# 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:

1. 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.
2. 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?
3. 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.
• I don't have the book in front of me to check, but I suggest looking in the book "Uhlenbeck compactness" by Katrin Wehrheim. – Andy Sanders Mar 8 '17 at 1:22
• Some pseudo-differential operator knowledge could be helpful to understand this issue. – Hu xiyu Dec 28 '17 at 1:11

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.)

• I knew both the references you gave well. Unfortunately, I did not find any verification of the ADN condition for the boundary conditions ( which is essentially equivalent to the Green's Kernel bound ). Can you be more specific? As far as I know, apart from Morrey ( who also does not verify the conditions, but uses a different argument ), I know of only Gunther Schwarz springer.com/de/book/9783540600169 and Taylor springer.com/de/book/9781441970541 who gives some argument as to why the BVP is elliptic. – Swarnendu Sil Apr 28 '18 at 8:44
• I was actually looking for a way to deduce the case I mentioned below, with a matrix A. None of the methods actually has an obvious generalisation to that case. I obtained the general result for this paper, using a different proof. link.springer.com/epdf/10.1007/… – Swarnendu Sil Apr 28 '18 at 8:46

Try Morrey's "Multiple integrals in the Calculus of Variations"

• It is preferable to have self-contained answers, with specific page or theorem numbers in the references. – Todd Trimble Nov 8 '15 at 3:01
• Yes, Morrey's book is indeed the first place I started to look, as most references actually refer to this. But in the book Morrey never really verified the Agmon-Douglis-Nirenberg condition. Another place is Lemma 1.6.5 of the book "Hodge Decomposition-A method for Solving Boundary Value Problems" by Gunter Schwarz. But there Schwarz assumes that the symbol is for the Laplacian, and the boundary operators takes the special their special form in half-spaces. The trouble is, to get the nice form of the boundary oprators, I need to flatten the boundary, which alters the symbol for the laplacian. – Swarnendu Sil Nov 9 '15 at 10:57
• So, it looks to me that this verifies the condition only when the boundary is flat... the same goes for the verification in "Partial Differential Equations I:Basic Theory" by M.E. Taylor, proposition 11.12, since his verification of the Neumann condition ( proposition 11.11 ) and his remark just before proposition 11.12 holds only when the boundary has been flattened already. – Swarnendu Sil Nov 9 '15 at 11:07
• Also, for my question (2), any help is appreciated. In fact, I saw that I dont need the estimate I mentioned. It will be enough for me to have an estimate $\lVert \nabla \omega \rVert_{L^p} \leq \lVert F \rVert_{L^p}$ when $\omega$ satisfies, $\Delta \omega = \delta F$ in $\Omega$, $\nu\wedge\omega = 0$ on $\partial\Omega$, $\nu\wedge\delta\omega = 0$ on $\partial\Omega.$ – Swarnendu Sil Nov 9 '15 at 11:10
• Such an estimate is clearly true in for the system with Dirichlet boundary condition, i.e $\Delta u = \operatorname*{div} F$ in $\Omega$, $u = 0$ on $\partial\Omega$. – Swarnendu Sil Nov 9 '15 at 11:14