Comparison of a Viscosity Subsolution with a Strict Classical Supersolution

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, and 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 of $v|_{\Omega}$ at $x$ and $D^2v(x)$ for its Hessian matrix at $x$; by Equality of Mixed Second Partial Derivatives and Symmetry of the Hessian the latter lies in $\mathcal{S}(n)$, so the quadruple $(x,v(x),Dv(x),D^2v(x))$ lies in the domain of $F$.

Assume the following two conditions.

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

2. (Boundary inequality) $u(x)\le v(x)$ for every $x\in\partial_{\mathbb{R}^n}\Omega$.

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