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

In the setting of Second-Order Equations on Euclidean Open Sets and of Bounded Open Domain in Euclidean Space, let $F$ be a second-order equation operator on $\Omega$, let $\gamma\in\mathbb{R}$ be positive, 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 $X,Y\in\mathcal{S}(n)$,

$$F(x,r,p,X)=F(x,r,p,Y).$$

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 positive $\beta\in\mathbb{R}$,

$$F\bigl(y,r,\beta(x-y),X\bigr)-F\bigl(x,r,\beta(x-y),X\bigr)\le\omega\bigl(\beta\,d_{E}(x,y)^{2}+d_{E}(x,y)\bigr),$$

where $\beta(x-y)$ is the scalar multiple by $\beta$ of the difference $x-y$ in $\mathbb{R}^{n}$, 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

$$u(x)\le v(x)\qquad\text{for every }x\in\partial\Omega .$$

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