Solve

in the first quadrant with on the boundaries in 2 dimensions.

Now I know the solution to this problem for the half-plane is determined by the fact that

where

and

which leads to

So to solve this problem, however, I think I need to add more image sources, so at a guess I would have...

and then due to superposition, Green's function would be given by

or something to that effect. Am I on the right track with this?