= Comparison principle for harmonic functions
{title2=$v\geq u\text{ on }\partial D\Longrightarrow v\geq u\text{ in }D$}
If $u,v$ are <harmonic functions> on a bounded domain and continuous on its closure, their difference is a <harmonic function>. Its minimum is attained on the boundary by the <maximum principle for harmonic functions> applied to its negative. Nonnegative boundary difference therefore implies nonnegative interior difference. Applying the comparison in both directions proves uniqueness for the <Dirichlet problem> whenever a classical solution exists. This is an order statement and does not itself establish existence.
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