Neumann Poisson problem ($-\Delta u=f,{\partial }_{n}u=0$)

On a smooth bounded connected domain, the homogeneous Neumann problem seeks $u\in H^1(U)$ such that $\int \nabla u\cdot \nabla v=\int fv$ for every $v\in H^1(U)$. For $f\in L^2(U)$ it is solvable exactly when $\int f=0$, and the weak solution is unique up to constants. The Poincare-Wirtinger inequality and the Riesz representation theorem give existence on the mean-zero Sobolev space. The test function one gives necessity of the compatibility condition.

In the upper half-space $z>0$, the displayed decaying harmonic function has outward normal derivative ${\partial }_n\Phi =-{\partial }_z\Phi =z/[2\pi (x^2+y^2+z^2{)}^{3/2}]$. This is the Poisson kernel for the upper half-space, whose integral on a horizontal plane is one and whose limit as $z\downarrow 0$ is the two-dimensional Dirac delta. Thus ${\partial }_n\Phi =\delta (x)\delta (y)$ in the distributional boundary sense. The compensating flux through a large hemisphere is minus one. This illustrates why a nonzero total boundary source is possible on an unbounded domain even when the interior source is zero.