Uniqueness of solutions

20.7 UNIQUENESS OF SOLUTIONS
Equation type
hyperbolic
parabolic
elliptic
Boundary
open
open
closed
Conditions
Cauchy
Dirichlet or Neumann
Dirichlet or Neumann
Table 20.1 The appropriate boundary conditions for different types of partial
differential equation.
conditions u and ∂u/∂t on the line segment t = 0, x = 0 to L, the solution is
specified in the shaded region.
As in the case of first-order PDEs, however, problems can arise. For example,
if for a hyperbolic equation the boundary curve intersects any characteristic
more than once then Cauchy conditions along C can overdetermine the problem,
resulting in there being no solution. In this case either the boundary curve C
must be altered, or the boundary conditions on the offending parts of C must be
relaxed to Dirichlet or Neumann conditions.
The general considerations involved in deciding which boundary conditions are
appropriate for a particular problem are complex, and we do not discuss them
any further here.§ We merely note that whether the various types of boundary
condition are appropriate (in that they give a solution that is unique, sometimes
to within a constant, and is well defined) depends upon the type of second-order
equation under consideration and on whether the region of solution is bounded
by a closed or an open curve (or a surface if there are more than two independent
variables). Note that part of a closed boundary may be at infinity if conditions
are imposed on u or ∂u/∂n there.
It may be shown that the appropriate boundary-condition and equation-type
pairings are as given in table 20.1.
For example, Laplace’s equation ∇2 u = 0 is elliptic and thus requires either
Dirichlet or Neumann boundary conditions on a closed boundary which, as we
have already noted, may be at infinity if the behaviour of u is specified there
(most often u or ∂u/∂n → 0 at infinity).
20.7 Uniqueness of solutions
Although we have merely stated the appropriate boundary types and conditions
for which, in the general case, a PDE has a unique, well-defined solution, sometimes to within an additive constant, it is often important to be able to prove
that a unique solution is obtained.
§
For a discussion the reader is referred, for example, to P. M. Morse and H. Feshbach, Methods of
Theoretical Physics, Part I (New York: McGraw-Hill, 1953), chap. 6.
705
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 into ρ. We aim to show that, to within an unimportant constant, the
solution of (20.46) is unique if either the potential u or its normal derivative
∂u/∂n is specified on all surfaces bounding a given region of space (including, if
necessary, a hypothetical spherical surface of indefinitely large radius on which u
or ∂u/∂n is prescribed to have an arbitrarily small value). Stated more formally
this is as follows.
Uniqueness theorem. If u is real and its first and second partial derivatives are
continuous in a region V and on its boundary S, and ∇2 u = ρ in V and either
u = f or ∂u/∂n = g on S, where ρ, f and g are prescribed functions, then u is
unique (at least to within an additive constant).
Prove the uniqueness theorem for Poisson’s equation.
Let us suppose on the contrary that two solutions u1 (r) and u2 (r) both satisfy the conditions
given above, and denote their difference by the function w = u1 − u2 . We then have
∇2 w = ∇2 u1 − ∇2 u2 = ρ − ρ = 0,
so that w satisfies Laplace’s equation in V . Furthermore, since either u1 = f = u2 or
∂u1 /∂n = g = ∂u2 /∂n on S, we must have either w = 0 or ∂w/∂n = 0 on S.
If we now use Green’s first theorem, (11.19), for the case where both scalar functions
are taken as w we have
2
∂w
dS.
w∇ w + (∇w) · (∇w) dV = w
∂n
V
S
However, either condition, w = 0 or ∂w/∂n = 0, makes the RHS vanish whilst the first
term on the LHS vanishes since ∇2 w = 0 in V . Thus we are left with
|∇w|2 dV = 0.
V
Since |∇w|2 can never be negative, this can only be satisfied if
∇w = 0,
i.e. if w, and hence u1 − u2 , is a constant in V .
If Dirichlet conditions are given then u1 ≡ u2 on (some part of) S and hence u1 = u2
everywhere in V . For Neumann conditions, however, u1 and u2 can differ throughout V
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.
706