Green's Function - PDE

Question

I am completely stumped on Green's function regarding PDE's. There are barely any examples in my book on how to apply it. For example, if a question asks, "find the Green's function ..." to some specified domain, what am I looking for? What does it mean to find Green's function? Is there a formula to use? Green's function seems so abstract to me. It's just not clicking.

Answer 1

Let $\Omega$ be a domain in the plane (open connected set) and $p$ a point in $\Omega$. A Green function, $G_p$ is a smooth function defined on $\Omega \cup \partial \Omega$ such that:

(i) $G_p$ is harmonic on $\Omega \setminus \{p\}$

(ii) $G_p$ is continuous on $(\Omega \cup \partial \Omega)\setminus \{p\}$

(iii) $G_p$ is zero on the boundary $\partial \Omega$

(iv) $G_p$ has a simple pole at $p$ with reside $\frac{1}{4\pi}$

My Reference is the PDE book by Walter Strauss. His treatment of Green's functions is excellent. You should look him up if you want to understand this topic better.