Proof of Uniqueness and Continuity on Energy Sublevel Sets of Bounded Viscosity Solutions on the Wasserstein Space

Each result cited is universally quantified over the data in its own statement, and is applied to the data named here. $|s|$ is the absolute value of $s\in\mathbb{R}$ and $W=W_{2}$.

Claim 1. Let $b$ and $b''$ be bounds for $u$ and $v$; by claim 6 of Properties of the Absolute Value in an Ordered Field, $-b\le u(\mu)\le b$ and $-b''\le v(\mu)\le b''$ for every $\mu\in\mathcal{D}$. By Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §solution, $u$ and $v$ are each a viscosity subsolution and a viscosity supersolution of $F$ relative to the penalty pair. A Comparison Principle for Viscosity Solutions on the Wasserstein Space §comparison, applied to the subsolution $u$, bounded above by $b$, and the supersolution $v$, bounded below by $-b''$, gives $u(\mu)\le v(\mu)$ for every $\mu\in\mathcal{D}$; applied to the subsolution $v$, bounded above by $b''$, and the supersolution $u$, bounded below by $-b$, it gives $v(\mu)\le u(\mu)$. By antisymmetry of the order of $\mathbb{R}$ (Total Order on a Set), $u(\mu)=v(\mu)$.

Claim 2. Let $b$ be a bound for $u$, so that $-b\le u(\mu)\le b$ for every $\mu\in\mathcal{D}$ (claim 6 of Properties of the Absolute Value in an Ordered Field), and let $c\in\mathbb{R}$. As in Claim 1, $u$ is a viscosity subsolution bounded above by $b$ and a viscosity supersolution bounded below by $-b$. For positive $\delta$, Basic Properties of the Delta-Envelopes on the Wasserstein Space §semicontinuity gives that $u^{-}_{\delta}$ is upper and $u^{+}_{\delta}$ lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W)$, with

For $\nu\in K_{c}$ one has $\mathcal{E}(\nu)\le c\le|c|$ (claim 3 of Properties of the Absolute Value in an Ordered Field).

Continuity on $K_{c}$. Let $\mu\in K_{c}$ and let $\varepsilon$ be positive; put $\theta=\varepsilon/3$. By A Comparison Principle for Viscosity Solutions on the Wasserstein Space §envelopes, applied with $v=u$ (and the bounds $b$ and $-b$), there is a positive $\delta_{0}$ with $u^{-}_{\delta}(\rho)-u^{+}_{\delta}(\rho)\le\theta$ for every $\rho\in\mathcal{D}$ and every $\delta$ with $0<\delta<\delta_{0}$. Let $\delta$ be half the least of $\delta_{0}$ and $\theta\bigl(2|c|+1\bigr)^{-1}$; then $0<\delta<\delta_{0}$ and $2\delta|c|\le\theta$. By the upper semicontinuity of $u^{-}_{\delta}$ at $\mu$ (Upper Semicontinuous Function on a Subset of a Metric Space) and the lower semicontinuity of $u^{+}_{\delta}$ at $\mu$ (Lower Semicontinuous Function on a Subset of a Metric Space), there is a positive $r$ such that every $\nu\in\mathcal{D}$ with $W(\nu,\mu)<r$ satisfies $u^{-}_{\delta}(\nu)<u^{-}_{\delta}(\mu)+\theta$ and $u^{+}_{\delta}(\mu)-\theta<u^{+}_{\delta}(\nu)$ (take the least of the two radii, claim 9 of Elementary Order Arithmetic in an Ordered Field). Let $\nu\in K_{c}$ with $W(\nu,\mu)<r$. Using (2a), these two bounds, $u^{-}_{\delta}(\mu)\le u^{+}_{\delta}(\mu)+\theta$, and $\delta\,\mathcal{E}(\nu)\le\delta|c|$, $\delta\,\mathcal{E}(\mu)\le\delta|c|$ (claim 5 of Elementary Arithmetic in an Ordered Field),

So $|u(\nu)-u(\mu)|<\varepsilon$ by claim 9 of Properties of the Absolute Value in an Ordered Field (the first chain is strict at its second step, the second at its second step). Hence the restriction of $u$ to $K_{c}$ is continuous on $K_{c}$.

Uniform continuity. The set $K_{c}$ is sequentially compact in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W)$ by Wasserstein-Coercive Penalty Pairs §coercive, hence compact in the topology of that metric space by A Sequentially Compact Subset of a Metric Space is Compact. By Heine-Cantor Theorem: Continuity on a Compact Subset Implies Uniform Continuity, with $X=\mathcal{P}_{2}(\mathbb{R}^{d})$, $K=K_{c}$, $Y=\mathbb{R}$ and $f$ the restriction of $u$ to $K_{c}$, that restriction is uniformly continuous on $K_{c}$.