Definition
Poisson equation
A Laplacian equation with a prescribed source.
The Poisson equation, with the sign convention used here, is
on a specified open set, where is the Laplacian and is prescribed. It may be interpreted classically or in distributions, with the solution class stated separately.
Uniqueness and conventions
The difference of two solutions with the same source is harmonic. Boundary conditions, decay, a norm class, or a normalization are therefore needed to select a unique solution. Some authors instead call the Poisson equation; their fundamental solution has the opposite sign.