The Discounted Stationary Hopf-Lax 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, let $\rho$ be a positive real, and let $f:\Omega\to\mathbb{R}$. The function $H_{\rho,f}:\Omega\times\mathbb{R}\times T\to\mathbb{R}$,

is a Hamiltonian on $\Omega$: for $p,p'\in T$ with $p\le p'$, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $p^{2}\le p'^{2}$, hence $H_{\rho,f}(x,r,p)\le H_{\rho,f}(x,r,p')$.

¶The discounted stationary Hopf--Lax equation with discount rate $\rho$ and right-hand side $f$ in $\Omega$ is

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