Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 3 / 14A (Methods)

Apply Green second identity to $u$ and the free-space Green function $G_{fs}$ on $V$ with a small ball about ${\mathbf{r}}_0$ removed. Both functions are harmonic there, so only boundary terms remain. The contribution from the small sphere tends to $u({\mathbf{r}}_0)$ becausehas unit delta source, while the remaining small-sphere term vanishes. Taking the radius to zero gives Green's third identity

Integrate ${\nabla }^{2}u=0$ over a large half-ball and apply the divergence theorem. The hemispherical contribution vanishes by the assumed decay. On $z=0$, the outward normal is $-{\mathbf{e}}_z$, so $\partial u/\partial n=-p$. Hence compatibility requiresThis is the usual solvability condition for the Neumann problem.

Reflect the source point ${\mathbf{r}}_0=(x_0,y_0,z_0)$ across the plane to ${\mathbf{r}}_0^{\ast }=(x_0,y_0,-z_0)$. Equal-sign source and image make the normal derivatives cancel on the plane. Thus the method of images gives the Neumann Green function

Green's identity with $\partial G/\partial n=0$ leaves only the prescribed normal derivative. Since $\partial u/\partial n=-p$ and, on the plane,the decaying solution isThus

Use the large-$z_0$ expansionThe first term vanishes by the compatibility condition. For the given $p$, symmetry leavesConsequently