Uniqueness theorem for Poisson's equation
The uniqueness theorem for Poisson's equation states that, for a large class of boundary conditions, the equation may have many solutions, but the gradient of every solution is the same. In the case of electrostatics, this means that there is a unique electric field derived from a potential function satisfying Poisson's equation under the boundary conditions.
The uniqueness of the gradient of the solution (the uniqueness of the electric field) can be proven for a large class of boundary conditions in the following way.
Suppose that there are two solutions and . One can then define which is the difference of the two solutions. Given that both and satisfy Poisson's equation, must satisfy
Using the identity
And noticing that the second term is zero, one can rewrite this as
Taking the volume integral over all space specified by the boundary conditions gives
Applying the divergence theorem, the expression can be rewritten as
where are boundary surfaces specified by boundary conditions.
Since and , then must be zero everywhere (and so ) when the surface integral vanishes.
This means that the gradient of the solution is unique when
The boundary conditions for which the above is true include:
- Dirichlet boundary condition: is well defined at all of the boundary surfaces. As such so at the boundary and correspondingly the surface integral vanishes.
- Neumann boundary condition: is well defined at all of the boundary surfaces. As such so at the boundary and correspondingly the surface integral vanishes.
- Modified Neumann boundary condition (also called Robin boundary condition – conditions where boundaries are specified as conductors with known charges): is also well defined by applying locally Gauss's Law. As such, the surface integral also vanishes.
- Mixed boundary conditions (a combination of Dirichlet, Neumann, and modified Neumann boundary conditions): the uniqueness theorem will still hold.
The boundary surfaces may also include boundaries at infinity (describing unbounded domains) – for these the uniqueness theorem holds if the surface integral vanishes, which is the case (for example) when at large distances the integrand decays faster than the surface area grows.