Comparison Principle for First-Order Strictly Proper Equations by Doubling of Variables

Let $n\ge1$ be a natural number and let $\mathbb{R}$ be the ordered field of real numbers.

Equip Euclidean space $\mathbb{R}^n$ with the Euclidean distance $d_E$, a metric by Euclidean Distance is a Metric on $\mathbb{R}^n$, and with the topology $\mathcal{T}_{d_E}$ of its metric-open subsets, a topology by Metric Open Sets Form a Topology that agrees with Euclidean openness by Euclidean Openness Agrees with Metric Openness on $\mathbb{R}^n$. Semicontinuity and local extrema refer to $d_E$, and the closure $\operatorname{cl}_{\mathbb{R}^n}$ and boundary $\partial_{\mathbb{R}^n}$ are taken in $(\mathbb{R}^n,\mathcal{T}_{d_E})$.

Let $\Omega\subseteq\mathbb{R}^n$ be nonempty, open and bounded, and write $\overline{\Omega}=\operatorname{cl}_{\mathbb{R}^n}(\Omega)$. Let $\mathcal{S}(n)$ be the set of symmetric real $n\times n$ matrices, let $F$ be a second-order equation operator on $\Omega$, let $\gamma\in\mathbb{R}$ satisfy $0<\gamma$, and let $\omega$ be a modulus of continuity. Assume the following three conditions.

1. (Independence of the matrix argument) For every $x\in\Omega$, every $r\in\mathbb{R}$, every $p\in\mathbb{R}^n$ and all $X,Y\in\mathcal{S}(n)$,

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

3. (Structure condition) For all $x,y\in\Omega$, every $r\in\mathbb{R}$, every $X\in\mathcal{S}(n)$ and every $\beta\in\mathbb{R}$ with $0<\beta$,

where $\beta(x-y)$ is the scalar multiple by $\beta$ of the difference of points of $\mathbb{R}^n$, and the argument of $\omega$ is nonnegative.

Let $u:\overline{\Omega}\to\mathbb{R}$ be a viscosity subsolution of $F$ up to the boundary of $\Omega$, let $v:\overline{\Omega}\to\mathbb{R}$ be a viscosity supersolution of $F$ up to the boundary of $\Omega$, and assume

Then $u(x)\le v(x)$ for every $x\in\overline{\Omega}$.