## 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
$\mathbf{\nabla \phi} =0 \rightarrow \phi = CONSTANT$