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

$${\nabla }^2\varphi (𝐫)=\rho (𝐫)$$

where ${\nabla }^2$ is the Laplacian and $\rho :D\to ℝ$, often called a source , is a given function on some subset $D$ of ${ℝ}^n$. 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 ${\nabla }^2{\varphi }_1={\rho }_1$ and ${\nabla }^2{\varphi }_2={\rho }_2$, then ${\nabla }^2({\varphi }_1+{\varphi }_2)={\rho }_1+{\rho }_2$. 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 ${ℝ}^3$, and $\varphi (𝐫)\to 0$ as $|𝐫|\to \mathrm{\infty }$. The general solution is then given by

$$\varphi (𝐫)=-\frac{1}{4\pi }{\int }_{{ℝ}^3}\frac{\rho ({𝐫}^{\prime })}{|𝐫-{𝐫}^{\prime }|}{\mathrm{d}}^3{𝐫}^{\prime }.$$