Comparison of a Viscosity Subsolution with a Strict Classical Supersolution

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 $\Omega$ that is proper.

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 lower semicontinuous on $\overline{\Omega}$, and let $v|_{\Omega}:\Omega\to\mathbb{R}$, the function whose value at $x\in\Omega$ is $v(x)$, be of class $C^{2}$ on $\Omega$. For $x\in\Omega$ write $Dv(x)$ for the gradient and $D^{2}v(x)$ for the Hessian of $v|_{\Omega}$ at $x$; by that clause $D^{2}v(x)\in\mathcal{S}(n)$, so the quadruple $(x,v(x),Dv(x),D^{2}v(x))$ lies in the domain of $F$.

Assume the following two conditions.

1. (Strict supersolution inequality) $0<F(x,v(x),Dv(x),D^{2}v(x))$ for every $x\in\Omega$.

2. (Boundary inequality) $u(x)\le v(x)$ for every $x\in\partial\Omega$.

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