I would like to solve the following problems:
$$-\Delta G(x)= \delta (x) - 1 , $$ and after that $$-\Delta G(x) + \kappa^2 G(x)= \delta (x) - 1 .$$
on the square $[-\frac{1}{2},\frac{1}{2}]\times [-\frac{1}{2},\frac{1}{2}]$ with periodic boundary conditions. I have seen this domain called a torus I don't know why. I am confused because I know how to solve the free space problem $\Delta G(x)= \delta (x)$ using the symmetry of Laplace's operator. I can solve Laplace equation $\Delta u(x)= 0$ on a bounded domain using separation of variables, but here the mixing of: Green's function + square domain + periodic boundary conditions gives me troubles. Could you please give me some hints on which direction to go? Thank you!
