show that if $v$ is harmonic on $\overline{\Omega}$ and $v$ is constant on $\partial \Omega$ then $v$ is constant on $\Omega$

Question

Let $\Omega \subset \mathbf{R^2}$ be bounded.Show that if $v$ is harmonic on $\overline{\Omega}$ and $v$ is constant on $\partial \Omega$ then $v$ is constant on $\Omega$

Answer 1

Hint: Maximum and minimum principle for harmonic functions.

Answer 2

Leg $v$ be harmonic on $\overline{\Omega}$ such that $v$ is constant on $\partial{\Omega}$ say $v \equiv c$ on $\partial{\Omega}$. Then as $v$ is harmonic, it must attain its max and min on $\partial{\Omega}$ *(Refer to Evans PDE section on the LaPlace PDE for a proof of this; it is very similar to the proof of the maximum modulus principle of a holomorphic function).

In particular, as the only value on $\partial{\Omega}$ is $c$, we see its max and min is $c$. This is possible if and only if $v \equiv c$, so $v$ is constant as desired.