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.

In Gaussian units, the general expression for Poisson's equation in electrostatics is

Here ${\displaystyle \varphi }$ is the electric potential and ${\displaystyle \mathbf {E} =-\mathbf {\nabla } \varphi }$ is the electric field.

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.

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