Proof of Perron's Method on a Hilbert Triple: Existence of a Viscosity Solution Between a Subsolution and a Supersolution

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 $d_{H}(x,y)=|x-y|_{H}$, by Real Inner Product Space §distance, and use the triangle inequality of The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle and the symmetry of $d_{H}$ from the metric axioms without further mention. Being a viscosity subsolution, $f$ is bounded above near each point of $U$; being a viscosity supersolution, $g$ is bounded below near each point of $U$.

Claim 1. Let $x\in U$. Since $f\in\mathcal{G}$, the number $f(x)$ belongs to $\{v(x):v\in\mathcal{G}\}$, of which $u(x)$ is an upper bound, so $f(x)\le u(x)$ by Upper Bound and Least Upper Bound; and $g(x)$ is an upper bound of that set while $u(x)$ is its least upper bound, so $u(x)\le g(x)$.

Since $g$ is bounded above near each point of $U$, for $x\in U$ there are $c\in\mathbb{R}$ and a positive $r$ with $g(y)\le c$ for every $y\in U$ with $d_{H}(y,x)\le r$; then $u(y)\le g(y)\le c$ for such $y$, so $u$ is bounded above near each point of $U$ by Upper and Lower Semicontinuous Envelopes of a Real-Valued Function §near-bounds. The same argument with $f$ and the reversed inequalities shows that $u$ is bounded below near each point of $U$.

Claim 2, the subsolution property. The set $\mathcal{G}$ is nonempty and each of its members is a viscosity subsolution of $F$ on $U$. It is locally uniformly bounded above in the sense of The Pointwise Supremum of a Locally Uniformly Bounded Family of Viscosity Subsolutions on a Hilbert Triple is a Viscosity Subsolution §locally-bounded: given $x\in U$, the pair $c,r$ furnished above by the local upper bound for $g$ satisfies $v(y)\le g(y)\le c$ for every $v\in\mathcal{G}$ and every $y\in U$ with $d_{H}(y,x)\le r$. The function whose value at $x$ is $\sup\{v(x):v\in\mathcal{G}\}$ is $u$, so claim 2 of The Pointwise Supremum of a Locally Uniformly Bounded Family of Viscosity Subsolutions on a Hilbert Triple is a Viscosity Subsolution shows that $u$ is a viscosity subsolution of $F$ on $U$.

Claim 2, the supersolution property. Suppose, seeking a contradiction, that $u$ is not a viscosity supersolution of $F$ on $U$. By claim 1 the local boundedness below required by Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple §supersolution holds, so the failure is of the quantified condition: there are a real $\delta>0$, a function $\varphi\in C^{2}(U)$, a point $\hat{x}\in V\cap U$ at which the function $V\cap U\to\mathbb{R}$ with value $u^{+}_{\delta}(x)-\varphi(x)$ at $x$ has a local minimum relative to $V\cap U$, and a real $\varepsilon>0$, such that no $y\in W$, $s\in\mathbb{R}$, $q\in H$ and $Y\in\mathrm{Sym}(H)$ satisfy all six conditions there. Since the order of $\mathbb{R}$ is total, this says:

the five closeness conditions being $|y-\hat{x}|_{H}<\varepsilon$, $|u^{+}_{\delta}(y)-u^{+}_{\delta}(\hat{x})|<\varepsilon$, $|s-u^{+}_{\delta}(\hat{x})|<\varepsilon$, $|q-D\varphi(\hat{x})|_{H}<\varepsilon$ and $\lVert Y-D^{2}\varphi(\hat{x})\rVert<\varepsilon$. Let $\tau$ be a positive real number witnessing the local minimum as in Local Minimum of a Function Relative to a Subset of a Metric Space, so that

The function $\varphi$ is continuous on $U$ and its gradient and Hessian maps are continuous on $U$, by claim 2 of Basic Properties of Differentiability on an Open Subset of a Real Inner Product Space and The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space §c1, The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space §c2. Finally, $u(x)\le g(x)$ for every $x\in U$ and both functions are bounded below near each point of $U$, so claim 5 of Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity gives $u^{+}_{\delta}(x)\le g^{+}_{\delta}(x)$ for every $x\in V\cap U$.

Step 1: a strict gap at $\hat{x}$. We show that $u^{+}_{\delta}(\hat{x})<g^{+}_{\delta}(\hat{x})$. Suppose instead that $u^{+}_{\delta}(\hat{x})=g^{+}_{\delta}(\hat{x})$. For $x\in V\cap U$ with $d_{H}(\hat{x},x)<\tau$,

so the function with value $g^{+}_{\delta}(x)-\varphi(x)$ at $x$ has a local minimum at $\hat{x}$ relative to $V\cap U$. By Continuous Map Between Metric Spaces there is a positive $\sigma_{1}\le\tau$ such that every $z\in U$ with $d_{H}(\hat{x},z)<\sigma_{1}$ satisfies $|\varphi(z)-\varphi(\hat{x})|<\varepsilon$; put $\varepsilon_{1}=\min\{\varepsilon,\sigma_{1}\}$, 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 §supersolution to the viscosity supersolution $g$, with $\delta$, $\varphi$, $\hat{x}$ and the tolerance $\varepsilon_{1}$, we obtain $y\in W$, $s\in\mathbb{R}$, $q\in H$ and $Y\in\mathrm{Sym}(H)$ with

and $-\varepsilon_{1}\le F^{+}_{\delta}(y,s,q,Y)$. Since $\varepsilon_{1}\le\varepsilon$ and $g^{+}_{\delta}(\hat{x})=u^{+}_{\delta}(\hat{x})$, four of the five closeness conditions of $(\ast)$ hold at once. For the remaining one, $y\in W\subseteq V\cap U$ and $d_{H}(\hat{x},y)<\varepsilon_{1}\le\sigma_{1}\le\tau$, so on the one hand

and on the other hand $u^{+}_{\delta}(y)\ge u^{+}_{\delta}(\hat{x})+\varphi(y)-\varphi(\hat{x})>u^{+}_{\delta}(\hat{x})-\varepsilon$ by the local minimum and claim 3 of Properties of the Absolute Value in an Ordered Field; so $|u^{+}_{\delta}(y)-u^{+}_{\delta}(\hat{x})|<\varepsilon$ by claim 9 of that lemma. Then $(\ast)$ gives $F^{+}_{\delta}(y,s,q,Y)<-\varepsilon\le-\varepsilon_{1}$, contradicting $-\varepsilon_{1}\le F^{+}_{\delta}(y,s,q,Y)$. Hence $u^{+}_{\delta}(\hat{x})<g^{+}_{\delta}(\hat{x})$, and we set $G=g^{+}_{\delta}(\hat{x})-u^{+}_{\delta}(\hat{x})$, a positive real number by claim 3 of Elementary Arithmetic in an Ordered Field.

Step 2: the radius and the parameters. By Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity §semicontinuity, $g^{+}_{\delta}$ is lower semicontinuous on $V\cap U$. Using Continuous Map Between Metric Spaces for $\varphi$, $D\varphi$ and $D^{2}\varphi$ at $\hat{x}$ and Lower Semicontinuous Function on a Subset of a Metric Space for $g^{+}_{\delta}$ at $\hat{x}$, and shrinking finitely many radii by claim 2 of Elementary Properties of the Minimum of Two Elements, choose a positive $\gamma$ with $\gamma\le\tau$, $\gamma\le\varepsilon$ and $B_{d_{H}}(\hat{x},\gamma)\subseteq U$, the last being possible by Open Subset of a Metric Space, such that every $z\in U$ with $d_{H}(\hat{x},z)<\gamma$ satisfies

and every $z\in V\cap U$ with $d_{H}(\hat{x},z)<\gamma$ satisfies $g^{+}_{\delta}(\hat{x})-\tfrac{G}{4}<g^{+}_{\delta}(z)$. Put

with $\eta$ and $\kappa$ positive. By claim 1 of Elementary Properties of the Minimum of Two Elements and claim 5 of Elementary Arithmetic in an Ordered Field we record

The set $U'$ is open in $H$ by Open Ball in a Metric Space is Open, contains $\hat{x}$, and is contained in $U$.

Let $\psi:U\to\mathbb{R}$ be given by $\psi(x)=\varphi(x)+c+\kappa-\eta\,|x-\hat{x}|_{H}^{2}$. By claim 3 of Affine and Quadratic Functions on a Real Hilbert Space are of Class $C^2$, applied with $\alpha=-2\eta$ and $y_{0}=\hat{x}$, and by claim 4 of that lemma, the function $x\mapsto-\eta|x-\hat{x}|_{H}^{2}$ restricted to $U$ belongs to $C^{2}(U)$, with gradient $-2\eta(x-\hat{x})$ and Hessian $-2\eta I_{H}$; so by claims 1 and 2 of Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space, $\psi\in C^{2}(U)$ with

and $\psi(\hat{x})=\varphi(\hat{x})+c+\kappa=u^{+}_{\delta}(\hat{x})+\kappa$.

Step 3: the bump on $U'$. Write $F'=F|_{U'}$ and $W'=D(A)\cap U'$. By claim 1 of Restriction of the Equation and the Locality of the Viscosity Sub- and Supersolution Properties on a Hilbert Triple, $F'$ is a degenerate elliptic second-order equation operator on $U'$ relative to $(H,V,A)$ whose $\delta$-shifts are the restrictions of those of $F$; by claim 2 of that lemma, $u|_{U'}$ is a viscosity subsolution of $F'$ on $U'$; and $\psi|_{U'}\in C^{2}(U')$ with the same gradients and Hessians, by claim 4 of Basic Properties of Differentiability on an Open Subset of a Real Inner Product Space.

We verify the hypothesis The Bump Construction on a Hilbert Triple: the Maximum of a Viscosity Subsolution and a Penalised $C^2$ Function §condition for these data with $\delta$ in the role of its $\mu$. Let $x\in W'$ satisfy $u(x)<\psi(x)-\delta h(x)$. Then $x\in W$ and $d_{H}(\hat{x},x)<\gamma$, so we may test $(\ast)$ with $(y,s,q,Y)=(x,\psi(x),D\psi(x),D^{2}\psi(x))$: First, $|x-\hat{x}|_{H}<\gamma\le\varepsilon$.

Secondly, by claim 1 of Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity and the hypothesis on $x$, and since $0\le\eta|x-\hat{x}|_{H}^{2}$,

while $x\in W'\subseteq V\cap U$ and $d_{H}(\hat{x},x)<\gamma\le\tau$ give $u^{+}_{\delta}(x)\ge u^{+}_{\delta}(\hat{x})+\varphi(x)-\varphi(\hat{x})>u^{+}_{\delta}(\hat{x})-\tfrac{\varepsilon}{4}$. Hence $|u^{+}_{\delta}(x)-u^{+}_{\delta}(\hat{x})|<\varepsilon$ by claim 9 of Properties of the Absolute Value in an Ordered Field. Thirdly, $|\psi(x)-u^{+}_{\delta}(\hat{x})|\le|\varphi(x)-\varphi(\hat{x})|+\kappa+\eta|x-\hat{x}|_{H}^{2}<\tfrac{\varepsilon}{4}+\tfrac{\varepsilon}{16}+\tfrac{\varepsilon}{4}<\varepsilon$, using claim 5 of Properties of the Absolute Value in an Ordered Field, $|x-\hat{x}|_{H}<\gamma$ and claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. Fourthly, $|D\psi(x)-D\varphi(\hat{x})|_{H}\le|D\varphi(x)-D\varphi(\hat{x})|_{H}+2\eta\,|x-\hat{x}|_{H}<\tfrac{\varepsilon}{2}+2\eta\gamma\le\tfrac{\varepsilon}{2}+\tfrac{\varepsilon}{4}<\varepsilon$. Finally, $\lVert D^{2}\psi(x)-D^{2}\varphi(\hat{x})\rVert\le\lVert D^{2}\varphi(x)-D^{2}\varphi(\hat{x})\rVert+\lVert 2\eta I_{H}\rVert<\tfrac{\varepsilon}{2}+2\eta\le\tfrac{3\varepsilon}{4}<\varepsilon$, by claim 3 of Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity, where $\lVert 2\eta I_{H}\rVert\le 2\eta$ because $|2\eta\langle x',y'\rangle_{H}|\le 2\eta|x'|_{H}|y'|_{H}$ 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.

Therefore $F^{+}_{\delta}(x,\psi(x),D\psi(x),D^{2}\psi(x))<-\varepsilon<0$, and this number equals $(F')^{+}_{\delta}(x,\psi(x),D\psi(x),D^{2}\psi(x))$ by claim 1 of Restriction of the Equation and the Locality of the Viscosity Sub- and Supersolution Properties on a Hilbert Triple. The hypothesis The Bump Construction on a Hilbert Triple: the Maximum of a Viscosity Subsolution and a Penalised $C^2$ Function §condition thus holds.

Let $w':U'\to\mathbb{R}$ be the function of The Bump Construction on a Hilbert Triple: the Maximum of a Viscosity Subsolution and a Penalised $C^2$ Function for these data, namely $w'(x)=\max\{\psi(x)-\delta h(x),u(x)\}$ for $x\in V\cap U'$ and $w'(x)=u(x)$ for $x\in U'\setminus V$. By claim 3 of that lemma, $w'$ is a viscosity subsolution of $F'$ on $U'$, and by claim 1 of it, $u(x)\le w'(x)$ on $U'$.

Step 4: gluing. Define $\tilde{w}:U\to\mathbb{R}$ by $\tilde{w}(x)=w'(x)$ for $x\in U'$ and $\tilde{w}(x)=u(x)$ for $x\in U\setminus U'$.

We first check that $\tilde{w}(x)=u(x)$ whenever $x\in U$ and $\tfrac{\gamma}{2}\le d_{H}(\hat{x},x)$. This is clear if $x\notin U'$, and if $x\in U'\setminus V$. Let then $x\in V\cap U'$ with $\tfrac{\gamma}{2}\le d_{H}(\hat{x},x)$. By claim 1 of Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity, $u^{+}_{\delta}(x)\le u(x)+\delta h(x)$, and by the local minimum, $d_{H}(\hat{x},x)<\gamma\le\tau$ giving $u^{+}_{\delta}(x)\ge u^{+}_{\delta}(\hat{x})+\varphi(x)-\varphi(\hat{x})=\varphi(x)+c$; hence $u(x)\ge\varphi(x)+c-\delta h(x)$. By claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field applied to $0\le\tfrac{\gamma}{2}\le|x-\hat{x}|_{H}$ and claim 5 of Elementary Arithmetic in an Ordered Field, $\kappa=\tfrac{\eta\gamma^{2}}{4}\le\eta\,|x-\hat{x}|_{H}^{2}$, so

and therefore $w'(x)=\max\{\psi(x)-\delta h(x),u(x)\}=u(x)$ by Maximum of Two Elements of a Totally Ordered Set.

Put $U''=\{x\in U:\tfrac{\gamma}{2}<d_{H}(\hat{x},x)\}$, the intersection of $U$ with the complement of the closed ball $\bar{B}_{d_{H}}(\hat{x},\tfrac{\gamma}{2})$. That ball is closed in $H$ by claim 3 of Elementary Properties of the Closed Ball in a Metric Space, so its complement is open by Closed Subset of a Topological Space, and $U''$ is open in $H$ by Metric Open Sets Form a Topology. Every $x\in U$ lies in $U'$ or in $U''$: if $d_{H}(\hat{x},x)<\gamma$ then $x\in U'$, and otherwise $\tfrac{\gamma}{2}<\gamma\le d_{H}(\hat{x},x)$ by claim 8 of Elementary Order Arithmetic in an Ordered Field, so $x\in U''$. Moreover $\tilde{w}$ restricted to $U'$ is $w'$, a viscosity subsolution of $F|_{U'}$ on $U'$; and if $U''$ is nonempty then $\tilde{w}$ restricted to $U''$ is $u$ restricted to $U''$ by the previous paragraph, a viscosity subsolution of $F|_{U''}$ on $U''$ by claim 2 of Restriction of the Equation and the Locality of the Viscosity Sub- and Supersolution Properties on a Hilbert Triple. By claim 3 of that lemma, $\tilde{w}$ is a viscosity subsolution of $F$ on $U$.

Step 5: $\tilde{w}$ belongs to $\mathcal{G}$. For $x\in U$ we have $u(x)\le\tilde{w}(x)$, with equality off $U'$ and by claim 1 of The Bump Construction on a Hilbert Triple: the Maximum of a Viscosity Subsolution and a Penalised $C^2$ Function on $U'$; hence $f(x)\le u(x)\le\tilde{w}(x)$ by claim 1 of the present theorem.

For the upper bound, let $x\in U$. If $x\notin U'$, or if $x\in U'\setminus V$, then $\tilde{w}(x)=u(x)\le g(x)$. Let then $x\in V\cap U'$. By claim 1 of Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity applied to $g$, $g^{+}_{\delta}(x)\le g(x)+\delta h(x)$, and by the choice of $\gamma$ in Step 2,

whereas, using $0\le\eta|x-\hat{x}|_{H}^{2}$, $|\varphi(x)-\varphi(\hat{x})|<\tfrac{G}{4}$ and $\kappa\le\tfrac{G}{4}$,

Since $\tfrac{G}{2}<\tfrac{3G}{4}$, we get $\psi(x)-\delta h(x)<g(x)$; together with $u(x)\le g(x)$ this gives $\tilde{w}(x)=\max\{\psi(x)-\delta h(x),u(x)\}\le g(x)$ by claim 3 of Elementary Properties of the Maximum of Two Elements. Hence $\tilde{w}\in\mathcal{G}$.

Step 6: the contradiction. Since $\tilde{w}\in\mathcal{G}$, the number $\tilde{w}(x)$ belongs to $\{v(x):v\in\mathcal{G}\}$, of which $u(x)$ is an upper bound, so $\tilde{w}(x)\le u(x)$ for every $x\in U$; with Step 5 this gives $\tilde{w}=u$ on $U$. In particular, for $x\in V\cap U'$,

by claim 1 of Elementary Properties of the Maximum of Two Elements, that is, $\psi(x)\le u(x)+\delta h(x)$.

By claim 1 of The $\delta$-Envelopes on an Open Subset, under Penalisation of a Continuous Function, and on a Closed Subset, $\bigl(u|_{U'}\bigr)^{+}_{\delta}(\hat{x})=u^{+}_{\delta}(\hat{x})$, and by The $\delta$-Envelopes $u^-_\delta$ and $u^+_\delta$ of a Function on an Open Subset of a Hilbert Triple §plus the left-hand side is the value at $\hat{x}$ of the lower semicontinuous envelope of the function $V\cap U'\to\mathbb{R}$ with value $u(x)+\delta h(x)$ at $x$. By claim 6 of Properties of the Lower Semicontinuous Envelope, by Duality, applied to that function on the nonempty set $V\cap U'$, there is a sequence $(x_{k})_{k\in\mathbb{N}}$ in $V\cap U'$ converging to $\hat{x}$ in $(H,d_{H})$ such that $(u(x_{k})+\delta h(x_{k}))_{k\in\mathbb{N}}$ converges to $u^{+}_{\delta}(\hat{x})$. Since $\psi$ is continuous at $\hat{x}$ relative to $U'$, being continuous on $U$, and $(x_{k})$ is a sequence in $U'$ converging to $\hat{x}$, Continuity Between Metric Spaces is Equivalent to Sequential Continuity shows that $(\psi(x_{k}))_{k\in\mathbb{N}}$ converges to $\psi(\hat{x})$. As $\psi(x_{k})\le u(x_{k})+\delta h(x_{k})$ for every $k$, claim 1 of Order Properties of Limits of Real Sequences gives

But $\psi(\hat{x})=u^{+}_{\delta}(\hat{x})+\kappa$, so claim 3 of Elementary Arithmetic in an Ordered Field gives $\kappa\le 0$, contradicting $0<\kappa$.

This contradiction shows that $u$ is a viscosity supersolution of $F$ on $U$. Being also a viscosity subsolution, and bounded above and below near each point of $U$ by claim 1, $u$ is a viscosity solution of $F$ on $U$ by Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple §solution.