Proof of The Bump Construction on a Hilbert Triple: the Maximum of a Viscosity Subsolution and a Penalised $C^2$ Function

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 $d_{H}(x,y)=|x-y|_{H}$, as in Real Inner Product Spaces (iii). Since $0<\mu$ and $0<\delta$ we have $0<\mu+\delta$ by claim 3 of Elementary Order Arithmetic in an Ordered Field. Being a viscosity subsolution of $F$ on $U$, the function $v$ is bounded above near each point of $U$, so $v^{-}_{\delta}$ is defined for every real $\delta>0$; and $\psi$, belonging to $C^{2}(U)$, is continuous on $U$ by claim 2 of Basic Properties of Differentiability on an Open Subset of a Real Inner Product Space, hence bounded above and bounded below near each point of $U$ by claim 2 of The $\delta$-Envelopes on an Open Subset, under Penalisation of a Continuous Function, and on a Closed Subset.

We use twice the following elementary identity: for $a,b,c\in\mathbb{R}$,

$$\max\{a,b\}-c=\max\{a-c,\,b-c\}.$$

Indeed, by claim 3 of Elementary Arithmetic in an Ordered Field, $a\le b$ holds if and only if $a-c\le b-c$, because $(b-c)-(a-c)=b-a$; so the two applications of Maximum of Two Elements of a Totally Ordered Set select corresponding branches, and the values agree.

Claim 1. Let $x\in U$. If $x\in V\cap U$, then $v(x)\le\max\{\psi(x)-\mu h(x),v(x)\}=w(x)$ by claim 1 of Elementary Properties of the Maximum of Two Elements; if $x\in U\setminus V$, then $w(x)=v(x)$. In both cases $v(x)\le w(x)$.

For the local bound, let $x\in U$. By Upper and Lower Semicontinuous Envelopes of a Real-Valued Function (i) there are $c_{1},c_{2}\in\mathbb{R}$ and positive $r_{1},r_{2}\in\mathbb{R}$ with $\psi(y)\le c_{1}$ for every $y\in U$ with $d_{H}(y,x)\le r_{1}$, and $v(y)\le c_{2}$ for every $y\in U$ with $d_{H}(y,x)\le r_{2}$. Put $c=\max\{c_{1},c_{2}\}$ and $r=\min\{r_{1},r_{2}\}$, the latter positive by claim 2 of Elementary Properties of the Minimum of Two Elements; by claim 1 of that lemma and claim 1 of Elementary Properties of the Maximum of Two Elements, together with transitivity of the order, every $y\in U$ with $d_{H}(y,x)\le r$ satisfies $\psi(y)\le c$ and $v(y)\le c$. Let such a $y$ be given. If $y\notin V$ then $w(y)=v(y)\le c$. If $y\in V$ then $0\le h(y)$ by claim 1 of The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds, so $0\le\mu h(y)$ by claim 5 of Elementary Arithmetic in an Ordered Field, whence $\psi(y)-\mu h(y)\le\psi(y)\le c$ by claim 3 of Elementary Arithmetic in an Ordered Field, and therefore $w(y)\le c$ by claim 3 of Elementary Properties of the Maximum of Two Elements. Thus $c\in A_{w}(x)$, and $w$ is bounded above near each point of $U$.

Claim 2. Let $\delta>0$ and write $P,Q:V\cap U\to\mathbb{R}$ for the functions with values $P(x)=\psi(x)-(\mu+\delta)h(x)$ and $Q(x)=v(x)-\delta h(x)$. For $x\in V\cap U$, the identity above gives

$$w(x)-\delta h(x)=\max\{\psi(x)-\mu h(x),v(x)\}-\delta h(x)=\max\{P(x),Q(x)\},$$

since $(\psi(x)-\mu h(x))-\delta h(x)=\psi(x)-(\mu+\delta)h(x)$. So the function $V\cap U\to\mathbb{R}$ with value $w(x)-\delta h(x)$ at $x$ is $P\vee Q$ in the notation of The Semicontinuous Envelopes of a Pointwise Maximum and of a Pointwise Minimum.

By claim 2 of The $\delta$-Envelopes on an Open Subset, under Penalisation of a Continuous Function, and on a Closed Subset, applied to the continuous function $\psi$ with $\mu+\delta$ in the role of its $\lambda$, the function $P$ is bounded above near each point of $V\cap U$ and $P^{*}=P$. By The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds (vi), $Q$ is bounded above near each point of $V\cap U$, and $Q^{*}=v^{-}_{\delta}$ by The $\delta$-Envelopes $u^-_\delta$ and $u^+_\delta$ of a Function on an Open Subset of a Hilbert Triple (i). Claim 2 of The Semicontinuous Envelopes of a Pointwise Maximum and of a Pointwise Minimum, applied in $(H,d_{H})$ with the nonempty set $V\cap U$ and the pair $P$, $Q$, therefore gives, for every $x\in V\cap U$,

$$w^{-}_{\delta}(x)=(P\vee Q)^{*}(x)=\max\{P^{*}(x),Q^{*}(x)\}=\max\bigl\{\psi(x)-(\mu+\delta)h(x),\,v^{-}_{\delta}(x)\bigr\},$$

the first equality by The $\delta$-Envelopes $u^-_\delta$ and $u^+_\delta$ of a Function on an Open Subset of a Hilbert Triple (i) applied to $w$, which is legitimate by claim 1.

Claim 3. 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 $w^{-}_{\delta}(x)-\varphi(x)$ at $x$ has a local maximum relative to $V\cap U$, and let $\varepsilon>0$. Let $\tau$ be a positive real number witnessing this local maximum as in Local Maximum of a Function Relative to a Subset of a Metric Space, so that

$$w^{-}_{\delta}(x)-\varphi(x)\le w^{-}_{\delta}(\hat{x})-\varphi(\hat{x})\qquad\text{for every }x\in V\cap U\text{ with }d_{H}(\hat{x},x)<\tau .$$

Keep the notation $P$ of claim 2, so that $w^{-}_{\delta}=\max\{P,v^{-}_{\delta}\}$ pointwise on $V\cap U$; in particular $P(x)\le w^{-}_{\delta}(x)$ and $v^{-}_{\delta}(x)\le w^{-}_{\delta}(x)$ for every $x\in V\cap U$, by claim 1 of Elementary Properties of the Maximum of Two Elements. Since $v^{-}_{\delta}(\hat{x})\le w^{-}_{\delta}(\hat{x})$, exactly one of the following two cases occurs.

Case 1: $w^{-}_{\delta}(\hat{x})=v^{-}_{\delta}(\hat{x})$. For $x\in V\cap U$ with $d_{H}(\hat{x},x)<\tau$,

$$v^{-}_{\delta}(x)-\varphi(x)\le w^{-}_{\delta}(x)-\varphi(x)\le w^{-}_{\delta}(\hat{x})-\varphi(\hat{x})=v^{-}_{\delta}(\hat{x})-\varphi(\hat{x}),$$

so the function with value $v^{-}_{\delta}(x)-\varphi(x)$ at $x$ has a local maximum at $\hat{x}$ relative to $V\cap U$, with the same witnessing radius $\tau$.

Since $\varphi$ is continuous at $\hat{x}$ relative to $U$, Continuous Map Between Metric Spaces applied with the positive number $\tfrac{\varepsilon}{2}$ provides a positive $\sigma_{0}$ such that every $z\in U$ with $d_{H}(\hat{x},z)<\sigma_{0}$ satisfies $|\varphi(z)-\varphi(\hat{x})|<\tfrac{\varepsilon}{2}$. Put $\varepsilon''=\min\{\varepsilon,\sigma_{0},\tau\}$, positive by claim 2 of Elementary Properties of the Minimum of Two Elements. Applying Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple (i) to the viscosity subsolution $v$, with $\delta$, $\varphi$, $\hat{x}$ and the tolerance $\varepsilon''$, we obtain $y\in W$, $s\in\mathbb{R}$, $q\in H$ and $Y\in\mathrm{Sym}(H)$ with

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

All six conditions required of $y,s,q,Y$ for $w$ at tolerance $\varepsilon$ follow, since $\varepsilon''\le\varepsilon$ and $w^{-}_{\delta}(\hat{x})=v^{-}_{\delta}(\hat{x})$, except for the bound on $|w^{-}_{\delta}(y)-w^{-}_{\delta}(\hat{x})|$, which we check now. On the one hand $y\in W\subseteq V\cap U$ and $d_{H}(\hat{x},y)=|y-\hat{x}|_{H}<\varepsilon''\le\tau$, so

$$w^{-}_{\delta}(y)-w^{-}_{\delta}(\hat{x})\le\varphi(y)-\varphi(\hat{x})<\tfrac{\varepsilon}{2}<\varepsilon$$

by the local maximum, by $d_{H}(\hat{x},y)<\varepsilon''\le\sigma_{0}$ and claim 3 of Properties of the Absolute Value in an Ordered Field, and by claim 8 of Elementary Order Arithmetic in an Ordered Field. On the other hand

$$w^{-}_{\delta}(y)\ge v^{-}_{\delta}(y)>v^{-}_{\delta}(\hat{x})-\varepsilon''\ge w^{-}_{\delta}(\hat{x})-\varepsilon .$$

Claim 9 of Properties of the Absolute Value in an Ordered Field now gives $|w^{-}_{\delta}(y)-w^{-}_{\delta}(\hat{x})|<\varepsilon$.

Case 2: $v^{-}_{\delta}(\hat{x})<w^{-}_{\delta}(\hat{x})$. By claim 2 of Elementary Properties of the Maximum of Two Elements, $w^{-}_{\delta}(\hat{x})$ is $P(\hat{x})$ or $v^{-}_{\delta}(\hat{x})$; the second is excluded, so

$$w^{-}_{\delta}(\hat{x})=P(\hat{x})=\psi(\hat{x})-(\mu+\delta)h(\hat{x}).$$

For $x\in V\cap U$ with $d_{H}(\hat{x},x)<\tau$ we therefore have

$$P(x)-\varphi(x)\le w^{-}_{\delta}(x)-\varphi(x)\le w^{-}_{\delta}(\hat{x})-\varphi(\hat{x})=P(\hat{x})-\varphi(\hat{x}),$$

so the function $V\cap U\to\mathbb{R}$ with value $(\psi-\varphi)(x)-(\mu+\delta)h(x)$ at $x$ has a local maximum at $\hat{x}$ relative to $V\cap U$. The function $\psi-\varphi$ belongs to $C^{2}(U)$ by claim 4 of Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space, with $D(\psi-\varphi)(x)=D\psi(x)-D\varphi(x)$ and $D^{2}(\psi-\varphi)(x)=D^{2}\psi(x)-D^{2}\varphi(x)$. Claim 1 of First- and Second-Order Conditions at a Local Extremum of a $C^2$ Function Penalised by $h$ on the Small Space, applied to $\psi-\varphi$ with $\mu+\delta$ in the role of its $\lambda$, therefore gives $\hat{x}\in W$ together with

$$(\mu+\delta)A\hat{x}=D\psi(\hat{x})-D\varphi(\hat{x}),\qquad \bigl(D^{2}\psi(\hat{x})-D^{2}\varphi(\hat{x})\bigr)\big|_{V}\preceq(\mu+\delta)I_{V}.$$

Next we check the hypothesis The Bump Construction on a Hilbert Triple: the Maximum of a Viscosity Subsolution and a Penalised $C^2$ Function (i) at $\hat{x}$. By claim 1 of Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity, $v(\hat{x})-\delta h(\hat{x})\le v^{-}_{\delta}(\hat{x})<\psi(\hat{x})-(\mu+\delta)h(\hat{x})$, so

$$v(\hat{x})<\psi(\hat{x})-(\mu+\delta)h(\hat{x})+\delta h(\hat{x})=\psi(\hat{x})-\mu h(\hat{x}).$$

Since $\hat{x}\in W$, the hypothesis gives $F^{+}_{\mu}\bigl(\hat{x},\psi(\hat{x}),D\psi(\hat{x}),D^{2}\psi(\hat{x})\bigr)\le 0$, that is, by Second-Order Equation Operator on an Open Subset of a Hilbert Triple and Its $\delta$-Shifts (ii),

$$F\bigl(\hat{x},\ \psi(\hat{x})-\mu h(\hat{x}),\ D\psi(\hat{x})-\mu A\hat{x},\ D^{2}\psi(\hat{x})|_{V}-\mu I_{V}\bigr)\le 0 .$$

We now show that $y=\hat{x}$, $s=w^{-}_{\delta}(\hat{x})$, $q=D\varphi(\hat{x})$ and $Y=D^{2}\varphi(\hat{x})$ are witnesses. By Second-Order Equation Operator on an Open Subset of a Hilbert Triple and Its $\delta$-Shifts (ii),

$$F^{-}_{\delta}\bigl(\hat{x},w^{-}_{\delta}(\hat{x}),D\varphi(\hat{x}),D^{2}\varphi(\hat{x})\bigr)=F\bigl(\hat{x},\ w^{-}_{\delta}(\hat{x})+\delta h(\hat{x}),\ D\varphi(\hat{x})+\delta A\hat{x},\ D^{2}\varphi(\hat{x})|_{V}+\delta I_{V}\bigr).$$

Its first two arguments after $\hat{x}$ coincide with those of the previous display:

$$w^{-}_{\delta}(\hat{x})+\delta h(\hat{x})=\psi(\hat{x})-(\mu+\delta)h(\hat{x})+\delta h(\hat{x})=\psi(\hat{x})-\mu h(\hat{x}),$$ $$D\varphi(\hat{x})+\delta A\hat{x}=D\psi(\hat{x})-(\mu+\delta)A\hat{x}+\delta A\hat{x}=D\psi(\hat{x})-\mu A\hat{x}.$$

As for the forms, claim 10 of Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity gives $\bigl(D^{2}\psi(\hat{x})-D^{2}\varphi(\hat{x})\bigr)|_{V}=D^{2}\psi(\hat{x})|_{V}-D^{2}\varphi(\hat{x})|_{V}$, so the order relation above reads $D^{2}\psi(\hat{x})|_{V}-D^{2}\varphi(\hat{x})|_{V}\preceq(\mu+\delta)I_{V}$. Adding the form $D^{2}\varphi(\hat{x})|_{V}-\mu I_{V}$ to both sides, which preserves the order by claim 8 of Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity, and using $(\mu+\delta)I_{V}-\mu I_{V}=\delta I_{V}$, which holds in the vector space $\mathrm{Sym}(V)$ by Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (iv) and claim 1 of Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity, we obtain

$$D^{2}\psi(\hat{x})|_{V}-\mu I_{V}\preceq D^{2}\varphi(\hat{x})|_{V}+\delta I_{V}.$$

Since $F$ is degenerate elliptic, Degenerate Elliptic Second-Order Equation Operator on a Hilbert Triple (i), applied at the point $\hat{x}\in W$ with the real number $\psi(\hat{x})-\mu h(\hat{x})$, the vector $D\psi(\hat{x})-\mu A\hat{x}$ and this pair of forms, gives

$$F^{-}_{\delta}\bigl(\hat{x},w^{-}_{\delta}(\hat{x}),D\varphi(\hat{x}),D^{2}\varphi(\hat{x})\bigr)\le F\bigl(\hat{x},\psi(\hat{x})-\mu h(\hat{x}),D\psi(\hat{x})-\mu A\hat{x},D^{2}\psi(\hat{x})|_{V}-\mu I_{V}\bigr)\le 0\le\varepsilon .$$

The remaining five conditions hold because the corresponding differences vanish: $|\hat{x}-\hat{x}|_{H}=0<\varepsilon$, $|w^{-}_{\delta}(\hat{x})-w^{-}_{\delta}(\hat{x})|=0<\varepsilon$, $|s-w^{-}_{\delta}(\hat{x})|=0<\varepsilon$, $|q-D\varphi(\hat{x})|_{H}=0<\varepsilon$ and $\lVert Y-D^{2}\varphi(\hat{x})\rVert=0<\varepsilon$, the last because the difference is the zero form, whose norm is $0$ by claim 3 of Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity.

In both cases witnesses exist for the given data. As $\delta$, $\varphi$, $\hat{x}$ and $\varepsilon$ were arbitrary, and $w$ is bounded above near each point of $U$ by claim 1, the function $w$ is a viscosity subsolution of $F$ on $U$.