# The $p$-CurlCurl : Spaces, traces, coercivity and a Helmholtz decomposition
in $L^{p}$

Marc Laforest École Polytechnique de Montréal, Département de mathématiques et
génie industriel, C.P. 6079 succ. Centre-ville, Montréal, Québec, Canada, H3C
3A7 (Marc.Laforest@polymtl.ca).

###### Abstract

This work provides the foundation for the finite element analysis of an
elliptic problem which is the rotational analogue of the $p$-Laplacian and
which appears as a model of the magnetic induction in a high-temperature
superconductor operating near it’s critical current. Whereas the function
theory for the $p$-Laplacian requires standard results in $L^{p}$ Sobolev
spaces, this problem requires an extension to $L^{p}$ spaces of the well-known
$L^{2}$ theory for divergence free vector fields, as used in the finite
element method applied to incompressible flows and electromagnetic radiation.
Among other things, the analysis requires extensions to $L^{p}$ of the well-
known $H(\operatorname{div};\Omega)$ and $H(\operatorname{curl};\Omega)$,
extensions of traces and Green’s theorem, a Helmholtz decomposition and
finally a Friedrich’s inequality. In this paper, we provide a proof of the
existence and uniqueness of weak solutions of our so-called $p$-CurlCurl
problem. In a companion paper, the analysis is extended to treat continuous
and finite element solutions of the nonlinear parabolic problem whose spatial
term is the $p$-CurlCurl operator. Many of the results presented here are
either already known, known in slightly different forms or are proven with the
help of techniques that are already well-known. The main novelty of this paper
appears to be the structured form of this $L^{p}$ theory and our form of the
Helmholtz decomposition and of the Friedrich’s inequality. In this respect, we
note that some of these results can be found in the works of M. Dauge, M.
Mitrea, I. Mitrea and …

###### keywords:

$L^{p}$ theory, $p$-Laplacian, Helmholtz decomposition, Hodge decomposition,
nonlinear, degenerate diffusion, magnetic resistivity, weak solution,
rotational, divergence free, superconductor, electromagnetism.

###### AMS:

78M10; 35J60; 35J92; 35J50.

##  1 Introduction

The objective of this paper is to study the existence and uniqueness of weak
solutions of a nonlinear elliptic problem from applied superconductivity [7,
Sirois]. With respect to the magnetic flux
$\mathbf{B}:\Omega\to\mathbb{R}^{3},$ defined over a bounded domain $\Omega$,
the stationnary problem takes the form

$\displaystyle\nabla\times\Big{(}\big{|}\nabla\times\mathbf{B}\big{|}^{p-2}\nabla\times\mathbf{B}\Big{)}$
$\displaystyle=\mathbf{S},\text{ over $\Omega$,}$ (1)
$\displaystyle\nabla\cdot\mathbf{B}$ $\displaystyle=0,\text{ over $\Omega$,}$
(2) $\displaystyle\mathbf{n}\times\mathbf{B}$ $\displaystyle=0,\text{ over
$\partial\Omega$,}$ (3)

where $p\in(1,\infty)$, $\mathbf{n}$ is the outwards normal along the boundary
$\partial\Omega$, $\mathbf{S}$ is given and satisfies some additional
conditions, and the boundary $\partial\Omega$ is sufficiently smooth. It is
quite obvious that this problem is closely related to the $p$-Laplacian whose
highest order term is of the form
$\nabla\cdot(|\nabla\mathbf{u}|^{p-2}\nabla\mathbf{u})$ and for which a well-
developed theory already exists [19, 35, 4, 15]. For this reason, we have
taken the liberty of naming this problem the elliptic $p$-CurlCurl.

In this paper, we tackle problem (1)-(3) by constructing extensions of
standard variational techniques from electromagnetics [27]. Roughly speaking,
these extensions take the form of generalizations of results from $L^{2}$ to
$L^{p}$ where $p\in(1,\infty)$ but in practice very large. Although many of
the results described in this paper are scattered throughout the literature,
we have found it useful to write them in a self-contained form for future
reference. For the engineering community which might not have access to the
mathematics literature, we have endeavoured to provide detailed references and
attributions to the results presented, while still providing a complete
account of those results. One reason why such a theory did not already exist
is probably that a problem requiring it, like the $p$-CurlCurl, had yet to
come to the attention of the applied mathematics community.

The main contributions of this paper, besides the existence and uniqueness
result for the elliptic $p$-CurlCurl problem, are the forms of the Helmholtz
decomposition, Theorem 31, and Friedrich’s inequality, Theorem
LABEL:Friedrich_inequality2. Nečas [29], Grisvard [20] and Taylor [34] have
already given detailed presentations of elliptic regularity results in
$L^{p}$, in the spirit of the groundbreaking work for smooth domains of Agmon,
Douglis and Nirenberg [2]. The earliest references to the $L^{p}$ analogues of
$H(\text{div})$ and $H(\text{curl})$ we have found were those in Mitrea [25,
26, 26] but they have also appeared implicitly in the work of Dauge [12, 13].
For smooth domains, the Helmholtz decomposition in $W^{s,p}$ can be found in
[8] while for $C^{1}$ domains and Besov spaces, the question has recently been
studied in [17]. Our Helmholtz decomposition is slightly different then those
in either reference, mostly because the decomposition is not done over $L^{p}$
but over the $L^{p}$ analogue of $H(\text{div})$. We were unable to find a
published account of Friedrich’s inequality in $L^{p}$ even though in theory,
it very closely related to the Helmholtz decomposition.

The theory developed in this paper is presented under the strong assumption
that the domain $\Omega$ be bounded with a $C^{1}$ boundary. The boundedness
of the domain is only present to simplify the proofs while the constraint on
the boundary is impossible to overcome for the large values of $p$, between
$5$ and $100$ [Sirois], which one typically encounters in the engineering
problems underlying the $p$-CurlCurl. This strong assumption requires some
explanation since it excludes the type of domain which one would typically
encounter in finite element discretizations. Going back to counter-examples of
Dahlberg [11] and the work of Jerison and Kenig on the Poisson problem [21,
22], it is known that singularities in the smoothness of the boundary impose
restrictions on the existence theory in $L^{p}$ spaces. This work has very
recently been extended to general mixed boundary conditions for elliptic
boundary value problems by Mitrea, … [Mit] where they show there exists a
neighborhood of $(2,2)$ for the values $(s,p)$, where $s$ the regularity being
$s$ and $p$ the summability, for which the problem is well-posed; see Section
3. Mention that Mitrea uses different techniques. As such, the constraint
imposed on the smoothness on the boundary is a fundamental obstacle which will
require the engineering community to improve their model for the resistivity.
Interesting work in that direction is being done by [Sirois].

For those interested in the modeling assumptions underlying the use of the
$p$-CurlCurl in applied superconductivity, we refer the reader to either
Section 2 of the companion paper [23] or to … Mention analytic solutions
available in the literature for the time-dependent problem.

This paper is organized along the lines of Chapter 3 and 4 of Monk’s recent
monograph [27] treating finite element methods in electromagnetics, which
itself borrowed heavily from [3] and [18]. Section 2 reviews definitions of
Sobolev and Besov spaces and presents precise statements concerning a few
basic elliptic problems with boundary conditions in Besov spaces. Section 3
introduces the $L^{p}$ analogues of $H(\text{div};\Omega)$ and
$H(\text{curl};\Omega)$ and proves Green’s formulas for them. The fourth
section construct scalar and vector potentials in $W^{1,p}$ for vector fields
that are respectively curl and divergence free. The fifth section deals with
Friedrich’s inequality and the Helmholtz decomposition. Section 6 proves that
there exists a weak solution to the $p$-CurlCurl problem.

##  2 Elliptic regularity results

This section presents a summary of some necessary existence results for
elliptic boundary value problems, in particular for the Poisson problem. We
also include a description of Sobolev and Besov spaces and a list of their
main properties. With the exception of Theorem 10, the results in this section
are known and presented without proof. We refer the reader to the original
paper of Agmon et al. [2], the monograph of Nečas for $W^{k,p}$ with $k$
integral [29] and Grisvard [20] or Taylor [34] for $W^{s,p}$ with $s$ real but
greater than $2$. Although Jerison and Kenig [21, 22] have focused on
Lipschitz domains, their results are presented in forms that are closer to
those which we have presented below and therefore represent a good reference.
The results are presented for domains with $C^{1}$ boundaries and will later
briefly comment on the nature of the restrictions on $p$ which one would
encounter if more general domains were considered. For people familiar with
elliptic problems, this section can be safely skipped.

Consider an open and bounded domain $\Omega\subset\mathbb{R}^{n}$ whose
boundary $\partial\Omega$ possesses Lipschitz regularity; see Definition 1.
For $1\leq p<\infty$, recall the definition of the $L^{p}$-norm

$\|u\|_{L^{p}(\Omega)}:=\left\\{\int_{\Omega}|u|^{p}\,dx\right\\}^{1/p}$

and the Banach space of functions over $\Omega$, $L^{p}(\Omega):=\\{u\text{
measurable}\,|\,\|u\|_{p}<\infty\\}$. Using the $L^{p}$-norm to control
regularity, for each multi-index
$\boldsymbol{\alpha}:=(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})\in\mathbb{N}^{n}$,
we define $|\boldsymbol{\alpha}|=\alpha_{1}+\alpha_{2}+\cdots+\alpha_{n}$ and
a semi-norm

$|u|_{W^{{k},{p}}(\Omega)}:=\left\\{\sum_{\boldsymbol{\alpha}:|\boldsymbol{\alpha}|=k}\left|\frac{\partial^{k}u}{\partial^{\alpha_{1}}x_{1}\cdots\partial^{\alpha_{n}}x_{n}}\right|_{p}^{p}\right\\}^{1/p}.$

We can thus construct a norm by using all semi-norms for derivatives less than
or equal to a fixed order, namely

$\|u\|_{W^{{k},{p}}(\Omega)}:=\left\\{\sum_{i=0}^{k}|u|_{i,p}^{p}\right\\}^{1/p},$

and thus, by completion with respect to smooth functions over $\Omega$, the
so-called Sobolev space

$\displaystyle W^{{k},{p}}(\Omega)=\Big{\\{}u\text{
measurable}\Big{|}\|u\|_{k,p}<\infty\Big{\\}}.$ (4)

The spaces (4) are Banach spaces and when $p=2$, they are Hilbert spaces.

To extend these spaces to all real positive real values of $s$, we define the
norm

$\|u\|_{W^{{s},{p}}(\Omega)}:=\left\\{\|u\|_{W^{{m},{p}}(\Omega)}^{p}+\sum_{|\boldsymbol{\alpha}|=m}\int_{\Omega}\int_{\Omega}\frac{\partial^{\boldsymbol{\alpha}}u(\mathbf{x})-\partial^{\boldsymbol{\alpha}}u(\mathbf{y})}{|\mathbf{x}-\mathbf{y}|^{n+\sigma
p}}\,d\mathbf{x}d\mathbf{y}\right\\}^{1/p},$

where $m$ is the positive integer and $\sigma\in[0,1)$ the real number
satisfying $s=m+\sigma$. In [29], it is shown that the completition of smooth
functions $C^{\infty}(\overline{\Omega})$ with respect to the norm
$\|\cdot\|_{W^{{s},{p}}(\Omega)}$, $1<p<\infty$, gives rise to a reflexive
Banach space.

######  Definition 1.

We say that a bounded domain $\Omega$ is Lipschitz if at every point
$\mathbf{x}\in\partial\Omega$, there exits a neighborhood
$V\subset\mathbb{R}^{n}$ of $\mathbf{x}$ such that $\partial\Omega\cap V$ can
be described as the graph of a Lipschitz function. Similarly, we say that the
domain is $C^{m}$ is it’s boundary can be represented locally as the graph of
a function which is $m$ times differentiable with continuous $m$-th order
derivatives.

It is a non-trivial result that for Lipschitz domains,
$C^{\infty}(\overline{\Omega})$ is dense in $W^{{k},{p}}(\Omega)$ for
$k\in\mathbb{N}$, while this follows by construction when
$s\in\mathbb{R}^{+}\setminus\mathbb{N}$. When $s$ is a non-negative real
number and $1\leq p<\infty$, it is well-known that when the boundary of the
domain is Lipschitz, then the set of restrictions to $\Omega$ of functions in
$C^{\infty}_{0}(\mathbb{R}^{n})$ is dense in $W^{{s},{p}}(\Omega)$; Theorem
3.22 of [1]. This suggests that we define the spaces $W^{s,p}_{0}(\Omega)$ to
be the closure of the set $C_{0}^{\infty}(\Omega)$ in the norm of
$W^{{s},{p}}(\Omega)$. For any normed topological vector space $V$, we define
its dual $V^{\prime}$ to be the set of continuous linear functionals
$\ell:V\longrightarrow\mathbb{R}$, which is also a normed topological vector
space with respect to the induced norm

$\|\ell\|_{V^{\prime}}:=\sup_{v\in V:\|v\|_{V}\leq 1}\big{|}\ell(v)\big{|}.$
(5)

When $1<p<\infty$, then we define the conjugate exponent to $p$ to be $q$
satisfying $1=1/p+1/q$. The Riesz representation theorem states that
$L^{p}(\Omega)^{\prime}=L^{q}(\Omega)$. When $s$ is negative, we define
$W^{{s},{p}}(\Omega)\equiv W^{-s,q}_{0}(\Omega)^{\prime}$ becomes a Banach
spaces; see [1, 29] for more information on those spaces. Below, we will use
the symbols $(u,v)$ to denote $\int uv\,dx$, where the subscript $\Omega$ will
be included if the space over which the integration occurs is not obvious. The
bracket notation $\langle\cdot,\cdot\rangle:V\times
V^{\prime}\longrightarrow\mathbb{R}$ will be reserved for pairings between $V$
and its dual $V^{\prime}$.

One of the facts that makes handling Sobolev spaces difficult in the $L^{p}$
setting is that their traces do not belong to the same family of spaces. This
requires us to introduce the so-called Besov spaces
$B^{{s}}_{{p},{p^{\prime}}}(\partial\Omega)$ where $s$ is a measure of the
regularity and $p$ and $p^{\prime}$ are two exponents in $[1,\infty)$ which
might or might not be related. We will never need the explicit definition of
these spaces, only the existence of these spaces and the existence of
continuous maps to these spaces. The difficult definition of these spaces is
therefore omitted and we refer the reader to Chapter 7 of [1] or Treibel
[Treibel]. The fundamental properties of these fractional order spaces are the
same as those of Sobolev spaces, namely completeness and density of the subset
of smooth functions. Moreover, if $B^{s}_{p,p;0}(\Omega)$ is the closure of
$C^{\infty}_{0}(\Omega)$ in the Besov space $B^{s}_{p,p}(\Omega)$, then the
negative norm Besov spaces will be defined by duality as
$B^{s}_{p,p}(\Omega):=(B^{-s}_{q,q;0}(\Omega))^{\prime}$, where $p$ and $q$
are conjugate to each other. We now include two fundamental results for these
spaces.

######  Lemma 2 (Lemma 7.41 of [1]).

Assume that $\Omega$ is a bounded Lipschitz domain and $1<p<\infty$. Then the
restriction operator $\gamma_{0}:C^{\infty}(\overline{\Omega})\longrightarrow
C^{\infty}(\partial\Omega)$ extends continuously to a surjective map
$\gamma_{0}:W^{{1},{p}}(\Omega)\longrightarrow
B^{{1-1/p}}_{{p},{p}}(\partial\Omega)$.

######  Lemma 3 (Poincaré’s inequality, Remark 7.5 of [29]).

Assume that $\Omega$ is a bounded Lipschitz domain and $1<p<\infty$. Then
there exists a constant $C=C(\Omega)$ such that

$\|u\|_{W^{{1},{p}}(\Omega)}\leq C\left\\{\|\nabla
u\|_{L^{p}(\Omega)}^{p}+\left|\int_{\Omega}u\,d\mathbf{x}\right|^{p}\right\\}^{1/p}.$

We now present the fundamental well-posedness results we will be using. In the
following theorems, we will be assuming that the domain $\Omega$ is bounded
with a smooth $C^{1}$ boundary. Removing this constraint would require us to
significantly reduce the range of the exponent which would be unacceptable,
under the current modeling assumptions leading to the problem (1)-(3). briefly
addressing this issue at the end of the current section. The following results
are parts of what is usually referred to as the shift theorem for elliptic
partial differential equations; see Theorem 5.4 [34]. In the form given below,
these results are consequences of the work of Agmon, Douglis and Nirenberg [2]
where they assumed throughout a smooth domain and $L^{p}$-type boundary
conditions. Below we give specific references, that are neither original nor
optimal, but have the benefit of being in precisely the same form as we have
stated.

######  Theorem 4.

Consider a bounded domain with a $C^{1}$ boundary, an exponent $1<p<\infty$.
Then for any $\mu\in B^{{1-1/p}}_{{p},{p}}(\partial\Omega)$ and $f\in
W^{{-1},{p}}(\Omega)$, there exists a unique weak solution $\phi\in
W^{{1},{p}}(\Omega)$ of

$-\Delta\phi+\phi=f,\quad\text{in
$\Omega$}\qquad\text{and}\qquad\phi=\mu\quad\text{on $\partial\Omega$}.$

Moreover, there is a constant $C$ such that

$\|\phi\|_{W^{{1},{p}}(\Omega)}\leq
C\Big{(}\|\mu\|_{B^{{1-1/p}}_{{p},{p}}(\partial\Omega)}+\|f\|_{W^{{-1},{p}}(\Omega)}\Big{)}.$
(6)

######  Theorem 5 (Theorem 6.1, [22]).

Consider a bounded domain with a $C^{1}$ boundary, an exponent $1<p<\infty$.
Then for any $\mu\in B^{{1-1/p}}_{{p},{p}}(\partial\Omega)$ and $f\in
W^{{-1},{p}}(\Omega)$, there exists a unique weak solution $\phi\in
W^{{1},{p}}(\Omega)$ of

$-\Delta\phi=f,\quad\text{in
$\Omega$}\qquad\text{and}\qquad\phi=\mu\quad\text{on $\partial\Omega$}.$

Moreover, there is a constant $C$ such that

$\|\phi\|_{W^{{1},{p}}(\Omega)}\leq
C\Big{(}\|\mu\|_{B^{{1-1/p}}_{{p},{p}}(\partial\Omega)}+\|f\|_{W^{{-1},{p}}(\Omega)}\Big{)}.$
(7)

######  Theorem 6.

Consider a bounded domain with a $C^{1}$ boundary, unit outwards normal
$\mathbf{n}$, and an exponent $1<p<\infty$. Then for any $\mu\in
B^{{-1/p}}_{{p},{p}}(\partial\Omega)$ and $f\in W^{{1},{q}}(\Omega)^{\prime}$,
there exists a unique weak solution $\phi\in W^{{1},{p}}(\Omega)$ of

$-\Delta\phi+\phi=f,\quad\text{in
$\Omega$}\qquad\text{and}\qquad\mathbf{n}\cdot\nabla\phi=\mu\quad\text{on
$\partial\Omega$}.$

Moreover, there is a constant $C$ such that

$\|\phi\|_{W^{{1},{p}}(\Omega)}\leq
C\Big{(}\|\nu\|_{B^{{-1/p}}_{{p},{p}}(\partial\Omega)}+\|f\|_{W^{{1},{q}}(\Omega)^{\prime}}\Big{)}.$
(8)

######  Theorem 7 (Theorem 9.2, [16]).

Consider a bounded domain with a $C^{1}$ boundary, unit outwards normal
$\mathbf{n}$, an exponent $1<p<\infty$, and any two functions $\mu\in
B^{{-1/p}}_{{p},{p}}(\partial\Omega)$ and $f\in W^{{1},{q}}(\Omega)^{\prime}$
satisfying the compatibility condition

$\int_{\Omega}f\,d\mathbf{x}+\int_{\partial\Omega}\mu\,d\boldsymbol{\sigma}=0.$

Then there exists a weak solution $\phi\in W^{{1},{p}}(\Omega)$, unique up to
an additive constant, of the problem

$-\Delta\phi=f,\quad\text{in
$\Omega$}\qquad\text{and}\qquad\mathbf{n}\cdot\nabla\phi=\mu\quad\text{on
$\partial\Omega$}.$

Moreover, there is a constant $C$ such that

$\|\phi\|_{W^{{1},{p}}(\Omega)}\leq
C\Big{(}\|\mu\|_{B^{{-1/p}}_{{p},{p}}(\partial\Omega)}+\|f\|_{W^{{1},{q}}(\Omega)^{\prime}}\Big{)}.$
(9)

We will also be needing the following additional regularity results.

######  Theorem 8 (Theorem , [20]).

Consider a bounded domain with a $C^{1}$ boundary, outwards normal
$\mathbf{n}$, and an exponent $1<p<\infty$. For any $\mu\in
B^{{1}}_{{p},{p}}(\partial\Omega)$, there exists a unique weak solution
$\phi\in W^{{1+1/p},{p}}(\Omega)$ of $\Delta\phi=0$ in $\Omega$ and $\phi=\mu$
on $\partial\Omega$. Moreover, there exists a constant $C$ for which

$\|\phi\|_{W^{{1+1/p},{p}}(\Omega)}\leq C\,\|\mu\|_{W^{1,p}(\partial\Omega)}.$
(10)

On the other hand, for any $\mu\in B^{{1-1/p}}_{{p},{p}}(\partial\Omega)$
satisfying

$\int_{\partial\Omega}\mu\,d\boldsymbol{\sigma}=0,$

there exists a unique weak solution $\phi\in W^{{1+1/p},{p}}(\Omega)$ of
$\Delta\phi=0$ in $\Omega$ and $\mathbf{n}\cdot\nabla\phi=\mu$ on
$\partial\Omega$. Moreover, for some constant $C$,

$\|\phi\|_{W^{{1+1/p},{p}}(\Omega)}\leq
C\,\|\mu\|_{B^{{1-1/p}}_{{p},{p}}(\partial\Omega)}.$ (11)

Include a discussion of the results of Mitrea.

Later in this paper, we will be needing three additional technical results.
The first of these is demonstrated in $L^{2}$ in Section 2.2 of [18] using two
deep functional analytic results, the first from Peetre [30] and Tartar [33],
the second from Nečas [28]. Although we have not found the next lemma, as
stated, within the literature, in fact both the result of Peetre and Tartar
(stated for Banach spaces) and the one of Nečas (proved for $W^{k,p}$, $k$
integer and $1<p<\infty$) are sufficiently general that the proof of the next
lemma, as given in [18], applies verbatim. Since the result is rather
tangential, technical and in any case, somewhat natural, we do not attempt a
proof here.

######  Lemma 9.

Assume that $\Omega\subset\mathbb{R}^{n}$ has a Lipschitz boundary and
$1<p<\infty$. If $\phi\in L^{p}_{\operatorname{loc}}(\Omega)$ and
$\nabla\phi\in W^{{-1},{p}}(\Omega)$ then $\phi\in L^{p}(\Omega)$.

The second technical result is an extension of a density result of Ben
Belgacem et al. [6] based on a proof by Michel Crouzeix.

######  Theorem 10.

Consider a bounded domain $\Omega\subset\mathbb{R}^{n}$ with Lipschitz
boundary and $1<p<\infty$. Then $C^{\infty}(\overline{\Omega})$ is dense in
the space

$G=\left\\{u\in W^{{1},{p}}(\Omega)\big{|}\gamma_{0}(u)\in
W^{1,p}(\partial\Omega)\right\\},$

with norm
$\|u\|_{G}=\|u\|_{W^{{1},{p}}(\Omega)}+\|\gamma_{0}(u)\|_{W^{1,p}(\partial\Omega)}$.

###### Proof.

The proof in [6] applies mutatis mutandis.

In a few words, the proof proceeds as follows. The boundary of the domain
$\partial\Omega$ is covered by a finite set of open starlike sets
$\\{\Omega_{k}\\}_{k}$ each of which can therefore be parametrized by
spherical coordinates $(r,\theta)$. The boundary is then described by
$\partial\Omega_{k}=\\{(r,\theta)|r=R_{k}(\theta)\\}$ where $R_{k}$ is a
Lipschitz function. Choose $u\in G$ and, using the fact that $\gamma_{0}(u)\in
W^{1,p}(\partial\Omega)$, construct $u_{k}\in W^{1,p}(\Omega_{k})$ such that
$u_{k}=u$ on $\partial\Omega_{k}\cap\partial\Omega$ (for example, take
$u_{k}(r,\theta)=\alpha_{k}(r)u(R_{k}(\theta),\theta)$ where $\alpha_{k}\equiv
1$ but vanishes in a neighborhood of $r=0$).

With the help of a partition of unity for the covering $\\{\Omega_{k}\\}$, a
function $u_{b}\in W^{1,p}(\Omega)$ can be constructed such that
$\gamma_{0}(u_{b})=\gamma_{0}(u)$. Then $u=u_{b}+u_{0}$ where $u_{0}\in
W^{1,p}_{0}(\Omega)$ can be approximated in $C^{\infty}_{0}(\Omega)$. Using
the same construction as for $u_{b}$, we can approximate $\gamma_{0}(u)$ in
$C^{\infty}(\partial\Omega)$ and extend the approximation into $\Omega$, thus
providing an approximation of $u_{b}$. ∎

The final ingredient is well-known in the literature as the Babuška-Lax-
Milgram Theorem . We provide a complete statement of the result because it
hinges on two important conditions which we would need to define in any case.

######  Theorem 11 (Babuška-Lax-Milgram).

Let $X$ and $Y$ be reflexive Banach spaces and consider $B:X\times
Y\longrightarrow\mathbb{R}$ a continuous, bilinear form satisfying the
following two conditions :

  * (i)

$B$ is non-degenrate with respect to the second variable, that is to say, for
each non-zero $y\in Y$ there exists $x\in X$ such that $B(x,y)\neq 0$ ;

  * (ii)

$B$ satisfies an $\inf$-$\sup$ condition, that is there exists a strictly
positive constant $c$ such that

$c\leq\inf_{x\in X}\sup_{y\in
Y}\frac{\big{|}B(x,y)\big{|}}{\|x\|_{X}\|y\|_{Y}}.$ (12)

Then for every $\ell\in Y^{\prime}$, there exists a unique solution $x\in X$
to

$B(x,y)=\ell(y),\qquad\forall y\in Y.$

Moreover, $c\|x\|_{X}\leq\|\ell\|_{Y^{\prime}}$.

##  3 Spaces and traces

In this section, we present some basic function theoretic results in function
spaces typically encountered in electromagnetism but usually in an $L^{2}$
setting [27]. The proofs of the results in this section can sometimes be found
in the literature but the proofs are often only briefly sketched and/or appear
buried deep in the papers themselves. In particular, we refer the interested
reader to either the short paper [25] or Section 2 of [26], where the
definitions are outlined for Riemannian manifolds, or Section 9 of [16] where
the results are presented without the geometry but more succinctly. At the
risk of appearing pedantic, we have chosen to present these concepts in a
logically complete fashion since this paper as a whole is likely to interest
some engineers with less experience with the mathematical literature.

For electromagnetic problems, the divergence and the curl of a vector field
must be well-defined. This implies that one must understand the following two
spaces

$\displaystyle W^{p}(\text{div};\Omega)$ $\displaystyle=\Big{\\{}\mathbf{u}\in
L^{p}(\Omega)^{3}\Big{|}\nabla\cdot\mathbf{u}\in L^{p}(\Omega)\Big{\\}},$ (13)
$\displaystyle W^{p}(\text{curl};\Omega)$
$\displaystyle=\Big{\\{}\mathbf{u}\in
L^{p}(\Omega)^{3}\Big{|}\nabla\times\mathbf{u}\in L^{p}(\Omega)^{3}\Big{\\}}.$
(14)

It is important to observe that these are again Banach spaces with norms,
respectively

$\begin{array}[]{rcl}\|\mathbf{u}\|_{W^{p}(\text{div};\Omega)}&:=&\|\mathbf{u}\|_{L^{p}(\Omega)}+\|\nabla\cdot\mathbf{u}\|_{L^{p}(\Omega)},\\\
\|\mathbf{u}\|_{W^{p}(\text{curl};\Omega)}&:=&\|\mathbf{u}\|_{L^{p}(\Omega)}+\|\nabla\times\mathbf{u}\|_{L^{p}(\Omega)},\end{array}$
(15)

and that there exist continuous injections $W^{1,p}(\Omega)^{3}\hookrightarrow
W^{p}(\text{div};\Omega)$ and $W^{1,p}(\Omega)^{3}\hookrightarrow
W^{p}(\text{curl};\Omega)$. On the other hand, these injections are not
surjective. We define the closure of $C^{\infty}_{0}(\Omega)^{3}$ in
$W^{p}(\text{div};\Omega)$ and $W^{p}(\text{curl};\Omega)$ to be respectively
$W^{p}_{0}({\text{div}};\Omega)$ and $W^{p}_{0}({\text{curl}};\Omega)$.

We need to know how to interpret the values along the boundary of functions in
either $W^{p}(\text{div};\Omega)$ or $W^{p}(\text{curl};\Omega)$. We begin the
analysis of $W^{p}(\text{div};\Omega)$ with a density result which will allow
us to extends many operators quite naturally to all elements in these spaces.
Theorem 12 is similar to Theorem 3.22 of [27] and its proof is simply an
adaptation to $L^{p}$ spaces.

######  Theorem 12.

Assuming that $\Omega$ has a Lipschitz boundary and $1<p<\infty$, then the set
$C^{\infty}(\overline{\Omega})^{3}$ is dense in
$W^{p}(\text{\emph{div}};\Omega)$.

The proof of this result requires three elementary lemmas.

######  Lemma 13.

For $1<p<\infty$, the space $C^{\infty}_{0}(\mathbb{R}^{3})^{3}$ is dense in
both $W^{p}(\text{\emph{div}};\mathbb{R}^{3})$ and
$W^{p}(\text{\emph{curl}};\mathbb{R}^{3})$.

###### Proof.

It suffices to observe that the norms in $W^{p}(\text{div};\mathbb{R}^{3})$
and $W^{p}(\text{curl};\mathbb{R}^{3})$ are bounded by the norm in
$W^{1,p}(\mathbb{R}^{3})$. The result then follows by the density of
$C^{\infty}_{0}(\mathbb{R}^{3})$ in $W^{1,p}(\mathbb{R}^{3})$, Corollary 3.23
of [1]. ∎

######  Lemma 14.

For $1<p<\infty$ and any $\mathbf{u}\in
W^{p}(\text{\emph{div}};\mathbb{R}^{3})$, $\phi\in
C^{\infty}_{0}(\mathbb{R}^{3})$, we have

$\int_{\mathbb{R}^{3}}\nabla\cdot\mathbf{u}\,\phi\,d\mathbf{x}+\int_{\mathbb{R}^{3}}\mathbf{u}\,\nabla\phi\,d\mathbf{x}=0.$
(16)

For any $\mathbf{u}\in W^{p}(\text{\emph{curl}};\mathbb{R}^{3})$ and
$\boldsymbol{\phi}\in C^{\infty}_{0}(\mathbb{R}^{3})^{3}$, we have

$\int_{\mathbb{R}^{3}}\nabla\times\mathbf{u}\,\boldsymbol{\phi}\,d\mathbf{x}-\int_{\mathbb{R}^{3}}\mathbf{u}\,\nabla\times\boldsymbol{\phi}\,d\mathbf{x}=0.$
(17)

###### Proof.

The result follows immediately by density using the classical Divergence
Theorem. ∎

We now have all the tools to tackle the proof of the density of
$C^{\infty}(\overline{\Omega})$ inside $W^{p}(\text{div};\Omega)$. The main
idea of the proof is that $C^{\infty}_{0}(\Omega)$ will always be dense in the
dual of a ”smooth” function space. The proof proceeds by identifying this fact
without having to obtain a complete characterization of the dual.

###### Proof of Theorem 12.

The idea for this proof is taken from [10] where it was used to study a
slightly different space. The techniques presented here will be reused in the
proof of Theorem 18. We will show that if $\ell\in
W^{p}(\text{div};\Omega)^{\prime}$ is identically zero over
$C^{\infty}(\overline{\Omega})^{3}$ then $\ell\equiv 0$. By Theorem 3.5 of
[31], this suffices to show that the closure of
$C^{\infty}(\overline{\Omega})^{3}$ equals $W^{p}(\text{div};\Omega)$.

Define $\Omega^{(4)}$ to be the union of four distinct copies of $\Omega$ and
consider the obvious embedding

$\displaystyle I:W^{p}(\text{div};\Omega)$ $\displaystyle\longrightarrow
L^{p}(\Omega^{(4)})$ $\displaystyle\mathbf{u}$
$\displaystyle\longmapsto(\mathbf{u},\nabla\cdot\mathbf{u}).$

This embedding is an isometry onto it’s image. Any linear functional $\ell$
over $W^{p}(\text{div};\Omega)$ defines a linear functional over the closed
image $I(W^{p}(\text{div};\Omega))$ and therefore, by the Hahn-Banach Theorem,
extends to a linear functional over all of $L^{p}(\Omega^{(4)})$. Since the
dual of $L^{p}$ is $L^{q}$, for any $\mathbf{u}\in W^{p}(\text{div};\Omega)$
this extension can be written in the form

$\ell(\mathbf{u})=(\mathbf{u},\mathbf{v})_{\Omega}+(\nabla\cdot\mathbf{u},w)_{\Omega}$

for some (non-unique) $\mathbf{v}\in L^{q}(\Omega)^{3}$ and $w\in
L^{q}(\Omega)$. Let $E_{0}$ be the extension by zero of any element in
$L^{q}(\Omega)$ to $L^{q}(\mathbb{R}^{3})$ (Theorem 2.30 of [1]) and let $E$
be any extension operator from $L^{p}(\Omega)$ to $L^{p}(\mathbb{R}^{3})$,
then

$\ell(\mathbf{u})=(E\mathbf{u},E_{0}\mathbf{v})_{\mathbb{R}^{3}}+(\nabla\cdot\mathbf{u},w)_{\Omega}.$

Assuming that $\ell(\boldsymbol{\phi})=0$ for all $\boldsymbol{\phi}\in
C^{\infty}(\overline{\Omega})^{3}$, then for all $\boldsymbol{\psi}\in
C^{\infty}_{0}(\mathbb{R}^{3})^{3}$ we have

$\displaystyle(\boldsymbol{\psi},E_{0}\mathbf{v})_{\mathbb{R}^{3}}+(\nabla\cdot\boldsymbol{\psi},E_{0}w)_{\mathbb{R}^{3}}=0.$

In a distributional sense, we therefore have $E_{0}\mathbf{v}=\nabla E_{0}w$.
This allows us to first recognize that $E_{0}w\in W^{1,q}(\mathbb{R}^{3})$.
Moreover, both $E_{0}w$ and $\nabla E_{0}w=E_{0}\mathbf{v}$ vanish outside of
$\Omega$ and therefore by Theorem 5.29 of [1], $E_{0}w\big{|}_{\Omega}=w\in
W^{1,q}_{0}(\Omega)$. This implies that there exists a sequence $w_{n}\in
C^{\infty}_{0}(\Omega)$ converging to $w$ in the space $W^{1,q}_{0}(\Omega)$.
Using identity (16), we can now conclude that for any $\mathbf{u}\in
W^{p}(\text{div};\Omega)$

$\displaystyle\ell(\mathbf{u})=$
$\displaystyle(E\mathbf{u},E_{0}\mathbf{v})_{\mathbb{R}^{3}}+(E(\nabla\cdot\mathbf{u}),E_{0}w)_{\mathbb{R}^{3}}$
$\displaystyle=$ $\displaystyle\lim_{n\to\infty}(E\mathbf{u},\nabla
w_{n})_{\mathbb{R}^{3}}+(E(\nabla\cdot\mathbf{u}),w_{n})_{\mathbb{R}^{3}}$
$\displaystyle=$ $\displaystyle\lim_{n\to\infty}(\mathbf{u},\nabla
w_{n})_{\Omega}+(\nabla\cdot\mathbf{u},w_{n})_{\Omega}=0$

where the second to last step required that $w_{n}\in C^{\infty}_{0}(\Omega)$.
∎

The next step is to study the traces of functions in
$W^{p}(\text{div};\Omega)$. For all $\boldsymbol{\phi}\in
C^{\infty}(\overline{\Omega})^{3}$, we can define the normal trace
$\gamma_{n}(\boldsymbol{\phi})=\boldsymbol{\phi}\cdot\mathbf{n}$ where we have
assumed that the domain is sufficiently smooth to define the unit outwards
normal $\mathbf{n}$. The question is to know where this quantity will belong
when we extend this to the weaker space $W^{p}(\text{div};\Omega)$.

######  Theorem 15.

Let $\Omega$ be a bounded $C^{1}$ domain in $\mathbb{R}^{3}$, unit outwards
normal $\mathbf{n}$, and $1<p<\infty$. Then

  * (i)

the mapping $\gamma_{n}$ defined on $C^{\infty}(\overline{\Omega})^{3}$ can be
extended to a continuous and surjective map
$\gamma_{n}:W^{p}(\text{\emph{div}};\Omega)\longrightarrow
B^{{1-1/q}}_{{q},{q}}(\partial\Omega)^{\prime}\equiv
B^{{-1/p}}_{{p},{p}}(\partial\Omega)$.

  * (ii)

the following Green’s formula holds for all $\mathbf{u}\in
W^{p}(\text{\emph{div}};\Omega)$ and $v\in W^{{1},{q}}(\Omega)$,

$(\mathbf{u},\nabla
v)+(\nabla\cdot\mathbf{u},v)=\langle\gamma_{n}(\mathbf{u}),\gamma_{0}(v)\rangle.$
(18)

###### Proof.

Given the elliptic regularity results in $L^{p}$, the proof is a natural
adaptation of the argument used in [27] for the proof of their Theorem 3.24.

We begin by examining Green’s formula. For any $\phi\in
C^{\infty}(\overline{\Omega})$ and $\boldsymbol{\psi}\in
C^{\infty}(\overline{\Omega})^{3}$, we have

$(\boldsymbol{\psi},\nabla\phi)+(\nabla\cdot\boldsymbol{\psi},\phi)=\langle\gamma_{n}(\boldsymbol{\psi}),\gamma_{0}({\phi})\rangle.$

By density, this must hold for all $\phi\in W^{1,q}(\Omega)$. For $v\in
W^{{1},{q}}(\Omega)$, using the previous identity as a weak definition of
$\gamma_{n}(\boldsymbol{\psi})$, we therefore have

$\displaystyle\big{|}\langle\gamma_{n}(\boldsymbol{\psi}),\gamma_{0}(v)\rangle\big{|}\leq$
$\displaystyle\|\boldsymbol{\psi}\|_{L^{p}(\Omega)}\cdot\|\nabla
v\|_{L^{q}(\Omega)}+\|\nabla\cdot\boldsymbol{\psi}\|_{L^{p}(\Omega)}\cdot\|v\|_{L^{q}(\Omega)}$
$\displaystyle\leq$
$\displaystyle\|\boldsymbol{\psi}\|_{W^{p}(\text{div};\Omega)}\cdot\|v\|_{W^{{1},{q}}(\Omega)},\qquad\forall\boldsymbol{\psi}\in
C^{\infty}(\Omega)^{3}.$

Pick any $\mu\in B^{{1-1/q}}_{{q},{q}}(\partial\Omega)$ and consider $v\in
W^{{1},{q}}(\Omega)$ the unique weak solution to

$-\Delta v+v=0\quad\text{on $\Omega$}\qquad\gamma_{0}(v)=\mu\quad\text{on
$\partial\Omega$,}$

guaranteed by Theorem 4. Using the fact that this solution depends
continuously on the boundary conditions, estimate (6), we have the bound

$\big{|}\langle\gamma_{n}(\boldsymbol{\psi}),\mu\rangle\big{|}\leq
C\|\boldsymbol{\psi}\|_{W^{p}(\text{div};\Omega)}\|\mu\|_{B^{{1-1/q}}_{{q},{q}}(\partial\Omega)}.$

According to the definition of the dual norm (5), we may conclude that the map
$\gamma_{n}$ can be extended by continuity to all of
$W^{p}(\text{div};\Omega)$ with image in the dual of
$B^{{1-1/q}}_{{q},{q}}(\partial\Omega)$, also known as
$B^{{-1/p}}_{{p},{p}}(\partial\Omega)$ (since $\partial\Omega$ is closed).
Green’s formula (18) follows immediately by density since
$\gamma_{0}(v)=\mu\in B^{{1-1/q}}_{{q},{q}}(\partial\Omega)$.

To prove the surjectivity of $\gamma_{n}$, choose any $\eta\in
B^{{-1/p}}_{{p},{p}}(\partial\Omega)$ and construct the unique weak solution
$\phi$ to

$-\Delta\phi+\phi=0\quad\text{on
$\Omega$}\qquad\mathbf{n}\cdot\nabla\phi=\eta\quad\text{on $\partial\Omega$.}$
(19)

By Theorem 6, the solution $\phi$ exists and belongs to $W^{{1},{p}}(\Omega)$.
If we define $\mathbf{u}=\nabla\phi\in L^{p}(\Omega)^{3}$ then the weak form
of (19) gives the identity

$(\mathbf{u},\nabla\psi)+(\phi,\psi)=0,\qquad\forall\psi\in
C^{\infty}_{0}(\Omega).$

In a distributional sense, we therefore have $\nabla\cdot\mathbf{u}=\phi\in
L^{p}(\Omega)$ and $\mathbf{u}\in W^{p}(\text{div};\Omega)$. Moreover,
$\gamma_{n}(\mathbf{u})=\mathbf{n}\cdot\mathbf{u}=\mathbf{n}\cdot\nabla\phi=\eta$
and the map is surjective. ∎

######  Lemma 16.

For a bounded $C^{1}$ domain $\Omega\subset\mathbb{R}^{3}$ and $1<p<\infty$,
we have

$W^{p}_{0}(\text{\emph{div}};\Omega)=\Big{\\{}\mathbf{u}\in
W^{p}(\text{\emph{div}};\Omega)\,\Big{|}\,\gamma_{n}(\mathbf{u})=0\Big{\\}}.$

###### Proof.

The proof given here is a non-trivial adaptation of the argument used in [27],
where duality was simpler to handle because Hilbert spaces were involved. The
trace $\gamma_{n}$ clearly vanishes on $C^{\infty}_{0}(\Omega)^{3}$ and
therefore, by density, $\gamma_{n}$ vanishes on
$W^{p}_{0}(\text{div};\Omega)$. This proves the inclusion of
$W^{p}_{0}(\text{div};\Omega)$ into the set on the right. The other inclusion
can be demonstrated if we show that when $\ell\in
W^{p}(\text{div};\Omega)^{\prime}$ vanishes on $C^{\infty}_{0}(\Omega)$, then
$\ell$ must also vanish on any $\mathbf{u}\in W^{p}(\text{div};\Omega)$ for
which $\gamma_{n}(\mathbf{u})=0$.

Repeating the trick used in the proof of Theorem 12, we exploit the embedding
of $W^{p}(\text{div};\Omega)$ into $L^{p}(\Omega^{(4)})$ and the induced
representation of every $\ell\in W^{p}(\text{div};\Omega)^{\prime}$ as

$\ell(\boldsymbol{\psi})=(\boldsymbol{\psi},\mathbf{v})+(\nabla\cdot\boldsymbol{\psi},w)$

for some $\mathbf{v}\in L^{q}(\Omega)^{3}$ and $w\in L^{q}(\Omega)$. For all
$\boldsymbol{\phi}\in C^{\infty}_{0}(\Omega)^{3}$, $\ell(\boldsymbol{\phi})=0$
and therefore $\mathbf{v}=\nabla w$ in a distributional sense. This implies
that $w\in W^{{1},{q}}(\Omega)$ and that $\ell$ over
$W^{p}(\text{div};\Omega)$ can be written as

$\displaystyle\ell(\boldsymbol{\psi})=$
$\displaystyle(\boldsymbol{\psi},\nabla
w)+(\nabla\cdot\boldsymbol{\psi},w)=\langle\gamma_{n}(\boldsymbol{\psi}),\gamma_{0}(w)\rangle.$

If $\mathbf{u}\in W^{p}(\text{div};\Omega)$ is such that
$\gamma_{n}(\mathbf{u})=0$ then $\ell(\mathbf{u})=0$ and $\mathbf{u}$ belongs
to the closure of $C^{\infty}_{0}(\Omega)^{3}$. ∎

We now turn to the set of spaces $W^{p}(\text{curl};\Omega)$ and
$W^{p}_{0}(\text{curl};\Omega)$. The analysis is very similar to the one used
for $W^{p}(\text{div};\Omega)$ and in fact, a suitable generalization of the
differential operators involved here would have allowed us to treat both cases
simultaneously, as Mitrea and Mitrea did in [25]. The objective is to prove
Green’s formula over these spaces and, in particular, to characterize the
traces of these spaces. In the case of $W^{p}(\text{curl};\Omega)$, we will
show that the tangential trace operators, defined for all
$\boldsymbol{\phi}\in C^{\infty}(\overline{\Omega})^{3}$ as
$\gamma_{t}(\boldsymbol{\phi})=\mathbf{n}\times\boldsymbol{\phi}$ and
$\gamma_{T}(\boldsymbol{\phi})=(\mathbf{n}\times\boldsymbol{\phi})\times\mathbf{n}$,
are well-defined. We begin with a simple extension of Theorem 5.29 from [1].

######  Lemma 17.

Assume $\Omega$ is a bounded Lipschitz domain, $1<p<\infty$ and $E_{0}u$ is
the extension by zero of $u$ to all of $\mathbb{R}^{3}$. Then $u$ belongs to
$W^{p}_{0}(\text{\emph{curl}};\Omega)$ if and only if $E_{0}u$ belongs to
$W^{p}(\text{\emph{curl}};\mathbb{R}^{3})$.

###### Proof.

Theorem 5.29 in [1] is stated only for $W^{{s},{p}}(\Omega)$ but the same
proof shows that this result also holds for $W^{p}(\text{curl};\Omega)$. ∎

######  Theorem 18.

Consider a bounded Lipschitz domain $\Omega\subset\mathbb{R}^{3}$ and
$1<p<\infty$. Then the set $C^{\infty}(\overline{\Omega})^{3}$ is dense in
$W^{p}(\text{\emph{curl}};\Omega)$.

###### Proof.

We will show that if $\ell\in W^{p}(\text{curl};\Omega)^{\prime}$ vanishes
over $C^{\infty}(\overline{\Omega})^{3}$ then $\ell$ must be identically zero.

As we did in the proof of Theorem 12, there exists an isometric embedding
$I:W^{p}(\text{curl};\Omega)\longrightarrow L^{p}(\Omega^{(6)})$,
$I(\mathbf{u})=(\mathbf{u},\nabla\times\mathbf{u})$. Given any linear
functional over $I(W^{p}(\text{curl};\Omega))\subset L^{p}(\Omega^{(6)})$, the
Hahn-Banach Theorem states that can it be extended to a linear functional over
$L^{p}(\Omega^{(6)})$, that is to an element in $L^{q}(\Omega^{(6)})$. The
linear functional can therefore be represented as

$\ell(\mathbf{u})=(\mathbf{u},\mathbf{v})_{\Omega}+(\nabla\times\mathbf{u},\mathbf{w})_{\Omega},\quad\forall\mathbf{u}\in
W^{p}(\text{curl};\Omega),$

for some $\mathbf{v},\mathbf{w}\in L^{q}(\Omega)^{3}$.

Let $E_{0}$ be the extension by zero operator and let $E$ be any extension
operator over $L^{p}(\Omega)^{3}$. Then

$\ell(\mathbf{u})=(E\mathbf{u},E_{0}\mathbf{v})_{\mathbb{R}^{3}}+(E(\nabla\times\mathbf{u}),E_{0}\mathbf{w})_{\mathbb{R}^{3}}.$

For any $\boldsymbol{\psi}\in C^{\infty}_{0}(\mathbb{R}^{3})^{3}$, Lemma 14
implies that

$(\boldsymbol{\psi},E_{0}\mathbf{v})_{\mathbb{R}^{3}}+(\nabla\times\boldsymbol{\psi},E_{0}\mathbf{w})_{\mathbb{R}^{3}}=0.$

In other words, $E_{0}\mathbf{v}=-\nabla\times E_{0}\mathbf{w}$ in a
distributional sense. Not only does this imply that $E_{0}\mathbf{w}\in
W^{q}(\text{curl};\mathbb{R}^{3})$, but Lemma 17 shows that $\mathbf{w}\in
W^{q}_{0}(\text{curl};\Omega)$.

We can therefore construct a sequence of $\mathbf{w}_{n}\in
C^{\infty}_{0}(\Omega)^{3}$ which converges to $\mathbf{w}$ with respect to
the norm in $W^{q}(\text{div};\Omega)$. For any $\mathbf{u}\in
W^{p}(\text{div};\Omega)$, we combine these facts with formula (17) to deduce

$\displaystyle\ell(\mathbf{u})=$
$\displaystyle(E\mathbf{u},E_{0}\mathbf{v})_{\mathbb{R}^{3}}+(E(\nabla\times\mathbf{u}),E_{0}\mathbf{w})_{\mathbb{R}^{3}}$
$\displaystyle=$
$\displaystyle\lim_{n\to\infty}-(E\mathbf{u},\nabla\times\mathbf{w}_{n})_{\mathbb{R}^{3}}+(E(\nabla\times\mathbf{u}),\mathbf{w}_{n})_{\mathbb{R}^{3}}$
$\displaystyle=$
$\displaystyle\lim_{n\to\infty}-(\mathbf{u},\nabla\times\mathbf{w}_{n})_{\Omega}+(\nabla\times\mathbf{u},\mathbf{w}_{n})_{\Omega}=0.$

This shows that the closure of $C^{\infty}(\overline{\Omega})^{3}$ must be
$W^{p}(\text{curl};\Omega).$ ∎

######  Theorem 19.

Consider a bounded $C^{1}$ domain $\Omega\subset\mathbb{R}^{3}$ and
$1<p<\infty$. Then

  * (i)

the mapping $\gamma_{t}$ defined on $C^{\infty}(\overline{\Omega})$ can be
extended to a continuous map
$\gamma_{t}:W^{p}(\text{\emph{curl}};\Omega)\longrightarrow(B^{{1-1/q}}_{{q},{q}}(\partial\Omega)^{3})^{\prime}\equiv
B^{{-1/p}}_{{p},{p}}(\partial\Omega)^{3}$.

  * (ii)

the following Green’s formula holds for all $\mathbf{u}\in
W^{p}(\text{\emph{curl}};\Omega)$ and $\mathbf{v}\in W^{{1},{q}}(\Omega)^{3}$,

$(\nabla\times\mathbf{u},\mathbf{v})-(\mathbf{u},\nabla\times\mathbf{v})=\langle\gamma_{t}(\mathbf{u}),\gamma_{0}(\mathbf{v})\rangle.$
(20)

###### Proof.

The proof is a straightforward extension of the one used in [27]. For any
$\boldsymbol{\phi},\boldsymbol{\psi}\in C^{\infty}(\overline{\Omega})^{3}$, we
have

$(\nabla\times\boldsymbol{\phi},\boldsymbol{\psi})-(\boldsymbol{\phi},\nabla\times\boldsymbol{\psi})=\langle\gamma_{t}(\boldsymbol{\phi}),\gamma_{0}(\boldsymbol{\psi})\rangle.$
(21)

By the density of $C^{\infty}(\overline{\Omega})^{3}$ in
$W^{1,q}(\Omega)^{3}$, this formula holds for all $\boldsymbol{\psi}\in
W^{{1},{q}}(\Omega)^{3}$. For any $\mu\in
B^{{1-1/q}}_{{q},{q}}(\partial\Omega)^{3}$, let $v\in W^{1,q}(\Omega)$ be the
weak solution to $-\Delta v+v=0,$ $\gamma_{0}(v)=\mu$, as provided by Theorem
4. Using (21) and estimate (6), for all $\boldsymbol{\phi}\in
C^{\infty}(\overline{\Omega})^{3}$ we have

$\big{|}\langle\gamma_{t}(\boldsymbol{\phi}),\gamma_{0}(\mu)\rangle\big{|}\leq\|\boldsymbol{\phi}\|_{W^{p}(\text{curl};\Omega)}\cdot\|v\|_{W^{{1},{q}}(\Omega)}\leq
C\|\boldsymbol{\phi}\|_{W^{p}(\text{curl};\Omega)}\|\mu\|_{B^{{1-1/q}}_{{q},{q}}(\partial\Omega)}.$

According to the definition of the dual norm (5), the previous estimate
implies that
$\gamma_{t}:W^{p}(\text{curl};\Omega)\longrightarrow(B^{{1-1/q}}_{{q},{q}}(\partial\Omega)^{3})^{\prime}=B^{{-1/p}}_{{p},{p}}(\partial\Omega)^{3}$
can be defined by continuous extension and that

$\|\gamma_{t}(\boldsymbol{\phi})\|_{B^{{-1/p}}_{{p},{p}}(\partial\Omega)}\leq
C\|\boldsymbol{\phi}\|_{W^{p}(\text{curl};\Omega)}.$

Since $C^{\infty}(\overline{\Omega})^{3}$ is dense in
$W^{p}(\text{curl};\Omega)$, we conclude that the identity (21) continues to
hold for all $\mathbf{u}\in W^{p}(\text{curl};\Omega)$ and
$\boldsymbol{\psi}\in W^{{1},{q}}(\Omega)^{3}$. ∎

An intrinsic definition of $\gamma_{t}(W^{p}(\text{curl};\Omega))$ exists and
is described in Mitrea [Mitrea]. For our purposes, a simpler approach will be
sufficient. Writing
$Y^{p}(\partial\Omega)=\gamma_{t}(W^{p}(\text{curl};\Omega))$, we define the
following norm on the image

$\|\mathbf{w}\|_{Y^{p}(\partial\Omega)}=\inf\left\\{\|\mathbf{u}\|_{W^{p}(\text{curl};\Omega)}\,\big{|}\,\mathbf{u}\in
W^{p}(\text{curl};\Omega)\text{ and
}\gamma_{t}(\mathbf{u})=\mathbf{w}\right\\}.$ (22)

Although the previous Green’s formula (20) used $\gamma_{0}$, we will be using
the more precise expression $\gamma_{T}$ since, for smooth functions, only the
tangential component of $\mathbf{v}$ appears in the inner product and
$\langle\gamma_{t}(\mathbf{u}),\gamma_{0}(\mathbf{v})\rangle=\langle\gamma_{t}(\mathbf{u}),\gamma_{T}(\mathbf{v})\rangle$.

######  Theorem 20.

Consider a bounded $C^{1}$ domain $\Omega$ and $1<p<\infty$. Then
$Y^{p}(\partial\Omega)$ is a normed topological vector space, the map
$\gamma_{t}:W^{p}(\text{\emph{curl}};\Omega)\longrightarrow
Y^{p}(\partial\Omega)$ is continuous and surjective and
$\gamma_{T}:W^{q}(\text{\emph{curl}};\Omega)\longrightarrow
Y^{p}(\partial\Omega)^{\prime}$ is well-defined and continuous. Moreover, for
all $\mathbf{u}\in W^{p}(\text{\emph{curl}};\Omega)$, $\mathbf{v}\in
W^{q}(\text{\emph{curl}};\Omega)$, we have

$(\nabla\times\mathbf{u},\mathbf{v})-(\mathbf{u},\nabla\times\mathbf{v})=\langle\gamma_{t}(\mathbf{u}),\gamma_{T}(\mathbf{v})\rangle.$
(23)

###### Proof.

The image $Y^{p}(\partial\Omega)$ equipped with the norm (22) is easily seen
to be a normed topological vector space since $\gamma_{t}$ is linear. To show
that $\gamma_{t}$ is continuous, it suffices to observe that for any
$\mathbf{u}\in W^{p}(\text{curl};\Omega)$

$\|\gamma_{t}(\mathbf{u})\|_{Y^{p}(\partial\Omega)}=\inf_{\begin{subarray}{c}\mathbf{v}\in
W^{p}(\text{curl};\Omega)\\\
\gamma_{t}(\mathbf{v})=\gamma_{t}(\mathbf{u})\end{subarray}}\|\mathbf{v}\|_{W^{p}(\text{curl};\Omega)}\leq\|\mathbf{u}\|_{W^{p}(\text{curl};\Omega)}.$

We now prove that $\gamma_{T}$ is continuous. First of all, note that for any
$s\in Y^{p}(\partial\Omega)$ and any $\mathbf{u}\in W^{p}(\text{curl};\Omega)$
such that $s=\gamma_{t}(\mathbf{u})$, we have

$\langle
s,\gamma_{T}(\boldsymbol{\phi})\rangle=(\nabla\times\mathbf{u},\boldsymbol{\phi})-(\mathbf{u},\nabla\times\boldsymbol{\phi}),\qquad\forall\boldsymbol{\phi}\in
C^{\infty}(\overline{\Omega})^{3}.$

Note that the right hand side is well-defined for all $\mathbf{u}\in
W^{p}(\text{curl};\Omega)$ and $\boldsymbol{\phi}\in
W^{q}(\text{curl};\Omega)$. Moreover, the value on the right-hand side is
independent of the choice of $\mathbf{u}\in W^{p}(\text{curl};\Omega)$ so long
as $\gamma_{t}(\mathbf{u})=s$. Therefore, for any fixed $s\in
Y^{p}(\partial\Omega)$, the linear functional

$L(\boldsymbol{\phi})=\langle s,\gamma_{T}(\boldsymbol{\phi})\rangle$

is well-defined for all $\boldsymbol{\phi}\in W^{q}(\text{curl};\Omega)$. In
fact,

$\displaystyle\big{|}L(\boldsymbol{\phi})\big{|}=$
$\displaystyle\big{|}(\nabla\times\mathbf{u},\boldsymbol{\phi})-(\mathbf{u},\nabla\times\boldsymbol{\phi})\big{|}$
$\displaystyle=$ $\displaystyle\inf_{\begin{subarray}{c}\mathbf{v}\in
W^{p}(\text{curl};\Omega)\\\
\gamma_{t}(\mathbf{v})=s\end{subarray}}\big{|}(\nabla\times\mathbf{v},\boldsymbol{\phi})-(\mathbf{v},\nabla\times\boldsymbol{\phi})\big{|}$
$\displaystyle\leq$ $\displaystyle\inf_{\begin{subarray}{c}\mathbf{v}\in
W^{p}(\text{curl};\Omega)\\\
\gamma_{t}(\mathbf{v})=s\end{subarray}}\|\mathbf{v}\|_{W^{p}(\text{curl};\Omega)}\|\boldsymbol{\phi}\|_{W^{q}(\text{curl};\Omega)}$
$\displaystyle=$
$\displaystyle\|s\|_{Y^{p}(\partial\Omega)}\|\boldsymbol{\phi}\|_{W^{q}(\text{curl};\Omega)},$

which shows that $L$ is continuous over $W^{q}(\text{curl};\Omega)$.
Considering the definition of the norm on $Y^{p}(\partial\Omega)^{\prime}$,
equation (5), we immediately see that the previous identity also implies that

$\|\gamma_{T}(\boldsymbol{\phi})\|_{Y^{p}(\partial\Omega)^{\prime}}\leq\|\boldsymbol{\phi}\|_{W^{q}(\text{curl};\Omega)}.$

Hence, the map $\gamma_{T}$ is continuous with image in
$Y^{p}(\partial\Omega)^{\prime}$. ∎

######  Lemma 21.

For a bounded $C^{1}$ domain $\Omega\subset\mathbb{R}^{3}$ and $1<p<\infty$,
we have

$W^{p}_{0}(\text{\emph{curl}};\Omega)=\Big{\\{}\mathbf{u}\in
W^{p}(\text{\emph{curl}};\Omega)\,\Big{|}\,\gamma_{t}(\mathbf{u})=0\Big{\\}}.$

###### Proof.

Since $\gamma_{t}(\boldsymbol{\phi})=0$ for all $\boldsymbol{\phi}\in
C^{\infty}_{0}(\Omega)^{3}$, it suffices to prove the opposite inclusion.
Given that $W^{p}_{0}(\text{curl};\Omega)$ is the intersection of all $\ell\in
W^{p}(\text{curl};\Omega)^{\prime}$ which vanish on
$C^{\infty}_{0}(\Omega)^{3}$, we must therefore show that such an $\ell$ also
vanishes on $\mathbf{u}\in W^{p}(\text{curl};\Omega)$ when
$\gamma_{t}(\mathbf{u})=0.$

As we did in the proof of Lemma 16, any functional $\ell\in
W^{p}(\text{curl};\Omega)^{\prime}$ can be written in the form

$\ell(\mathbf{u})=(\mathbf{g},\mathbf{u})+(\mathbf{h},\nabla\times\mathbf{u}),\qquad\forall\mathbf{u}\in
W^{p}(\text{curl};\Omega),$

for some $\mathbf{g},\mathbf{h}\in L^{q}(\Omega)^{3}$. If $\ell$ vanishes for
all $\boldsymbol{\psi}\in C^{\infty}_{0}(\Omega)^{3}$, then this
representation shows that $\mathbf{g}=-\nabla\times\mathbf{h}$ in a
distributional sense and therefore that $\mathbf{h}\in
W^{q}(\text{curl};\Omega)$. For $\mathbf{u}\in W^{p}(\text{curl};\Omega)$ such
that $\gamma_{t}(\mathbf{u})=0$, we then apply identity (23) to deduce that

$\displaystyle\ell(\mathbf{u})=$
$\displaystyle-(\nabla\times\mathbf{h},\mathbf{u})+(\mathbf{h},\nabla\times\mathbf{u})=\langle\gamma_{t}(\mathbf{u}),\gamma_{T}(\mathbf{h})\rangle=0.$

Hence $\mathbf{u}$ belongs to the closure of $C^{\infty}_{0}(\Omega)^{3}$. ∎

We complete this section with a simple result concerning the spaces which we
have introduced.

######  Lemma 22.

For a bounded Lipschitz domain $\Omega\subset\mathbb{R}^{3}$ and $1<p<\infty$,
the spaces $W^{p}(\text{\emph{div}};\Omega)$ and
$W^{p}(\text{\emph{curl}};\Omega)$ are locally convex and reflexive Banach
spaces.

###### Proof.

Local convexity is a triviality since the spaces are normed. Recalling the
isometries

$\displaystyle I_{\text{div}}(\mathbf{u})=$
$\displaystyle(\mathbf{u},\nabla\cdot\mathbf{u})\in L^{p}(\Omega^{(4)},$
$\displaystyle I_{\text{curl}}(\mathbf{u})=$
$\displaystyle(\mathbf{u},\nabla\times\mathbf{u})\in L^{p}(\Omega^{(6)},$

introduced in the proofs of Theorems 12 and 18, then we see that
$W^{p}(\text{div};\Omega)$ and $W^{p}(\text{curl};\Omega)$ is isometric to a
closed subspace of a reflexive space and are therefore, by Theorem 1.22 of
[1], also reflexive. ∎

##  4 Divergence and rotational free vector fields

The objective of this section is to derive fundamental results concerning the
existence of divergence free and rotational free vector fields. These results
appear to be well-known in the literature but we have either not found the
results in the form we needed or within the same reference. The fundamental
result of this section is a Friedrich’s inequality for problems with
homogeneous Dirichlet boundary conditions. This result is the only one which
appears to be entirely new to the literature. Give more detailed citations for
the two theorems in this section : Dauge …

######  Theorem 23.

Let $\Omega$ be a bounded simply connected Lipschitz domain in
$\mathbb{R}^{3}$. Suppose that $\mathbf{u}\in L^{p}(\Omega)^{3}$. Then
$\nabla\times\mathbf{u}=0$ in $\Omega$ if and only if there exists $\phi\in
W^{{1},{p}}(\Omega)$ such that $\mathbf{u}=\nabla\phi$. If this is the case,
then $\phi$ is unique up to an additive constant and

$\|\phi\|_{W^{{1},{p}}(\Omega)\setminus\mathbb{R}}\leq
C\|\mathbf{u}\|_{L^{p}(\Omega)^{3}}.$

###### Proof.

The proof follows the one for Theorem 3.37 from [27] which itself goes back to
a proof in [3]. The only differences are the use of convolutions in $L^{p}$
and of the $L^{p}$ version of Lemma 9.

Given that $\mathbf{u}\in L^{p}(\Omega)^{3}$, $\mathbf{u}$ can be extended by
zero to all of $\mathbb{R}^{3}$, which we denote by $\mathbf{\widetilde{u}}$.
With the help of a mollifier $\rho_{\epsilon}$, a suitably smooth
approximation can be constructed

$\mathbf{\widetilde{u}}_{\epsilon}=\rho_{\epsilon}\star\mathbf{\widetilde{u}}\in
L^{p}(\mathbb{R}^{3})^{3}\cap C^{\infty}_{0}(\mathbb{R}^{3})^{3};$

see Lemma 3.16 in [1]. Moreover,
$\mathbf{\widetilde{u}}_{\epsilon}\to\mathbf{\widetilde{u}}$ as $\epsilon\to
0$ in $L^{p}(\mathbb{R}^{3})^{3}$.

Under our assumptions on $\Omega$, it is possible to produce a sequence of
simply connected Lipschitz domains $\\{\mathcal{O}_{j}\\}_{j}$ satisfying
$\mathcal{O}_{j}\subset\mathcal{O}_{j+1}$,
$\overline{\mathcal{O}_{j}}\subset\Omega$ and
$\Omega=\cup_{j=1}^{\infty}\mathcal{O}_{j}$. Fixing $\mathcal{O}_{j}$, since
$\mathbf{\widetilde{u}}_{\epsilon}$ is smooth and

$\nabla\times\mathbf{\widetilde{u}}_{\epsilon}=\nabla\times\big{(}\rho_{\epsilon}\star\mathbf{\widetilde{u}}\big{)}=\rho_{\epsilon}\star\big{(}\nabla\times\mathbf{\widetilde{u}}\big{)}=\rho_{\epsilon}\star\big{(}\nabla\times\mathbf{u}\big{)}=0,$

by integration we can construct $\phi^{j}_{\epsilon}\in C^{\infty}(\Omega)$
such that

$\mathbf{\widetilde{u}}_{\epsilon}=\nabla\phi^{j}_{\epsilon},\qquad\int_{\mathcal{O}_{j}}\phi^{j}_{\epsilon}\,d\mathbf{x}=0,\qquad\text{and
}\qquad\|\phi^{j}_{\epsilon}\|_{W^{1,p}(\mathcal{O}_{j})}\leq
C\|\mathbf{u}\|_{L^{p}(\mathcal{O}_{j})^{3}}.$

We now observe that since
$\nabla\phi^{j}_{\epsilon}=\mathbf{\widetilde{u}}_{\epsilon}\to\mathbf{\widetilde{u}}$
in $L^{p}(\Omega)^{3}$, then Poincaré’s inequality implies that
$\phi^{j}_{\epsilon}\to\phi^{j}\in W^{1,p}(\mathcal{O}_{j})$. Even in the
limit, we have $\|\phi^{j}\|_{W^{1,p}(\mathcal{O}_{j})}\leq
C\|\mathbf{u}\|_{L^{p}(\Omega)^{3}}$. The average of $\phi^{j}$ over
$\mathcal{O}_{j}$ also vanishes. At the cost of modifying $\phi^{j}$ by a
constant, we may assume that $\phi^{j}=\phi^{j-1}$ over $\mathcal{O}_{j-1}$.
For any compact $K\subset\Omega$, there exists $k$ such that
$K\subset\mathcal{O}_{k}$ and therefore by setting $\phi=\phi_{k}$ over
$\mathcal{O}_{k}$, we can define $\phi\in L^{p}_{\text{loc}}(\Omega)$. In
fact, since $\mathbf{u}|_{\mathcal{O}_{j}}=\nabla\phi^{j}$ we have that
$\nabla\phi=\mathbf{u}\in L^{p}(\Omega)^{3}\subset W^{{-1},{p}}(\Omega)^{3}$.
Using Lemma 9, we conclude that $\phi\in L^{p}(\Omega)$. ∎

######  Theorem 24.

Consider an exponent $1<p<\infty$ and a bounded $C^{1}$ domain
$\Omega\subset\mathbb{R}^{3}$ such that the boundary
$\partial\Omega=\overline{\Gamma_{D}\cup\Gamma_{N}}$ with $\Gamma_{D}$ and
$\Gamma_{N}$ open and connected subset of $\partial\Omega$. If $\mathbf{u}\in
W^{p}(\text{\emph{div}};\Omega)$ satisfies

$\nabla\cdot\mathbf{u}=0\qquad\text{on $\Omega$, and
}\qquad\int_{\Gamma_{i}}\mathbf{n}\cdot\mathbf{u}\,d\mathbf{x}=0,\,\,i=D,N,$
(24)

then there exists a vector potential $\mathbf{A}\in W^{1,p}(\Omega)^{3}$ such
that $\mathbf{u}=\nabla\times\mathbf{A}$ and $\nabla\cdot\mathbf{A}=0.$
Moreover, there exists a constant $C$ for which

$\|\mathbf{A}\|_{W^{{1},{p}}(\Omega)^{3}}\leq
C\|\mathbf{u}\|_{W^{p}(\text{\emph{div}};\Omega)}.$

###### Proof.

The proof is identical to the one given for Theorem 3.38 in [27] except for
the application of the Marcinkiewicz-Mihlin multiplier theorem; see Chapter 4,
subsection 3.2 of [32]. This ingredient is not new since it was already used
by Dauge [12] in a proof of the same theorem for domains with polyhedral
boundaries. Again for completeness, we provide the proof.

The problem will be solved using Fourier transforms and therefore, it will be
necessary to construct an extension of $\mathbf{u}\in
W^{p}(\text{div};\Omega).$ Our assumptions on the domain imply that
$\mathbb{R}^{3}\setminus\Omega$ is formed of two components $\Omega_{0}$ and
$\Omega_{1}$, one of which is unbounded, say $\Omega_{0}$. Consider the
following problem over $\Omega_{1}$ :

$\Delta v_{1}=0\qquad\text{on $\Omega_{1}$, and }\qquad\mathbf{n}\cdot\nabla
v_{1}=\mathbf{n}\cdot\mathbf{u}\qquad\text{on $\partial\Omega_{1}$.}$

By Theorem 15, the boundary condition $\mathbf{n}\cdot\mathbf{u}\in
B^{-1/p}_{p,p}(\partial\Omega_{1})$ possesses sufficient regularity to
guarantee, by Theorem 7, a solution $v_{1}\in W^{1,p}(\Omega_{1})$. Clearly
$\nabla v_{1}$ is divergence free inside $\Omega_{1}$.

To extend $\mathbf{u}$ into $\Omega_{0}$, choose a large closed ball $K$ whose
interior contains the closure of $\Omega$. Over $K\setminus\Omega$, consider
the problem

$\begin{array}[]{rcll}\Delta v_{0}&=&0&\text{on $K\setminus\Omega$,}\\\
\mathbf{n}\cdot\nabla v_{0}&=&\mathbf{n}\cdot\mathbf{u}&\text{on
$\partial\Omega_{0}$,}\\\ \mathbf{n}\cdot\nabla v_{0}&=&0&\text{on $\partial
K$.}\end{array}$

Again, there exists a unique solution $v_{0}\in W^{1,p}(K\setminus\Omega)$.
The new vector field $\mathbf{\widetilde{u}}$ obtained by patching together
$\mathbf{u},\nabla v_{0}$ and $\nabla v_{1}$ according to

$\mathbf{\widetilde{u}}(\mathbf{x})=\begin{cases}\mathbf{u}(\mathbf{x})&\text{if
$\mathbf{x}\in\Omega$,}\\\ \mathbf{n}\cdot\nabla v_{0}(\mathbf{x})&\text{if
$\mathbf{x}\in\Omega_{0}$,}\\\ \mathbf{n}\cdot\nabla
v_{1}(\mathbf{x})&\text{if $\mathbf{x}\in(K\setminus\Omega)$,}\\\ 0&\text{if
$\mathbf{x}$ is in the complement of $K$,}\end{cases}$

then belongs to $W^{p}(\text{div};\Omega)$ since the normal components are
equal along the boundaries. The proof that this new function belongs to
$L^{p}$ requires an obvious extension of Lemma 5.3 in [27]. Furthermore, this
extension of $\mathbf{u}$ to all of $\mathbb{R}^{3}$ satisfies
$\nabla\times\mathbf{\widetilde{u}}=0$.

With the extension $\mathbf{\widetilde{u}}$ belonging to $L^{p}(\Omega)\subset
L^{1}(\Omega)$, the Fourier transform of $\mathbf{\widetilde{u}}$ is well-
defined. As in the $L^{2}$ case, we want to consider the vector potential
$\mathbf{A}$ defined by

$\widehat{A}_{1}=\frac{\xi_{3}\widehat{u}_{2}-\xi_{2}\widehat{u}_{3}}{|\boldsymbol{\xi}|^{2}},\qquad\widehat{A}_{2}=\frac{\xi_{1}\widehat{u}_{3}-\xi_{3}\widehat{u}_{1}}{|\boldsymbol{\xi}|^{2}},\qquad\widehat{A}_{3}=\frac{\xi_{2}\widehat{u}_{1}-\xi_{1}\widehat{u}_{2}}{|\boldsymbol{\xi}|^{2}},$

but a priori, it is not clear that the inverse Fourier transform of
$\mathbf{A}$ can be taken. In this vein, write

$\widehat{\mathbf{A}}(\boldsymbol{\xi})=\chi(\boldsymbol{\xi})\widehat{\mathbf{A}}(\boldsymbol{\xi})+\big{(}1-\chi(\boldsymbol{\xi})\big{)}\widehat{\mathbf{A}}(\boldsymbol{\xi})$

where $\chi\in C^{\infty}(\mathbb{R}^{3})$ is identically $1$ outside a large
ball centered at the origin while being identically $0$ in a neighborhood of
the origin. The inverse Fourier transform of $(1-\chi)\widehat{\mathbf{A}}$ is
analytic in $\mathbb{R}^{3}$ and therefore in $L^{p}(\Omega)^{3}$. On the
other hand, the inverse Fourier transform of $\chi\widehat{\mathbf{A}}$ and
$\xi_{j}\chi(\boldsymbol{\xi})\widehat{\mathbf{A}}_{i}$ can be written as

$\displaystyle\big{(}\chi(\boldsymbol{\xi})\widehat{\mathbf{A}}(\boldsymbol{\xi})\big{)}\check{}=\big{(}W(\boldsymbol{\xi})\widehat{\mathbf{u}}(\boldsymbol{\xi})\big{)}\check{}$
$\displaystyle\big{(}\xi_{j}\chi(\boldsymbol{\xi})\widehat{A}_{i}(\boldsymbol{\xi})\big{)}\check{}=\big{(}\xi_{j}W(\boldsymbol{\xi})\widehat{\mathbf{u}}(\boldsymbol{\xi})\big{)}\check{}$

where the matrix $W(\boldsymbol{\xi})$ possesses terms that are asymptotically
of the form $\xi_{k}/|\boldsymbol{\xi}|^{2}$. A simple computation shows that
the multipliers $W(\boldsymbol{\xi})$ and $\xi_{j}W(\boldsymbol{\xi})$
satisfies the conditions of the Marcinkiewicz-Mihlin multiplier theorem and
therefore, that there exists a constant $C$ such that

$\|\big{(}\chi(\boldsymbol{\xi})\widehat{\mathbf{A}}(\boldsymbol{\xi})\big{)}\check{}\|_{L^{p}(\Omega)^{3}}\leq
C\|\mathbf{u}\|_{L^{p}(\Omega)^{3}}\quad\text{and}\quad\|\big{(}\xi_{j}\chi(\boldsymbol{\xi})\widehat{A}_{i}(\boldsymbol{\xi})\big{)}\check{}\|_{L^{p}(\Omega)}\leq
C\|\mathbf{u}\|_{L^{p}(\Omega)^{3}}.$

This shows that $\mathbf{A}$ in fact belongs to $W^{{1},{p}}(\Omega)^{3}$.
Using the fact that $\mathbf{u}$ is divergence free, it is easy to verify that
$\mathbf{A}$ is also divergence free and that
$\mathbf{u}=\nabla\times\mathbf{A}$. ∎

##  5 Friedrich’s inequality

The objective of this section is to provide detailed estimates concerning the
regularity of the spaces described in the preceding sections, in particular
with regard to the boundary conditions. We will show that the boundary
condition $\mathbf{n}\times\mathbf{B}=0\in L^{p}(\partial\Omega)$ provides
some additional smoothness inside the domain. The most important result will
be the Friedrich’s inequality, Theorem 27.

We begin our analysis with the following space

$W^{p}_{\text{imp}}(\text{curl};\Omega)=\left\\{\mathbf{u}\in
W^{p}(\text{curl};\Omega)\,\Big{|}\,\mathbf{n}\times\mathbf{u}\in
L^{p}_{t}(\partial\Omega)\right\\},$ (25)

where $L^{p}_{t}(\partial\Omega)=\\{\mathbf{u}\in
L^{p}(\Omega)^{3}|\,\mathbf{n}\cdot\mathbf{u}=0\\}$. The norm over
$W^{p}_{\text{imp}}(\text{curl};\Omega)$ will be

$\|\mathbf{u}\|_{W^{p}_{\text{imp}}(\text{curl};\Omega)}:=\|\mathbf{u}\|_{L^{p}(\Omega)^{3}}+\|\nabla\times\mathbf{u}\|_{L^{p}(\Omega)^{3}}+\|\mathbf{n}\times\mathbf{u}\|_{L^{p}(\partial\Omega)^{3}}.$

The next result shows that this strengthened requirement at the boundary
produces a non-trivial amount of regularity inside the domain. The result
comes from Costabel [9] where it was presented in $L^{2}$. We also add that
this result bears a close resemblance to a result mentioned in Remark 2.19 of
[3] and attributed to Dauge [13]. Although the next bound is interesting in
it’s own right, its true purpose is to demonstrate Corollary 26.

######  Theorem 25.

Consider a bounded $C^{1}$ domain $\Omega\subset\mathbb{R}^{3}$ and
$1<p<\infty$. If $\mathbf{u}\in W^{p}(\text{\emph{div}};\Omega)\cap
W^{p}(\text{\emph{curl}};\Omega)$ and $\mathbf{n}\times\mathbf{u}\in
L^{p}(\Omega)^{3}$, then $\mathbf{u}\in W^{{1/p},{p}}(\Omega)^{3}$ and for
some constant $C$,

$\|\mathbf{u}\|_{W^{{1/p},{p}}(\Omega)^{3}}\leq
C\left\\{\|\mathbf{u}\|_{L^{p}(\Omega)^{3}}+\|\nabla\cdot\mathbf{u}\|_{L^{p}(\Omega)}+\|\nabla\times\mathbf{u}\|_{L^{p}(\Omega)^{3}}+\|\mathbf{u}\times\mathbf{n}\|_{L^{p}(\partial\Omega)^{3}}\right\\}.$
(26)

###### Proof.

The proof is identical to the one given by Monk [27] for his Theorem 3.47
except that his elliptic regularity result, Theorem 3.17, must be replaced by
the generalization, Theorem 8.

To prove estimate (26), it suffices to prove a local version of this estimate
and then to use a partition of unity to extend the estimate to all of
$\Omega$. Following [27], the regularity of the boundary allows us to assume
that locally $\Omega$ is a bounded simply connected domain with a connected
$C^{1}$ boundary. Since $\mathbf{u}\in W^{p}(\text{curl};\Omega)$ and
$\nabla\cdot\nabla\times\mathbf{u}=0$, then there exists a vector potential
$\mathbf{w}\in W^{{1},{p}}(\Omega)^{3}$ such that
$\nabla\times\mathbf{w}=\nabla\times\mathbf{u}$ and $\nabla\cdot\mathbf{w}=0$
(Note that the constraint on the boundary is satisfied because of formula
(23)). Secondly, for $\mathbf{z}=\mathbf{u}-\mathbf{w}\in
W^{p}(\text{div};\Omega)$, we have $\nabla\times\mathbf{z}=0$ and by Theorem
23 there exists $\xi\in W^{{1},{p}}(\Omega)$ such that $\mathbf{z}=\nabla\xi$.
Finally, Theorem 5.4 of [34] states that we can construct $\eta\in
W^{{2},{p}}(\Omega)$ satisfying $\Delta\eta=\nabla\cdot\mathbf{z}\in
L^{p}(\Omega)$ (in constrast to the previous $\xi$, to obtain $\eta$ we need
to extend $\nabla\cdot\mathbf{z}$ by zero outside of $\Omega$ and thereby use
smoother boundary conditions).

To obtain the increased reguarity, we will find some additional regularity at
the boundary and invoke Theorem 8 to lift the regularity to all of $\Omega$.
Since $\mathbf{n}\times\mathbf{u}\in L^{p}_{t}(\partial\Omega)^{3}$ and
$\mathbf{w}\in W^{{1},{p}}(\Omega)^{3}$, then

$\mathbf{n}\times\mathbf{z}=\mathbf{n}\times(\mathbf{u}-\mathbf{w})\in
L^{p}_{t}(\partial\Omega).$

Using the fact that $\xi\in W^{{1},{p}}(\Omega)$ and $\eta\in
W^{{2},{p}}(\Omega)$ we have that the function $\nu=\xi-\eta$ satisfies
$\mathbf{n}\times\mathbf{z}=\mathbf{n}\times\nabla\xi=\mathbf{n}\times\nabla(\eta+\nu)$
and $\nabla\nu\times\mathbf{n}\in L^{p}_{t}(\partial\Omega)$. Therefore
$\nu\in W^{1,p}(\partial\Omega)$,
$\Delta\nu=\nabla\cdot(\nabla\xi-\nabla\eta)=0$ and Theorem 8 implies that
$\nu\in W^{{1+1/p},{p}}(\Omega)$.

We begin with an estimate for $\nu$. First of all, notice that since $\xi$ is
a consequence of Theorem 23, it is therefore defined up a to constant. The
function $\nu$ is thus also defined up to a constant and we may assume that
it’s average value along the boundary vanishes. Using the elliptic regularity
result, Theorem 8, and Poincaré’s inequality on the boundary, we deduce the
following.

$\displaystyle\|\nabla\nu\|_{W^{{1/p},{p}}(\Omega)}\leq$
$\displaystyle\|\nu\|_{W^{{1+1/p},{p}}(\Omega)^{3}}$ $\displaystyle\leq$
$\displaystyle
C\left\\{\|\nu\|_{L^{p}(\partial\Omega)}+\|\mathbf{n}\times\nabla\nu\|_{L^{p}(\partial\Omega)^{3}}\right\\}$
$\displaystyle\leq$ $\displaystyle
C\left\\{\|\mathbf{n}\times\nabla\nu\|_{L^{p}(\partial\Omega)^{3}}+\left|\int_{\partial\Omega}\nu\,d\boldsymbol{\sigma}\right|\right\\}=C\|\mathbf{n}\times\nabla\nu\|_{L^{p}(\partial\Omega)^{3}}$

Using the identity $\mathbf{u}=\mathbf{w}+\nabla\eta+\nabla\nu$ and the
stability estimates of Theorems 20, 24 and 23 for $\mathbf{w}$ and $\eta$, we
further obtain

$\displaystyle\|\nabla\nu\|_{W^{{1/p},{p}}(\Omega)^{3}}\leq$ $\displaystyle
C\left\\{\|\mathbf{n}\times\mathbf{u}\|_{L^{p}(\partial\Omega)^{3}}+\|\mathbf{n}\times\mathbf{w}\|_{L^{p}(\partial\Omega)^{3}}+\|\mathbf{n}\times\nabla\eta\|_{L^{p}(\partial\Omega)^{3}}\right\\}$
$\displaystyle\leq$ $\displaystyle
C\left\\{\|\mathbf{n}\times\mathbf{u}\|_{L^{p}(\partial\Omega)^{3}}+\|\mathbf{w}\|_{W^{{1},{p}}(\Omega)^{3}}+\|\nabla\eta\|_{W^{p}(\text{curl};\Omega)}\right\\}$
$\displaystyle\leq$ $\displaystyle
C\left\\{\|\mathbf{n}\times\mathbf{u}\|_{L^{p}(\partial\Omega)^{3}}+\|\nabla\times\mathbf{u}\|_{L^{p}(\Omega)^{3}}+\|\nabla\cdot\mathbf{u}\|_{L^{p}(\Omega)}\right\\}.$

It is already known that $\mathbf{u}\in W^{{1/p},{p}}(\Omega)^{3}$ and we can
estimate its norm by exploiting the previous estimate.

$\displaystyle\|\mathbf{u}\|_{W^{{1/p},{p}}(\Omega)^{3}}\leq$ $\displaystyle
C\left\\{\|\mathbf{w}\|_{W^{{1/p},{p}}(\Omega)^{3}}+\|\nabla\eta\|_{W^{{1/p},{p}}(\Omega)^{3}}+\|\nabla\nu\|_{W^{{1/p},{p}}(\Omega)^{3}}\right\\}$
$\displaystyle\leq$ $\displaystyle
C\left\\{\|\nabla\times\mathbf{u}\|_{L^{p}(\Omega)^{3}}+\|\nabla\cdot\mathbf{u}\|_{L^{p}(\Omega)}+\|\mathbf{n}\times\mathbf{u}\|_{L^{p}(\partial\Omega)^{3}}\right\\}$

This completes the proof of the theorem. ∎

We introduce the spaces

$\displaystyle X^{p}_{N}=$ $\displaystyle\left\\{\mathbf{u}\in
W^{p}(\text{div};\Omega)\cap
W^{p}(\text{curl};\Omega)\,\big{|}\,\mathbf{n}\times\mathbf{u}=0\text{ on
$\partial\Omega$}\right\\},$ (27) $\displaystyle X^{p}_{N,0}=$
$\displaystyle\left\\{\mathbf{u}\in
X^{p}_{N}\,\big{|}\,\nabla\cdot\mathbf{u}=0\text{ in $\Omega$}\right\\},$ (28)
$\displaystyle W^{p}_{N}=$ $\displaystyle\left\\{\mathbf{u}\in
W^{p}(\text{div};\Omega)\cap
W^{p}(\text{curl};\Omega)\,\big{|}\,\mathbf{n}\times\mathbf{u}\in
L^{p}_{t}(\partial\Omega)\text{ and $\nabla\cdot\mathbf{u}=0$}\right\\},$ (29)
$\displaystyle X^{p}_{T}=$ $\displaystyle\left\\{\mathbf{u}\in
W^{p}(\text{div};\Omega)\cap
W^{p}(\text{curl};\Omega)\,\big{|}\,\mathbf{n}\cdot\mathbf{u}=0\text{ on
$\partial\Omega$}\right\\},$ (30) $\displaystyle X^{p}_{T,0}=$
$\displaystyle\left\\{\mathbf{u}\in
X^{p}_{T}\,\big{|}\,\nabla\cdot\mathbf{u}=0\text{ in $\Omega$}\right\\},$ (31)
$\displaystyle W^{p}_{T}=$ $\displaystyle\left\\{\mathbf{u}\in
W^{p}(\text{div};\Omega)\cap
W^{p}(\text{curl};\Omega)\,\big{|}\,\mathbf{n}\cdot\mathbf{u}\in
L^{p}(\partial\Omega)\text{ and $\nabla\cdot\mathbf{u}=0$}\right\\}.$ (32)

The previous theorem implies that these spaces imbed continuously into
$W^{{1/p},{p}}(\Omega)^{3}$. Theorem 1.4.3.2 in [20] states that
$W^{{1/p},{p}}(\Omega)$ imbeds compactly into $L^{p}(\Omega)$ and therfore, we
have the following important corollary which generalizes Corollary 3.49 of
Monk [27].

######  Corollary 26.

Consider a bounded $C^{1}$ domain $\Omega\subset\mathbb{R}^{3}$ and
$1<p<\infty$. Then the spaces
$X^{p}_{N},X^{p}_{N,0},W^{p}_{N},X^{p}_{T},X^{p}_{T,0}$ and $W^{p}_{T}$ imbed
compactly into $L^{p}(\Omega)^{3}.$

######  Theorem 27 (Friedrich’s inequality).

Consider a bounded $C^{1}$ domain $\Omega\subset\mathbb{R}^{3}$ with a
connected boundary. Assume the exponent satisfies $1<p<\infty$. Then there
exists a constant $C$ such that for all $\mathbf{u}\in W_{N}^{p}$

$\|\mathbf{u}\|_{L^{p}(\Omega)^{3}}\leq
C\left\\{\|\nabla\times\mathbf{u}\|_{L^{p}(\Omega)^{3}}+\|\mathbf{n}\times\mathbf{u}\|_{L^{p}_{t}(\partial\Omega)}\right\\}.$

Similarly, there exists a constant $C$ such that for all $\mathbf{u}\in
W^{p}_{T}$

$\|\mathbf{u}\|_{L^{p}(\Omega)^{3}}\leq
C\left\\{\|\nabla\times\mathbf{u}\|_{L^{p}(\Omega)^{3}}+\|\mathbf{n}\cdot\mathbf{u}\|_{L^{p}(\partial\Omega)}\right\\}.$

Note: to prove the second estimate, we need the analogue of Theorem 25 when
$\mathbf{n}\cdot\mathbf{u}\in L^{p}(\partial\Omega)$.

###### Proof.

The proof is an extension of the one given for Corollary 3.51 in [27].

Proceeding by contradiction with the first inequality. Suppose that there
exists a sequence $\mathbf{u}_{n}$ of functions in $W^{p}_{N}$ such that

$\|\nabla\times\mathbf{u}_{n}\|_{L^{p}(\Omega)^{3}}+\|\mathbf{n}\times\mathbf{u}\|_{L^{p}(\partial\Omega)^{3}}\leq
1/n,$

while $\|\mathbf{u}_{n}\|_{L^{p}(\Omega)^{3}}=1$ for all $n$. Since
$W^{p}_{N}$ imbeds compactly into $L^{p}(\Omega)^{3}$, then we can extract a
subsequence (renamed $\mathbf{u}_{n}$) which converges to some $\mathbf{u}\in
L^{p}(\Omega)^{3}$. On the other hand, for all $\boldsymbol{\phi}\in
C^{\infty}_{0}(\Omega)^{3}$ the curl of $\mathbf{u}$ as a distribution
satisfies

$\displaystyle\big{|}L_{\nabla\times\mathbf{u}}(\boldsymbol{\phi})\big{|}=\big{|}(\mathbf{u},\nabla\times\boldsymbol{\phi})\big{|}=\lim_{n\to\infty}\big{|}(\mathbf{u}_{n},\nabla\times\boldsymbol{\phi})\big{|}=\lim_{n\to\infty}\big{|}(\nabla\times\mathbf{u}_{n},\boldsymbol{\phi})\big{|}=0.$

In other words, $\mathbf{u}\in L^{p}(\Omega)^{3}$ is such that
$\nabla\times\mathbf{u}=0$.

According to Theorem 23, there exists $\psi\in W^{{1},{p}}(\Omega)$ such that
$\mathbf{u}=\nabla\psi$. Since $\nabla\cdot\mathbf{u}=0,$ then $\Delta\psi=0$
and along the boundary

$\nabla_{\partial\Omega}\psi:=\mathbf{n}\times\nabla\psi=\mathbf{n}\times\mathbf{u}=0.$

In conclusion, $\psi$ is constant along the connected boundary and, by
uniqueness of solutions to Poisson’s problem, $\psi=0$ and $\mathbf{u}=0$.
This contradicts our earlier hypothesis concerning the first inequality. A
similar argument could be used to demonstrate the estimate over $W^{p}_{T}$. ∎

Rather than attempt to immediately demonstrate that
$C^{\infty}(\overline{\Omega})^{3}$ is dense in
$W^{p}_{\text{imp}}(\text{curl};\Omega)$, we will first show that
$C^{\infty}(\overline{\Omega})^{3}$ is dense in the auxiliary space

$\widetilde{W}^{p}_{\text{{imp}}}(\text{{curl}};\Omega)=\left\\{\mathbf{u}\in
W^{p}_{\text{imp}}(\text{curl};\Omega)\,\,\big{|}\,\,\mathbf{n}\cdot(\nabla\times\mathbf{u})=0\text{
on $\partial\Omega$}\right\\}.$

######  Lemma 28.

The functions in the subset
$\widetilde{W}^{p}_{\text{\emph{imp}}}(\text{\emph{curl}};\Omega)$ of
$W^{p}_{\text{\emph{imp}}}(\text{\emph{curl}};\Omega)$ can be approximated to
arbitrary accuracy by functions in $C^{\infty}(\overline{\Omega})^{3}$.

We now show how this lemma, whose proof we defer to the end of this section,
can be used to prove the next theorem.

######  Theorem 29.

The set $C^{\infty}(\overline{\Omega})^{3}$ is dense in
$W^{p}_{\text{\emph{imp}}}(\text{\emph{curl}};\Omega)$.

###### Proof.

The proof given here is a straightforward adaptation of Theorem 3.54 in [27].
As in the proof of Theorem 25, it suffices to demonstrate the density over
subsets that are simply connected with connected boundaries.

Choose $\mathbf{u}\in W^{p}_{\text{imp}}(\text{curl};\Omega)$ and consider the
variational problem : find $\phi\in W^{{1},{p}}(\Omega)/\mathbb{R}$ such that

$\int_{\Omega}\nabla\phi\cdot\nabla\psi\,d\mathbf{x}=\int_{\Omega}\nabla\times\mathbf{u}\cdot\nabla\psi\,d\mathbf{x},\quad\forall\psi\in
W^{{1},{q}}(\Omega)/\mathbb{R}.$

It is an exercise to show that the bilinear form
$B:W^{{1},{p}}(\Omega)/\mathbb{R}\times
W^{{1},{q}}(\Omega)/\mathbb{R}\to\mathbb{R}$

$B(\phi,\psi)=\int_{\Omega}\nabla\phi\cdot\nabla\psi\,d\mathbf{x}$

is continuous, non-degenerate and satisfies an $\inf$-$\sup$ condition of the
form (12). By the Babuška-Lax-Milgram Theorem, Theorem 11, this problem
possesses a unique solution.

Evaluating the divergence of $\nabla\phi$ in weak form, we find that

$-(\nabla\phi,\nabla\xi)=-(\nabla\times\mathbf{u},\nabla\xi)=0,\quad\forall\xi\in
C^{\infty}_{0}(\Omega),$

or in other words $\nabla\cdot\nabla\phi=0$. By adjusting the constant term of
$\phi$, we can satisfy condition (24) of Theorem 24 and deduce the existence
of $\mathbf{A}\in W^{{1},{p}}(\Omega)^{3}$ such that
$\nabla\times\mathbf{A}=\nabla\phi$, $\nabla\cdot\mathbf{A}=0$. Since
$C^{\infty}(\overline{\Omega})^{3}$ is dense in $W^{{1},{p}}(\Omega)^{3}$, the
same smooth functions which approximate $\mathbf{A}$ can be used in
$W^{p}_{\text{imp}}(\text{curl};\Omega)$.

Finally, we will show that the second term in the decomposition
$\mathbf{u}=\mathbf{A}+(\mathbf{u}-\mathbf{A})$ belongs to
$\widetilde{W}^{p}_{\text{imp}}(\text{curl};\Omega)$ and therefore, by Lemma
28, can also be approximated by functions in
$C^{\infty}(\overline{\Omega})^{3}$. For any $\psi\in W^{{1},{q}}(\Omega)$,
Green’s formula (20) can be used to show

$\displaystyle\int_{\partial\Omega}\mathbf{n}\cdot\nabla\times(\mathbf{u}-\mathbf{A})\psi\,d\mathbf{x}=$
$\displaystyle\int_{\Omega}\nabla\times(\mathbf{u}-\mathbf{a})\cdot\nabla\psi\,d\mathbf{x}=\int_{\Omega}(\nabla\times\mathbf{u}-\nabla\phi)\cdot\nabla\psi\,d\mathbf{x}=0,$

and therefore $\mathbf{n}\cdot\nabla\times(\mathbf{u}-\mathbf{A})=0$ on
$\partial\Omega$ and
$\mathbf{u}-\mathbf{A}\in\widetilde{W}^{p}_{\text{imp}}(\text{curl};\Omega)$.
∎

We now return to the proof of Lemma 28 whose proof is more involved in the
$L^{p}$ setting than the corresponding Lemma 3.53 in [27]. We therefore
propose to begin with the following important preliminary result.

######  Lemma 30.

Consider a bounded simply connected domain $\Omega\subset\mathbb{R}^{3}$ with
a connected $C^{1}$ boundary. Assume the exponent satisfies $1<p<\infty$. Then
the bilinear form $B:X^{p}_{T,0}\times X^{q}_{T,0}\to\mathbb{R}$ defined by

$B(\mathbf{u},\mathbf{v}):=\int_{\Omega}\nabla\times\mathbf{u}\cdot\nabla\times\mathbf{v}\,d\mathbf{x},$

is continuous, non-degenerate and satisfies the $\inf$-$\sup$ condition (12).

###### Proof.

The bilinear form is easily seen to be continuous since, by Theorem (27),
$\|\nabla\times\mathbf{u}\|_{L^{p}(\Omega)^{3}}$ and
$\|\nabla\times\mathbf{v}\|_{L^{q}(\Omega)^{3}}$ are norms over respectively
$X^{p}_{T,0}$ and $X^{q}_{T,0}$.

We will only demonstrate the $\inf$-$\sup$ condition because the ”symmetry” of
$B$ will imply the non-degeneracy. We will proceed by contradiction and
therefore, we assume that there exists a sequence $\mathbf{A}_{n}\in
X^{p}_{T,0}$ for which

$\sup_{{\boldsymbol{\phi}\in
X^{q}_{T,0}:\|\boldsymbol{\phi}\|=1}}\big{|}(\nabla\times\mathbf{A}_{n},\nabla\times\boldsymbol{\phi})\big{|}<1/n,$
(33)

while $\|\nabla\times\mathbf{A}_{n}\|_{L^{p}(\Omega)^{3}}=1$.

Pick any $\boldsymbol{\psi}\in C^{\infty}_{0}(\Omega)^{3}$ and construct
$\xi\in C^{\infty}_{0}(\overline{\Omega})$ satisfying

$\Delta\xi=\nabla\cdot\boldsymbol{\psi}\text{ on $\Omega$, and }\xi=0\text{ on
$\partial\Omega$.}$

Now let $\boldsymbol{\chi}$ be the function in
$C^{\infty}(\overline{\Omega})^{3}$ given by
$\nabla\times\boldsymbol{\chi}=\boldsymbol{\psi}-\nabla\xi$ and
$\nabla\cdot\boldsymbol{\chi}=0$, as guaranteed by Theorem 24. Returning to
property (33), we have that for any $\boldsymbol{\psi}\in
C^{\infty}_{0}(\Omega)^{3}$,

$\displaystyle\big{|}(\nabla\times\mathbf{A}_{n},\boldsymbol{\psi})_{\Omega}\big{|}=$
$\displaystyle\big{|}(\nabla\times\mathbf{A}_{n},\nabla\times\boldsymbol{\chi}+\nabla\xi)_{\Omega}\big{|}$
$\displaystyle=$
$\displaystyle\big{|}(\nabla\times\mathbf{A}_{n},\nabla\times\boldsymbol{\chi})_{\Omega}-(\nabla\cdot\nabla\times\mathbf{A}_{n},\xi)_{\Omega}+(\mathbf{n}\cdot\nabla\times\mathbf{A}_{n},\xi)_{\partial\Omega}\big{|}$
$\displaystyle=$
$\displaystyle\big{|}(\nabla\times\mathbf{A}_{n},\nabla\times\boldsymbol{\chi})_{\Omega}\big{|}.$

By the density result of $C^{\infty}_{0}(\Omega)^{3}$ in $L^{q}(\Omega)^{3}$
and Lemma 2.7 of [1], this implies that
$\|\nabla\times\mathbf{A}_{n}\|_{L^{p}(\Omega)^{3}}<1/n$. This contradicts our
earlier assumption that $\|\nabla\times\mathbf{A}_{n}\|_{L^{p}(\Omega)^{3}}=1$
and proves the result. ∎

We complete this section with a proof of Lemma 28.

###### Lemma 28.

As we mentionned earlier in Theorem 25, the proof of a density result can be
demonstrated locally and we may assume that $\Omega$ is a bounded simply
connected domain with a connected $C^{1}$ boundary. Choose
$\mathbf{u}\in\widetilde{W}^{p}_{\text{imp}}(\text{curl};\Omega)$. Lemma 30
states that $B$ satisfies the conditions of the Babuška-Lax-Milgram theorem
and therefore, that there exists a unique solution $\mathbf{A}\in X^{p}_{T,0}$
to

$\int_{\Omega}\nabla\times\mathbf{A}\cdot\nabla\times\boldsymbol{\phi}\,d\mathbf{x}=\int_{\Omega}\mathbf{u}\cdot\nabla\times\boldsymbol{\phi}-\nabla\times\mathbf{u}\cdot\boldsymbol{\phi}\,d\mathbf{x},\quad\forall\boldsymbol{\phi}\in
W^{q}_{T,0}.$ (34)

For any $\boldsymbol{\psi}\in X^{q}_{T}$, Green’s formula shows that

$\int_{\Omega}\nabla\cdot\boldsymbol{\psi}\,d\mathbf{x}=\int_{\partial\Omega}\mathbf{n}\cdot\boldsymbol{\psi}\,d\boldsymbol{\sigma}=0.$

Hence, Theorem 23 states that we can construct $\xi\in W^{{1},{q}}(\Omega)$
satisfying

$\Delta\xi=\nabla\cdot\boldsymbol{\psi}\quad\text{on $\Omega$, and
}\quad\mathbf{n}\cdot\nabla\xi=0\quad\text{on $\partial\Omega$.}$

Then any $\boldsymbol{\psi}\in X^{q}_{T}$ can be decomposed as
$\nabla\xi+\big{(}\boldsymbol{\psi}-\nabla\xi\big{)}$ where the two two
components are such that $\nabla\xi\in X^{q}_{T}$ and
$\boldsymbol{\psi}-\nabla\xi\in X^{q}_{T,0}$. In fact, if $\mathbf{A}$
satisfies equation (34), then exploiting the previous decomposition we find
that for all $\boldsymbol{\psi}\in X^{q}_{T}$,

$\displaystyle\int_{\Omega}\nabla\times\mathbf{A}\cdot\nabla\times\boldsymbol{\psi}\,d\mathbf{x}=$
$\displaystyle\int_{\Omega}\nabla\times\mathbf{A}\cdot\nabla\times\big{(}\boldsymbol{\psi}-\nabla\xi\big{)}\,d\mathbf{x}$
$\displaystyle=$
$\displaystyle\int_{\Omega}\mathbf{u}\cdot\nabla\times\big{(}\boldsymbol{\psi}-\nabla\xi\big{)}-\nabla\times\mathbf{u}\cdot\big{(}\boldsymbol{\psi}-\nabla\xi\big{)}\,d\mathbf{x}$
$\displaystyle=$
$\displaystyle\int_{\Omega}\mathbf{u}\cdot\nabla\times\boldsymbol{\psi}-\nabla\times\mathbf{u}\cdot\boldsymbol{\psi}\,d\mathbf{x}$
$\displaystyle+\int_{\Omega}\mathbf{u}\cdot\nabla\times\big{(}-\nabla\xi\big{)}\,d\mathbf{x}+\int_{\partial\Omega}\mathbf{n}\times\mathbf{u}\cdot(-\nabla\xi)\,d\boldsymbol{\sigma}$
$\displaystyle=$
$\displaystyle\int_{\Omega}\mathbf{u}\cdot\nabla\times\boldsymbol{\psi}-\nabla\times\mathbf{u}\cdot\boldsymbol{\psi}\,d\mathbf{x}.$
(35)

By Theorems 12 and 18, constraint (34) must also be satisfied by all
$\boldsymbol{\psi}\in C^{\infty}_{0}(\Omega)^{3}$. Furthermore, Green’s
formula (23) applied to the right-hand side of (34) provides the identity

$\int_{\Omega}\nabla\times\mathbf{A}\cdot\nabla\times\boldsymbol{\psi}\,d\mathbf{x}=\langle\gamma_{t}(\mathbf{u}),\gamma_{T}(\boldsymbol{\psi})\rangle_{\partial\Omega}=0,$

and therefore, in a distributional sense,
$\nabla\times(\nabla\times\mathbf{A})=0$.

For the original
$\mathbf{u}\in\widetilde{W}^{p}_{\text{imp}}(\text{curl};\Omega)$, we write
$\mathbf{u}=\nabla\times\mathbf{A}+(\mathbf{u}-\nabla\times\mathbf{A})$.
Applying Green’s formula (20) to both sides of (35), we find that

$\langle\gamma_{t}(\nabla\times\mathbf{A}),\gamma_{T}(\boldsymbol{\psi})\rangle_{\partial\Omega}=\langle\gamma_{t}(\mathbf{u}),\gamma_{T}(\boldsymbol{\psi})\rangle_{\partial\Omega},$

that is $\mathbf{n}\times(\mathbf{u}-\nabla\times\mathbf{A})=0$. In
particular, $\mathbf{u}-\nabla\times\mathbf{A}$ belongs to
$W^{p}_{0}(\text{curl};\Omega)$ and can be approximated by functions in
$C^{\infty}_{0}(\Omega)^{3}$.

Continuing with the decomposition
$\mathbf{u}=\nabla\times\mathbf{A}+(\mathbf{u}-\nabla\times\mathbf{A})$ of
$\mathbf{u}\in\widetilde{W}^{p}_{\text{imp}}(\text{curl};\Omega)$, we now show
that $\nabla\times\mathbf{A}$ can be approximated by smooth functions. Since
$\nabla\times(\nabla\times\mathbf{A})=0$, there exists $\xi\in
W^{{1},{p}}(\Omega)$ for which $\nabla\times\mathbf{A}=\nabla\xi$. The
gradient along the boundary is given by

$\nabla_{\partial\Omega}\xi=(\mathbf{n}\times\nabla\xi)\times\mathbf{n}=(\mathbf{n}\times\nabla\times\mathbf{A})\times\mathbf{n}=(\mathbf{n}\times\mathbf{u})\times\mathbf{n}\in
L^{p}_{t}(\partial\Omega).$

Therefore, $\gamma_{0}(\xi)\in W^{1,p}(\partial\Omega)$. Recalling Theorem 10,
we conclude that $\xi$ can be approximated by functions in
$C^{\infty}(\overline{\Omega})$. Say that $\xi_{n}\in
C^{\infty}(\overline{\Omega})$ is a sequence converging to $\xi$ in the graph
space $G$, previously defined in the statement of Theorem 10. Then the
convergence inside the domain gives us that

$\displaystyle\nabla\xi_{n}$ $\displaystyle\to\nabla\times\mathbf{A}\text{ in
}L^{p}(\Omega)^{3},$ $\displaystyle\nabla\times\nabla\xi_{n}$
$\displaystyle\to\nabla\times(\nabla\times\mathbf{A})\text{ in
}L^{p}(\Omega)^{3},$

while the convergence along the boundary guarantees that

$\mathbf{n}\times\nabla\xi_{n}\to\mathbf{n}\times(\nabla\times\mathbf{A})\text{
in }L^{p}_{t}(\partial\Omega).$

According to the definition of the norm over
$W^{p}_{\text{imp}}(\text{curl};\Omega)$, the three previous limits imply that
$\nabla\xi_{n}\to\nabla\times\mathbf{A}$ in
$\widetilde{W}^{p}_{\text{imp}}(\text{curl};\Omega)$. ∎

######  Theorem 31 (Helmholtz decomposition).

Consider a domain $\Omega$ with a smooth $C^{1}$ boundary and an exponent
$2\leq p<\infty$. Then $W^{p}_{0}(\text{\emph{curl}};\Omega)$ has the
following direct sum decomposition

$W^{p}_{0}(\text{\emph{curl}};\Omega)=X_{0}\oplus\nabla W^{1,p}_{0}(\Omega),$

where

$X_{0}=\left\\{\mathbf{u}\in
W^{p}_{0}(\text{\emph{curl}};\Omega)\,\Big{|}\,(\mathbf{u},\nabla\phi)\text{
for all $\phi\in W^{1,p}_{0}(\Omega)$}\right\\}.$

###### Proof.

∎

##  6 Applications to the analysis of a nonlinear elliptic problem

With the results given so far, we are in a position to study the existence and
uniqueness of solutions to the problem (1)-(3). In anticipation of our
analysis of the associated homogeneous parabolic problem, we will consider the
problem with a source term $\mathbf{S}$ satisfying $\nabla\cdot\mathbf{S}=0$
and $\mathbf{n}\times\mathbf{S}=0$ along the boundary $\partial\Omega$. The
elliptic $p$-CurlCurl written in weak form is therefore : given $\mathbf{S}\in
X_{0}$, find $\mathbf{u}\in X_{0}$ such that

$\left(|\nabla\times\mathbf{u}|^{p-2}\nabla\times\mathbf{u},\nabla\times\mathbf{v}\right)_{\Omega}=\left(\mathbf{S},\mathbf{v}\right)_{\Omega},\,\,\forall\mathbf{v}\in
X_{0}.$ (36)

This suggests we consider the nonlinear differential operator
$A:W^{p}(\text{{curl}};\Omega)\to W^{p}(\text{{curl}};\Omega)^{\prime}$,
defined by the pairing

$\left(A\mathbf{u},\mathbf{v}\right)=\int_{\Omega}|\nabla\times\mathbf{u}|^{p-2}\nabla\times\mathbf{u}\cdot\nabla\times\mathbf{v}\,d\mathbf{x}.$

The next two vector inequalities are well-known and demonstrated in [5]. In
fact, a generalization can also be found in [24].

######  Lemma 32 ([19, 5]).

For all $p\in(1,\infty)$ and $\delta\geq 0$ there exists positive constants
$a_{1}(p,n)$ and $a_{2}(p,n)$ such that for all $\xi,\eta\in\mathbb{R}^{n}$,

$||\xi|^{p-2}\xi-|\eta|^{p-2}\eta|\leq
a_{1}|\xi-\eta|^{1-\delta}\big{(}|\xi|+|\eta|\big{)}^{p-2+\delta},$ (37)

and

$|\xi-\eta|^{2+\delta}\big{(}|\xi|+|\eta|\big{)}^{p-2-\delta}\leq
a_{2}\big{(}|\xi|^{p-2}\xi-|\eta|^{p-2}\eta\big{)}\cdot(\xi-\eta).$ (38)

######  Lemma 33.

Assume that $\Omega$ is bounded domain with a $C^{1}$ boundary and that $2\leq
p<\infty$. Then, the operator $A$ is

  * (i)

well-defined and hemicontinuous over $X_{0}$ : for some $C_{1}$

$\|A\mathbf{u}\|_{X_{0}^{\prime}}\leq
C_{1}\|\mathbf{u}\|_{X_{0}}^{p-1},\qquad\forall\mathbf{u}\in X_{0};$ (39)

  * (ii)

strictly monotone over $X_{0}$ : for some $C_{2}$

$\|\mathbf{u}-\mathbf{v}\|_{X_{0}}^{p}\leq
C_{2}\left(A\mathbf{u}-A\mathbf{v},\mathbf{u}-\mathbf{v}\right)_{\Omega},\qquad\forall\mathbf{u},\mathbf{v}\in
X_{0};$ (40)

  * (iii)

stable over $X_{0}$ : for some $C_{3}$

$\|A\mathbf{u}-A\mathbf{v}\|_{X_{0}^{\prime}}\leq
C_{3}\|\mathbf{u}-\mathbf{v}\|_{X_{0}}\left(\|\mathbf{u}\|_{X_{0}}+\|\mathbf{v}\|_{X_{0}}\right)^{p-2},\qquad\forall\mathbf{u},\mathbf{v}\in
X_{0};$ (41)

  * (iv)

coercive over $X_{0}$,

$\big{|}(A\mathbf{u},\mathbf{u})\big{|}\to\infty,\text{ if
$\|\mathbf{u}\|_{X_{0}}\to\infty$.}$ (42)

In fact, (i) and (iii) also hold over $W^{p}_{0}(\text{curl},\Omega)$.

###### Proof.

To prove that $A$ is well-defined and hemicontinuous, it suffices to observe
that

$\displaystyle\big{|}(A\mathbf{u},\mathbf{v})\big{|}\leq$
$\displaystyle\int_{\Omega}|\nabla\times\mathbf{u}|^{p-1}|\nabla\times\mathbf{v}|\,d\mathbf{x}$
$\displaystyle\leq$
$\displaystyle\||\nabla\times\mathbf{u}|^{p-1}\|_{L^{q}(\Omega)^{3}}\cdot\|\nabla\times\mathbf{v}\|_{L^{p}(\Omega)^{3}}$
$\displaystyle=$
$\displaystyle\big{(}\|\nabla\times\mathbf{u}\|_{L^{p}(\Omega)^{3}}\big{)}^{p-1}\cdot\|\nabla\times\mathbf{v}\|_{L^{p}(\Omega)^{3}}\leq\big{(}\|\mathbf{u}\|_{W^{p}(\text{curl};\Omega)}\big{)}^{p-1}\|\mathbf{v}\|_{W^{p}(\text{curl};\Omega)}.$

According to the definition of the dual norm, (5), this implies (39). To prove
(40), we use (38) with $\delta=p-2\geq 0$ and Friedrich’s inequality to write

$\displaystyle\big{(}A\mathbf{u}-A\mathbf{v},\mathbf{u}-\mathbf{v}\big{)}=$
$\displaystyle\int_{\Omega}\Big{[}|\nabla\times\mathbf{u}|^{p-2}\nabla\times\mathbf{u}-|\nabla\times\mathbf{v}|^{p-2}\nabla\times\mathbf{v}\Big{]}\cdot\nabla\times(\mathbf{u}-\mathbf{v})\,d\mathbf{x}$
$\displaystyle\geq$ $\displaystyle
a_{2}\int_{\Omega}\big{|}\nabla\times\mathbf{u}-\nabla\times\mathbf{v}\big{|}^{p}\,d\mathbf{x}=a_{2}\|\mathbf{u}-\mathbf{v}\|^{p}_{X_{0}}.$

To prove (41), we begin with the application of Hölder’s inequality

$\displaystyle\|A\mathbf{u}-A\mathbf{v}$
$\displaystyle\|_{W^{p}(\text{curl};\Omega)^{\prime}}=\sup_{\boldsymbol{\phi}\in
W^{p}(\text{curl};\Omega)}\frac{(A\mathbf{u}-A\mathbf{v},\boldsymbol{\phi})}{\|\boldsymbol{\phi}\|_{W^{p}(\text{curl};\Omega)}}$
$\displaystyle=$
$\displaystyle\sup_{\boldsymbol{\phi}}\frac{1}{\|\boldsymbol{\phi}\|}\Bigg{|}\int_{\Omega}\Big{[}|\nabla\times\mathbf{u}|^{p-2}\nabla\times\mathbf{u}-|\nabla\times\mathbf{v}|^{p-2}\nabla\times\mathbf{v}\Big{]}\cdot\nabla\times\boldsymbol{\phi}\,d\mathbf{x}\Bigg{|}$
$\displaystyle\leq$
$\displaystyle\sup_{\boldsymbol{\phi}}\frac{1}{\|\boldsymbol{\phi}\|}\Bigg{|}\int_{\Omega}\left||\nabla\times\mathbf{u}|^{p-2}\nabla\times\mathbf{u}-|\nabla\times\mathbf{v}|^{p-2}\nabla\times\mathbf{v}\right|^{q}\,d\mathbf{x}\Bigg{|}^{1/q}\cdot\|\nabla\times\boldsymbol{\phi}\|_{L^{p}(\Omega)^{3}}$
$\displaystyle\leq$
$\displaystyle\Bigg{|}\int_{\Omega}\left||\nabla\times\mathbf{u}|^{p-2}\nabla\times\mathbf{u}-|\nabla\times\mathbf{v}|^{p-2}\nabla\times\mathbf{v}\right|^{q}\,d\mathbf{x}\Bigg{|}^{1/q}$

At this point, we recall that $q=p/(p-1)$, we exploit (37) with $\delta=0$ and
then follow up with another application of Hölder’s inequality

$\displaystyle\|A\mathbf{u}-A\mathbf{v}\|_{W^{p}(\text{curl};\Omega)^{\prime}}\leq$
$\displaystyle\,a_{1}\left\\{\int_{\Omega}\left|\nabla\times\mathbf{u}-\nabla\times\mathbf{v}\right|^{q}\cdot\big{(}|\nabla\times\mathbf{u}|+|\nabla\times\mathbf{v}|\big{)}^{(p-2)q}\right\\}^{1/q}$
$\displaystyle\leq$
$\displaystyle\,a_{1}\Bigg{\\{}\bigg{\\{}\int_{\Omega}\big{|}\nabla\times\mathbf{u}-\nabla\times\mathbf{v}\big{|}^{\frac{p}{p-1}\cdot(p-1)}\,d\mathbf{x}\bigg{\\}}^{\frac{1}{p-1}}$
$\displaystyle\phantom{\,C\Bigg{\\{}}\times\bigg{\\{}\int_{\Omega}\big{(}|\nabla\times\mathbf{u}|+|\nabla\times\mathbf{v}|\big{)}^{(p-2)\cdot\frac{p}{p-1}\cdot\frac{p-1}{p-2}}\bigg{\\}}^{\frac{p-2}{p-1}}\Bigg{\\}}^{\frac{p-1}{p}}$
$\displaystyle\leq$
$\displaystyle\,a_{1}\|\mathbf{u}-\mathbf{v}\|_{X_{0}}^{\frac{p}{p-1}\frac{p-1}{p}}\times\||\nabla\times\mathbf{u}|+|\nabla\times\mathbf{v}|\|_{X_{0}}^{p\frac{p-2}{p-1}\frac{p-1}{p}}$
$\displaystyle\leq$
$\displaystyle\,a_{1}\|\mathbf{u}-\mathbf{v}\|_{W^{p}(\text{curl};\Omega)}\cdot\Big{(}\|\mathbf{u}\|_{W^{p}(\text{curl};\Omega)}+\|\mathbf{v}\|_{W^{p}(\text{curl};\Omega)}\Big{)}^{p-2}.$

Finally, Friedrich’s inequality implies (42) because of the identity

$\big{|}(A\mathbf{u},\mathbf{u})\big{|}=\int_{\Omega}|\nabla\times\mathbf{u}|^{p}\,d\mathbf{x}=\|\nabla\times\mathbf{u}\|_{L^{p}(\Omega)^{3}}^{p}.$

∎

######  Theorem 34.

Assume that $\Omega$ is bounded domain with a $C^{1}$ boundary and that $2\leq
p<\infty$. For any $\mathbf{S}\in X_{0}$, the problem (1)-(3) has a unique
weak solution $\mathbf{u}\in X_{0}$. Moreover, the inverse operator $A^{-1}$
is continuous.

###### Proof.

Note that $S\in X_{0}^{\prime}$ because $p\geq 2$. Lemma 33 states that $A$
satisfies all the conditions Theorem 12.1 from [14], and therefore $A$ is onto
$X_{0}^{\prime}$. Strict monotonicity implies that the solution is unique. The
existence of a smooth inverse follows from the remarks following Theorem 12.1
in [14]. ∎

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