Slope-Based Viscosity Solutions on a Metric Space: Standing Notation

This setting fixes the standing notation for slope-based viscosity solutions on a metric space. It introduces no new concepts. The conventions of The Real Numbers: Standing Notation and Background are in force.

1. (Space)¶ $(X,d)$ is a metric space; $B_{d}(x,r)$ is the open ball with center $x\in X$ and radius $r>0$; openness of a subset of $X$ is that of Open Subset of a Metric Space; and for a nonempty $A\subseteq X$ and $x\in X$, $\operatorname{dist}(x,A)$ is the distance from $x$ to $A$, with the properties listed in The Distance to a Set is Nonexpansive. Whether $(X,d)$ is complete or has interpolation points is stated by each result that uses it.

2. (Functions)¶ For $A\subseteq X$, upper and lower semicontinuity on $A$, local maxima and local minima relative to $A$, and uniform continuity on $A$ of a map into $(\mathbb{R},d_{\mathbb{R}})$ are those of the cited definitions; boundedness above and below of a real function is that of The Real Numbers: Standing Notation and Background §bounds.

3. (Slopes)¶ For an open $\Omega\subseteq X$, a function $\psi:\Omega\to\mathbb{R}$ locally Lipschitz on $\Omega$ and $x\in\Omega$, $|\nabla\psi|(x)$, $|\nabla^{+}\psi|(x)$, $|\nabla^{-}\psi|(x)$ and $|\nabla\psi|^{*}(x)$ are the local slope, super-slope, sub-slope and upper envelope of the slope; $\underline{\mathcal{C}}(\Omega)$ and $\overline{\mathcal{C}}(\Omega)$ are the sub-slope and super-slope test classes.

4. (Solutions)¶ For an open $\Omega\subseteq X$, $T=\{p\in\mathbb{R}:0\le p\}$; Hamiltonians on $\Omega$ and s-subsolutions, s-supersolutions and s-solutions of $H=0$ in $\Omega$ are those of Slope-Based Viscosity Subsolutions, Supersolutions and Solutions on a Metric Space.

5. (Background)¶ The following results are in force by reference: Elementary Properties of Local Slopes: Order, Negation, Bounds from Difference Quotients, Lipschitz Functions and Squared Distances, Ekeland's Variational Principle for Upper Semicontinuous Functions on a Complete Metric Space and The Distance to a Set is Nonexpansive.