Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 3 / 14A / b / i / Solution

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.