 Mixed Dirichlet-Neumann Green function in a quadrant

For the quadrant $x,y>0$, the displayed Green function satisfies $\Delta G={\delta }_{(\xi ,\eta )}$, zero value on $y=0$ and zero outward normal derivative on $x=0$. The method of images uses the logarithmic fundamental solution of the Laplace equation with coefficient $1/(2\pi )$: keep the source sign when reflecting across the Neumann side and reverse it when reflecting across the Dirichlet side. The four image signs are therefore $+,+,-,-$ for sources $(\xi ,\eta ),(-\xi ,\eta ),(\xi ,-\eta ),(-\xi ,-\eta )$.

On $y=0$, equal numerator and denominator give zero. Evenness in $x$ gives the Neumann condition at $x=0$. The outward normal derivative on the Dirichlet side is the positive kernelGreen's second identity consequently gives $u(P)={\int }_{D}Gf+{\int }_{y=0}g{\partial }_{n}G$ for $\Delta u=f$ with these mixed boundary data, assuming any remote-boundary contribution vanishes. The sign convention is $\Delta G=+\delta$; changing to $-\Delta G=\delta$ changes both the Green function and the representation signs.