Let $\Omega := (a,b)\times \mathbb{R}^n$, I'm asked to prove or disprove a uniqueness result for this problem in the class $C^2(\Omega)\cup C(\bar{\Omega})$ $$ \begin{cases} \Delta u=0 & for \ (t,x)\in \Omega\\ u(t,x)=0 & for \ (t,x)\in\partial\Omega \end{cases} $$
I know that a non-zero solution of this problem when $n=1$ and $(a,b)=(0,2\pi)$ is the function $u(t,x)=\sin(t)e^{-x}$, so the uniqueness doesn't hold. However, this solution is not bounded. Also, since it's smooth up to the boundary one can extend it (by repeatedly odd-reflecting it) to a $C^2$ function defined on the whole $\mathbb{R}^{2}$.
But what if we also ask the solution to be bounded? Reasoning as above one could expect to employ Liouville's theorem after extending the solution to the whole space to conclude uniqueness, but one would need additional requirements on the regularity of the solution (i.e. of being $C^2(\bar\Omega)$).
$\textbf{Question}$
Is this additional requirement redundant, meaning that being harmonic, bounded and $C(\bar\Omega)$ implies being $C^2(\bar\Omega)$? If not, what is a counterexample?
Can I extend the solution anyways to obtain a function of class $C^2$ except that on a null-measure set (the hyperplanes $\{a+k(b-a)\}\times \mathbb{R}^n, k\ge0$) which solves the Laplace equation weakly on $\mathbb{R}^{n+1}$ and then apply Weyl's lemma together with continuity of the solution to recover regularity and therefore unicity via Liouville's theorem?
Furthermore, I was wondering if there is maybe a variational approach to the problem which guarantees regularity via some kind of bootstrap argument (I have not investigated this possibility yet)