Elliptic equations

While modeling steady state phenomena, we often come across elliptic equations e.g. in heat conduction phenomena at steady state situation in which temperature $u$ does not vary with time, the governing equation in $2-D$ is

$$\Delta^2u=u_{xx}+u_{yy}=0.$$

This equation is known as Laplace equation. This kind of equation is also satisfied by electric field and gravitational potential functions and also known as potential functions.