# Harmonic functions which are constant on boundary

I know that a harmonic function $f\in C^2(\Omega)\cap C(\bar\Omega)$ which is constant on boundary is constant, if $\Omega\subset\mathbb{R}^N$ is a bounded and connected domain. Also, if $\Omega$ is not bounded, this property does no longer hold (e.g. $f:\mathbb{\bar H}^2=\{(x,y)\in\mathbb{R}^2\ \vert\ y\geq0\}\rightarrow\mathbb{R}$, $f(x,y)=y$).

I wonder if there are some unbounded domains for which this property remains true? For example, if $\Omega=\{(x,y)\in\mathbb{R}^2\ \vert\ |y|<1\}$, it looks that this may be true, provided we prove $f$ should be bounded.

Is there an unbounded $\Omega$ such that a harmonic function vanishing on the boundary must vanish? I don't know, but I seriously doubt it.
I'm not sure what you mean by "...provided we prove $f$ should be bounded". In case you didn't know, if $f$ is a bounded harmonic function in a half plane that vanishes on the boundary then $f=0$.