Let $\Sigma_{u}$ and $\Sigma_{v}$ be the singular sets of $u$ and $v$ respectively, and let $A=\Omega-(\Sigma_{u}\cup\Sigma_{v})$. Since $A$ is dense in $\Omega$, it suffices to show that $u=v$ on $A$.

By Caffarelli’s theory ([C1]), $A$ is an open set, and by Theorem 1.1, $A$ is connected. By Theorem 6.2, the difference $u-v$ satisfies the linear equation

on $A$, where $a^{ij}$ are locally uniformly elliptic and $C^{1,\alpha}$ in $A$. The conclusion follows from Theorem 6.1. ∎