Newtonian potential in three dimensions

For $f\in C_c^\infty(\mathbb R^3)$ , its Newtonian potential is

u(x)=(\Gamma*f)(x)=\int_{\mathbb R^3}\frac{f(y)}{4\pi|x-y|}\,dy.

The kernel $\Gamma$ is the fundamental solution of $-\Delta$ .

Equation and regularity

The kernel is locally integrable and the source is compactly supported. Distributional differentiation gives $-\Delta u=f$ . Derivatives can be transferred to $f$ , making $u$ smooth and the equality classical. At large $|x|$ , $u=O(|x|^{-1})$ . Other sources require their own convergence conditions; the displayed integral is not automatically finite for every distribution or every integrable function at every point.

References

Hunter, Laplace equation, §2.6.