Definition: The Green’s function G(x) for the operator $\Delta$ and the domain $D$ at the point $x_0 ∈ D$ is a function defined for $x ∈D$ such that:
(i) $G(x)$ possesses continuous second derivatives and $\Delta G = 0$ in D, except at the point x = x0.
(ii) $G(x) = 0$ for $x ∈$ bdy $D$.
(iii) The function $G(x) + \frac{1}{4π|x − x_0|}$ is finite at $x_0$ and has continuous second derivatives everywhere and is harmonic at x_0.
Let $Ω = \{(x, y) ∈ R^2 | y > 0, x > y\}$. Give an example of a Green’s function for Ω. I am new to the subject. I appreciate clear explanation of the example.