# Poisson’s equation

Poisson’s equation is a second-order partial differential equation which arises in physical problems such as finding the electric potential of a given charge distribution. Its general form in $n$ is

 $\nabla^{2}\phi(\mathbf{r})=\rho(\mathbf{r})$

where $\nabla^{2}$ is the Laplacian and $\rho:D\to\mathbb{R}$, often called a source , is a given function on some subset $D$ of $\mathbb{R}^{n}$. If $\rho$ is identically zero, the Poisson equation reduces to the Laplace equation.

The Poisson equation is linear, and therefore obeys the superposition principle: if $\nabla^{2}\phi_{1}=\rho_{1}$ and $\nabla^{2}\phi_{2}=\rho_{2}$, then $\nabla^{2}(\phi_{1}+\phi_{2})=\rho_{1}+\rho_{2}$. This fact can be used to construct solutions to Poisson’s equation from fundamental solutions, or Green’s functions, where the source distribution is a delta function.

A very important case is the one in which $n=3$, $D$ is all of $\mathbb{R}^{3}$, and $\phi(\mathbf{r})\to 0$ as $|\mathbf{r}|\to\infty$. The general solution is then given by

 $\phi(\mathbf{r})=-\frac{1}{4\pi}\int_{\mathbb{R}^{3}}\frac{\rho(\mathbf{r^{% \prime}})}{|\mathbf{r}-\mathbf{r^{\prime}}|}\mathrm{d}^{3}\mathbf{r^{\prime}}.$
Title Poisson’s equation PoissonsEquation 2013-03-22 13:38:28 2013-03-22 13:38:28 pbruin (1001) pbruin (1001) 6 pbruin (1001) Definition msc 35J05 HelmholtzDifferentialEquation LaplacesEquation GreensFunction