- If is harmonic in the connected domain and is not constant, then is open in .
- Suppose is bounded and that its boundary is connected. If is harmonic in , then .
- A radial harmonic function on is constant.
- A positive harmonic function on is constant.
Suppose is harmonic in some domain in . Then
is also harmonic in a suitable domain.
For , find the Green’s function for the Laplace operator on the first quadrant.