Regularity of Hodge Laplacian on bounded domains 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.

 A: Morrey's book [2] mentioned in another answer covers the $L^2$ theory only. It is surprisingly difficult to find good, detailed and reliable references for the Hodge decomposition in $L^p$ for $1<p<\infty$. 
The $L^p$ Hodge decomposition on compact manifolds is done carefully in the papers [1,3]. Although the authors do not discuss directly the higher order regularity,  it is shown in [3] (see Definition 5.23) that the Green operator maps $L^p$ to $W^{2,p}$. The paper [1] says very little about the boundary conditions (which for forms are far from obvious), but the boundary conditions are carefully studied in [1].
[1] T. Iwaniec, C. Scott, B. Stroffolini, 
Nonlinear Hodge theory on manifolds with boundary.
Ann. Mat. Pura Appl. (4) 177 (1999), 37–115. 
(MathSciNet review.)
[2] C. B. Morrey, Jr., Multiple integrals in the calculus of variations. Die Grundlehren der mathematischen Wissenschaften, Band 130 Springer-Verlag New York, Inc., New York 1966. 
(MathSciNet review.)
[3] C. Scott, $L^p$ theory of differential forms on manifolds.
Trans. Amer. Math. Soc. 347 (1995), no. 6, 2075–2096. 
(MathSciNet review.)
A: Try Morrey's "Multiple integrals in the Calculus of Variations"
