This might be a known problem, but I could not find a precise answer.
I have the following Laplace equation
\begin{equation}
\begin{cases}
-\Delta u = f & x \in \Omega;\\
\quad\: u = g & x \in \partial \Omega,
\end{cases}
\end{equation}
on a GENERAL polyhedral domain $\Omega \subset\mathbb{R}^3$ (the only assumption is that $\Omega$ has finitely many faces), with $f\in\mathcal{C}^{\infty}(\Omega)$ and $g\in\mathcal{C}^0(\partial\Omega)$. Assume that there exists $\varepsilon > 0$ such that, for each face $F$ of $\partial\Omega$, the restriction $g_{|F} \in H^{1+\varepsilon}(F)$. Two questions:
- Does there exist $\eta > 0$ such that $u\in H^{\frac{3}{2}+\eta}(\Omega)$?
- Is it true that $u\in\mathcal{C}^0(\Omega)$?
Note that claim 1 implies 2.