4 Laplace’s equation

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

u t = k 2 u x 2 + 2 u y 2

it is clear that if the heat flow is steady, i.e. time independent, then u t = 0 so the temperature u ( x , y ) is a solution of

2 u x 2 + 2 u y 2 = 0 (8)

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

2 u x 2 + 2 u y 2 + 2 u 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 u 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)