Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PDES: GENERAL AND PARTICULAR SOLUTIONS

As an important example let us consider Poisson’s equation in three dimensions,

∇^2 u(r)=ρ(r), (20.46)

with either Dirichlet or Neumann conditions on a closed boundary appropriate

to such an elliptic equation; for brevity, in (20.46), we have absorbed any physical

constants intoρ. We aim to show that, to within an unimportant constant, the

solution of (20.46) isuniqueif either the potentialuor its normal derivative

∂u/∂nis specified on all surfaces bounding a given region of space (including, if

necessary, a hypothetical spherical surface of indefinitely large radius on whichu

or∂u/∂nis prescribed to have an arbitrarily small value). Stated more formally

this is as follows.

Uniqueness theorem.Ifuis real and its first and second partial derivatives are

continuous in a regionVand on its boundaryS, and∇^2 u=ρinVand either

u=for∂u/∂n=gonS,whereρ,fandgare prescribed functions, thenuis

unique (at least to within an additive constant).

Prove the uniqueness theorem for Poisson’s equation.

Let us suppose on the contrary that two solutionsu 1 (r)andu 2 (r) both satisfy the conditions
given above, and denote their difference by the functionw=u 1 −u 2. We then have

∇^2 w=∇^2 u 1 −∇^2 u 2 =ρ−ρ=0,

so thatwsatisfies Laplace’s equation inV. Furthermore, since eitheru 1 =f=u 2 or
∂u 1 /∂n=g=∂u 2 /∂nonS, we must have eitherw=0or∂w/∂n=0onS.
If we now use Green’s first theorem, (11.19), for the case where both scalar functions
are taken aswwe have
∫

V

[

w∇^2 w+(∇w)·(∇w)

]

dV=

∫

S

w

∂w

∂n

dS.

However, either condition,w=0or∂w/∂n= 0, makes the RHS vanish whilst the first
term on the LHS vanishes since∇^2 w=0inV. Thus we are left with
∫

V

|∇w|^2 dV=0.

Since|∇w|^2 can never be negative, this can only be satisfied if

∇w= 0 ,

i.e. ifw, and henceu 1 −u 2 , is a constant inV.
If Dirichlet conditions are given thenu 1 ≡u 2 on (some part of)Sand henceu 1 =u 2
everywhere inV. For Neumann conditions, however,u 1 andu 2 can differ throughoutV
by an arbitrary (but unimportant) constant.

The importance of this uniqueness theorem lies in the fact that if a solution to

Poisson’s (or Laplace’s) equation that fits the given set of Dirichlet or Neumann

conditions can be found by any means whatever, then that solution is the correct

one, since only one exists. This result is the mathematical justification for the

method of images, which is discussed more fully in the next chapter.