Distance Functions Give Explicit Slope-Based Solutions of the Eikonal and Discounted Hopf-Lax Equations

In the setting of Slope-Based Viscosity Solutions on a Metric Space: Standing Notation, suppose that $(X,d)$ has interpolation points. Let $K\subseteq X$ be nonempty and write $D(x)=\operatorname{dist}(x,K)$ for $x\in X$.

1. (Eikonal equation)¶ Suppose that $\Omega=X\setminus K$ is open, and let $\mathbf{1}:\Omega\to\mathbb{R}$ be the constant function with value $1$. Then the restriction of $D$ to $\Omega$ is an s-solution of the eikonal equation $|\nabla u|=\mathbf{1}$ in $\Omega$.

2. (Discounted Hopf--Lax equation)¶ Let $\rho$ be a positive real and $b$ a nonnegative real, and let

where $\sqrt{\rho^{2}+8b}$ is the nonnegative square root of the nonnegative real $\rho^{2}+8b$. Then $a\ge0$ and $\rho a+2a^{2}=b$, and the function $x\mapsto a\,D(x)^{2}$ on $X$ is an s-solution of the discounted stationary Hopf--Lax equation $\rho\,u+\frac{1}{2}|\nabla u|^{2}=f$ in $X$ with right-hand side $f(x)=b\,D(x)^{2}$ (the equation being taken with $\Omega=X$).