Proof That Harmonic Function Equal on a Closed Surface are Equal on the Enclosed Volume

Theorem
If

\[\phi , \psi\]

are harmonic functions on a region

\[V\]

with surface

\[S\]

and

\[\phi = \psi\]

on

\[S\]

then

\[\phi = psi\]

on

\[V\]

. Proof
If

\[\phi , \psi\]

are harmonic on

\[V\]

then

\[\nabla^2 \phi = \nabla^2 \psi =0 \rightarrow \nabla^2 (\phi - \psi ) =0\]

on

\[V\]

.
Hence

\[\phi - \psi\]

is harmonic on

\[V\]

and if

\[\phi = \psi \]

on

\[S\]

then

\[\phi - \psi =0\]

on

\[S\]

.
Apply thisTheorem to the function

\[\phi - \psi\]

and the theorem is proved.