# Difference between the fundamental solution and the Green function

A lot of sources (books, internet courses, articles etc.) deal with just one of the two: Green function, and the fundamental solution. I wasn't able to find a distinction, but I suppose there is one. However both seem to be defined for a differential operator $D$ as follows:

$G(x-y)$ such, that: $DG(x-y)=\delta(y)$

So what's the difference?

## 1 Answer

Fundamental solution solves $DG(x-y)=\delta(y)$ on the entire space $\mathbb R^n$.

Green's function solves $DG(x-y)=\delta(y)$ on some domain $\Omega\subset \mathbb R^n$, and satisfies some homogeneous boundary condition. (Most often, but not always, the Dirichlet condition).

The lecture notes by Sijue Wu present the derivation of Green functions for the Laplacian from the fundamental solution. You should be able to see how these objects are related, and how they are different: in a nutshell, to find Green's function we add a smooth solution of the PDE to the fundamental solution, so that the sum satisfies the boundary condition.