Laplace's Equation

where the function is often referred to as a potential. Suppose that we wish to find a solution to this equation in some finite volume , bounded by a closed surface , subject to the boundary condition

when lies on . Consider the vector identity

(129) |

Integrating this expression over , making use of the divergence theorem, we obtain

It follows from Equations (127) and (128) that

(131) |

which implies that throughout and on . Hence, Equation (128) yields

throughout and on . We conclude that the only solution to Laplace's equation, (127), subject to the boundary condition (128), is the trivial solution (132). Finally, if we let the surface tend to infinity then we deduce that the only solution to Laplace's equation, (127), subject to the boundary condition

as | (133) |

is

(134) |

for all .

Consider a potential that satisfies Laplace's equation, (127), in some finite volume , bounded by the closed surface , subject to the boundary condition

(135) |

when lies on . Here, is a known surface distribution. We can demonstrate that this potential is unique. Let and be two supposedly different potentials that both satisfy Laplace's equation throughout , as well as the previous boundary condition on . Let us form the difference . This function satisfies Laplace's equation throughout , subject to the boundary condition

(136) |

when lies on . However, as we have already seen, this implies that throughout and on . Hence, and are identical, and the potential is therefore unique.