Existence, Penalty Bounds and Monotonicity of the Maximum of a Two-Space Doubled Difference with a Continuous Link

In the setting of N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level. For $i\in\{1,2\}$ let $n_{i}$ be one of the two natural numbers $d$ and $dN$, and let $(\mathcal{D}_{i},\mathcal{D}_{i,\Sigma},\mathcal{E}_{i},\Sigma_{i})$ be a Wasserstein-coercive penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{n_{i}})$, read at the configuration level when $n_{i}=dN$. Fix $e_{0}\in\mathbb{R}$ with $e_{0}\le\mathcal{E}_{1}(\mu)$ for every $\mu\in\mathcal{D}_{1}$ and $e_{0}\le\mathcal{E}_{2}(\nu)$ for every $\nu\in\mathcal{D}_{2}$, as provided by Basic Properties of a Wasserstein-Coercive Penalty Pair §bounded-below for each pair and claim 9 of Elementary Order Arithmetic in an Ordered Field. Let $u:\mathcal{D}_{1}\to\mathbb{R}$, $v:\mathcal{D}_{2}\to\mathbb{R}$ and $b,b'\in\mathbb{R}$ satisfy $u(\mu)\le b$ for every $\mu\in\mathcal{D}_{1}$ and $b'\le v(\nu)$ for every $\nu\in\mathcal{D}_{2}$; for positive $\delta\in\mathbb{R}$ the $\delta$-envelopes $u^{-}_{\delta}$ of $u$ relative to the first pair and $v^{+}_{\delta}$ of $v$ relative to the second pair are defined by The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Coercive Penalty Pair §growth. Let $\kappa_{1},\kappa_{2}\in\mathbb{R}$ be positive, and let $L:\mathcal{P}_{2}(\mathbb{R}^{n_{1}})\times\mathcal{P}_{2}(\mathbb{R}^{n_{2}})\to\mathbb{R}$ satisfy $0\le L(\mu,\nu)$ for all $(\mu,\nu)$ and be continuous in the sense that $\bigl(L(\mu_{j},\nu_{j})\bigr)_{j\in\mathbb{N}}$ converges to $L(\mu,\nu)$ whenever $\bigl(W_{2}(\mu_{j},\mu)\bigr)_{j\in\mathbb{N}}$ and $\bigl(W_{2}(\nu_{j},\nu)\bigr)_{j\in\mathbb{N}}$ converge to $0$. For positive $\delta,\alpha\in\mathbb{R}$ let $\Psi_{\delta,\alpha}:\mathcal{D}_{1}\times\mathcal{D}_{2}\to\mathbb{R}$ have the value

and write $|s|$ for the absolute value of $s\in\mathbb{R}$. Then the following hold for all positive $\delta,\alpha\in\mathbb{R}$.

1. (A maximiser exists)¶ Every value of $\Psi_{\delta,\alpha}$ is at most $\kappa_{1}b-\kappa_{2}b'-(\kappa_{1}+\kappa_{2})\,\delta\,e_{0}$; the supremum $M(\delta,\alpha)$ of these values is a real number; and there is $(\hat{\mu},\hat{\nu})\in\mathcal{D}_{1}\times\mathcal{D}_{2}$ with $\Psi_{\delta,\alpha}(\hat{\mu},\hat{\nu})=M(\delta,\alpha)$, called a maximising pair of $\Psi_{\delta,\alpha}$.

2. (The penalty where the doubled difference is nonnegative)¶ If $(\mu,\nu)\in\mathcal{D}_{1}\times\mathcal{D}_{2}$ satisfies $0\le\Psi_{\delta,\alpha}(\mu,\nu)$, then, with $B_{\delta}=\kappa_{1}|b|+\kappa_{2}|b'|+(\kappa_{1}+\kappa_{2})\,\delta\,|e_{0}|$,

3. (Decreasing the penalty weight)¶ If $\delta'\in\mathbb{R}$ satisfies $0<\delta'<\delta$ and $(\hat{\mu},\hat{\nu})$ is a maximising pair of $\Psi_{\delta,\alpha}$, then

4. (Decreasing the link strength)¶ If $\alpha'\in\mathbb{R}$ satisfies $0<\alpha'<\alpha$ and $(\hat{\mu},\hat{\nu})$ is a maximising pair of $\Psi_{\delta,\alpha}$, then