= Solution
The difference $w=v-u$ is a <harmonic function>, because the <Laplacian> is a <linear differential operator>:
$$
\Delta w=\Delta v-\Delta u=0.
$$
The boundary inequality says $w\geq0$ on $\partial D$. The minimum principle from part (ii) gives
$$
\boxed{v(p)-u(p)=w(p)\geq\min_{\partial D}w\geq0\quad(p\in D).}
$$
This is the <comparison principle for harmonic functions>. Applying it in both directions also shows why fixed <Dirichlet boundary conditions> determine at most one <harmonic function>.
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