Uniqueness Theorem
There is a uniqueness theorem for Laplace's equation such that if a
solution is found, by whatever means, it is the solution. The proof
follows a proof by contradiction.
Suppose that, in a given finite volume V bounded by the closed
surface S, we have
and f is given on the surface S. Now we assume that there are two
different functions f1 and f2 satisfying these conditions. Then,
in V
on S
Let us now calculate the integral
vanishes on S
vanishes in V
This means that f must be a constant. However, since it vanishes
on S, it must be zero!
A matter of terminology - specification of either the value or the
derivative of the potential at the boundary of the region is an example
of a boundary condition.
A boundary value problem is a differential equation together with a set
of additional restraints, called the boundary conditions. A solution to a
boundary value problem is a solution to the differential equation which
also satisfies the boundary conditions. To be useful in applications, a
boundary value problem should be well posed. This means that given
the input to the problem there exists a unique solution, which depends
continuously on the input.
If the boundary gives a value to the normal derivative of the problem
then it is a Neumann boundary condition.
If the boundary gives a value to the problem then it is a Dirichlet
boundary condition.
If the boundary has the form of a curve or surface that gives a value
to the normal derivative and the problem itself then it is a Cauchy
boundary condition. It corresponds to imposing both a Dirichlet and a
Neumann boundary condition.
Uniqueness Theorem for Poisson Equation
Suppose we know that some region contains a charge density r.
Suppose we also know the value of the electrostatic potential f on
the boundary of this region. The uniqueness theorem then
guarantees that there is only one function f which describes the
potential in that region. This means that no matter how we figure out
f’s value — guessing, computer aided numerical computation,
demonic invocation — the function φ we find is guaranteed to be the
one we want.
Let us assume that two functions, f1 and f2 both satisfy Poisson’s
equation:
Both of these functions must satisfy the same boundary condition.
on boundary
in V
An electrostatic potential is completely determined within a
region once its value is known on the region’s boundary
Example 1: Randomly shaped conductor
Example 2: Nested concentric spherical shells
Example 3: Point charge and flat plane conductor