Proof of Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality

Each result cited is universally quantified over the data in its own statement. Throughout, the weight of Comparison Principle with a Weighted Penalty for Operators Convex in the Value, Gradient and Matrix Variables and 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 is taken to be $w=0$; for $f:D\to\mathbb{R}$ the function $f-0P$ is $f$ itself, since $0\cdot P(x)=0$ by claim 1 of Zero Products and Elementary Identities in a Field.

Step 1: the operator hypotheses. Since $P$ is of class $C^{2}$ on $D$ (Penalty on an Open Subset of Euclidean Space §regularity), the map $b=DP:D\to\mathbb{R}^{n}$ is continuous by Continuity of the Value, Gradient and Hessian Maps of a Differentiable Function §hessian, and hypothesis Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality §one-sided is the one-sided bound 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 §one-sided for this $b$. By The Penalty-Drift Hamilton-Jacobi Equation on an Open Subset of Euclidean Space §operator, $F$ 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 drift $b=DP$ and $G=g$. That lemma shows that $F$ satisfies hypotheses 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.

Step 2: a classical subsolution of negative weight. Let $\varepsilon,C$ be as in hypothesis Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality §dissipation, put $\sigma=\varepsilon(\theta+1)^{-1}$ and $s=-\sigma$, and let $K=\lambda^{-1}(\sigma|C|+M)$. Then $0<\sigma$ (claims 7 and 5 of Elementary Order Arithmetic in an Ordered Field), so $s<0$; and $\theta\sigma\le\varepsilon$, because $\theta\le\theta+1$ gives $\theta\varepsilon\le(\theta+1)\varepsilon$ and multiplying by $(\theta+1)^{-1}$ preserves this (claim 5 of Elementary Arithmetic in an Ordered Field); hence $\tfrac{\theta\sigma}{2}\le\varepsilon$. The function $\varphi=sP-K$ is of class $C^{2}$ on $D$ with $D\varphi=sDP$ and $D^{2}\varphi=sD^{2}P$, by claims 1 to 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set. For $x\in D$, expanding the operator of The Penalty-Drift Hamilton-Jacobi Equation on an Open Subset of Euclidean Space §operator with the field axioms (and $\operatorname{tr}(sX)=s\operatorname{tr}(X)$ by claim 1 of Basic Properties of the Trace),

Since $1-\varepsilon\le1-\tfrac{\theta\sigma}{2}$ and $0\le\lVert DP(x)\rVert^{2}$, the bracket is at most $\tfrac{\kappa}{2}\operatorname{tr}(D^{2}P(x))-(1-\varepsilon)\lVert DP(x)\rVert^{2}-\lambda P(x)$, which is at most $C$ by the dissipation hypothesis. Multiplying by $\sigma\ge0$ and using $C\le|C|$, $-g(x)\le M$ (claim 3 of Properties of the Absolute Value in an Ordered Field) and $\lambda K=\sigma|C|+M$,

So $\varphi$ is a classical subsolution of $F$ on $D$ with $s<0$: this is hypothesis Comparison Principle with a Weighted Penalty for Operators Convex in the Value, Gradient and Matrix Variables §barrier with $w=0$.

Step 3: comparison. By Steps 1 and 2 all five hypotheses of Comparison Principle with a Weighted Penalty for Operators Convex in the Value, Gradient and Matrix Variables hold with $w=0$ and $\gamma=\lambda$. For $u$ and $v$ as in claim 1, $u-0P=u$ and $v-0P=v$ have the required growth, so that theorem gives $u\le v$ on $D$. This is claim 1.

Step 4: existence and uniqueness. For $c\in\mathbb{R}$ the constant function $c$ on $D$ is of class $C^{2}$ with vanishing gradient and Hessian (claims 1 and 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set), so $F(x,c,0,0)=\lambda c-g(x)$ for $x\in D$. With $c_{-}=-\lambda^{-1}M$ this equals $-M-g(x)\le0$, and with $c_{+}=\lambda^{-1}M$ it equals $M-g(x)\ge0$, by claim 3 of Properties of the Absolute Value in an Ordered Field. Thus $c_{-}$ is a classical subsolution and $c_{+}$ a classical supersolution of $F$ on $D$; as $F$ is degenerate elliptic (condition 1 of Strictly Proper Second-Order Equation Operator, Step 1), they are viscosity subsolution and supersolution by Classical Sub- and Supersolutions of a Degenerate Elliptic Operator are Viscosity Sub- and Supersolutions. Each constant $c$ has $P$-subordinate growth from above and from below: let $m$ be a lower bound of $P$ (Basic Properties of the Sublevel Sets of a Penalty §bounded-below); for positive $\delta$, $-\delta|m|\le\delta m\le\delta P(x)$, so $c\le(|c|+\delta|m|)+\delta P(x)$ and $-(|c|+\delta|m|)-\delta P(x)\le c$, which are Penalty-Subordinate Growth of a Function on an Open Subset of Euclidean Space §above and Penalty-Subordinate Growth of a Function on an Open Subset of Euclidean Space §below. Hence $c_{-}$ and $c_{+}$ are of weight $0$ in the sense 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, whose hypotheses on $F$ hold by Steps 1 and 2. Its claims 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, 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 and 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 give exactly one function that is both a viscosity subsolution and a viscosity supersolution of $F$ on $D$ with $P$-subordinate growth from above and from below, and show that it is continuous with $c_{-}\le u\le c_{+}$. This is claim 2.