Sign conventions for a Dirichlet Green function ($\Delta G=\delta ⟹P={\partial }_{n}G$)

Let $G$ vanish on the boundary of a domain and satisfy $\Delta G(\cdot ,y)={\delta }_y$. Green's second identity givesThus the Poisson kernel is $+{\partial }_{n}G$ for this convention. If instead $-\Delta K=\delta$, then $K=-G$ and the same kernel is $-{\partial }_{n}K$. At the boundary of $x>0$, the outward normal is $-e_x$, so the first convention gives $P=-{\partial }_{x}G$. 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.