I’m trying to calculate the form of the Green function in a half-disk with a slit and I can not figure out the specific formula.
I tried 2 ways, first, I tried to use the equation
$$ \nabla^2 \phi(x,y)=0 $$ $$ \phi(x,y)=-\frac{1}{2\pi}log |x-y| $$
But it doesn’t seem to work because I could not figure out what exactly the solution is with a formula.
Second, I tried to use the eigenvectors to calculate it but stilll, nothing ever gets done ‘cause I do not know how to calculate it.
Maybe we could use a martingale to solve it? As I have no clue, I don’t know how to do it.
I put a picture below, the red curves are the boundary of the domain I’m working about.
