Uniqueness for the Dirichlet Problem for Second-Order Equations

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 that $F$ is continuous and strictly proper with constant $\gamma$, and that $F$ and $\omega$ satisfy the structure condition of the comparison principle.

Let $u,v:\overline{\Omega}\to\mathbb{R}$ be continuous on $\overline{\Omega}$, as maps into the metric space $(\mathbb{R},d_{\mathbb{R}})$ of The Absolute Value Metric on the Real Numbers, and let $u|_{\Omega},v|_{\Omega}:\Omega\to\mathbb{R}$ be the functions whose values at $x\in\Omega$ are $u(x)$ and $v(x)$. Assume that $u|_{\Omega}$ and $v|_{\Omega}$ are viscosity solutions of $F$ on $\Omega$ and that

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

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