Comparison Principle with a Weighted Penalty for Operators Convex in the Value, Gradient and Matrix Variables

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 $D_{R}=\{x\in D:P(x)<R\}$, a bounded open set by Basic Properties of the Sublevel Sets of a Penalty §sublevel-sets. Let $w\in\mathbb{R}$, let $\gamma\in\mathbb{R}$ be positive and let $F$ be a second-order equation operator on $D$. Assume the following five conditions.

1. (Continuity)¶ $F$ is continuous.

2. (Strict properness)¶ $F$ is strictly proper with constant $\gamma$.

3. (Convexity in $(r,p,X)$)¶ $F$ is convex in $(r,p,X)$.

4. (Structure condition on the sublevel sets of $P$)¶ For every $R\in\mathbb{R}$ with $D_{R}$ nonempty there is a modulus of continuity $\omega_{R}$ such that the restriction $F|_{D_{R}}$ of Restriction of a Second-Order Equation Operator to an Open Subset §operator and $\omega_{R}$ satisfy the structure condition of the comparison principle for the Dirichlet problem with $D_{R}$ in place of $\Omega$.

5. (A classical subsolution of weight below $w$)¶ There are $s,K\in\mathbb{R}$ with $s<w$ such that the function $sP-K$ on $D$, with value $sP(x)-K$ at $x$, is a classical subsolution of $F$ on $D$; it is of class $C^{2}$ on $D$ by Penalty on an Open Subset of Euclidean Space §regularity and Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set.

Let $u:D\to\mathbb{R}$ be a viscosity subsolution of $F$ on $D$ such that $u-wP$ has $P$-subordinate growth from above, and let $v:D\to\mathbb{R}$ be a viscosity supersolution of $F$ on $D$ such that $v-wP$ has $P$-subordinate growth from below.

Then $u(x)\le v(x)$ for every $x\in D$.