Proof of Perron's Method with a Weighted Penalty for Operators Convex in the Value, Gradient and Matrix Variables: Existence and Uniqueness of a Viscosity Solution of Given Weight

Each result cited is universally quantified over the data in its own statement. The data $D,P,w,\gamma,F$ satisfy the five hypotheses of Comparison Principle with a Weighted Penalty for Operators Convex in the Value, Gradient and Matrix Variables; we refer to its conclusion as the comparison principle: every viscosity subsolution $z$ of $F$ on $D$ with $z-wP$ of $P$-subordinate growth from above satisfies $z\le y$ on $D$ for every viscosity supersolution $y$ with $y-wP$ of $P$-subordinate growth from below. Semicontinuity, envelopes and local extrema are taken in $(\mathbb{R}^{n},d_{E})$ relative to $D$, and $P$ is continuous at every point of $D$ by Continuity of the Value, Gradient and Hessian Maps of a Differentiable Function §value.

Step 0: local bounds. Let $f:D\to\mathbb{R}$ be such that $f-wP$ has $P$-subordinate growth from above, and let $x\in D$. With $\delta=1$ there is $C$ with $f\le C+(w+1)P$ on $D$, and by continuity of $P$ at $x$ there is a positive $\eta$ with $|P(y)-P(x)|<1$ for $y\in D$ with $d_{E}(y,x)\le\eta$. For such $y$, $(w+1)P(y)\le|w+1|\,|P(y)|\le|w+1|(|P(x)|+1)$ by claims 3, 4 and 5 of Properties of the Absolute Value in an Ordered Field, so $f(y)\le C+|w+1|(|P(x)|+1)$. Thus $f$ is bounded above on the set of $y\in D$ with $d_{E}(y,x)\le\eta$, and in particular bounded above near each point of $D$. Symmetrically, if $f-wP$ has $P$-subordinate growth from below, $f$ is bounded below near each point of $D$.

Step 1: the Perron family. By the comparison principle applied to $\underline{u}$ and $\overline{u}$, $\underline{u}\le\overline{u}$ on $D$. Let $\mathcal{G}$ be the set of viscosity subsolutions $z:D\to\mathbb{R}$ of $F$ on $D$ with $\underline{u}\le z\le\overline{u}$ on $D$; it contains $\underline{u}$. For $x\in D$ the set $\{z(x):z\in\mathcal{G}\}$ is nonempty and bounded above by $\overline{u}(x)$, so it has a least upper bound (Least Upper Bound Property of the Real Numbers); let $W(x)$ be it. Then $\underline{u}\le W\le\overline{u}$. By Step 0 applied to $\overline{u}$, each $x\in D$ has $c$ and a positive $r$ with $z(y)\le\overline{u}(y)\le c$ for all $z\in\mathcal{G}$ and $y\in D$ with $d_{E}(y,x)\le r$; so $\mathcal{G}$ is locally uniformly bounded above, and since $D$ is open and nonempty and $F$ is continuous, The Upper Semicontinuous Envelope of a Supremum of Viscosity Subsolutions is a Viscosity Subsolution §subsolution shows that the upper semicontinuous envelope $W^{*}$ is a viscosity subsolution of $F$ on $D$.

Step 2: $W=W^{*}$. Let $\delta$ be positive and let $C$ satisfy $\overline{u}\le h_{\delta}$ on $D$, where $h_{\delta}=C+(w+\delta)P$ (Penalty-Subordinate Growth of a Function on an Open Subset of Euclidean Space §above for $\overline{u}-wP$). The function $h_{\delta}$ is of class $C^{2}$ on $D$ by claims 2 and 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, hence continuous by Continuity of the Value, Gradient and Hessian Maps of a Differentiable Function §value and upper semicontinuous by claim 2 of Semicontinuity Under Negation and Characterization of Continuity, and $W\le h_{\delta}$, so $W^{*}\le h_{\delta}$ by Properties of the Upper Semicontinuous Envelope §least; that is, $W^{*}-wP\le C+\delta P$. Hence $W^{*}-wP$ has $P$-subordinate growth from above, and the comparison principle applied to $W^{*}$ and $\overline{u}$ gives $W^{*}\le\overline{u}$. Also $\underline{u}\le W\le W^{*}$ by Properties of the Upper Semicontinuous Envelope §bounds. So $W^{*}\in\mathcal{G}$, whence $W^{*}\le W$, and therefore $W=W^{*}$. In particular $W$ is a viscosity subsolution of $F$ on $D$, and $W-wP$ has $P$-subordinate growth from above.

Step 3: the lower envelope is a supersolution. By Step 0 applied to $\underline{u}\le W$, $W$ is bounded below near each point of $D$, so its lower semicontinuous envelope $W_{*}$ is defined; it is lower semicontinuous on $D$ and $W_{*}\le W$ (Properties of the Lower Semicontinuous Envelope, by Duality §lsc, Properties of the Lower Semicontinuous Envelope, by Duality §bounds). For positive $\delta$ let $C'$ satisfy $g_{\delta}\le\underline{u}$ on $D$ with $g_{\delta}=(w-\delta)P-C'$ (Penalty-Subordinate Growth of a Function on an Open Subset of Euclidean Space §below for $\underline{u}-wP$); $g_{\delta}$ is continuous and hence lower semicontinuous, exactly as for $h_{\delta}$, and $g_{\delta}\le W$, so $g_{\delta}\le W_{*}$ by Properties of the Lower Semicontinuous Envelope, by Duality §greatest. Hence $W_{*}-wP$ has $P$-subordinate growth from below.

Suppose that $W_{*}$ is not a viscosity supersolution of $F$ on $D$. Then there are $\varphi:D\to\mathbb{R}$ of class $C^{2}$ on $D$ and $\hat x\in D$ such that $W_{*}-\varphi$ has a local minimum at $\hat x$ relative to $D$ and $F(\hat x,W_{*}(\hat x),D\varphi(\hat x),D^{2}\varphi(\hat x))<0$. Put $R=P(\hat x)+1$. By Basic Properties of the Sublevel Sets of a Penalty §sublevel-sets, $D_{R}=\{y\in D:P(y)<R\}$ is open and contains $\hat x$, so there is a positive $\kappa$ such that every $y$ with $d_{E}(y,\hat x)<\kappa$ lies in $D_{R}$. The operator $F$ is continuous and degenerate elliptic (condition 1 of Strictly Proper Second-Order Equation Operator), and $W$ is a viscosity subsolution bounded below near each point of $D$, so The Bump Construction: Raising a Subsolution Whose Lower Envelope Fails the Supersolution Inequality gives $U:D\to\mathbb{R}$ with: $U$ is a viscosity subsolution of $F$ on $D$ (The Bump Construction: Raising a Subsolution Whose Lower Envelope Fails the Supersolution Inequality §subsolution); $W\le U$ on $D$ and $W(x_{1})<U(x_{1})$ for some $x_{1}\in D$ (The Bump Construction: Raising a Subsolution Whose Lower Envelope Fails the Supersolution Inequality §above); and $U(y)=W(y)$ whenever $\kappa\le d_{E}(y,\hat x)$ (The Bump Construction: Raising a Subsolution Whose Lower Envelope Fails the Supersolution Inequality §localisation).

$U-wP$ has $P$-subordinate growth from above. The set $K=\overline{D_{R}}$ is compact, nonempty and contained in $D$ by Basic Properties of the Sublevel Sets of a Penalty §sublevel-sets; $U$ is upper semicontinuous on $D$, hence on $K$ (claim 4 of Semicontinuity and Continuity Under Composition with a Continuous Map), so by claim 1 of Semicontinuous Functions Attain Their Extrema on a Compact Set there is $M$ with $U\le M$ on $K$. Let $m$ be a lower bound of $P$ on $D$ (Basic Properties of the Sublevel Sets of a Penalty §bounded-below). Fix a positive $\delta$ and let $C$ be as in Step 2. If $d_{E}(y,\hat x)<\kappa$, then $y\in D_{R}\subseteq K$, so $U(y)\le M$ and $m\le P(y)<R$, whence $|P(y)|\le|m|+|R|$ and $-\delta P(y)\le\delta|m|$ (claims 3 and 6 of Properties of the Absolute Value in an Ordered Field); thus $U(y)-wP(y)\le M+|w|(|m|+|R|)+\delta|m|+\delta P(y)$. If $\kappa\le d_{E}(y,\hat x)$, then $U(y)-wP(y)=W(y)-wP(y)\le C+\delta P(y)$ by Step 2. Taking the larger of the two constants (claim 1 of Elementary Properties of the Maximum of Two Elements), $U-wP\le C''+\delta P$ on $D$.

The comparison principle applied to $U$ and $\overline{u}$ gives $U\le\overline{u}$; with $\underline{u}\le W\le U$ this puts $U$ in $\mathcal{G}$, so $U\le W$ on $D$, contradicting $W(x_{1})<U(x_{1})$. Hence $W_{*}$ is a viscosity supersolution of $F$ on $D$.

Step 4: existence and continuity. The comparison principle applied to the subsolution $W$ (Step 2) and the supersolution $W_{*}$ (Step 3) gives $W\le W_{*}$, so $W=W_{*}$. Thus $W$ is upper semicontinuous ($W=W^{*}$) and lower semicontinuous ($W=W_{*}$) on $D$, and it is both a viscosity subsolution and a viscosity supersolution of $F$ on $D$. By Step 2, $W-wP$ has $P$-subordinate growth from above, and by Step 3 and $W=W_{*}$ it has $P$-subordinate growth from below; that is, $W$ is of weight $w$. This proves claim 1 with $u=W$. For $x\in D$ and positive $\varepsilon$, upper and lower semicontinuity at $x$ give positive $\delta_{1},\delta_{2}$ with $W(y)<W(x)+\varepsilon$ and $W(x)-\varepsilon<W(y)$ when $d_{E}(y,x)$ is less than $\delta_{1}$, respectively $\delta_{2}$; for $d_{E}(y,x)$ less than the least of them, $|W(y)-W(x)|<\varepsilon$ by claim 9 of Properties of the Absolute Value in an Ordered Field. So $W$ is continuous on $D$.

Step 5: uniqueness. Let $u_{1},u_{2}$ both have the properties of claim 1. The comparison principle applied to the subsolution $u_{1}$ and the supersolution $u_{2}$ gives $u_{1}\le u_{2}$, and with the roles exchanged $u_{2}\le u_{1}$; so $u_{1}=u_{2}$. This is claim 2, and by it the function of claim 1 is $W$, which by Steps 1 and 4 is continuous with $\underline{u}\le W\le\overline{u}$: claim 3.