Theorem
If
\[\phi\]
is a harmonic function on a volume \[V\]
with surface \[S\]
on which \[\phi =0\]
then \[\phi =0\]
on \[V\]
.Proof
By Green's First Theorem for Harmonic Functions, if
\[\psi = \phi\]
we have\[\int \int_V (\mathbf{\nabla \phi} ) \cdot (\mathbf{\nabla \phi} ) dV = \int \int_S \phi \frac{\partial \phi}{\partial n} dS \]
By the statement in the theorem
\[\phi =0\]
on \[S\]
. \[\int \int_V (\mathbf{\nabla \phi} ) \cdot (\mathbf{\nabla \phi} ) dV = 0\]
Hence
\[\phi = CONSTANT\]
on \[S\]
.But
\[\phi\]
is continuous and twice differentiable in \[V\]
, so \[\phi =0\]
throughout \[V\]
.