Green's function in 2D

Question

How does one compute the Green's function of the laplacian on $ \mathbb{R}^2 $? Can someone point me to a reference? In particular, the fourier transform: $$ \int_0^\infty \int_0^\infty \frac{e^{i m x} e^{i m y}}{m^2+n^2} $$ does not converge. Is there some standard way to regulate this?

Answer 1

The integral diverges for a good reason: $\mathbb R^2$ does not have Green's function. In potential-theory-speak, $\mathbb R^2$ is not a Greenian domain. More precisely, a domain $\Omega\subset \mathbb R^2$ is Greenian if and only if its complement has positive logarithmic capacity. This is discussed in all potential theory books, such as Armitage-Gardiner.

The closest thing to Green's function for the plane is $\log\sqrt{x^2+y^2}$, which inverts the Laplacian but does not have zero boundary values (or constant sign).