Proof of Restriction of the Equation and the Locality of the Viscosity Sub- and Supersolution Properties on a Hilbert Triple

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. Throughout, $U'$ denotes a nonempty subset of $U$ open in $H$, and $W'=D(A)\cap U'$. Recall that $d_{H}(x,y)=|x-y|_{H}$ and that $d_{H}$ is symmetric, by Real Inner Product Spaces (iii) and the metric axioms.

Claim 1. Since $U'\subseteq U$ we have $W'=D(A)\cap U'\subseteq D(A)\cap U=W$. The function $F|_{U'}$ maps $W'\times\mathbb{R}\times H\times\mathrm{Sym}(V)$ into $\mathbb{R}$, so it is a second-order equation operator on $U'$ relative to $(H,V,A)$ by Second-Order Equation Operator on an Open Subset of a Hilbert Triple and Its $\delta$-Shifts (i), $U'$ being nonempty and open in $H$.

Let $\delta>0$ and $(x,r,p,Y)\in W'\times\mathbb{R}\times H\times\mathrm{Sym}(H)$. Then $x\in W$, so both sides below are defined, and by Second-Order Equation Operator on an Open Subset of a Hilbert Triple and Its $\delta$-Shifts (ii), applied first to $F|_{U'}$ and then to $F$,

$$\bigl(F|_{U'}\bigr)^{-}_{\delta}(x,r,p,Y)=F|_{U'}\bigl(x,r+\delta h(x),p+\delta Ax,Y|_{V}+\delta I_{V}\bigr)=F\bigl(x,r+\delta h(x),p+\delta Ax,Y|_{V}+\delta I_{V}\bigr)=F^{-}_{\delta}(x,r,p,Y),$$

the middle equality holding because $F|_{U'}$ takes the value of $F$ at every point of its domain. The identity for the $+$ shifts is obtained in the same way. Finally, if $F$ is degenerate elliptic and $x\in W'$, $r\in\mathbb{R}$, $p\in H$ and $X,Y\in\mathrm{Sym}(V)$ satisfy $X\preceq Y$, then $x\in W$, so $F(x,r,p,Y)\le F(x,r,p,X)$ by Degenerate Elliptic Second-Order Equation Operator on a Hilbert Triple (i), and these are the values of $F|_{U'}$ at the same data; hence $F|_{U'}$ is degenerate elliptic.

Claim 2. Suppose first that $u$ is a viscosity subsolution of $F$ on $U$; in particular $u$ is bounded above near each point of $U$, so by claim 1 of The $\delta$-Envelopes on an Open Subset, under Penalisation of a Continuous Function, and on a Closed Subset the function $u|_{U'}$ is bounded above near each point of $U'$ and

$$\bigl(u|_{U'}\bigr)^{-}_{\delta}(x)=u^{-}_{\delta}(x)\qquad\text{for every }\delta>0\text{ and every }x\in V\cap U' .$$

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 $\bigl(u|_{U'}\bigr)^{-}_{\delta}(x)-\varphi'(x)$ at $x$ has a local maximum relative to $V\cap U'$, and let $\varepsilon>0$. Since $U'$ is open in $H$ and $\hat{x}\in U'$, there is a positive $\rho\in\mathbb{R}$ with $B_{d_{H}}(\hat{x},\rho)\subseteq U'$, by Open Subset of a Metric Space. Put

$$\eta=\tfrac{\varepsilon}{4},\qquad \varepsilon'=\min\bigl\{\tfrac{\varepsilon}{2},\rho\bigr\},$$

both positive, with $\varepsilon'\le\tfrac{\varepsilon}{2}$ and $\varepsilon'\le\rho$ by claims 1 and 2 of Elementary Properties of the Minimum of Two Elements, and $2\eta=\tfrac{\varepsilon}{2}$.

Let $T:H\to\mathbb{R}$ be the function of Quadratic Test Functions: a Global $C^2$ Majorant and Minorant Matching a $C^2$ Function to Second Order at a Point formed from the open set $U'$, the function $\varphi'$, the point $\hat{x}$ and the number $\eta$. By claim 1 of that lemma, the restriction of $T$ to $U$ belongs to $C^{2}(U)$, with $DT(\hat{x})=D\varphi'(\hat{x})$ and $\lVert D^{2}T(\hat{x})-D^{2}\varphi'(\hat{x})\rVert\le 2\eta$; here $\hat{x}\in U'\subseteq U$ and the gradients and Hessians of the restriction agree with those of $T$.

By claim 4 of Quadratic Test Functions: a Global $C^2$ Majorant and Minorant Matching a $C^2$ Function to Second Order at a Point, applied with the set $A=V\cap U'$, which is contained in $U'$ and contains $\hat{x}$, and with $g=\bigl(u|_{U'}\bigr)^{-}_{\delta}$, the function $V\cap U'\to\mathbb{R}$ with value $u^{-}_{\delta}(x)-T(x)$ at $x$ has a local maximum at $\hat{x}$ relative to $V\cap U'$. By claim 4 of Local Bounds, Semicontinuous Envelopes and Local Extrema Are Unchanged on the Trace of an Open Set, applied in $(H,d_{H})$ with $S=V\cap U$, $O=U'$, so that $S\cap O=V\cap U'$, and with the function on $V\cap U$ whose value at $x$ is $u^{-}_{\delta}(x)-T(x)$, that function has a local maximum at $\hat{x}$ relative to $V\cap U$.

Since $u$ 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 $T$ restricted to $U$, the point $\hat{x}$ and the tolerance $\varepsilon'$, yields $y\in W$, $s\in\mathbb{R}$, $q\in H$ and $Y\in\mathrm{Sym}(H)$ with

$$|y-\hat{x}|_{H}<\varepsilon',\quad |u^{-}_{\delta}(y)-u^{-}_{\delta}(\hat{x})|<\varepsilon',\quad |s-u^{-}_{\delta}(\hat{x})|<\varepsilon',$$ $$|q-DT(\hat{x})|_{H}<\varepsilon',\quad \lVert Y-D^{2}T(\hat{x})\rVert<\varepsilon',\quad F^{-}_{\delta}(y,s,q,Y)\le\varepsilon' .$$

From $d_{H}(\hat{x},y)=|y-\hat{x}|_{H}<\varepsilon'\le\rho$ and claim 2 of Elementary Order Arithmetic in an Ordered Field we get $y\in B_{d_{H}}(\hat{x},\rho)\subseteq U'$, so $y\in W'$ and in particular $y\in V\cap U'$; hence $\bigl(u|_{U'}\bigr)^{-}_{\delta}(y)=u^{-}_{\delta}(y)$ and $\bigl(u|_{U'}\bigr)^{-}_{\delta}(\hat{x})=u^{-}_{\delta}(\hat{x})$. Since $DT(\hat{x})=D\varphi'(\hat{x})$ and

$$\lVert Y-D^{2}\varphi'(\hat{x})\rVert\le\lVert Y-D^{2}T(\hat{x})\rVert+\lVert D^{2}T(\hat{x})-D^{2}\varphi'(\hat{x})\rVert<\varepsilon'+2\eta\le\tfrac{\varepsilon}{2}+\tfrac{\varepsilon}{2}=\varepsilon$$

by claim 3 of Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity and claim 3 of Elementary Order Arithmetic in an Ordered Field, and since $\bigl(F|_{U'}\bigr)^{-}_{\delta}(y,s,q,Y)=F^{-}_{\delta}(y,s,q,Y)\le\varepsilon'\le\varepsilon$ by claim 1, the six conditions of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple (i) hold for $u|_{U'}$, $F|_{U'}$, $\delta$, $\varphi'$, $\hat{x}$ and $\varepsilon$, all the remaining ones because $\varepsilon'\le\varepsilon$. As the data were arbitrary, $u|_{U'}$ is a viscosity subsolution of $F|_{U'}$ on $U'$.

Now suppose $u$ is a viscosity supersolution of $F$ on $U$, and let $\tilde{F}$ be the operator of Sign Reversal for Second-Order Equations on a Hilbert Triple: Subsolutions of $F$ are Supersolutions of $\tilde F$. By claim 5 of that lemma, $-u$ is a viscosity subsolution of $\tilde{F}$ on $U$, so by the case just proved $(-u)|_{U'}$ is a viscosity subsolution of $\tilde{F}|_{U'}$ on $U'$. The operator obtained from $F|_{U'}$ by the construction of that lemma is $\tilde{F}|_{U'}$, since both are the function on $W'\times\mathbb{R}\times H\times\mathrm{Sym}(V)$ with value $-F(x,-r,-p,-X)$ at $(x,r,p,X)$, and $(-u)|_{U'}$ is the function $-\bigl(u|_{U'}\bigr)$. Applying claim 5 of Sign Reversal for Second-Order Equations on a Hilbert Triple: Subsolutions of $F$ are Supersolutions of $\tilde F$ to the operator $F|_{U'}$ on $U'$ and the function $u|_{U'}$ therefore shows that $u|_{U'}$ is a viscosity supersolution of $F|_{U'}$ on $U'$.

Claim 3. Assume the hypothesis for subsolutions and let $x\in U$, with $U_{x}$ as given.

Local bounds. Being a viscosity subsolution of $F|_{U_{x}}$ on $U_{x}$, the function $u|_{U_{x}}$ is bounded above near each point of $U_{x}$, so by Upper and Lower Semicontinuous Envelopes of a Real-Valued Function (i) there are $c\in\mathbb{R}$ and a positive $r_{0}\in\mathbb{R}$ with $u(z)\le c$ for every $z\in U_{x}$ with $d_{H}(z,x)\le r_{0}$. Since $U_{x}$ is open in $H$ and contains $x$, there is a positive $\rho$ with $B_{d_{H}}(x,\rho)\subseteq U_{x}$. Put $r=\min\{r_{0},\tfrac{\rho}{2}\}$, positive by claim 2 of Elementary Properties of the Minimum of Two Elements, with $r\le r_{0}$ and $r\le\tfrac{\rho}{2}<\rho$ by claim 1 of that lemma and claim 8 of Elementary Order Arithmetic in an Ordered Field. Every $z\in U$ with $d_{H}(z,x)\le r$ satisfies $d_{H}(x,z)=d_{H}(z,x)\le r<\rho$, hence $z\in B_{d_{H}}(x,\rho)\subseteq U_{x}$ by claim 2 of Elementary Order Arithmetic in an Ordered Field, and $d_{H}(z,x)\le r_{0}$, hence $u(z)\le c$. So $c\in A_{u}(x)$ and, $x$ being arbitrary, $u$ is bounded above near each point of $U$.

The subsolution property. 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 $\varepsilon>0$. Put $U'=U_{\hat{x}}$, so that $\hat{x}\in U'$ and $W'=D(A)\cap U'\subseteq W$. By claim 1 of The $\delta$-Envelopes on an Open Subset, under Penalisation of a Continuous Function, and on a Closed Subset, $\bigl(u|_{U'}\bigr)^{-}_{\delta}=u^{-}_{\delta}$ on $V\cap U'$. By claim 4 of Basic Properties of Differentiability on an Open Subset of a Real Inner Product Space, the restriction $\varphi|_{U'}$ belongs to $C^{2}(U')$ and has the gradient $D\varphi(z)$ and the Hessian $D^{2}\varphi(z)$ at every $z\in U'$. By claim 4 of Local Bounds, Semicontinuous Envelopes and Local Extrema Are Unchanged on the Trace of an Open Set, applied with $S=V\cap U$, $O=U'$ and the function with value $u^{-}_{\delta}(x)-\varphi(x)$ at $x$, the restriction of that function to $V\cap U'$ has a local maximum at $\hat{x}$ relative to $V\cap U'$; and that restriction is the function with value $\bigl(u|_{U'}\bigr)^{-}_{\delta}(x)-\varphi|_{U'}(x)$ at $x$.

Applying Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple (i) to the viscosity subsolution $u|_{U'}$ of $F|_{U'}$ on $U'$, with $\delta$, $\varphi|_{U'}$, $\hat{x}$ and $\varepsilon$, we obtain $y\in W'$, $s\in\mathbb{R}$, $q\in H$ and $Y\in\mathrm{Sym}(H)$ satisfying the six conditions there. Now $y\in W'\subseteq W$ and $y,\hat{x}\in V\cap U'$, so $\bigl(u|_{U'}\bigr)^{-}_{\delta}$ may be replaced by $u^{-}_{\delta}$ at both points; $D\varphi|_{U'}(\hat{x})=D\varphi(\hat{x})$ and $D^{2}\varphi|_{U'}(\hat{x})=D^{2}\varphi(\hat{x})$; and $F^{-}_{\delta}(y,s,q,Y)=\bigl(F|_{U'}\bigr)^{-}_{\delta}(y,s,q,Y)\le\varepsilon$ by claim 1. Hence the six conditions of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple (i) hold for $u$, $F$, $\delta$, $\varphi$, $\hat{x}$ and $\varepsilon$, and $u$ is a viscosity subsolution of $F$ on $U$.

Finally, assume the hypothesis for supersolutions, and let $\tilde{F}$ be as in Sign Reversal for Second-Order Equations on a Hilbert Triple: Subsolutions of $F$ are Supersolutions of $\tilde F$. For each $x\in U$, claim 5 of that lemma applied to $F|_{U_{x}}$ on $U_{x}$ shows that $-\bigl(u|_{U_{x}}\bigr)=(-u)|_{U_{x}}$ is a viscosity subsolution of the operator obtained from $F|_{U_{x}}$ by that construction, which as above is $\tilde{F}|_{U_{x}}$. By the part of this claim already proved, applied to $\tilde{F}$ and $-u$, the function $-u$ is bounded above near each point of $U$ and is a viscosity subsolution of $\tilde{F}$ on $U$. By Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity (ii), $u$ is then bounded below near each point of $U$, and by claim 5 of Sign Reversal for Second-Order Equations on a Hilbert Triple: Subsolutions of $F$ are Supersolutions of $\tilde F$, $u$ is a viscosity supersolution of $F$ on $U$.