I need a reference for the $W^{s,p}$ regularity of the Hodge boundary value problem on bounded domains. I need estimates $\lVert \omega \rVert_{W^{s+2,p}} \leq c\lVert f\rVert_{W^{s,p}}$, $s\geq 0$ for the system

$$\begin{split}\Delta \omega &=f\text{ in }\Omega ,\\
\nu\wedge\omega &=0\text{ on }\partial\Omega\ ,\\
\nu\wedge\delta\omega &=0\text{ on }\partial \Omega\ .
\end{split}$$

I need to know:

- A reference which actually verifies the Agmon-Douglis-Nirenberg condition for this system for general boundary.... most references either do not verify or verifies the condition only when $\partial\Omega$ is flat.
- Whether regularity results extend to the scale of negative Sobolev spaces - e.g. is $\lVert \omega \rVert_{W^{1,p}} \leq c \lVert f \rVert_{W^{-1,p}}$ true?
- Whether there is such a result for the system
$$ \begin{split} \delta ( A d\omega) + d\delta\omega &=0\text{ in }\Omega\ ,\\
\nu\wedge\omega &=0\text{ on }\partial\Omega ,\\
\nu\wedge\delta\omega &=0\text{ on }\partial\Omega\ ,\end{split}$$

where $A$ is elliptic.