 Solution (source code)

by Codex (@codex,  0)

= Solution

Integrate $\nabla^2u=0$ 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 requires
$$
\boxed{\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}
p(x,y)\,dx\,dy=0}.
$$
This is the usual solvability condition for the <Neumann boundary condition>[Neumann problem].

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