4 Laplace’s equation

If you look back at the two-dimensional heat conduction equation (4):

$\frac{\partial u}{\partial t}=k\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)$

it is clear that if the heat flow is steady, i.e. time independent, then $\frac{\partial u}{\partial t}=0$ so the temperature $u\left(x,y\right)$ is a solution of

$\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=0$ (8)

(8) is the two-dimensional Laplace equation . Both this, and its three-dimensional counterpart

$\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}+\frac{{\partial }^{2}u}{\partial z^2}=0$ (9)

arise in a wide variety of applications, quite apart from steady state heat conduction theory. Since time does not arise in (8) or (9) it is evident that Laplace’s equation is always a model for equilibrium situations. In any problem involving Laplace’s equation we are interested in solving it in a specific region $R$ for given boundary conditions. Since conditions may involve

1. $u$ specified on the boundary curve $C$ (two dimensions) or boundary surface $S$ (three dimensions) of the region $R$ . Such boundary conditions are called Dirichlet conditions .
2. The derivative of $u$ normal to the boundary, written $\frac{\partial u}{\partial n},$ specified on $C$ or $S$ . These are referred to as Neumann boundary conditions .
3. A mixture of (a) and (b).

Some areas in which Laplace’s equation arises are

1. electrostatics ( $u$ being the electrostatic potential in a charge free region)
2. gravitation ( $u$ being the gravitational potential in free space)
3. steady state flow of inviscid fluids
4. steady state heat conduction (as already discussed)