Neumann boundary-value problem

A boundary value problem with prescribed outward normal derivative, or the associated weighted flux, on the boundary. Its boundary data are Neumann boundary data. For an elliptic divergence-form operator without a zeroth-order term, integration over the domain imposes a compatibility condition between total source and boundary flux. A connected domain generally leaves a constant undetermined, fixed by a mean or reference value.

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.

On a connected bounded domain, two harmonic potentials with the same normal derivative differ by a constant. Their difference $u$ has ${\nabla }^{2}u=0$ and ${\partial }_{n}u=0$. Green's first identity gives $\int |\nabla u{|}^{2}dV={\int }_{\partial \Omega }u{\partial }_{n}udA=0$, so the gradient vanishes. In a multiply connected magnetic core, also fix the circulation around each independent hole: a curl-free magnetic field intensity with nonzero circulation need not be the gradient of a single-valued scalar potential. After those circulations have been matched, apply the scalar-potential uniqueness argument to the difference field.