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