Positive Harmonic functions that can be extended to boundary

Question

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain with smooth boundary $\partial \Omega$. Let $u \in C^\infty(\Omega)$ be a positive harmonic function. I wonder under which assumption, we can find a $\tilde{u} \in C^\infty(\Omega) \cap C(\partial \Omega)$ harmonic, such that $\tilde{u}=u$ in $\Omega$? Could anyone give me some examples or counterexamples?

Answer 1

This works iff $u$ is uniformly continuous.

If $u\colon\Omega\to(0,\infty)$ is uniformly continuous, then it extends to a continuous function on the completion $\tilde{u}\in C(\overline{\Omega};[0,\infty))\subset C(\overline{\Omega})$.

Conversely, since $\Omega$ is bounded, $\overline{\Omega}$ is compact so any continuous $\tilde{u}$ is uniformly continuous, hence $u=\tilde{u}\vert_\Omega$ is too.

Note there isn't really any need to restrict to $u$ positive, or $\partial\Omega$ smooth.