Green's Function within the the given "quarter-space"

Solution 1:

The volume of interest is a quarter of the space as you defined it. You need a Green's function that vanishes on the 2 boundaries that you specificed. To interpret it, I'll use electrostatic notation (familiar to physicists - hope it will be also clear to you). If $$ G_1=+\frac1{4\pi|{\bf r}-{\bf r'}|} $$ is the potential due to a "unit charge" locates at ${\bf r'}=(x',y',z')$, then you need 3 more charges to make the total potential (the Green's function) vanish on $V$ - they are located symmetrically to the charge with respect to the 2 boundaries and have opposite charge: $$ G=G_1+G_2+G_3+G_4 $$ where: $$ G_2=-\frac1{4\pi|{\bf r}-{\bf r'}+2x'{\bf{\hat x}}|} $$ $$ G_3=-\frac1{4\pi|{\bf r}-{\bf r'}+2y'{\bf{\hat y}}|} $$ $$ G_4=+\frac1{4\pi|{\bf r}-{\bf r'}+2x'{\bf{\hat x}}+2y'{\bf{\hat y}}|} $$ where $\bf{\hat x},\bf{\hat y}$ are unit vectors along the respective axes. You should check $G$ vanishes on the boundary, that is: $$ G(0,y,z)=G(x,0,z)=0 $$