= Zero-boundary harmonic uniqueness
{title2=$\Delta u=0,\quad u|_{\partial V}=0\ \Longrightarrow\ u=0$}
For a bounded regular region and a sufficiently smooth real <harmonic function>, apply the <divergence theorem> to $u\nabla u$. Its divergence is $|\nabla u|^2+u\Delta u$. The boundary term vanishes, hence the integral of the nonnegative gradient square is zero. The gradient vanishes and the boundary values fix each component's constant as zero. Taking the difference of two solutions gives uniqueness for the <Poisson equation> with <Dirichlet boundary data>. Unbounded regions require an additional condition at infinity.
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