{x})|_{H}\le|q-D\psi(\bar{x})|_{H}+2\mu\,|\bar{x}-\bar{y}|_{H}+|D\varphi(\bar{x})-D\varphi(\hat{x})|_{H}<\varepsilon'+16\mu\lambda+\tfrac{\varepsilon}{8}\le\tfrac{\varepsilon}{8}+\tfrac{\varepsilon}{8}+\tfrac{\varepsilon}{8}<\varepsilon ,$$

using $D\psi(\bar{x})-D\varphi(\bar{x})=2\mu(\bar{x}-\bar{y})$ and $|\bar{x}-\bar{y}|_{H}\le 8\lambda$.

(v) Likewise, by claim 3 of Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity,

$$\lVert Y-D^{2}\varphi(\hat{x})\rVert\le\lVert Y-D^{2}\psi(\bar{x})\rVert+\lVert 2\mu I_{H}\rVert+\lVert D^{2}\varphi(\bar{x})-D^{2}\varphi(\hat{x})\rVert<\varepsilon'+2\mu+\tfrac{\varepsilon}{8}\le\tfrac{3\varepsilon}{8}<\varepsilon ,$$

where $\lVert 2\mu I_{H}\rVert\le 2\mu$ because $|2\mu\langle x',y'\rangle_{H}|\le 2\mu|x'|_{H}|y'|_{H}$ for all $x',y'\in H$ by The Cauchy–Schwarz Inequality in a Real Inner Product Space and claim 4 of Properties of the Absolute Value in an Ordered Field, so that claim 2 of Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity applies.

(vi) $F^{-}_{\delta}(y,s,q,Y)\le\varepsilon'\le\varepsilon$.

As $\delta$, $\varphi$, $\hat{x}$ and $\varepsilon$ were arbitrary, and $u$ is bounded above near each point of $U$ by claim 1, the function $u$ is a viscosity subsolution of $F$ on $U$.