Uniqueness of Solutions to the Poisson Equation

We show that the Poisson equation has at most one solution where M is open and bounded. As intermediate steps we also show two properties for harmonic functions. The first that for a harmonic function v, the value at any given point will equal the mean value of v in a ball centered around that point, and the second that the maximum and minimum value of any harmonic function can be found on the boundary ∂ M of M has at most one solution where M is open and bounded.

uniqueness_of_solutions_to_the_poisson_equation_emil_hossjer