= Potential of a uniform spherical source
{title2=$\Delta f=\rho_0\mathbf1_{\{|x|<a\}}$}
For a uniform spherical source of radius $a$, the origin-regular radial <Poisson equation> gives $\nabla f=\rho_0x/3$ inside and $\nabla f=\rho_0a^3x/(3|x|^3)$ outside. Integrating radially, imposing $f\to0$ at infinity and continuity at $|x|=a$, gives
$$
f(x)=\begin{cases}\rho_0(|x|^2-3a^2)/6,&|x|\le a,\\-\rho_0a^3/(3|x|),&|x|\ge a.\end{cases}
$$
The potential and its normal derivative are continuous at the interface, so its distributional <Laplacian> has no surface delta contribution. The classical equation holds off the interface and the source equation holds almost everywhere and distributionally.
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