= Solution
If $u_1,u_2$ satisfy the same <boundary value problem>, their difference $w=u_1-u_2$ satisfies $Lw=0$ in $D$ and $w=0$ on the boundary, by linearity. Part (b)(ii) gives
$$
w\leq\max_{\partial D}w=0.
$$
Apply it again to $-w$, which also satisfies $L(-w)=0$ and zero <Dirichlet boundary conditions>. This gives $-w\leq0$. Hence $w=0$ throughout $\overline D$, proving
$$
\boxed{u_1=u_2.}
$$
This proves uniqueness whenever a solution exists; existence is the separately supplied assumption. When the drift coefficients vanish, the same argument uses part (a).
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