Green's function for the quarter plane

Question

Let $Ω = \{(x_1, x_2)∈\mathbb R^2\mid x_1, x_2 > 0\}$. Given a $y = (y_1, y_2) \in Ω$ put $r_1(y) = (y_1, −y_2)$ (reflection about the $x_1$-axis), $r_2(y) = (−y_1, y_2)$ (reflection about the $x_2$-axis), and $r_{12}(y) = (−y_1, −y_2)$

I know that the Green's function for the quarter plane is given by:

$$G(x,y) = \frac{1}{2π}(\ln\|x − y\| − \ln\|x − r_1(y)\| − \ln\|x − r_2(y)\| + \ln\|x − r_{12}(y)\|). $$

I want to compute $\frac{dG}{dn}$ which is equal to gradient of $G \cdot n(y)$ where $n(y)$ is the unit at normal.

My question is what is the normal that I need to use in order to compute this derivative?

Any help would be appreciated, thank you!

Answer 1

It is the outer unit normal to $\Omega$, which is $-e_1 = (-1,0)$ along the $y$ axis and $-e_2=(0,-1)$ along the $x$ axis.

For example, $\frac{\partial G^y}{\partial n}(0,x_2) = -\frac{\partial G^y}{\partial x_1}(0, x_2)$, where $G^y(x) = G(x,y)$.