Laplace's Equation

Laplace's equation is written

$$\nabla^{\,2} \phi({\bf r}) = 0,$$ (127)

where the function $\phi({\bf r})$ is often referred to as a potential. Suppose that we wish to find a solution to this equation in some finite volume $V$ , bounded by a closed surface $S$ , subject to the boundary condition

$$\phi({\bf r})= 0,$$ (128)

when ${\bf r}$ lies on $S$ . Consider the vector identity

$$\nabla\cdot (\phi\,\nabla \phi) \equiv \phi\,\nabla^{\,2} \phi + \nabla \phi\cdot \nabla \phi.$$ (129)

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

$$\int_S \phi\,\nabla\phi\cdot d{\bf S} = \int_V\left( \phi\,\nabla^{\,2} \phi + \nabla \phi\cdot \nabla \phi\right) dV.$$ (130)

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

$$\int_V \vert\nabla \phi\vert^{\,2} \,dV = 0,$$ (131)

which implies that $\nabla\phi={\bf0}$ throughout $V$ and on $S$ . Hence, Equation (128) yields

$$\phi({\bf r}) = 0$$ (132)

throughout $V$ and on $S$ . 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 $S$ tend to infinity then we deduce that the only solution to Laplace's equation, (127), subject to the boundary condition

$$\phi({\bf r}) \rightarrow 0 \vert{\bf r}\vert\rightarrow \infty,$$ (133)

is

$$\phi({\bf r}) = 0$$ (134)

for all ${\bf r}$ .

Consider a potential $\phi({\bf r})$ that satisfies Laplace's equation, (127), in some finite volume $V$ , bounded by the closed surface $S$ , subject to the boundary condition

$$\phi({\bf r})= \phi_S({\bf r}),$$ (135)

when ${\bf r}$ lies on $S$ . Here, $\phi_S({\bf r})$ is a known surface distribution. We can demonstrate that this potential is unique. Let $\phi_1({\bf r})$ and $\phi_2({\bf r})$ be two supposedly different potentials that both satisfy Laplace's equation throughout $V$ , as well as the previous boundary condition on $S$ . Let us form the difference $\phi_3({\bf r})=\phi_1({\bf r})-\phi_2({\bf r})$ . This function satisfies Laplace's equation throughout $V$ , subject to the boundary condition

$$\phi_3({\bf r}) = 0$$ (136)

when ${\bf r}$ lies on $S$ . However, as we have already seen, this implies that $\phi_3({\bf r})=0$ throughout $V$ and on $S$ . Hence, $\phi_1({\bf r})$ and $\phi_2({\bf r})$ are identical, and the potential $\phi({\bf r})$ is therefore unique.