Proof of Well-Posedness of the Singular-Cost Dyson Hamilton-Jacobi Equation above the Hardy Threshold

Each result cited is universally quantified over the data in its own statement.

Throughout, $H=H_{\beta}$, and $a_{kj}$, $S$ and $d_{N}$ are as in The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity; $W_{N}$ is open and nonempty by The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity §open, and $\Pi_{N}$ is as in The Logarithmic Energy of N Ordered Particles on the Weyl Chamber §energy. We write $n_{\Pi}=\sum_{(i,j)\in\Pi_{N}}1$, $N_{\mathbb{R}}=\sum_{k=1}^{N}1$ (a real number), $q_{0}(x)=\lVert x\rVert^{2}$, $q(x)=1+\lVert x\rVert^{2}$ and $l(x)=\log q(x)$. For a function $\varphi$ of class $C^{2}$ on $W_{N}$ we abbreviate $F(x,\varphi(x),D\varphi(x),D^{2}\varphi(x))$ to $F[\varphi](x)$. Elementary order arithmetic is by Elementary Order Arithmetic in an Ordered Field, Elementary Arithmetic in an Ordered Field and Properties of the Absolute Value in an Ordered Field.

Step 0 (conventions and elementary facts). (a) Signs. $0\le d_{N}$ and $0\le N_{\mathbb{R}}$, being finite sums of numbers equal to $0$ or $1$ (claim 5 of Properties of Finite Sums). $\Pi_{N}$ contains $(1,2)$, the sum of $1$ over $\{(1,2)\}$ is $1$ by claim 1 of Peeling, Splitting, and Interchange for Sums over a Finite Index Set, so $1\le n_{\Pi}$ by Nonnegativity and Monotonicity of a Sum over a Finite Index Set §monotone. For $x\in W_{N}$ we have $0\le\lVert x\rVert^{2}$, so $1\le q(x)$ and $0<q(x)^{-1}\le1$ by Order Reversal under Reciprocals, and Summability of the Reciprocals of the Squares §reciprocal. Finally $0\le|g(x)|\le M$ at any point $x$ of $W_{N}$, so $0\le M$.

(b) Logarithm. By The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §log, $1-t^{-1}\le\log t\le t-1$ for positive $t$, and $\log(st)=\log s+\log t$ for positive $s,t$ by The Natural Logarithm. Hence: (b1) if $0<y\le z$ then $\log y\le\log z$, because $\log z=\log y+\log(zy^{-1})$ and $\log(zy^{-1})\ge1-yz^{-1}\ge0$, as $yz^{-1}\le1$; (b2) $\log(t^{2})=2\log t$ for positive $t$; (b3) $0\le1-q(x)^{-1}\le l(x)$; (b4) $0<1-2^{-1}\le\log2$.

(c) Calculus. If $\varphi,\psi$ are of class $C^{2}$ on an open set and $c\in\mathbb{R}$, then $c\varphi+\psi$ is of class $C^{2}$, with gradient $cD\varphi+D\psi$ and Hessian $cD^{2}\varphi+D^{2}\psi$: this is claims 1 and 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, applied to first and to iterated partial derivatives, read through Gradient of a Real-Valued Function on a Euclidean Open Set and Hessian Matrix of a C^2 Function. A constant function is smooth (claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set), hence of class $C^{2}$ by Smooth Map on a Euclidean Open Set, and its partial derivatives vanish, its difference quotients in Partial Derivative on a Euclidean Open Set being $0$. Traces satisfy $\operatorname{tr}(cX+Y)=c\operatorname{tr}X+\operatorname{tr}Y$ by claim 1 of Basic Properties of the Trace. By claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, $\lVert z\rVert^{2}=z\cdot z$, so by Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$

(d) The squared norm. By claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, $q_{0}(x)=1\cdot d_{E}(x,0_{\mathbb{R}^{N}})^{2}$, so by A Scaled Squared Distance to a Point is of Class $C^2$, with Gradient and Hessian (with $c=1$ and $a$ the origin), on every open subset of $\mathbb{R}^{N}$ the function $q_{0}$ is of class $C^{2}$ with $Dq_{0}(x)=2x$ and $D^{2}q_{0}(x)=2I_{N}$. By (c), the same holds for $q=1+q_{0}$ on $W_{N}$.

Step 1 (the profile; claim 1). Let $r=\sqrt{\kappa^{2}+8\theta c}$ and $r'=\sqrt{\lambda^{2}+8\theta\omega}$, so $r^{2}=\kappa^{2}+8\theta c$ and $r'^{2}=\lambda^{2}+8\theta\omega$ by Existence and Uniqueness of the Nonnegative Square Root. Since $\beta=\tfrac12(\kappa+r)$ gives $\beta-\kappa=\tfrac12(r-\kappa)$,

Since $a=(4\theta)^{-1}(r'-\lambda)$ gives $2\theta a+\lambda=\tfrac12(r'+\lambda)$,

Let $V_{1}(t)=\theta a\,t^{2}$ for $t\in\mathbb{R}$. By A Scaled Squared Distance to a Point is of Class $C^2$, with Gradient and Hessian with $n=1$, $c=\theta a$ and $a$ the origin (and claims 1 and 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ in $\mathbb{R}^{1}$), $V_{1}$ is of class $C^{2}$ on $\mathbb{R}$ with $V_{1}'(t)=2\theta a\,t$ and $V_{1}''(t)=2\theta a$. Moreover $|t|^{2}=t^{2}$ (claims 1 and 4 of Properties of the Absolute Value in an Ordered Field), so

and dividing by $4\theta a>0$ gives $1\cdot|t|-(4\theta a)^{-1}\le V_{1}(t)$. Let $\widehat{P}(x)=H(x)+\sum_{k=1}^{N}V_{1}(x_{k})$. By claim 3 of Properties of Finite Sums and claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, $\sum_{k=1}^{N}V_{1}(x_{k})=\theta a\lVert x\rVert^{2}$, so $\widehat{P}=\theta P$. By The Confined Logarithmic Energy on the Weyl Chamber: Regularity, Penalty Property, Monotone Gradient and the Dissipation Inequality §penalty with $a_{0}=1$ and $b_{0}=(4\theta a)^{-1}$, $\widehat{P}$ is a penalty on $W_{N}$. Hence $P=\theta^{-1}\widehat{P}$ is of class $C^{2}$ by (c) and Penalty on an Open Subset of Euclidean Space §regularity; and for $t\in\mathbb{R}$, multiplying by $\theta$ and by $\theta^{-1}$ (claim 5 of Elementary Arithmetic in an Ordered Field) shows $\{x\in W_{N}:P(x)\le t\}=\{x\in W_{N}:\widehat{P}(x)\le\theta t\}$, which is compact by Penalty on an Open Subset of Euclidean Space §sublevel. So $P$ is a penalty on $W_{N}$, and claim 1 holds.

Step 2 (derivatives of $P$). By The Confined Logarithmic Energy on the Weyl Chamber: Regularity, Penalty Property, Monotone Gradient and the Dissipation Inequality §regularity with this $V_{1}$ and The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity §derivatives, $\partial_{k}\widehat{P}(x)=2\theta a\,x_{k}+\partial_{k}H(x)$ and $\operatorname{tr}(D^{2}\widehat{P}(x))=\beta S(x)+\sum_{k=1}^{N}2\theta a=\beta S(x)+2\theta a N_{\mathbb{R}}$. By (c),

By (c), The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity §calogero and The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity §euler,

Step 3 (the key identity). Let $s,k\in\mathbb{R}$ and $\varphi=sP+k$. By (c), $\varphi$ is of class $C^{2}$ with $D\varphi=sDP$ and $D^{2}\varphi=sD^{2}P$, and $\lVert sDP\rVert^{2}=s^{2}\lVert DP\rVert^{2}$. Insert Step 2 into the operator, write $\lambda sP=\lambda s\theta^{-1}H+\lambda sa\lVert x\rVert^{2}$, and use $c=(2\theta)^{-1}(\beta^{2}-\kappa\beta)$ from Step 1. Collecting the coefficients of $S(x)$ and of $\lVert x\rVert^{2}$ gives, for $x\in W_{N}$,

Here $e(1)=0$, and $f(1)=0$ by Step 1.

Step 4 (two bounds on $H$). (U) For every positive $\mu$ and $x\in W_{N}$,

Indeed, let $(i,j)\in\Pi_{N}$ and $t=x_{i}-x_{j}>0$, so $a_{ij}(x)=t^{-1}$. The lower bound of (b) at $\mu t^{2}$, together with (b2), gives $\log\mu+2\log t=\log(\mu t^{2})\ge1-\mu^{-1}a_{ij}(x)^{2}$, so $-\log t\le\tfrac12(\mu^{-1}a_{ij}(x)^{2}-1+\log\mu)$. Squares are nonnegative, so $a_{ij}(x)^{2}\le\sum_{j'=1}^{N}a_{ij'}(x)^{2}\le S(x)$ by claims 5 and 6 of Properties of Finite Sums. Hence $-\log(x_{i}-x_{j})\le\tfrac12(\mu^{-1}S(x)-1+\log\mu)$ for every $(i,j)\in\Pi_{N}$. Summing over $\Pi_{N}$ (Nonnegativity and Monotonicity of a Sum over a Finite Index Set §comparison, and claims 3 and 4 of Properties of a Sum over a Finite Index Set) and multiplying by $\beta$ gives (U).

(L) For every $x\in W_{N}$,

Indeed, for $(i,j)\in\Pi_{N}$, claims 2, 3 and 5 of Properties of the Absolute Value in an Ordered Field and claim 4 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ give $0<x_{i}-x_{j}\le|x_{i}|+|x_{j}|\le2\lVert x\rVert$, so $(x_{i}-x_{j})^{2}\le4\lVert x\rVert^{2}\le4q(x)$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. By (b1) and (b2), $2\log(x_{i}-x_{j})\le\log4+l(x)=2\log2+l(x)$. Summing over $\Pi_{N}$ as in (U) and multiplying by $-\beta$ gives (L).

Step 5 (classical subsolutions of weight below $1$). Let $s_{0}=2^{-1}\bigl(1+\kappa(2\beta)^{-1}\bigr)$. As stated in the theorem, $0\le\kappa<2\beta$, so $0\le\kappa(2\beta)^{-1}<1$ and, by claim 8 of Elementary Order Arithmetic in an Ordered Field, $0<s_{0}$ and $\kappa(2\beta)^{-1}<s_{0}<1$. Fix $s$ with $s_{0}\le s<1$; then $\tfrac{\kappa}{2}<s\beta<\beta$. Since $t^{2}-\kappa t-(u^{2}-\kappa u)=(t-u)(t+u-\kappa)$,

the first factor being negative and the second positive (as $s\beta>\tfrac{\kappa}{2}$ and $\beta>\tfrac{\kappa}{2}$); and

Put $\mu_{s}=\lambda s\theta^{-1}\beta n_{\Pi}\bigl(2(-e(s))\bigr)^{-1}$, which is positive since $n_{\Pi}\ge1$; $C_{s}=\lambda s\theta^{-1}\tfrac{\beta n_{\Pi}}{2}(\log\mu_{s}-1)$; and $K_{s}=\lambda^{-1}(|C_{s}|+M)\ge0$. Multiplying (U) with $\mu=\mu_{s}$ by $\lambda s\theta^{-1}>0$ gives $\lambda s\theta^{-1}H(x)\le-e(s)S(x)+C_{s}$. By Step 3 with $k=-K_{s}$, using $f(s)\lVert x\rVert^{2}\le0$, $-s^{2}a\beta d_{N}\le0$, $-\kappa saN_{\mathbb{R}}\le0$ (Step 0(a)) and $-g(x)\le M$,

So $sP-K_{s}$ is a classical subsolution of $F$ on $W_{N}$. With $s=s_{0}<1$ this is hypothesis Comparison Principle with a Weighted Penalty for Operators Convex in the Value, Gradient and Matrix Variables §barrier for $w=1$.

Step 6 (the operator hypotheses). Let $b:W_{N}\to\mathbb{R}^{N}$ be the constant map with value the origin and $G(x)=cS(x)+\omega q_{0}(x)+g(x)$. By claim 4 of Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$ (with $\mu=0$), $b(x)\cdot p=0$, so $F$ is the operator of The Viscous Hamilton-Jacobi Operator with a Drift that is One-Sided Lipschitz on Sublevel Sets Satisfies the Operator Hypotheses of the Weighted-Penalty Comparison Principle with $n=N$, $D=W_{N}$, the penalty $P$, and $\lambda,\theta,\kappa,b,G$. The map $b$ is continuous, and it is one-sided Lipschitz on sublevel sets with $c_{R}=0$, since $(b(x)-b(y))\cdot(x-y)=0$. For continuity of $G$: $H$ is of class $C^{2}$ by The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity §derivatives, so $DH$ is continuous on $W_{N}$ by Continuity of the Value, Gradient and Hessian Maps of a Differentiable Function §hessian; $q_{0}$ is continuous on $\mathbb{R}^{N}$ by The Squared Euclidean Norm is Smooth and claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous; so $q_{0}\circ DH$ and the restriction of $q_{0}$ to $W_{N}$ are continuous on $W_{N}$ by claims 3 and 4 of Semicontinuity and Continuity Under Composition with a Continuous Map. By The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity §calogero, $G=c\beta^{-2}(q_{0}\circ DH)+\omega q_{0}+g$. A combination $c_{1}f_{1}+c_{2}f_{2}+c_{3}f_{3}$ of real functions continuous on $W_{N}$ is continuous (Continuous Map Between Metric Spaces, with the metric of The Absolute Value Metric on the Real Line): given $x$ and $\varepsilon>0$, take for each $i$ a $\delta_{i}$ that works for $f_{i}$ at $x$ with $\varepsilon\bigl(3(1+|c_{i}|)\bigr)^{-1}$, and the least $\delta$ of the three (claim 9 of Elementary Order Arithmetic in an Ordered Field); by claims 4 and 5 of Properties of the Absolute Value in an Ordered Field the combination then varies by less than $\varepsilon$. Hence $G$ is continuous and, by The Viscous Hamilton-Jacobi Operator with a Drift that is One-Sided Lipschitz on Sublevel Sets Satisfies the Operator Hypotheses of the Weighted-Penalty Comparison Principle, $F$ satisfies Comparison Principle with a Weighted Penalty for Operators Convex in the Value, Gradient and Matrix Variables §continuity, Comparison Principle with a Weighted Penalty for Operators Convex in the Value, Gradient and Matrix Variables §strictly-proper with $\gamma=\lambda$, Comparison Principle with a Weighted Penalty for Operators Convex in the Value, Gradient and Matrix Variables §convex and Comparison Principle with a Weighted Penalty for Operators Convex in the Value, Gradient and Matrix Variables §structure. In particular $F$ is degenerate elliptic, by claim 1 of Strictly Proper Second-Order Equation Operator.

Step 7 (comparison; claim 2). Apply Comparison Principle with a Weighted Penalty for Operators Convex in the Value, Gradient and Matrix Variables with $D=W_{N}$, the penalty $P$ of Step 1, $w=1$ and $\gamma=\lambda$; its hypotheses 1 to 4 hold by Step 6, and hypothesis 5 by Step 5. Since $1\cdot P=P$, the growth assumptions of claim 2 are exactly those on $u-wP$ and $v-wP$, so $u(x)\le v(x)$ for every $x\in W_{N}$.

Step 8 (a viscosity subsolution of weight $1$). Let $m$ be as in Basic Properties of the Sublevel Sets of a Penalty §bounded-below, so $m\le P$. Let $\mathcal{F}=\{sP-K_{s}:s_{0}\le s<1\}$, a nonempty set. Each member is of class $C^{2}$ and a classical subsolution (Step 5), hence a viscosity subsolution by claim 1 of Classical Sub- and Supersolutions of a Degenerate Elliptic Operator are Viscosity Sub- and Supersolutions, $F$ being degenerate elliptic (Step 6). For $s_{0}\le s<1$ we have $0<1-s\le1$, so for $y\in W_{N}$

thus every $v\in\mathcal{F}$ satisfies $v\le P+|m|$. Fix $x\in W_{N}$. By Continuity of the Value, Gradient and Hessian Maps of a Differentiable Function §value, $P$ is continuous at $x$, so there is $\delta>0$ with $P(y)<P(x)+1$ whenever $d_{E}(x,y)<\delta$; with $r=\tfrac{\delta}{2}$, every $v\in\mathcal{F}$ satisfies $v(y)\le P(x)+1+|m|$ whenever $d_{E}(y,x)\le r$. So $\mathcal{F}$ is locally uniformly bounded above, and since $F$ is continuous (Step 6), The Upper Semicontinuous Envelope of a Supremum of Viscosity Subsolutions is a Viscosity Subsolution shows that $\underline{u}=\widetilde{u}^{*}$, where $\widetilde{u}(x)=\sup\{v(x):v\in\mathcal{F}\}$, is a viscosity subsolution of $F$ on $W_{N}$.

The function $P+|m|$ is of class $C^{2}$ by (c), hence continuous by Continuity of the Value, Gradient and Hessian Maps of a Differentiable Function §value, hence upper semicontinuous on $W_{N}$ (claim 3 of Properties of the Absolute Value in an Ordered Field). As $\widetilde{u}\le P+|m|$, Properties of the Upper Semicontinuous Envelope §least gives $\underline{u}\le P+|m|$. Growth from above: for $\delta>0$, $\delta P\ge\delta m\ge-\delta|m|$, so $\underline{u}-P\le|m|\le(1+\delta)|m|+\delta P$. Growth from below: by Properties of the Upper Semicontinuous Envelope §bounds, $\underline{u}\ge\widetilde{u}\ge sP-K_{s}$ for every $s\in[s_{0},1)$. If $0<\delta\le1-s_{0}$, take $s=1-\delta$: $\underline{u}-P\ge-K_{1-\delta}-\delta P$. If $\delta>\delta_{0}=1-s_{0}$, take $s=s_{0}$: $\underline{u}-P\ge-K_{s_{0}}-\delta_{0}P=-K_{s_{0}}-\delta P+(\delta-\delta_{0})P\ge-\bigl(K_{s_{0}}+(\delta-\delta_{0})|m|\bigr)-\delta P$. So $\underline{u}-P$ has $P$-subordinate growth from above and from below.

Step 9 (a viscosity supersolution of weight $1$). Let $L=\beta n_{\Pi}(2\theta)^{-1}>0$, $\Lambda=\theta^{-1}\beta n_{\Pi}\log2\ge0$ (by (b4)),

Derivatives of $l$. The set $V=(0,\infty)$ is open in $\mathbb{R}^{1}$ and $q(x)\in V$ for $x\in W_{N}$. By The Natural Logarithm, $\log$ is smooth on $V$ with derivative $\rho(t)=t^{-1}$; $\rho=\partial_{1}\log$ is smooth on $V$ by claim 3 of Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map, and $\partial_{1}\rho(t)=-(t^{-1})^{2}$ by claim 2 of Reciprocal Rule for One-Dimensional Derivatives (for a function of one variable on an open interval, the limit defining the partial derivative in Partial Derivative on a Euclidean Open Set is the one defining the derivative in Derivative at an Interior Point). By claims 1 and 2 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$ (smooth maps being of class $C^{k}$ for every $k$ by Smooth Map on a Euclidean Open Set) and Step 0(d), $l=\log\circ q$ is of class $C^{2}$ on $W_{N}$ with $\partial_{i}l=(\rho\circ q)\,\partial_{i}q$, i.e. $Dl(x)=2q(x)^{-1}x$; and $\rho\circ q$ is of class $C^{1}$ with $\partial_{i}(\rho\circ q)(x)=-q(x)^{-2}\,2x_{i}$. By the product rule (claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set) and $\partial_{i}\partial_{i}q=2$ (Step 0(d), Hessian Matrix of a C^2 Function, Identity Matrix),

by Trace of a Real Square Matrix, claims 2 and 3 of Properties of Finite Sums, claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and Step 0(a).

Supersolution. By (c), $\overline{u}$ is of class $C^{2}$ with $D\overline{u}=DP+L\,Dl$ and $D^{2}\overline{u}=D^{2}P+L\,D^{2}l$. Expanding $\lVert DP+L\,Dl\rVert^{2}$ by (c) and using Step 3 with $s=1$, $k=K'$ and $e(1)=f(1)=0$,

all functions evaluated at $x$. By (L), $\theta^{-1}H\ge-\Lambda-L\,l$, so $\lambda\theta^{-1}H+\lambda L\,l\ge-\lambda\Lambda$. By Step 2, Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$ and The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity §euler,

since $0<q^{-1}\le1$ and $0\le d_{N}$; so $\theta L\,DP\cdot Dl\ge-L\beta d_{N}$. Also $\tfrac{\theta}{2}L^{2}\lVert Dl\rVert^{2}\ge0$, $-\tfrac{\kappa}{2}L\operatorname{tr}(D^{2}l)\ge-\kappa LN_{\mathbb{R}}$ and $-g\ge-M$. Therefore

so $\overline{u}$ is a classical supersolution, hence a viscosity supersolution by claim 2 of Classical Sub- and Supersolutions of a Degenerate Elliptic Operator are Viscosity Sub- and Supersolutions.

Weight. $\overline{u}-P=L\,l+K'$. From below: for $\delta>0$, $-\delta|m|-\delta P\le0\le L\,l+K'$ by (b3). From above: fix $\delta>0$ and let $\eta=\delta a\bigl((1+\delta)L\bigr)^{-1}>0$. The upper bound of (b) at $\eta q(x)$ gives $\log\eta+l(x)\le\eta q(x)-1$, hence

By (L), $a\lVert x\rVert^{2}=P(x)-\theta^{-1}H(x)\le P(x)+\Lambda+L\,l(x)$, so $\delta a\lVert x\rVert^{2}\le\delta P(x)+\delta\Lambda+\delta L\,l(x)$. Combining and subtracting $\delta L\,l(x)$,

So $\overline{u}-P$ has $P$-subordinate growth from above and from below.

Step 10 (existence and uniqueness; claim 3). Apply 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 with $D=W_{N}$ (open and nonempty), the penalty $P$, $w=1$ and $\gamma=\lambda$: its operator hypotheses hold by Step 6 and its barrier by Step 5 ($s_{0}<1$). Since $1\cdot P=P$, a function $f$ is of weight $1$ in the sense of that theorem exactly when $f-P$ has $P$-subordinate growth from above and from below; so $\underline{u}$ (Step 8) and $\overline{u}$ (Step 9) are a viscosity subsolution and a viscosity supersolution of weight $1$. By 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 §existence there is a function $u$ with the properties of claim 3, by 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 §uniqueness any two such functions are equal, and by 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 §continuity it is continuous on $W_{N}$. This proves claim 3.