= Neumann compatibility condition for Laplace's equation
{c}
{title2=$\int_{\partial\Omega}g\,dS=0$}
For a smooth bounded region $\Omega$, prescribing the outward <normal derivative> $\partial_nu=g$ of a <harmonic function> requires
$$
\int_{\partial\Omega}g\,dS=0.
$$
This follows from the <divergence theorem> applied to the <gradient>:
$$
\int_{\partial\Omega}\partial_nu\,dS
=\int_\Omega\nabla\cdot\nabla u\,dV
=\int_\Omega\Delta u\,dV=0.
$$
On the unit <disk>, the condition becomes $\int_0^{2\pi}g(\theta)\,d\theta=0$. If $g=\sum_{n\geq1}(a_n\cos(n\theta)+b_n\sin(n\theta))$ is sufficiently regular with zero mean, its regular <harmonic expansion on a disk> is
$$
u(r,\theta)=C+\sum_{n\geq1}\frac{r^n}{n}(a_n\cos(n\theta)+b_n\sin(n\theta)).
$$
The 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.
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