The Bump Construction for Plan-Jet Viscosity Subsolutions on Square-Integrable Noncommutative Laws

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, let $\rho>0$ be real, let $\mathcal{H}:\Sigma^{2}_{2d}\to\mathbb{R}$ be uniformly continuous on bounded sets, and let $(\mathrm{E})$ be the discounted stationary Hamilton--Jacobi equation with discount rate $\rho$ and Hamiltonian $\mathcal{H}$; metrics are those of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §metrics. Let $g:\Sigma^{2}_{d}\to\mathbb{R}$ be bounded, lower semicontinuous and a plan-jet viscosity supersolution of $(\mathrm{E})$, and let $w:\Sigma^{2}_{d}\to\mathbb{R}$ be bounded, upper semicontinuous and a plan-jet viscosity subsolution of $(\mathrm{E})$ with $w\le g$. Let $w_{*}$ be the lower semicontinuous envelope of $w$, which exists because $w$ is bounded below.

¶If $w_{*}$ is not a plan-jet viscosity supersolution of $(\mathrm{E})$, then there is a bounded, upper semicontinuous plan-jet viscosity subsolution $u:\Sigma^{2}_{d}\to\mathbb{R}$ of $(\mathrm{E})$ with $w\le u\le g$ and $u(\nu)>w(\nu)$ for some $\nu\in\Sigma^{2}_{d}$.