Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 2 / 30C / i / Solution

A fundamental solution of a linear differential operator $P(D)$ is a distribution $E$ satisfying $P(D)E={\delta }_0$. For the three-dimensional negative Laplacian,Away from zero the function is harmonic. For a test function $\varphi$, integrate $E(-\Delta \varphi )$ outside a ball of radius $\epsilon$ and use Green's identity. The small inner sphere contributes $\varphi (0)+o(1)$ because the normal derivative of $E$ has total outward-from-the-ball flux $-1$; the other boundary term is $O(\epsilon )$. Letting $\epsilon \to 0$ verifies the distributional equation and its normalization.