The maximum-minimum principle says that
A harmonic function on a domain cannot attain its maximum or its minimum unless it is constant.
Here is my question:
If we restrict our attention in ${\mathbb R}^2$ or ${\mathbb R}^3$, what's the hypothesis for the domain? (bounded? closed? open?)
According to the proof of this principle, it seems that the domain is open. I could not find the context which may indicate the properties of the domain.