---
tag: 9Y01DS79PR
kind: definition
name: Fundamental solution of the Laplacian
msc: [35A08, 31B10, 35J05]
defines: BSJ8X2XRN9
uses: [K2NK9A8M5F, 0E0B6RA8K0, 0KPHENZ5S7, 0DEB0JMN0W]
sources:
  - url: https://en.wikipedia.org/wiki/Fundamental_solution
---

## Definition

Let $n \ge 2$ and let $\omega_n$ be the volume of the unit ball of $\R^n$.
The *fundamental solution of the Laplacian* on $\R^n$ is the function $\Phi \colon \R^n \setminus \set{0} \to \R$ given by
$$\Phi(x) = -\frac{1}{2\pi} \log \abs{x} \quad (n = 2), \qquad \Phi(x) = \frac{1}{n(n-2)\omega_n} \, \frac{1}{\abs{x}^{n-2}} \quad (n \ge 3).$$

## Remarks

The function $\Phi$ is locally integrable on $\R^n$, and in the sense of distributions it satisfies $-\Delta \Phi = \delta$, where $\Delta$ is the [[Laplacian|0E0B6RA8K0]] and $\delta$ is the [[Dirac delta|0KPHENZ5S7]] at $0$.
So $-\Phi$ is a [[fundamental solution|K2NK9A8M5F]] of $\Delta$ in the general sense; many texts use this sign convention so that $\Phi > 0$ near the origin.
Example: for $n = 3$, $\omega_3 = \frac{4\pi}{3}$ and $\Phi(x) = \frac{1}{4\pi\abs{x}}$.
For a radial function $f(r)$ with $r = \abs{x}$ one has $\Delta f = f'' + \frac{n-1}{r} f'$, and $f = r^{2-n}$ gives $(2-n)(1-n) r^{-n} + (n-1)(2-n) r^{-n} = 0$, so $\Phi$ is [[harmonic|0DEB0JMN0W]] on $\R^n \setminus \set{0}$.
Non-example: $\abs{x}^{-1}$ on $\R^4$ is not a multiple of $\Phi$, and it is not harmonic there, since $\Delta \abs{x}^{-1} = -\abs{x}^{-3}$.
The constant is chosen so that the flux of $-\nabla \Phi$ out of every sphere about $0$ equals $1$; here $n\omega_n$ is the area of the unit sphere.
Other normalisations, such as $\abs{x}^{2-n}$ or $\log(1/\abs{x})$ without constants, are common in potential theory.