Outdated Archival Version

These pages are not updated anymore. They reflect the state of 20 August 2005. For the current production of this journal, please refer to http://www.math.psu.edu/era/.

% **************************************************************************
% * The TeX source for AMS journal articles is the publisher's TeX code    *
% * which may contain special commands defined for the AMS production      *
% * environment.  Therefore, it may not be possible to process these files *
% * through TeX without errors.  To display a typeset version of a journal *
% * article easily, we suggest that you either view the HTML version or    *
% * retrieve the article in DVI, PostScript, or PDF format.                *
% **************************************************************************
%  Author Package
%% Translation via Omnimark script a2l, September 19, 1996 (all in one day!)
%\controldates{4-OCT-1996,4-OCT-1996,4-OCT-1996,11-OCT-1996}
\documentclass{era-l}
%\issueinfo{2}{2}{OCT}{1996}
\pagespan{92}{97}
\PII{S 1079-6762(96)00012-1}


%% Declarations:

\theoremstyle{plain}
\newtheorem*{theorem1}{Theorem 1}
\newtheorem*{theorem2}{Theorem 2}
\newtheorem*{theorem3}{Theorem 3}


%% User definitions:



\begin{document}

\title[Boundary integral methods for harmonic differential forms]{Boundary 
integral methods for harmonic differential forms
in Lipschitz domains}
\author{Dorina Mitrea}
\address{Department of Mathematics, University of
Missouri-Columbia, Columbia, MO 65211}
\email{dorina@sidon.math.missouri.edu}
\author{Marius Mitrea}
\address{Department of Mathematics, University of
Missouri-Columbia, Columbia, MO 65211}
\email{marius@msindy2.math.missouri.edu}
\subjclass{Primary 35J55; Secondary 42B20}
\keywords{Harmonic differential forms, Lipschitz domains, layer potentials}
\commby{Michael Taylor}
\date{July 25, 1996}
\begin{abstract}A layer potential based approach for boundary value problems
for harmonic 
differential forms in nonsmooth domains is developed. This allows a 
complete and unified treatment of several fundamental problems in 
potential theory.
\end{abstract}
\maketitle



\section*{\S 1. Introduction}In this note we report on recent progress in the 
study of
boundary value problems for harmonic differential forms on Lipschitz domains
by means of layer potential methods. 
The main issue which is addressed here is that of the effectiveness of the 
boundary 
integral methods in the higher degree context, arbitrary topology and in the 
presence 
of singularities.

Our treatment unifies and 
generalizes several major directions in potential theory. Most notably, our 
theory
encompasses both the classical theory of harmonic integrals (or generalized 
potential
theory) initiated by Hodge in the 1930's and studied at length by many authors 
thereafter (cf., e.g., the monographs \cite{Ho}, \cite{Mo}, \cite{Con}, 
\cite{Ta} 
and the references therein) as well as the more recent theory of boundary 
value 
problems for harmonic 
scalar-valued functions in Lipschitz domains emerging from Calder\'{o}n's 
program
in the 1960's; see \cite{Dah}, \cite{FaJoRi}, \cite{JeKe}, \cite{Ve}.

The rather very general setting within which we formulate and solve these 
problems
introduces new, significant difficulties which can be roughly categorized as 
having
(a) analytical nature, and (b) topological nature. Obstacles in the first 
category arise
as a result of allowing higher degree differential forms on domains with only 
Lipschitz 
continuous boundaries and square-integrable boundary data. Those in the 
second category
occur in connection with the process of ``patching'' together local results 
in order 
to deal with global versions of our BVP's. 

We develop a very effective, hands-on approach for these problems, 
and many of our results are new even in the smooth context. In fact, 
this allows us for the first time to undertake a systematic study of all
natural boundary conditions for the Laplace operator acting on differential 
forms. 
However, due to obvious space limitations, here we explain only one example,
playing a paradigm role for the entire theory. 

\section*{\S 2. Statement of a boundary value problem}We shall work with 
differential forms for which we employ fairly standard notation.
In particular, $d$, $\delta $, $\ast $ stand for the exterior derivative, 
co-derivative
and Hodge-$\ast $ operator, respectively. Also, $\wedge $ and $\vee $ denote, 
respectively,
the exterior and interior product of forms. 

Let $\Omega $ be an arbitrary bounded Lipschitz domain in ${\mathbb{R}}^{m}$ 
(occasionally denoted 
by $\Omega _{+}$), i.e. a domain whose boundary is locally given by graphs of 
Lipschitz 
functions. The outward unit normal 
$n$ will be canonically identified with the $1$-form $\sum _{j}n_{j}dx_{j}$.  
For each integer $0\leq l\leq m$, we introduce the space of {\em tangential} 
square-integrable (with respect to the surface measure $d\sigma $) $l$-forms 
defined on $\partial \Omega $ by  
\begin{equation*}L^{2}_{tan}(\partial \Omega ,\Lambda 
^{l}{\mathbb{R}}^{m}):=\{A\in L^{2}(\partial \Omega ,\Lambda 
^{l}{\mathbb{R}}^{m}),\,n\vee A=0\,\,
\text{a.e. on }\partial \Omega \}. 
\end{equation*}
Here ``a.e.'' is taken with respect to the canonical surface measure $d\sigma 
$.
Let $\langle \cdot ,\cdot \rangle $ stand for the usual (pointwise) Euclidean 
pairing 
of forms. An $l$-form $A$ in the space $L^{2}_{tan}(\partial \Omega ,\Lambda 
^{l}{\mathbb{R}}^{m})$ 
is said to have its 
{\em boundary (exterior) co-derivative} in $L^{2}$ if there exists an 
$(l-1)$-form in 
$L^{2}(\partial \Omega ,\Lambda ^{l-1}{\mathbb{R}}^{m})$, which we denote by 
$\delta _{\partial }A$, such that 
\begin{equation*}\int _{\partial \Omega }\langle d\psi ,A\rangle d\sigma 
=\int _{\partial \Omega }\langle \psi ,\delta _{\partial }A\rangle d\sigma 
\,\,\,\,\,\text{for any}\,\,\,\psi \in C^{\infty }({\mathbb{R}}^{m},\Lambda 
^{l-1}{\mathbb{R}}^{m}).
\end{equation*}
We set 
$L^{2,\delta }_{tan}(\partial \Omega ,\Lambda ^{l}{\mathbb{R}}^{m})
:=\{A\in L^{2}_{tan}(\partial \Omega ,\Lambda ^{l}{\mathbb{R}}^{m});\,\delta 
_{\partial }A\in L^{2}(\partial \Omega ,\Lambda ^{l-1}{\mathbb{R}}^{m})\}$, 
and equip it with the natural norm 
$\|A\|_{L_{tan}^{2,\delta }}:=\|A\|_{L^{2}(\partial \Omega )}
+\|\delta _{\partial }A\|_{L^{2}(\partial \Omega )}$.
We shall also make use of the (closed) subspace 
$L^{2,0}_{tan}(\partial \Omega ,\Lambda ^{l}{\mathbb{R}}^{m})$ of 
$L^{2,\delta }_{tan}(\partial \Omega ,\Lambda ^{l}{\mathbb{R}}^{m})$ given by
\begin{equation*}L^{2,0}_{tan}(\partial \Omega ,\Lambda 
^{l}{\mathbb{R}}^{m}):=\left \{A\in L^{2,\delta }_{tan}(\partial \Omega 
,\Lambda ^{l}{\mathbb{R}}^{m});\,\delta _{\partial }A=0\right \}.
\end{equation*}

Consider the boundary value problem
\begin{equation*}(BVP1_{l})\,
\begin{cases}F\in C^{\infty }(\Omega ,\Lambda ^{l}{\mathbb{R}}^{m}),\\
\triangle F=0 \,\,\text{in}\,\,\Omega ,\\
{\mathcal{N}}(F),\,{\mathcal{N}}(d F)\in L^{2}(\partial \Omega ),\\
n\vee F|_{\partial \Omega }=A\in L^{2}_{tan}(\partial \Omega ;\Lambda 
^{l-1}{\mathbb{R}}^{m}),\\
n\vee (dF)|_{\partial \Omega }=B\in L^{2}_{tan}(\partial \Omega ,\Lambda 
^{l}{\mathbb{R}}^{m}).
\end{cases}
\end{equation*}
Here $A$ and $B$ are some a priori given tangential forms on the
boundary and the restriction to the boundary is taken in the
nontangential pointwise sense. Also, ${\mathcal{N}}(\cdot )$ is the usual 
nontangential
maximal operator and $\triangle $ is the Laplacian in ${\mathbb{R}}^{m}$. 

In order to state our first result, for $0\leq l\leq m$ we introduce
\begin{multline*}
{\mathcal{H}}^{l}_{\vee }(\Omega ):=\{E\in C^{\infty }(\Omega ,\Lambda 
^{l}{\mathbb{R}}^{m});
\,{\mathcal{N}}(E)\in L^{2}(\partial \Omega ),\\
dE=0\,\,\text{and}\,\,\delta E=0\,\,
\text{in}\,\,\Omega ,\,n\vee E|_{\partial \Omega }=0\,\,\text{on}\,\,\partial 
\Omega \},
\end{multline*}
and recall that $S_{l}$ stands for the usual single layer potential 
operator for $\partial \Omega $ acting (componentwise) on $l$-forms on 
$\partial \Omega $. 

\begin{theorem1} With the above notation we have:
\begin{enumerate}
\item A solution of $(BVP1_{l})$ exists if and only if $B$ satisfies 
the compatibility condition $B\in \{E|_{\partial \Omega };
E\in {\mathcal{H}}_{\vee }^{l}(\Omega )\}^{\bot }$ . 

\item The dimension of the space of null-solutions for the homogeneous 
problem is 
$b_{l}(\Omega )$, the $l$-th Betti number of $\Omega $, and, in fact, this 
space coincides 
precisely with 
${\mathcal{H}}_{\vee }^{l}(\Omega )$. In particular, the boundary data 
determine 
$dF$ and $\delta F$ uniquely.
\end{enumerate}

If the compatibility condition is satisfied, then every solution has the form 
$F=S_{l}U_{1}+dS_{l-1}U_{2}+\delta S_{l+1}U_{3}$ with 
$U_{1}\in L^{2}(\partial \Omega ,\Lambda ^{l}{\mathbb{R}}^{m})$, 
$U_{2}\in L^{2}_{tan}(\partial \Omega ,\Lambda ^{l-1}{\mathbb{R}}^{m})$, 
$U_{3}\in \ast L^{2,0}_{tan}(\partial \Omega ,\Lambda 
^{m-l-1}{\mathbb{R}}^{m})$, 
and 
\begin{equation*}\|{\mathcal{N}}(dF)\|_{L^{2}(\partial \Omega )}
\leq C\|B\|_{L^{2}_{tan}(\partial \Omega ,\Lambda ^{l}{\mathbb{R}}^{m})}.
\end{equation*}
Moreover, we have the following regularity statements:

\begin{enumerate}
\item [3.] ${\mathcal{N}}(\delta dF)\in L^{2}(\partial \Omega )$ if and only if 
$B\in L^{2,\delta }_{tan}(\partial \Omega ,\Lambda ^{l}{\mathbb{R}}^{m})$. In 
this case 
\begin{equation*}\|{\mathcal{N}}(\delta dF)\|_{L^{2}(\partial \Omega )}
\leq C\|B\|_{L^{2,\delta }_{tan}(\partial \Omega ,\Lambda 
^{l}{\mathbb{R}}^{m})}.
\end{equation*}

\item [4.] $\delta dF=0$ in $\Omega $ if and only if 
$B\in L^{2,0}_{tan}(\partial \Omega ,\Lambda ^{l}{\mathbb{R}}^{m})$. In 
particular, 
for $B$ in $L^{2,0}_{tan}(\partial \Omega ,\Lambda ^{l}{\mathbb{R}}^{m})$ the 
problem 
$(BVP1_{l})$ becomes
\begin{equation*}(BVP2_{l})\,
\begin{cases}F\in C^{\infty }(\Omega ,\Lambda ^{l}{\mathbb{R}}^{m}),\\
\triangle F=0 \,\,\text{in}\,\,\Omega ,\\
\delta dF=0 \,\,\text{in}\,\,\Omega ,\\
{\mathcal{N}}(F),\,{\mathcal{N}}(dF)\in L^{2}(\partial \Omega ),\\
n\vee F|_{\partial \Omega }=A\in L^{2}_{tan}(\partial \Omega ;\Lambda 
^{l-1}{\mathbb{R}}^{m}),\\
n\vee (dF)|_{\partial \Omega }=B\in L^{2,0}_{tan}(\partial \Omega ,\Lambda 
^{l}{\mathbb{R}}^{m}).
\end{cases}
\end{equation*}
\item [5.] ${\mathcal{N}}(\delta F)\in L^{2}(\partial \Omega )$ if and only if 
$A\in L^{2,\delta }_{tan}(\partial \Omega ,\Lambda ^{l-1}{\mathbb{R}}^{m})$. 
Moreover, for 
$A\in L^{2,\delta }_{tan}(\partial \Omega ,\Lambda ^{l-1}{\mathbb{R}}^{m})$, 
there holds
\begin{equation*}\|{\mathcal{N}}(\delta F)\|_{L^{2}(\partial \Omega )}
\leq C\left (\|A\|_{L^{2,\delta }_{tan}(\partial \Omega ,\Lambda 
^{l-1}{\mathbb{R}}^{m})}+
\|B\|_{L^{2}_{tan}(\partial \Omega ,\Lambda ^{l}{\mathbb{R}}^{m})}\right ).
\end{equation*}
\item [6.] $dF=0$ in $\Omega $ if and only if $B=0$.
Furthermore, when $B=0$, we can prescribe periods and have genuine uniqueness. 
More precisely, for any
$\beta _{j}\in {\mathbb{R}}$, $j=1,...,b_{l}(\Omega )$, there exists a unique 
solution of
\begin{equation*}(BVP3_{l})\,
\begin{cases}F\in C^{\infty }(\Omega ,\Lambda ^{l}{\mathbb{R}}^{m}),\\
\triangle F=0 \,\,\text{in}\,\,\Omega ,\\
dF=0 \,\,\text{in}\,\,\Omega ,\\
{\mathcal{N}}(F)\in L^{2}(\partial \Omega ),\\
n\vee F|_{\partial \Omega }=A\in L^{2}_{tan}(\partial \Omega ;\Lambda 
^{l-1}{\mathbb{R}}^{m}),\\
\int _{\gamma _{j}}\iota ^{*}F=\beta _{j},\,\,j=1,...,b_{l}(\Omega ),
\end{cases}
\end{equation*}
where ${[\gamma _{j}]}_{j=1,...,b_{l}(\Omega )}$ is a basis of 
$H^{l}_{sing}(\Omega ;{\mathbb{R}})$, the 
$l$-th singular homology group of $\Omega $ over the reals, and $\iota 
:\gamma _{j}\to \Omega $
is the inclusion for all $j$'s.

\item [7.] $\delta F=0$ if and only if
$A\in L^{2,0}_{tan}(\partial \Omega ,\Lambda ^{l-1}{\mathbb{R}}^{m})\cap 
\{{\mathcal{H}}^{l-1}_{\vee }(\Omega )|_{\partial \Omega }\}^{\bot }$
and $B\in L^{2,0}_{tan}(\partial \Omega ,\Lambda ^{l}{\mathbb{R}}^{m})$.
In fact, if $A$ and $B$ are as above, then $(BVP1_{l})$ reduces to
\begin{equation*}(BVP4_{l})\,
\begin{cases}F\in C^{\infty }(\Omega ,\Lambda ^{l}{\mathbb{R}}^{m}),\\
\triangle F=0 \,\,\text{in}\,\,\Omega ,\\
\delta F=0 \,\,\text{in}\,\,\Omega ,\\
{\mathcal{N}}(F),\,{\mathcal{N}}(dF)\in L^{2}(\partial \Omega ),\\
n\vee F|_{\partial \Omega }=A\in L^{2,0}_{tan}(\partial \Omega ,\Lambda 
^{l-1}{\mathbb{R}}^{m}),\\
n\vee dF|_{\partial \Omega }=B\in L^{2,0}_{tan}(\partial \Omega ,\Lambda 
^{l}{\mathbb{R}}^{m}).
\end{cases}
\end{equation*}
In particular, $B=0$ forces $dF=0$, and we can prescribe periods,
in which case $(BVP4_{l})$ becomes
\begin{equation*}(BVP5_{l})\,
\begin{cases}F\in C^{\infty }(\Omega ,\Lambda ^{l}{\mathbb{R}}^{m}),\\
\delta F=0 \,\,\text{in}\,\,\Omega ,\\
dF=0 \,\,\text{in}\,\,\Omega ,\\
{\mathcal{N}}(F)\in L^{2}(\partial \Omega ),\\
n\vee F|_{\partial \Omega }=A\in L^{2,0}_{tan}(\partial \Omega ,\Lambda 
^{l-1}{\mathbb{R}}^{m}),\\
\int _{\gamma _{j}}\iota ^{*}F=\beta _{j},\,\,j=1,...,b_{l}(\Omega ).
\end{cases}
\end{equation*}
Formulated as such, the problem $(BVP5_{l})$ has a solution if and only if
$A\in \{{\mathcal{H}}^{l-1}_{\vee }(\Omega )|_{\partial \Omega }\}^{\bot }$ 
and the
solution is unique.
\end{enumerate}\end{theorem1}
There is an analogous statement for the dual problem of $(BVP1_{l})$, 
corresponding to an application of the
Hodge star isomorphism. Also, similar results are valid for the exterior domain
$\Omega _{-}:={\mathbb{R}}^{m}\setminus \bar {\Omega }$ (with appropriate 
decay 
conditions included). 

It is both rewarding and illuminating to point out that $(BVP1_{l})$ becomes 
the 
Dirichlet problem for the
Laplacian (in slight disguise) for $l=m$ and, further, its so-called regular
version if, in addition, 
$A\in L^{2,\delta }_{tan}(\partial \Omega ,\Lambda 
^{m}{\mathbb{R}}^{m})(=W^{1,2}(\partial \Omega )
d\text{Vol}_{m})$. Also, $(BVP1_{l})$ reduces precisely to the 
classical Neumann problem for the Laplacian in the case when $l=0$. 
For Lipschitz domains, these problems have been first addressed in the work of 
B. Dahlberg, E. Fabes, D. Jerison, C. Kenig, G. Verchota, among others. 
The family $(BVP_{l})_{0\leq l\leq m}$ also encompasses problems arising in 
static
electromagnetism for Lipschitz domains in ${\mathbb{R}}^{3}$. This 
identification 
takes place at the
level $l=1$, when one canonically identifies differential forms of degree one 
with 
vector fields. See \cite{MiMiPi}. 

For forms of arbitrary degree, so far the only reasonably well understood 
case of 
the problem $(BVP1_{l})$ is when the domain $\Omega $ has a smooth boundary 
and when 
the boundary data are smooth as well (see the excellent exposition in 
\cite{Ta} in this connection). 
Of course, in the smooth case, there are a number of specific techniques 
which are not available in the irregular setting like, e.g., a symbolic 
calculus
for pseudo-differential operators, as well as various regularity results near 
the boundary which readily open the door for the applicability of the De Rham 
cohomology machinery. 
As many major tools valid in the smooth case break down in the 
presence of singularities, the connection between potential theory and 
topology is 
considerably more difficult to explore in the present, more general context. 

\section*{\S 3. Boundary integral operators}In solving the above problems, we 
rely on boundary integral methods. 
A particularly important role is played by the family of operators
$\{\pm {{\textstyle {\frac{1}{2}}}}I+M_{l}\}_{0\leq l\leq m}$, where 
$I$ is the identity and $M_{l}$ is a singular integral operator defined by 
\begin{equation*}M_{l}A(P):=\lim _{\epsilon \to 0}\left (n(P)\vee \int 
_{\underset {{|P-Q|\geq \epsilon }}{Q\in \partial \Omega }}
(d\Gamma )(P-Q)\wedge A(Q)\,d\sigma (Q)\right),\quad P\in \partial \Omega .
\end{equation*}
Here $\Gamma (X)$ is the canonical fundamental solution for the Laplacian in 
${\mathbb{R}}^{m}$
and $A\in L^{2}_{tan}(\partial \Omega ,\Lambda ^{l}{\mathbb{R}}^{m})$. 
It is important to note that the family of singular integral 
operators $\{M_{l}\}_{0\leq l\leq m}$ encompasses both the classical double 
layer
potential operator $K$ as well as its adjoint $K^{*}$. Indeed, $M_{l}$ can be 
canonically
identified with $K$ and $K^{*}$ for $l=m-1$ and $l=0$, respectively. 

\begin{theorem2} Let $\Omega $ be a bounded Lipschitz domain in 
${\mathbb{R}}^{m}$ and 
$0\le l\le m$. Then the following statements are valid:

\begin{enumerate}
\item The operators $\pm {{\textstyle {\frac{1}{2}}}} +M_{l}$ are Fredholm 
with 
index zero on each of the 
spaces $L^{2}_{tan}(\partial \Omega ,\Lambda ^{l}{\mathbb{R}}^{m})$, 
$L^{2,\delta }_{tan}(\partial \Omega ,\Lambda ^{l}{\mathbb{R}}^{m})$, 
$L^{2,0}_{tan}(\partial \Omega ,\Lambda ^{l}{\mathbb{R}}^{m})$. 
Furthermore, their kernels on these spaces coincide and, in fact,
\begin{equation*}\text{Ker}\left (\pm {{\textstyle {\frac{1}{2}}}} I+M_{l};
L^{2}_{tan}(\partial \Omega ,\Lambda ^{l}{\mathbb{R}}^{m})\right )=
\left \{n\vee E|_{\partial \Omega _{\pm }};E\in {\mathcal{H}}_{\wedge }^{l+
1}(\Omega _{\pm })
\right \}. 
\end{equation*}
In particular, $\text{dim}\,\text{Ker}\left ({{\textstyle {\frac{1}{2}}}} I+
M_{l};
L^{2}_{tan}(\partial \Omega ,\Lambda ^{l}{\mathbb{R}}^{m})\right 
)=b_{m-l-1}(\Omega )$, and
\begin{equation*}\text{dim}\,\text{Ker}\left (-{{\textstyle {\frac{1}{2}}}} I+M_{l};
L^{2}_{tan}(\partial \Omega ,\Lambda ^{l}{\mathbb{R}}^{m})\right 
)=b_{l}(\Omega ).
\end{equation*} 
Also, 
\begin{align*}
&\text{Image}\left (\pm {{\textstyle {\frac{1}{2}}}}I+M_{l};
L^{2}_{tan}(\partial \Omega ,\Lambda ^{l}{\mathbb{R}}^{m})\right )\\
&\qquad=
\left \{A\in L^{2}_{tan}(\partial \Omega ,\Lambda ^{l}{\mathbb{R}}^{m});
\,A\bot E|_{\partial \Omega _{\mp }}\,\,\text{for any}\,\,E
\in {\mathcal{H}}_{\vee }^{l}(\Omega _{\mp })\right \}.
\end{align*}
Similar descriptions are valid for the images of 
$\pm {{\textstyle {\frac{1}{2}}}}I+M_{l}$ when acting on 
$L^{2,\delta }_{tan}(\partial \Omega ,\Lambda ^{l}{\mathbb{R}}^{m})$ and 
$L^{2,0}_{tan}(\partial \Omega ,\Lambda ^{l}{\mathbb{R}}^{m})$. 
\end{enumerate}Also, the following operators are isomorphisms on the 
indicated spaces:
\begin{enumerate}
\item [2.] $\pm {{\textstyle {\frac{1}{2}}}} I+M_{l}$ acting on $\delta 
_{\partial }\left [
L^{2,\delta }_{tan}(\partial \Omega ,\Lambda ^{l}{\mathbb{R}}^{m})\right ]$;
\item [3.] $\pm {{\textstyle {\frac{1}{2}}}} I+M_{l}$ acting on 
\begin{equation*}\frac{L^{2,\delta }_{tan}(\partial \Omega ,\Lambda 
^{l}{\mathbb{R}}^{m})}{L^{2,0}_{tan}(\partial \Omega ,\Lambda 
^{l}{\mathbb{R}}^{m})}\,\,\,\,\text{and}\,\,\,\,
\frac{L^{2}_{tan}(\partial \Omega ,\Lambda ^{l}{\mathbb{R}}^{m})}{L^{2,0}_{tan}
(\partial \Omega ,\Lambda ^{l}{\mathbb{R}}^{m})};
\end{equation*}
\item [4.] $\pm {{\textstyle {\frac{1}{2}}}} I+M_{l}$ acting on 
\begin{equation*}\frac{L^{2}_{tan}(\partial \Omega ,\Lambda 
^{l}{\mathbb{R}}^{m})}{\text{Ker}\,
(\pm {{\textstyle {\frac{1}{2}}}} I+M_{l})},\,\,\,\,
\frac{L^{2,\delta }_{tan}(\partial \Omega ,\Lambda 
^{l}{\mathbb{R}}^{m})}{\text{Ker}\,
(\pm {{\textstyle {\frac{1}{2}}}} I+M_{l})},\,
\,\,\,\text{and}\,\,\,\,
\frac{L^{2,0}_{tan}(\partial \Omega ,\Lambda 
^{l}{\mathbb{R}}^{m})}{\text{Ker}\,(\pm {{\textstyle {\frac{1}{2}}}} I+
M_{l})}.
\end{equation*}
\item [5.] $\pm {{\textstyle {\frac{1}{2}}}} I+M_{l}$ acting on 
$(n\vee \ast L^{2,0}_{tan}(\partial \Omega ,\Lambda 
^{m-l}{\mathbb{R}}^{m}))^{\bot }$;
\item [6.] $n\vee S_{l}+\left (\pm {{\textstyle {\frac{1}{2}}}}I+M_{l-1}\right 
)^{-1}
[\delta _{\partial }(n\vee S_{l+1})(n\wedge S_{l})]$ from 
$\ast \text{Ker}\,(\mp {{\textstyle {\frac{1}{2}}}}I+M_{m-l})$ onto 
$\text{Ker}\,(\pm {{\textstyle {\frac{1}{2}}}}I+M_{l-1})$, 
where the inverse operators $(\pm {{\textstyle {\frac{1}{2}}}}I+M_{l-1})^{-1}$ 
are considered on the space 
$\delta _{\partial }\left [L^{2,\delta }_{tan}(\partial \Omega ,\Lambda 
^{l-1}{\mathbb{R}}^{m})\right ]$ 
(cf. (2) above). 
\end{enumerate}\end{theorem2}


As this theorem shows, there are natural obstructions to inverting the 
boundary 
layer potential operators $\pm {{\textstyle {\frac{1}{2}}}} I+M_{l}$ 
on, e.g., $L^{2}_{tan}(\partial \Omega ,\Lambda ^{l}{\mathbb{R}}^{m})$ which 
are expressed
in the form of the nonvanishing of certain singular homology groups of the 
underlying domain. In particular, a too direct utilization of these operators 
in 
conjunction with $(BVP1_{l})$ appears to require suitable topological 
restrictions. 
In this light, it is rather remarkable that such ``side-effects'' can be 
avoided
making appropriate corrections (i.e., by adding further source terms). 

\section*{\S 4. A regularity theorem}A basic result which allows us 
to relate boundary integral operators to the mechanism of (interior) Hodge type
decompositions for forms with coefficients in $L^{2}(\Omega )$, is a certain 
regularity
theorem which is interesting in its own rights and which we now describe.

\begin{theorem3} For a differential 
form $E\in L^{2}(\Omega ,\Lambda ^{l}{\mathbb{R}}^{m})$ with $dE\in 
L^{2}(\Omega ,\Lambda ^{l+1}{\mathbb{R}}^{m})$, 
$\delta E\in L^{2}(\Omega ,\Lambda ^{l-1}{\mathbb{R}}^{m})$ ($d$ and 
$\delta $ are considered in some weak distributional sense) the following are 
equivalent:
\begin{enumerate}
\item $n\wedge E$, initially considered in the sense of distributions in the 
Sobolev 
space $W^{-{{\textstyle {\frac{1}{2}}}},2}(\partial \Omega ,\Lambda ^{l+
1}{\mathbb{R}}^{m})$, 
actually belongs to $L^{2}(\partial \Omega ,\Lambda ^{l+1}{\mathbb{R}}^{m})$;
\item $n\vee E$, initially considered in the sense of distributions in the 
Sobolev
space $W^{-{{\textstyle {\frac{1}{2}}}},2}(\partial \Omega ,\Lambda 
^{l-1}{\mathbb{R}}^{m})$,
actually belongs to $L^{2}(\partial \Omega ,\Lambda ^{l-1}{\mathbb{R}}^{m})$;
\item $E\in W^{\frac{1}{2},2}(\Omega ,\Lambda ^{l}{\mathbb{R}}^{m})$.
\end{enumerate}\end{theorem3}


There are also natural accompanying estimates in each case. In particular, 
\begin{align*}
\|E\|_{W^{\frac{1}{2},2}(\Omega ,\Lambda ^{l}{\mathbb{R}}^{m})}\leq & 
C\left (\|E\|_{L^{2}(\Omega ,\Lambda ^{l}{\mathbb{R}}^{m})}
+\|dE\|_{L^{2}(\Omega ,\Lambda ^{l+1}{\mathbb{R}}^{m})}
+\|\delta E\|_{L^{2}(\Omega ,\Lambda ^{l-1}{\mathbb{R}}^{m})}\right )\\
&+C\,\text{min}\left \{\|n\wedge E\|_{L^{2}(\partial \Omega ,\Lambda ^{l+
1}{\mathbb{R}}^{m})}\,,\,
\|n\vee E\|_{L^{2}(\partial \Omega ,\Lambda ^{l-1}{\mathbb{R}}^{m})}\right \}.
\end{align*}
The exponent ${{\textstyle {\frac{1}{2}}}}$ is sharp in the class of 
Lipschitz domains, and this is strongly
contrasting the smooth case where ${{\textstyle {\frac{1}{2}}}}$ may be 
replaced by $1$. 

\bibliographystyle{amsalpha}
\begin{thebibliography}{AVG222}



\bibitem[Con]{Con}
P. E. Conner, {\em The Neumann's problem for differential forms on Riemannian 
manifolds}, vol.~20, Memoirs of Amer. Math. Soc., (third printing), 1956.
 \MR{17:1197e}
\bibitem[Dah]{Dah}
J. E. Dahlberg, {\em Estimates of harmonic measure}, Arch. Rat. Mech. 
Anal. {\bf 65} (1977), 275--288.
 \MR{57:6470} 
\bibitem[FaJoRi]{FaJoRi}
E. Fabes, M. Jodeit and N. Rivi\`{e}re, {\em Potential techniques for 
boundary value problems on $C^{1}$ domains}, Acta Math. {\bf 141} (1978), 
165--186.
 \MR{80b:31006} 
\bibitem[Ho]{Ho}
W. V. D. Hodge, {\em The theory and applications of harmonic integrals}, 
Cambridge University Press, 1941.
 \MR{2:296d}
\bibitem[JeKe]{JeKe}
D. Jerison and C. Kenig, {\em The Neumann problem on Lipschitz domains}, 
Bull.  Amer.  Math. Soc. {\bf 4} (1981), 203--207.
 \MR{84a:35064}
\bibitem[MiMiPi]{MiMiPi}
D. Mitrea, M. Mitrea, and J. Pipher, {\em Vector potential theory on 
nonsmooth domains in ${\mathbb{R}}^{3}$ and applications to
electromagnetic scattering}, to appear in the Journal of Fourier Analysis and
Applications, Fall, (1995).
\bibitem[Mo]{Mo}
C. B. Morrey, Jr., {\em Multiple integrals in the calculus of variations}, 
Heidelberg, New York, Springer-Verlag, Berlin, 1966.
 \MR{34:2380}
\bibitem[Ta]{Ta}
M. E. Taylor, {\em Partial differential equations}, Springer-Verlag, 1996.
\CMP{96:14}
\bibitem[Ve]{Ve}
G. Verchota, {\em Layer potentials and regularity for the Dirichlet problem
for Laplace's equation in 
Lipschitz domains}, J. Funct. Anal. {\bf 59} (1984), 572--611.
 \MR{86e:35038}

\end{thebibliography}

\end{document}