Well-Posedness of the Eikonal and Discounted Hopf-Lax Equations on Noncommutative Laws with Distance Data

In the settings of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation and Slope-Based Viscosity Solutions on a Metric Space: Standing Notation, let $d\in\mathbb{N}$, let $R>0$ be real, and take for the metric space of the latter setting the space $(\Sigma_{d,R},W_{2})$ of The Noncommutative Laws with a Norm Bound Form a Complete Bounded Metric Space with Interpolation Points, so that its set is $\Sigma_{d,R}$ and its metric is $W_{2}$ (the letter $d$ keeps denoting the number of variables). Let $K\subseteq\Sigma_{d,R}$ be nonempty and such that $\lambda\in K$ whenever $\lambda\in\Sigma_{d,R}$ and some sequence in $K$ converges weak-star to $\lambda$; let $\Omega=\Sigma_{d,R}\setminus K$, which is open in $(\Sigma_{d,R},W_{2})$ by Moments of Noncommutative Laws are Lipschitz in the Wasserstein Distance §closed; and write $D(\lambda)=\operatorname{dist}(\lambda,K)$ for $\lambda\in\Sigma_{d,R}$.

1. (Eikonal equation)¶ Let $\mathbf{1}:\Omega\to\mathbb{R}$ be the constant function with value $1$. The restriction of $D$ to $\Omega$ is an s-solution of the eikonal equation $|\nabla u|=\mathbf{1}$ in $\Omega$. Moreover, if $U:\Sigma_{d,R}\to\mathbb{R}$ is bounded above and below and uniformly continuous on $\Sigma_{d,R}$, satisfies $U(\lambda)=0$ for every $\lambda\in K$, and its restriction to $\Omega$ is an s-solution of $|\nabla u|=\mathbf{1}$ in $\Omega$, then $U=D$.

2. (Discounted Hopf--Lax equation)¶ Let $\rho$ be a positive real, $b$ a nonnegative real, and $a=\frac{1}{4}\bigl(\sqrt{\rho^{2}+8b}-\rho\bigr)$ with $\sqrt{\cdot}$ the nonnegative square root. Then $a\,D^{2}:\lambda\mapsto a\,D(\lambda)^{2}$ is the only function $\Sigma_{d,R}\to\mathbb{R}$ bounded above and below that is an s-solution of the discounted stationary Hopf--Lax equation $\rho\,u+\frac{1}{2}|\nabla u|^{2}=b\,D^{2}$ in $\Sigma_{d,R}$.

3. (Commuting laws)¶ For $i,j\in[d]$ let $c_{ij}=x_{i}x_{j}-x_{j}x_{i}\in\mathcal{P}_{d}$. The set of commuting laws $\{\lambda\in\Sigma_{d,R}:\lambda(c_{ij}^{*}c_{ij})=0\text{ for all }i,j\in[d]\}$ is nonempty and contains every $\lambda\in\Sigma_{d,R}$ that is the weak-star limit of a sequence in it, so it may be taken for $K$ in claims 1 and 2.