The Eikonal Equation on an Open Subset of a Metric Space

In the setting of Slope-Based Viscosity Solutions on a Metric Space: Standing Notation, let $\Omega\subseteq X$ be open and let $f:\Omega\to\mathbb{R}$. The function $H_{f}:\Omega\times\mathbb{R}\times T\to\mathbb{R}$, $H_{f}(x,r,p)=p-f(x)$, is a Hamiltonian on $\Omega$, since $p\le p'$ gives $p-f(x)\le p'-f(x)$.

¶The eikonal equation with right-hand side $f$ in $\Omega$ is

by which is meant the equation $H_{f}=0$ in $\Omega$; its s-subsolutions, s-supersolutions and s-solutions are those of $H_{f}=0$ in $\Omega$ in the sense of Slope-Based Viscosity Solutions on a Metric Space: Standing Notation §solutions.