Proof of The Upper Semicontinuous Envelope of a Supremum of Viscosity Subsolutions is a Viscosity Subsolution

Conventions. From the setting we use the real numbers with their order and absolute value, Euclidean space with its distance $d_{E}$ and notion of openness, $\mathcal{S}(n)$ with its distance, and the notions of class $C^{2}$, gradient, Hessian, semicontinuity and local extrema; $\mathbb{R}$ carries the metric $d_{\mathbb{R}}$ of The Absolute Value Metric on the Real Line, so that $d_{\mathbb{R}}(s,t)=|s-t|$. Elementary order facts are those of Elementary Order Arithmetic in an Ordered Field, whose claim 1 is the compatibility of the strict order with addition, the non-strict law being an axiom of the ordered field $\mathbb{R}$, whose order $\le$ is a total order. We write $\iota$ for the canonical map of $\mathbb{N}$ into $\mathbb{R}$ of Properties of the Canonical Map from the Natural Numbers to an Ordered Field. Two points in the proof choose one object for each natural number; both are appeals to the axiom of countable choice, and it is used nowhere else.

Three remarks used repeatedly.

(R1) Negated strict inequalities. If $a,b\in\mathbb{R}$ and $a<b$ fails, then $b\le a$: by totality $a\le b$ or $b\le a$, and in the first case $a\ne b$ would give $a<b$, so $a=b$ and $b\le a$ by reflexivity.

(R2) Reciprocals. If $N,k\in\mathbb{N}$ and $N\le k$, then $\iota(k)^{-1}\le\iota(N)^{-1}$. Indeed $\iota(N)\le\iota(k)$, by claim 6 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field when $N<k$ and trivially when $N=k$; both are positive by claim 3 of that lemma, so $\iota(N)^{-1}\iota(k)^{-1}$ is positive by claims 7 and 5 of Elementary Order Arithmetic in an Ordered Field, and multiplying by it, using claim 5 of Elementary Arithmetic in an Ordered Field and the field identities $\iota(N)\iota(N)^{-1}=1$ and $\iota(k)\iota(k)^{-1}=1$, gives the claim.

(R3) Limits respect eventual bounds. If a sequence $(a_{m})_{m\in\mathbb{N}}$ of real numbers converges to $L$ and there are $M\in\mathbb{N}$ and $C\in\mathbb{R}$ with $a_{m}\le C$ for every $m\ge M$, then $L\le C$. Otherwise $C<L$ by (R1), and applying the definition of convergence with the positive real $L-C$ gives some $m\ge M$ with $|a_{m}-L|<L-C$; since $L-a_{m}\le|a_{m}-L|$ by claim 3 of Properties of the Absolute Value in an Ordered Field, this yields $L-a_{m}<L-C$ and hence $C<a_{m}$, contradicting $a_{m}\le C$.

Proof. By claim 2 of Properties of the Upper Semicontinuous Envelope the function $w^{*}$ is upper semicontinuous on $U$. Let $\varphi:U\to\mathbb{R}$ be of class $C^{2}$ on $U$ and let $z\in U$ be a point at which $w^{*}-\varphi$ has a local maximum relative to $U$. We must prove that

Step 1: a strict maximum on a compact ball. Let $\psi:U\to\mathbb{R}$ be given by $\psi(x)=\varphi(x)+\lVert x-z\rVert^{4}$. By claim 3 of The Quartic Bump: Making a Local Maximum Strict without Changing the Test Data, applied with $V=U$, $S=U$ and $h=w^{*}$, the function $\psi$ is of class $C^{2}$ on $U$, satisfies $D\psi(z)=D\varphi(z)$ and $D^{2}\psi(z)=D^{2}\varphi(z)$, and $w^{*}-\psi$ has a strict local maximum at $z$ relative to $U$. It therefore suffices to prove that $F(z,w^{*}(z),D\psi(z),D^{2}\psi(z))\le 0$.

By the definition of a strict local maximum there is a positive $\delta_{0}\in\mathbb{R}$ such that every $x\in U$ with $d_{E}(z,x)<\delta_{0}$ and $x\ne z$ satisfies $w^{*}(x)-\psi(x)<w^{*}(z)-\psi(z)$. Since $U$ is open, claim 4 of The Interior is the Largest Open Subset gives $\operatorname{int}_{\mathbb{R}^{n}}(U)=U$, so claim 1 of Interior Points in the Metric Topology are Exactly the Centres of Contained Closed Balls provides a positive $r_{0}\in\mathbb{R}$ with $\{x\in\mathbb{R}^{n}:d_{E}(x,z)\le r_{0}\}\subseteq U$. By claim 8 of Elementary Order Arithmetic in an Ordered Field the number $\tfrac{\delta_{0}}{2}$ is positive and smaller than $\delta_{0}$, and by claim 9 of that lemma there is a positive $\rho\in\mathbb{R}$ with $\rho\le r_{0}$ and $\rho\le\tfrac{\delta_{0}}{2}$. Put

the closed ball of centre $z$ and radius $\rho$ in $(\mathbb{R}^{n},d_{E})$. Then $z\in K$, so $K$ is nonempty; $K\subseteq U$ because $\rho\le r_{0}$; and $K$ is compact by claim 2 of A Closed Euclidean Ball is Convex and Compact. Every $x\in K$ with $x\ne z$ satisfies $d_{E}(z,x)=d_{E}(x,z)\le\rho\le\tfrac{\delta_{0}}{2}<\delta_{0}$, hence

Step 2: almost maximising members of the family. By claim 5 of Properties of the Upper Semicontinuous Envelope there is a sequence $(x_{k})_{k\in\mathbb{N}}$ in $U$ converging to $z$ in $(\mathbb{R}^{n},d_{E})$ such that $(w(x_{k}))_{k\in\mathbb{N}}$ converges to $w^{*}(z)$ in $(\mathbb{R},d_{\mathbb{R}})$. For each $k\in\mathbb{N}$ the real number $\iota(k)^{-1}$ is positive, so claim 3 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ provides some $v\in\mathcal{F}$ with $w(x_{k})-\iota(k)^{-1}<v(x_{k})$; choosing one such $v$ for each $k$ gives a sequence $(v_{k})_{k\in\mathbb{N}}$ in $\mathcal{F}$ with

the right-hand inequality because $w(x_{k})$ is an upper bound of $\{v(x_{k}):v\in\mathcal{F}\}$.

The sequence $(v_{k}(x_{k}))_{k\in\mathbb{N}}$ converges to $w^{*}(z)$. Indeed, let $\varepsilon\in\mathbb{R}$ be positive. There is $N_{1}\in\mathbb{N}$ with $|w(x_{k})-w^{*}(z)|<\tfrac{\varepsilon}{2}$ for every $k\ge N_{1}$, and by claim 3 of The Archimedean Property of the Real Numbers there is $N_{2}\in\mathbb{N}$ with $\iota(N_{2})^{-1}<\tfrac{\varepsilon}{2}$. Let $N$ be the greater of $N_{1}$ and $N_{2}$, which exists by the definition of the maximum because the order of $\mathbb{N}$ is a total order by claims 1, 2 and 3 of Properties of the Order on the Natural Numbers, and satisfies $N_{1}\le N$ and $N_{2}\le N$ by claim 1 of Elementary Properties of the Maximum of Two Elements. For $k\ge N$ we have $0\le w(x_{k})-v_{k}(x_{k})<\iota(k)^{-1}\le\iota(N_{2})^{-1}<\tfrac{\varepsilon}{2}$ by (2) and (R2), so $|v_{k}(x_{k})-w(x_{k})|<\tfrac{\varepsilon}{2}$ by claim 1 of Properties of the Absolute Value in an Ordered Field, and the triangle inequality of claim 5 of Properties of the Absolute Value in an Ordered Field gives

Step 3: maximisers on $K$. Fix $k\in\mathbb{N}$. Being a viscosity subsolution, $v_{k}$ is upper semicontinuous on $U$, hence its restriction to $K$ is upper semicontinuous on $K$ by claim 2 of Negation, Restriction, and Separated Differences of Semicontinuous Functions. By claim 1 of Continuity of the Value, Gradient and Hessian Maps of a Differentiable Function the function $\psi$ is continuous at every point of $U$ relative to $U$, hence lower semicontinuous there by claim 2 of Semicontinuity Under Negation and Characterization of Continuity, and its restriction to $K$ is lower semicontinuous on $K$ by claim 2 of Negation, Restriction, and Separated Differences of Semicontinuous Functions. Claim 3 of the latter lemma then shows that the function $K\to\mathbb{R}$ whose value at $x$ is $v_{k}(x)-\psi(x)$ is upper semicontinuous on $K$, so claim 1 of Semicontinuous Functions Attain Their Extrema on a Compact Set provides a point of $K$ at which it attains a maximum. Choosing one such point for each $k$ gives a sequence $(\hat y_{k})_{k\in\mathbb{N}}$ in $K$ with

Since $(x_{k})$ converges to $z$ and $\rho$ is positive, there is $N_{0}\in\mathbb{N}$ with $d_{E}(x_{k},z)<\rho$, hence $x_{k}\in K$, for every $k\ge N_{0}$.

Step 4: the maximisers converge to $z$. We verify the hypothesis of claim 2 of The Subsequence Criterion for Convergence in a Metric Space for the sequence $(\hat y_{k})_{k\in\mathbb{N}}$ and the point $z$. Let $(k_{l})_{l\in\mathbb{N}}$ be a strictly increasing sequence in $\mathbb{N}$. Since $K$ is compact and $(\hat y_{k_{l}})_{l\in\mathbb{N}}$ is a sequence in $K$, A Compact Subset of a Metric Space is Sequentially Compact provides a strictly increasing $(l_{m})_{m\in\mathbb{N}}$ in $\mathbb{N}$ and a point $y\in K$ such that $(\hat y_{k_{l_{m}}})_{m\in\mathbb{N}}$ converges to $y$ in $(\mathbb{R}^{n},d_{E})$. Put $\pi_{m}=k_{l_{m}}$; by claim 1 of The Subsequence Criterion for Convergence in a Metric Space the sequence $(\pi_{m})_{m\in\mathbb{N}}$ is strictly increasing, so $m\le\pi_{m}$ for every $m$ by Strictly Increasing Sequences of Natural Numbers Dominate Their Index, and by A Subsequence of a Convergent Sequence Has the Same Limit the sequences $(x_{\pi_{m}})_{m}$ and $(v_{\pi_{m}}(x_{\pi_{m}}))_{m}$ converge to $z$ and to $w^{*}(z)$ respectively.

We show $y=z$. Let $\varepsilon\in\mathbb{R}$ be positive. By claim 2 of Properties of the Upper Semicontinuous Envelope, by the lower semicontinuity of $\psi$ established in Step 3, and by claim 3 of Negation, Restriction, and Separated Differences of Semicontinuous Functions, the function $w^{*}-\psi$ is upper semicontinuous at $y$ relative to $U$, so there is a positive $\delta$ such that every $x\in U$ with $d_{E}(y,x)<\delta$ satisfies

There is $M_{1}\in\mathbb{N}$ with $d_{E}(\hat y_{\pi_{m}},y)<\delta$ for every $m\ge M_{1}$. Let $M$ be the greater of $M_{1}$ and $N_{0}$. For $m\ge M$ we have $\pi_{m}\ge m\ge N_{0}$, so $x_{\pi_{m}}\in K$ and (3) applies with $x=x_{\pi_{m}}$; moreover $v_{\pi_{m}}(\hat y_{\pi_{m}})\le w(\hat y_{\pi_{m}})\le w^{*}(\hat y_{\pi_{m}})$ by the definition of $w$ and claim 1 of Properties of the Upper Semicontinuous Envelope. Combining,

By claim 1 of Continuity Between Metric Spaces is Equivalent to Sequential Continuity the sequence $(\psi(x_{\pi_{m}}))_{m}$ converges to $\psi(z)$, so by claim 3 of Arithmetic of Limits of Real Sequences the left-hand side converges to $w^{*}(z)-\psi(z)$, and (R3) gives

As $\varepsilon$ was an arbitrary positive real, claim 1 of Comparison of Real Numbers with Arbitrary Positive Slack gives $w^{*}(z)-\psi(z)\le w^{*}(y)-\psi(y)$. If $y\ne z$ this contradicts (1) together with the mixed transitivity of claim 2 of Elementary Order Arithmetic in an Ordered Field and the irreflexivity of the strict order. Hence $y=z$.

Thus every subsequence of $(\hat y_{k})_{k\in\mathbb{N}}$ has in turn a subsequence converging to $z$, and claim 2 of The Subsequence Criterion for Convergence in a Metric Space shows that $(\hat y_{k})_{k\in\mathbb{N}}$ converges to $z$ in $(\mathbb{R}^{n},d_{E})$.

Step 5: the values at the maximisers converge. We show that $(v_{k}(\hat y_{k}))_{k\in\mathbb{N}}$ converges to $w^{*}(z)$. Let $\varepsilon\in\mathbb{R}$ be positive. By claim 1 of Continuity Between Metric Spaces is Equivalent to Sequential Continuity the sequences $(\psi(x_{k}))_{k}$ and $(\psi(\hat y_{k}))_{k}$ both converge to $\psi(z)$, so by claims 1 and 3 of Arithmetic of Limits of Real Sequences and Step 2 the sequence $b_{k}=v_{k}(x_{k})-\psi(x_{k})+\psi(\hat y_{k})$ converges to $w^{*}(z)-\psi(z)+\psi(z)=w^{*}(z)$; hence there is $N_{3}$ with $|b_{k}-w^{*}(z)|<\varepsilon$, and so $w^{*}(z)-\varepsilon<b_{k}$ by claim 9 of Properties of the Absolute Value in an Ordered Field, for every $k\ge N_{3}$. For $k\ge N_{0}$ we have $x_{k}\in K$, so (3) gives $b_{k}\le v_{k}(\hat y_{k})$. On the other hand $w^{*}$ is upper semicontinuous at $z$ relative to $U$, so there is a positive $\delta_{1}$ such that every $x\in U$ with $d_{E}(z,x)<\delta_{1}$ satisfies $w^{*}(x)<w^{*}(z)+\varepsilon$, and there is $N_{4}$ with $d_{E}(\hat y_{k},z)<\delta_{1}$ for $k\ge N_{4}$; for such $k$,

Taking $k$ at least each of $N_{0}$, $N_{3}$ and $N_{4}$ — a bound obtained by two applications of the maximum of two natural numbers — we get $w^{*}(z)-\varepsilon<v_{k}(\hat y_{k})<w^{*}(z)+\varepsilon$, hence $|v_{k}(\hat y_{k})-w^{*}(z)|<\varepsilon$ by claim 9 of Properties of the Absolute Value in an Ordered Field. This is the asserted convergence.

Step 6: passing to the limit in the viscosity inequality. Since $(\hat y_{k})$ converges to $z$ there is $N_{5}\in\mathbb{N}$ with $d_{E}(\hat y_{k},z)<\rho$ for every $k\ge N_{5}$. Fix such a $k$ and put $\delta_{2}=\rho-d_{E}(\hat y_{k},z)$, which is positive by claim 1 of Elementary Order Arithmetic in an Ordered Field. Every $x\in U$ with $d_{E}(\hat y_{k},x)<\delta_{2}$ satisfies, by the triangle inequality of a metric,

so $x\in K$ and (3) gives $v_{k}(x)-\psi(x)\le v_{k}(\hat y_{k})-\psi(\hat y_{k})$. Hence $v_{k}-\psi$ has a local maximum at $\hat y_{k}$ relative to $U$, and since $v_{k}$ is a viscosity subsolution of $F$ on $U$ and $\psi$ is of class $C^{2}$ on $U$,

By claims 2 and 3 of Continuity of the Value, Gradient and Hessian Maps of a Differentiable Function the gradient and Hessian maps of $\psi$ are continuous at $z$ relative to $U$, so claim 1 of Continuity Between Metric Spaces is Equivalent to Sequential Continuity shows that $(D\psi(\hat y_{k}))_{k}$ converges to $D\psi(z)$ in $(\mathbb{R}^{n},d_{E})$ and $(D^{2}\psi(\hat y_{k}))_{k}$ converges to $D^{2}\psi(z)$ in $(\mathcal{S}(n),d_{\mathcal{S}(n)})$. Together with Steps 4 and 5 and the continuity of $F$, Sequential Form of the Continuity of a Second-Order Equation Operator shows that the real sequence appearing in (5) converges to $F(z,w^{*}(z),D\psi(z),D^{2}\psi(z))$. By (R3),

and since $D\psi(z)=D\varphi(z)$ and $D^{2}\psi(z)=D^{2}\varphi(z)$ this is the required inequality. As $\varphi$ and $z$ were arbitrary, $w^{*}$ is a viscosity subsolution of $F$ on $U$. $\blacksquare$