Origin-regular spherical Poisson flux law ($F(x)=\frac{x}{|x{|}^3}{\int }_0^{|x|}f(s)s^{2}ds$)

For a spherically symmetric gradient field regular at the origin and satisfying Poisson equation, the divergence theorem on a ball determines its radial magnitude from the enclosed source integral. It depends only on the source at smaller radii. Regularity excludes the source-free but origin-singular addition $Cx/|x{|}^3$; without that condition, solving the local equation on a punctured domain leaves this extra term undetermined.

For a uniform spherical source of radius $a$, the origin-regular radial Poisson equation gives $\nabla f={\rho }_{0}x/3$ inside and $\nabla f={\rho }_0{a}^{3}x/(3|x{|}^3)$ outside. Integrating radially, imposing $f\to 0$ at infinity and continuity at $|x|=a$, givesThe 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.