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I am stuck on a problem that I am trying for exam practice and I would very much appreciate a hint to help me out, here is the section where I am stuck:

A solution is sought to the Neumann problem for $\nabla^2 u = 0$ in the half plane $z > 0$: $u = O(|x|^{−a}), \frac{\partial u}{\partial r} = O(|x|^{−a−1}) ~~ \mathrm{as} ~~ |x| → ∞,~~ \frac{\partial u}{\partial z} ~ = p(x, y) ~ on ~ z = 0, \mathrm{where}~ a > 0$. It is assumed that $\int_{\infty}^{\infty}\int_{-\infty}^{\infty} p(x, y) dx dy = 0$. Explain why this condition is necessary.

My feeling is that this is to do with Green's third identity and that we need the normal derivative in the $x-y$ plane to be integrable in order to find out solution with a Green's function, am I correct?

EDIT: the divergence theorem sorts this out.

user27182
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Let $f(z)=\int\int u_z$. Then $f_z=\int\int u_{zz} =\int\int (-u_{xx}-u_{yy})$. Consider this quantity and also $\lim _{z\to\infty} f(z)$. That should give you the result you want.

not all wrong
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