Slope-Based Viscosity Subsolutions, Supersolutions and Solutions on a Metric Space

In the setting of The Real Numbers: Standing Notation and Background, let $(X,d)$ be a metric space, let $\Omega\subseteq X$ be open in $(X,d)$, and let $T=\{p\in\mathbb{R}:0\le p\}$. For a function $\psi:\Omega\to\mathbb{R}$ locally Lipschitz on $\Omega$ and $x\in\Omega$, $|\nabla\psi|(x)$ and $|\nabla\psi|^{*}(x)$ are its local slope and upper envelope of the slope at $x$, both nonnegative reals; $\underline{\mathcal{C}}(\Omega)$ and $\overline{\mathcal{C}}(\Omega)$ are the sub-slope and super-slope test classes. Upper and lower semicontinuity on $\Omega$, and local maxima and local minima relative to $\Omega$, are those of the cited definitions; for functions $u,\psi_{1},\psi_{2}:\Omega\to\mathbb{R}$, $u-\psi_{1}-\psi_{2}$ is the function $x\mapsto u(x)-\psi_{1}(x)-\psi_{2}(x)$. For reals $a$ and $b$ with $0\le b$, $\max\{a,0\}\in T$ by claim 1 of Elementary Properties of the Maximum of Two Elements, and $a+b\in T$ whenever $a\in T$.

1. (Hamiltonian)¶ A Hamiltonian on $\Omega$ is a function $H:\Omega\times\mathbb{R}\times T\to\mathbb{R}$ such that $H(x,r,p)\le H(x,r,p')$ for all $x\in\Omega$, $r\in\mathbb{R}$ and $p,p'\in T$ with $p\le p'$.

For the rest of this definition let $H$ be a Hamiltonian on $\Omega$.

2. (Subsolution)¶ A function $u:\Omega\to\mathbb{R}$ that is upper semicontinuous on $\Omega$ is a slope-based subsolution, or s-subsolution, of $H=0$ in $\Omega$ if for every $\psi_{1}\in\underline{\mathcal{C}}(\Omega)$, every $\psi_{2}:\Omega\to\mathbb{R}$ locally Lipschitz on $\Omega$, and every $x\in\Omega$ at which $u-\psi_{1}-\psi_{2}$ has a local maximum relative to $\Omega$,

3. (Supersolution)¶ A function $v:\Omega\to\mathbb{R}$ that is lower semicontinuous on $\Omega$ is a slope-based supersolution, or s-supersolution, of $H=0$ in $\Omega$ if for every $\psi_{1}\in\overline{\mathcal{C}}(\Omega)$, every $\psi_{2}:\Omega\to\mathbb{R}$ locally Lipschitz on $\Omega$, and every $x\in\Omega$ at which $v-\psi_{1}-\psi_{2}$ has a local minimum relative to $\Omega$,

4. (Solution)¶ A function $u:\Omega\to\mathbb{R}$ is a slope-based solution, or s-solution, of $H=0$ in $\Omega$ if it is both an s-subsolution and an s-supersolution of $H=0$ in $\Omega$.