Neumann compatibility condition for Laplace's equation (${\int }_{\partial \Omega }gdS=0$)

For a smooth bounded region $\Omega$, prescribing the outward normal derivative ${\partial }_{n}u=g$ of a harmonic function requiresThis follows from the divergence theorem applied to the gradient:On the unit disk, the condition becomes ${\int }_0^{2\pi }g(\theta )d\theta =0$. If $g={\sum }_{n\ge 1}(a_n\cos (n\theta )+b_n\sin (n\theta ))$ is sufficiently regular with zero mean, its regular harmonic expansion on a disk isThe factor $1/n$ compensates for the radial derivative. The arbitrary $C$ reflects the uniqueness of the Neumann problem up to an additive constant on a connected region.