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

In the setting of Second-Order Equations on Euclidean Open Sets, let $n\ge1$ be a natural number, let $D\subseteq\mathbb{R}^{n}$ be open, let $P$ be a penalty on $D$, and for $R\in\mathbb{R}$ let $D_{R}=\{x\in D:P(x)<R\}$. We write $\operatorname{tr}$ for the trace of a square real matrix, abbreviate $\lVert z\rVert^{2}=\lVert z\rVert\lVert z\rVert$, and write $\tfrac{a}{2}$ for the product of $a\in\mathbb{R}$ with the multiplicative inverse of $2=1+1$.

Let $\lambda\in\mathbb{R}$ be positive and let $\theta,\kappa\in\mathbb{R}$ be nonnegative. Let $b:D\to\mathbb{R}^{n}$ and $G:D\to\mathbb{R}$ be continuous, with $D$ and $\mathbb{R}^{n}$ carrying the metric $d_{E}$ and $\mathbb{R}$ the metric $d_{\mathbb{R}}$ of the real line, and assume that $b$ is one-sided Lipschitz on sublevel sets:¶ for every $R\in\mathbb{R}$ there is a nonnegative $c_{R}\in\mathbb{R}$ such that

Let $F$ be the second-order equation operator on $D$ given by

Then $F$ satisfies the 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 of the comparison principle with a weighted penalty.