= Gradient pairing with a boundary-constant function and a harmonic function
{title2=$\int_V\nabla f\cdot\nabla g\,dV=0$}
If $f$ is constant on the boundary and $g$ is a regular <harmonic function> throughout a volume, the <divergence theorem> applied to $f\nabla g$ gives $\int_V\nabla f\cdot\nabla g=\int_{\partial V}f\partial_ng$. The constant boundary value factors out; a second application of the divergence theorem makes the remaining flux $\int_V\Delta g=0$. This requires enough regularity to use <Green's first identity> over the entire volume.
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