Proof of Perron's Method for Plan-Jet Viscosity Solutions of the Discounted Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws: Existence and Well-Posedness

Each result cited is universally quantified over the data in its own statement. The comparison principle used below is Comparison Principle for Plan-Jet Viscosity Solutions of the Discounted Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws §lipschitz if $\mathcal{H}$ is Lipschitz in the momentum with linear growth and Comparison Principle for Plan-Jet Viscosity Solutions of the Discounted Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws §quadratic if it is quadratic with a convex Lipschitz remainder; in either case $\mathcal{H}$ satisfies the structure condition, so for bounded $v_{1},v_{2}:\Sigma^{2}_{d}\to\mathbb{R}$ with $v_{1}$ upper semicontinuous and a subsolution and $v_{2}$ lower semicontinuous and a supersolution, $v_{1}\le v_{2}$. We call this (C).

Claim 1. Let $K_{1}\ge0$ bound $|g_{-}|$ and $|g_{+}|$. Let $\mathcal{F}$ be the set of upper semicontinuous subsolutions $v:\Sigma^{2}_{d}\to\mathbb{R}$ of $(\mathrm{E})$ with $g_{-}\le v\le g_{+}$; it contains $g_{-}$, and $v\le K_{1}$ for all $v\in\mathcal{F}$. Let $w(\nu)=\sup\{v(\nu):v\in\mathcal{F}\}$ (The Real Numbers: Standing Notation and Background §bounds), so $g_{-}\le w\le g_{+}$ and $|w|\le K_{1}$, and let $w^{*}$ and $w_{*}$ be its upper and lower semicontinuous envelopes (Upper and Lower Semicontinuous Envelopes of a Real-Valued Function §upper, Upper and Lower Semicontinuous Envelopes of a Real-Valued Function §lower).

$w$ is an upper semicontinuous subsolution. By The Upper Semicontinuous Envelope of a Supremum of Plan-Jet Viscosity Subsolutions is a Plan-Jet Viscosity Subsolution §sup, $w^{*}$ is a subsolution; it is upper semicontinuous with $w\le w^{*}\le K_{1}$ by Properties of the Upper Semicontinuous Envelope §usc, Properties of the Upper Semicontinuous Envelope §bounds and Properties of the Upper Semicontinuous Envelope §least, hence bounded. By (C) with $v_{1}=w^{*}$ and $v_{2}=g_{+}$, $w^{*}\le g_{+}$. Also $w^{*}\ge w\ge g_{-}$. Thus $w^{*}\in\mathcal{F}$, so $w^{*}\le w$, and $w^{*}=w$. In particular $w$ is upper semicontinuous and a subsolution.

$w_{*}$ is a supersolution. Otherwise The Bump Construction for Plan-Jet Viscosity Subsolutions on Square-Integrable Noncommutative Laws §bump (with $g=g_{+}$ and this $w$, which is bounded, upper semicontinuous, a subsolution and $\le g_{+}$) gives a bounded upper semicontinuous subsolution $u$ with $w\le u\le g_{+}$ and $u(\nu)>w(\nu)$ for some $\nu$. Since $g_{-}\le w\le u$, $u\in\mathcal{F}$, so $u(\nu)\le w(\nu)$, a contradiction.

Conclusion. $w_{*}$ is lower semicontinuous (Properties of the Lower Semicontinuous Envelope, by Duality §lsc) and $-K_{1}\le w_{*}\le w\le K_{1}$ (Properties of the Lower Semicontinuous Envelope, by Duality §bounds, Properties of the Lower Semicontinuous Envelope, by Duality §greatest with the constant $-K_{1}$). By (C) with $v_{1}=w$ and $v_{2}=w_{*}$, $w\le w_{*}$; hence $w=w_{*}$, and $w$ is a supersolution. So $w$ is a solution with $g_{-}\le w\le g_{+}$, bounded by $K_{1}$. Finally $w$ is both upper and lower semicontinuous: given $\nu$ and $\varepsilon>0$, Upper Semicontinuous Function on a Subset of a Metric Space and Lower Semicontinuous Function on a Subset of a Metric Space give $\delta_{1},\delta_{2}>0$ with $w(\nu)-\varepsilon<w(\nu')<w(\nu)+\varepsilon$ whenever $\widehat{W}_{2}(\nu,\nu')<\min\{\delta_{1},\delta_{2}\}$, which is continuity at $\nu$ in the sense of Continuous Map Between Metric Spaces for the metric of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §metrics on $\mathbb{R}$ (Properties of the Absolute Value in an Ordered Field claim 9). Take $u=w$.

Claim 2. Put $c=K/\rho$. For every tracial W*-probability space and $L^{2}$ $d$-tuple $X$, $\rho(-c)+\mathcal{H}_{M}(X,0)=\mathcal{H}_{M}(X,0)-K\le0$ and $\rho c+\mathcal{H}_{M}(X,0)=K+\mathcal{H}_{M}(X,0)\ge0$. By Constant Plan-Jet Viscosity Subsolutions and Supersolutions of the Discounted Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws §sub and Constant Plan-Jet Viscosity Subsolutions and Supersolutions of the Discounted Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws §super, the constant functions $g_{-}=-c$ and $g_{+}=c$ are a subsolution and a supersolution; they are bounded, continuous (hence upper and lower semicontinuous), and $g_{-}\le g_{+}$ as $c\ge0$. Claim 1 gives a bounded continuous solution. If $u$ and $v$ are bounded continuous solutions, (C) applied to $(u,v)$ and to $(v,u)$ gives $u\le v\le u$, so $u=v$.