 Neumann Green function with constant boundary flux

For a smooth bounded connected domain $V$ with boundary $S$, integrating the Green function equation and applying the divergence theorem forces ${\int }_S{\partial }_{n}GdS=1$. Thus zero Neumann boundary data are incompatible with an unmodified unit point source. The displayed constant boundary derivative has the required total flux. For $\Delta \Phi =\sigma$ and ${\partial }_n\Phi =f$, the compatibility condition is ${\int }_V\sigma dV={\int }_{S}fdS$. Green's second identity then yieldsThe last term is an undetermined additive constant. Adding a constant to $G$ does not change the other two terms together, because of the compatibility condition. This construction keeps the unit Dirac delta source; using a mean-subtracted source with zero normal derivative is a different normalization.