Condition for Harmonic Function to be Constant on a Region

Theorem
If  
\[\phi\]
  is harmonic on a region  
\[V\]
  with surface  
\[S\]
 and  
\[\frac{\partial \phi}{\partial n} =0\]
  the  
\[\phi\]
  is constant on  
\[V\]
.
Proof
Use Green's First Theorem for Harmonic Functions:
\[\int \int_V (\mathbf{\nabla \phi}) \cdot (\mathbf{\nabla \phi}) dV = \int \int_S \phi \frac{\partial \phi}{\partial n} dS \]

If  
\[\frac{\partial \phi}{\partial n} =0\]
  on  
\[S\]
  then
\[\int \int_V (\mathbf{\nabla \phi}) \cdot (\mathbf{\nabla \phi}) dV = 0\]

Hence  
\[\mathbf{\nabla \phi} =0 \rightarrow \phi = CONSTANT\]