= Sign conventions for a Dirichlet Green function
{title2=$\Delta G=\delta\ \Longrightarrow\ P=\partial_nG$}
Let $G$ vanish on the boundary of a domain and satisfy $\Delta G(\cdot,y)=\delta_y$. <Green's second identity> gives
$$
u(y)=\int_\Omega G(x,y)\Delta u(x)dx+\int_{\partial\Omega}u(x)\partial_{n_x}G(x,y)dS_x.
$$
Thus the <Poisson kernel> is $+\partial_nG$ for this convention. If instead $-\Delta K=\delta$, then $K=-G$ and the same kernel is $-\partial_nK$. At the boundary of $x>0$, the outward normal is $-e_x$, so the first convention gives $P=-\partial_xG$. Changing the delta-source sign or replacing the outward derivative by an inward coordinate derivative changes a sign; changing both leaves the physical kernel unchanged.
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