\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 196, pp. 1--28.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/196\hfil Stokes problem with several boundary conditions]
{Stokes problem with several types of boundary conditions in an exterior domain}

\author[C. Amrouche,  M. Meslameni \hfil EJDE-2013/196\hfilneg]
{Ch\'erif Amrouche,  Mohamed Meslameni}  % in alphabetical order

\address{Ch\'erif Amrouche \newline
Laboratoire de Math\'ematiques et de leurs Applications,
Pau - CNRS UMR 5142\\
Universit\'e de Pau et des Pays de l'Adour\\
IPRA, Avenue de l'Universit\'e - 64000 Pau - France}
\email{cherif.amrouche@univ-pau.fr}

\address{Mohamed Meslameni \newline
Laboratoire de Math\'ematiques et de leurs Applications,
Pau - CNRS UMR 5142\\
Universit\'e de Pau et des Pays de l'Adour\\
IPRA, Avenue de l'Universit\'e - 64000 Pau - France}
\email{mohamed.meslameni@univ-pau.fr}

\thanks{Submitted July 28, 2013. Published September 3, 2013.}
\subjclass[2000]{35J25, 35J50, 76M30}
\keywords{Stokes equations; exterior domain; weighted Sobolev spaces;
\hfill\break\indent  vector potentials; inf-sup conditions}

\begin{abstract}
 In this article, we solve the Stokes problem in an exterior domain of
 $\mathbb{R}^{3}$,  with non-standard boundary conditions.
 Our approach  uses  weighted Sobolev spaces to
 prove the existence, uniqueness of weak and strong solutions.
 This work is based on the vector potentials studied in
 \cite{vecteur potentiel vivet} for exterior domains,  and in
 \cite{bernardi} for bounded domains.
 This problem is well known in the classical Sobolev spaces $ W ^{m,2}(\Omega)$
 when $\Omega$ is bounded; see \cite{Nour,Nour1}.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction and functional setting}

Let  $\Omega'$  denotes a bounded open in $\mathbb{R}^{3}$ of class $C^{1,1}$,
simply-connected and with a connected boundary $\partial \Omega'=\Gamma$,
representing an obstacle and $\Omega $ is its complement;
i.e. $\Omega=\mathbb{R}^{3}\setminus\overline{\Omega'}$.
Then a unit exterior normal vector to the boundary can be defined almost
everywhere on $\Gamma$; it is denoted by $\mathbf{n}$.
The purpose of this paper is to solve the Stokes equation in $\Omega$,
with two types of non standard boundary conditions on $\Gamma$:
\begin{equation} \label{eST}
\begin{gathered}
-\Delta \mathbf{u}+\nabla \pi=\mathbf{f}\quad\text{and}\quad
\operatorname{div}\mathbf{u}=\chi \quad\text{in }\Omega,\\
\mathbf{u}\cdot\mathbf{n}=g\quad\text{and}\quad\operatorname{\bf curl}
\mathbf{u}\times\mathbf{n}=\mathbf{h}\times\mathbf{n}
\quad\text{on }\Gamma ,\\
\end{gathered}
\end{equation}
and
\begin{equation} \label{eSN}
\begin{gathered}
-\Delta \mathbf{u}+\nabla \pi=\mathbf{f}\quad\text{and}\quad
\operatorname{div}\mathbf{u}=\chi \quad\text{in } \Omega,\\
\pi=\pi_{0},\quad
\mathbf{u}\times\mathbf{n}=\mathbf{g}\times\mathbf{n}\quad\text{on}\quad
\Gamma \quad\text{and}\quad \int_{\Gamma}\mathbf{u}\cdot \mathbf{n}\,d\sigma=0.
\end{gathered}
\end{equation}
Since this problem is posed in an exterior domain, our approach is
to  use weighted Sobolev spaces. Let us begin by introducing these spaces.
 A point in $\Omega$ will be denoted by $\textbf{x}=(x_1, x_2, x_3)$
and its distance to the origin by $r=|\textbf{x}|=(x_1^2+x_2^2+x_3^2)^{1/2}$.
We will use the weights
\begin{equation*}
\rho=\rho(r)=(1+r^2)^{1/2}.
\end{equation*}
For all $m$ in $\mathbb{N}$ and all $k$ in $\mathbb{Z}$, we define the weighted space
\begin{equation*}
W_{k}^{m,2}(\Omega)=\{u \in \mathcal{D}'(\Omega):
\forall \lambda \in \mathbb{N}^{3}: 0\leq |\lambda| \leq m,\,
\rho(r)^{k-m+|\lambda|}D^{\lambda}u \in L^2(\Omega) \},
\end{equation*}
which is a Hilbert space with the norm
\begin{equation*}
\|u\|_{W_{k}^{m,2}(\Omega)}=\Big( \sum _{|\lambda| =0 }^{m}\|
\rho^{k-m+|\lambda|}D^{\lambda}u\|^2_{L^2(\Omega)} \Big) ^{1/2},
\end{equation*}
where $\|\cdot\|_{L^2(\Omega)}$ denotes the standard norm of $L^2(\Omega)$.
 We shall sometimes use  the seminorm
\begin{equation*}
|u|_{W_{k}^{m,2}(\Omega)}=\Big( \sum _{|\lambda| =m }\|\rho^{k}
D^{\lambda}u\|^2_{L^2(\Omega)} \Big) ^{1/2}.
\end{equation*}
In addition, it is established by Hanouzet in \cite{Hanouzet}, for domains
with a Lipschitz-continuous boundary, that $\mathcal{D}(\overline{\Omega})$
is dense in $W_{k}^{m,2}(\Omega)$. We set $\mathring{W}_{k}^{m,2}(\Omega)$
as the adherence of $\mathcal{D}(\Omega)$ for the norm
$\| \cdot \|_{W_{k}^{m,2}(\Omega)}$. Then, the dual space of
$\mathring{W}_{k}^{m,2}(\Omega)$, denoting  by $W_{-k}^{-m,2}(\Omega)$,
is a space of distributions.
 Furthermore, as in bounded domain, we have  for $m=1$ or $m=2$,
\begin{gather*}
 \mathring{W}_{k}^{1,2}(\Omega)=\{ v \in W_{k}^{1,2}(\Omega),\,\
 v=0\quad \text{on} \,\, \Gamma \}, \\
 \mathring{W}_{0}^{2,2}(\Omega)=\{ v \in W_{0}^{2,2}(\Omega),\,
 v=\frac{\partial v}{\partial \mathbf{n}}=0\quad \text{on }
 \Gamma \} ,
\end{gather*}
where $\frac{\partial v}{\partial \mathbf{n}}$ is the normal derivative of $v$.
As a consequence of Hardy's inequality, the following Poincar\'e inequality
holds: for $m=0$ or $m=1$ and for all $k$ in $\mathbb{Z}$ there exists a constant $C$ such that
\begin{equation}\label{egalite de hardy}
\forall   v \in \mathring{W}_{k}^{m,2}(\Omega),\quad
 \|v\|_{W_{k}^{m,2}(\Omega)}\leq C |v|_{{W_{k}^{m,2}(\Omega)}};
\end{equation}
i.e., the seminorm $|\cdot|_{W_{k}^{m,2}(\Omega)}$ is a norm on
$\mathring{W}_{k}^{m,2}(\Omega)$ equivalent to the norm
$\|\cdot\|_{W_{k}^{m,2}(\Omega)}$.


In the sequel, we shall use the following properties.
For all integers $m$ and $k$ in $\mathbb{Z}$, we have
\begin{equation}
\forall  n  \in \mathbb{Z} \quad \text{with}\quad n\leq m-k-2,\quad
\mathcal{P}_{n}\subset W_{k}^{m,2}(\Omega),
\end{equation}
where $\mathcal{P}_{n}$ denotes the space of all polynomials
(of three variables) of degree at most $n$, with the convention
that the space is reduced to zero when $n$ is negative.
Thus the difference $m-k$ is an important parameter of the space
$W_{k}^{m,2}(\Omega)$. We denote by $\mathcal{P}_{n}^{\Delta}$
the subspace of all harmonic polynomials of $\mathcal{P}_{n}$.

Using the derivation in the distribution sense, we can define the
operators $\operatorname{\bf curl}$ and
 $\operatorname{div}$ on $\mathbf{L}^2(\Omega)$.
Indeed, let $ \langle\cdot,\cdot\rangle$ denote the duality pairing between
$\mathcal{D}(\Omega)$ and its dual space $\mathcal{D}'(\Omega)$.
For any function $\mathbf{v}=(v_1, v_2, v_3) \in \mathbf{L}^2(\Omega)$,
 we have for any
$\boldsymbol{\varphi}=(\varphi_1, \varphi_2, \varphi_3)
\in \boldsymbol{\mathcal{D}}(\Omega)$,
\begin{align*}
\langle \operatorname{\bf curl}\mathbf{v},\boldsymbol{\varphi}\rangle
&= \int_{\Omega}\mathbf{v}\cdot \operatorname{\bf curl}
\boldsymbol{\varphi}\,d\mathbf{x}\\
&= \int_{\Omega}\Big( v_1(\frac{\partial \varphi_3}{\partial x_2}
-\frac{\partial \varphi_2}{\partial x_3})
+ v_2(\frac{\partial \varphi_1}{\partial x_3}
-\frac{\partial \varphi_3}{\partial x_1})
+v_3(\frac{\partial \varphi_2}{\partial x_1}
-\frac{\partial \varphi_1}{\partial x_2})\Big)\,d\mathbf{x},
\end{align*}
and for any $\varphi \in \mathcal{D}(\Omega)$,
$$
\langle \operatorname{div}\mathbf{v},
\varphi \rangle=-\int_{\Omega} \mathbf{v}\cdot \operatorname{\bf grad}
\varphi \,d\mathbf{x}=-\int_{\Omega}
\Big(v_1\frac{\partial \varphi}{\partial x_1}
+v_2\frac{\partial \varphi}{\partial x_2}
+v_3\frac{\partial \varphi}{\partial x_3} \Big)\,d\mathbf{x}.
$$
We note that the vector-valued Laplace operator of a vector field
$\mathbf{v}=(v_1,\,v_2,\,v_3)$ is equivalently defined by
\begin{equation}
\Delta \mathbf{v}=\operatorname{\bf grad}
(\operatorname{div}\mathbf{v})-\operatorname{\bf curl}\operatorname{\bf curl}
\mathbf{v}.
\end{equation}

This leads to the following definitions:

\begin{definition} \label{def1.1} \rm
For all integers $k \in \mathbb{Z}$, we define the space
\begin{equation*}
\mathbf{H}_{k}^2(\operatorname{\bf curl},\Omega)
=\{ \mathbf{v}\in\mathbf{W}_{k}^{0,2}(\Omega);\operatorname{\bf curl}
 \mathbf{v}\in\mathbf{W}_{k+1}^{0,2}(\Omega)\} \,,
\end{equation*}
 with the norm
$$
\|\mathbf{v}\|_{\mathbf{H}_{k}^2(\operatorname{\bf curl},\Omega)}
=\left( {\|\mathbf{v}\|^2}_{\mathbf{W}_{k}^{0,2}(\Omega)}
+{\|\operatorname{\bf curl}\, \mathbf{v}\|^2}_{\mathbf{W}_{k+1}^{0,2}(\Omega)}
\right) ^{1/2}.
 $$
Also we define the space
\begin{equation*}
 \mathbf{H}_{k}^2(\operatorname{div},\Omega)
=\{ \mathbf{v}\in\mathbf{W}_{k}^{0,2}(\Omega);\operatorname{div}
 \mathbf{v}\in W_{k+1}^{0,2}(\Omega)\} \,,
\end{equation*}
with the norm
$$
\|\mathbf{v}\|_{\mathbf{H}_{k}^2(\operatorname{div},\Omega)}
=\left( \|\mathbf{v}\|^2_{\mathbf{W}_{k}^{0,2}(\Omega)}
+\|\operatorname{div} \,\, \mathbf{v}\|^2_{{W}_{k+1}^{0,2}(\Omega)}\right)^{1/2}.
 $$
Finally, we set
\begin{equation*}
 \mathbf{X}_{k}^2(\Omega)=\mathbf{H}_{k}^2(\operatorname{\bf curl},\Omega)\cap\mathbf{H}_{k}^2(\operatorname{div},\Omega).
\end{equation*}
with the norm
$$
\mathbf{X}_{k}^2(\Omega)
=\left( \|\mathbf{v}\|^2_{\mathbf{W}_{k}^{0,2}(\Omega)}
+\|\operatorname{div} \mathbf{v}\|^2_{{W}_{k+1}^{0,2}(\Omega)}
+\|\operatorname{\bf curl} \mathbf{v}\|^2_{\mathbf{W}_{k+1}^{0,2}(\Omega)}\right)
^{1/2}.
$$
These definitions will also be used with $\Omega$ replaced by ${\mathbb{R}}^{3}$.
\end{definition}

The argument used by Hanouzet \cite{Hanouzet} to prove the denseness of
 $\mathcal{D}(\overline{\Omega})$ in $W_{k}^{m,2}(\Omega)$ can be easily
adapted to establish that $\boldsymbol{\mathcal{D}}(\overline{\Omega})$
is dense in the space $\mathbf{H}_{k}^2(\operatorname{div},\Omega)$ and in
the space $\mathbf{H}_{k}^2(\operatorname{\bf curl},\Omega)$ and so in $\mathbf{X}_{k}^2(\Omega)$.
Therefore, denoting by $\mathbf{n}$ the exterior unit normal to the boundary
 $\Gamma$, the normal trace $\mathbf{v}\cdot \mathbf{n}$ and the tangential
trace $\mathbf{v} \times \mathbf{n}$ can be defined respectively in
$H^{-1/2}(\Gamma)$ for the functions of
$\mathbf{H}_{k}^2(\operatorname{div},\Omega)$ and in
$\mathbf{H}^{-1/2}(\Gamma)$ for functions of
$\mathbf{H}_{k}^2(\operatorname{\bf curl},\Omega)$, where $H^{-1/2}(\Gamma)$ denotes
the dual space of $H^{1/2}(\Gamma)$. They satisfy the trace theorems;
 i.e, there exists a constant $C$ such that
\begin{gather}\label{theorem de trace de H div}
 \forall \mathbf{v} \in \mathbf{H}_{k}^2(\operatorname{div},\Omega),\quad
 \|\mathbf{v}\cdot \mathbf{n}\|_{ H^{-1/2}(\Gamma)}
\leq C\|\mathbf{v}\|_{\mathbf{H}_{k}^2(\operatorname{div},\Omega)},
\\ \label{theorem de trace de H curl}
 \forall \mathbf{v} \in \mathbf{H}_{k}^2(\operatorname{\bf curl},\Omega),\quad
 \|\mathbf{v}\times \mathbf{n}\|_{ \mathbf{H}^{-1/2}(\Gamma)}
\leq  C \|\mathbf{v}\|_{\mathbf{H}_{k}^2(\operatorname{\bf curl},\Omega)}
\end{gather}
and the following Green's formulas holds:
For any $\mathbf{v} \in \mathbf{H}_{k}^2(\operatorname{div},\Omega)$ and
$\varphi \in W_{-k}^{1,2}(\Omega)$
\begin{equation}\label{formule de green v.n}
\langle \mathbf{v}\cdot\mathbf{n}, \varphi\rangle_{\Gamma}
=\int_{\Omega}\mathbf{v}\cdot\nabla \varphi\,dx
+\int_{\Omega}\varphi \operatorname{div}\mathbf{v}\,dx,
\end{equation}
where $\langle,\rangle_{\Gamma}$ denotes the duality pairing between
 $H^{-1/2}(\Gamma)$ and $H^{1/2}(\Gamma)$.
For any $\mathbf{v} \in \mathbf{H}_{k}^2(\operatorname{\bf curl},\Omega)$
and $\boldsymbol{\varphi} \in \mathbf{W}_{-k}^{1,2}(\Omega)$
\begin{equation}\label{formule de green v*n}
\langle \mathbf{v}\times\mathbf{n}, \boldsymbol{\varphi}\rangle_{\Gamma}
=\int_{\Omega}\mathbf{v}\cdot\operatorname{\bf curl}\boldsymbol{\varphi}\,dx
-\int_{\Omega}\operatorname{\bf curl}\mathbf{v}\cdot\boldsymbol{\varphi}dx,
\end{equation}
where $\langle,\rangle_{\Gamma}$ denotes the duality pairing between
$\mathbf{H}^{-1/2}(\Gamma)$ and $\mathbf{H}^{1/2}(\Gamma)$.

\begin{remark} \rm
If $\mathbf{v}$ belongs to $\mathbf{H}_{k}^2(\operatorname{div},\Omega)$
for some integer $k\geq1$, then $\operatorname{div}\mathbf{v}$ is in
$L^{1}(\Omega)$ and Green's formula \eqref{formule de green v.n} yields
\begin{equation}\label{formule de la divergence et integrale}
 \langle \mathbf{v}\cdot\mathbf{n},1\rangle_{\Gamma}
=\int_{\Omega}\operatorname{div}\mathbf{v}\,dx
\end{equation}
But when $k\leq 0$, then $\operatorname{div}\mathbf{v}$ is not necessarily
in $L^{1}(\Omega)$ and \eqref{formule de la divergence et integrale}
is generally not valid. Note also that when $k\leq 0$,
$ W^{0,2}_{-k-1}(\Omega)$ does not contain the constants.
\end{remark}

The closures of $\boldsymbol{\mathcal{D}}(\Omega)$ in
$\mathbf{H}_{k}^2(\operatorname{div},\Omega)$ and in
$\mathbf{H}_{k}^2(\operatorname{\bf curl},\Omega)$
are denoted respectively by
$\mathring{\mathbf{H}}_{k}^2(\operatorname{\bf curl},\Omega)$ and
$\mathring{\mathbf{H}}_{k}^2(\operatorname{div},\Omega)$
and can be characterized respectively by
\begin{gather*}
 \mathring{\mathbf{H}}_{k}^2(\operatorname{\bf curl},\Omega)
=\{ \mathbf{v}\in  \mathbf{H}_{k}^2(\operatorname{\bf curl},\Omega):
\mathbf{v}\times \mathbf{n}=\mathbf{0} \text{ on } \Gamma\},
\\
 \mathring{\mathbf{H}}_{k}^2(\operatorname{div},\Omega)
=\{ \mathbf{v}\in \mathbf{H}_{k}^2(\operatorname{div},\Omega):
\mathbf{v}\cdot\mathbf{n}=0 \text{ on }\Gamma\}.
\end{gather*}
Their dual spaces are characterized by the following propositions:

\begin{proposition}\label{caracterisation dual de H^p_0(div)}
 A distribution $\mathbf{f}$ belongs to
$[\mathring{\mathbf{H}}_{k}^2(\operatorname{div},\Omega)]'$
if and only if there exist $\boldsymbol{\psi}\in\mathbf{W}_{-k}^{0,2}(\Omega)$
and $\chi\in W_{-k-1}^{0,2}(\Omega)$, such that
$\mathbf{f}=\boldsymbol{\psi}+\operatorname{\bf grad} \chi$.
Moreover
\begin{equation}\label{estimation d'un element dans le dual de H^2_0(div)}
\|\mathbf{f} \|_{[\mathring{\mathbf{H}}_{k}^{2}(\operatorname{div},\Omega)]'}
=\max \{\|\boldsymbol{\psi}\|_{\mathbf{W}_{-k}^{0,2}(\Omega)},\,
\|\chi\|_{W_{-k-1}^{0,2}(\Omega)}\} .
\end{equation}
\end{proposition}
\begin{proof}
Let $\boldsymbol{\psi}\in\mathbf{W}_{-k}^{0,2}(\Omega)$ and
$\chi\in W_{-k-1}^{0,2}(\Omega)$, we have
\begin{equation*}
 \forall \mathbf{v}\in{\boldsymbol{\mathcal{D}}}(\Omega),\quad
\langle\boldsymbol{\psi}+\operatorname{\bf grad } \chi,
\mathbf{v}\rangle_{{\boldsymbol{\mathcal{D}}}'(\Omega)
\times{\boldsymbol{\mathcal{D}}}(\Omega)}=\int_{\Omega}
(\boldsymbol{\psi}\cdot\mathbf{v}-\chi\,\operatorname{div}\mathbf{v})
\,\text{d}\mathbf{x}.
\end{equation*}
Therefore, the linear mapping
$\boldsymbol{\ell}: \mathbf{v}\,\longmapsto \,\int_{\Omega}
(\boldsymbol{\psi}\cdot\mathbf{v}-\chi\,\operatorname{div}\mathbf{v})
\text{d}\mathbf{x}$ defined on $\boldsymbol{\mathcal{D}}(\Omega)$
is continuous for the norm of
$\mathring{\mathbf{H}}^2_{k}(\operatorname{div},\Omega)$.
Since $\boldsymbol{\mathcal{D}}(\Omega)$ is dense in
$\mathring{\mathbf{H}}^2_{k}(\operatorname{div},\,\Omega)$,
$\boldsymbol{\ell}$ can be extended by continuity to a mapping still
called $\boldsymbol{\ell} \in
 [\mathring{\mathbf{H}}^2_{k}(\operatorname{div},\,\Omega)]'$.
Thus $\boldsymbol{\psi}+\operatorname{\bf grad} \chi$ is an element of
$[\mathring{\mathbf{H}}^2_{k}(\operatorname{div},\Omega)]'$.

Conversely, Let $E=\mathbf{W}_{k}^{0,2}(\Omega)\times W_{k+1}^{0,2}(\Omega)$
equipped by the following norm
 $$
\|\mathbf{v}\|_{E}=(\|\mathbf{v}\|^2_{\mathbf{W}_{k}^{0,2}(\Omega)}
+\|\operatorname{div}\mathbf{v}\|^2_{W_{k+1}^{0,2}(\Omega)})^{1/2}.
$$
The mapping $T: \mathbf{v} \in \mathring{\mathbf{H}}^{2}_{k}
(\operatorname{div},\Omega)\to(\mathbf{v},
\operatorname{div}\mathbf{v}) \in E$ is an isometry from
$\mathring{\mathbf{H}}^2_{k}(\operatorname{div},\Omega)$ in $E$.
Suppose $G=T(\mathring{\mathbf{H}}^2_{k}(\operatorname{div},\Omega))$
with the $E$-topology. Let $S=T^{-1}:G\to
 \mathring{\mathbf{H}}^2_{k}(\operatorname{div},\Omega)$.
Thus, we can define the following mapping:
\begin{equation*}
\mathbf{v}\in G\mapsto \langle \mathbf{f},S\mathbf{v}
\rangle _{[\mathring{\mathbf{H}}^2_{k}(\operatorname{div},\Omega)]'
\times\mathring{\mathbf{H}}^2_{k}(\operatorname{div},\Omega) }
\quad\text{for } \mathbf{f} \in [\mathring{\mathbf{H}}^2_{k}
(\operatorname{div},\Omega)]'
\end{equation*}
which is a linear continuous form on $G$. Thanks to Hahn-Banach's Theorem,
such form can be extended to a linear continuous form on $E$,
 denoted by $\Upsilon $ such that
\begin{equation}\label{estimation de Riesz}
\|\Upsilon\|_{E'}=\|\mathbf{f}\|_{[\mathring{\mathbf{H}}^2_{k}
(\operatorname{div},\Omega)]'}.
\end{equation}
 From the Riesz's Representation Lemma, there exist functions
$\boldsymbol{\psi}\in\mathbf{W}_{-k}^{0,2}(\Omega)$ and
 $\chi\in W_{-k-1}^{0,2}(\Omega)$, such that for any
$\mathbf{v}=(\mathbf{v}_1,v_2)\in E$,
\begin{equation*}
\langle \Upsilon, \mathbf{v} \rangle _{E'\times E}
=\int_{\Omega}\,\mathbf{v}_1\cdot \boldsymbol{\psi}\,dx
+\int_{\Omega}v_2 \chi\,dx,
\end{equation*}
with
$\|\Upsilon\|_{E'}=\text{max}\{ \|\boldsymbol{\psi}\|_{\mathbf{W}_{-k}^{0,2}
(\Omega)},\|\chi\|_{W_{-k-1}^{0,2}(\Omega)}\} $.
In particular,  if $\mathbf{v}=T\boldsymbol{\varphi}\in G$, where
$\boldsymbol{\varphi} \in \boldsymbol{\mathcal{D}}(\Omega)$, we have
\begin{equation*}
 \langle \mathbf{f},\boldsymbol{\varphi}\rangle _{[\mathring{\mathbf{H}}^2_{k}
(\operatorname{div},\,\Omega)]'\times\mathring{\mathbf{H}}^2_{k}
(\operatorname{div},\,\Omega) }=\langle \boldsymbol{\psi}
-\nabla \chi,\boldsymbol{\varphi}\rangle _{[\mathring{\mathbf{H}}^2_{k}
(\operatorname{div},\,\Omega)]'\times\mathring{\mathbf{H}}^2_{k}
(\operatorname{div},\,\Omega) },
\end{equation*}
and \eqref{estimation d'un element dans le dual de H^2_0(div)}
follows imeddiatly from \eqref{estimation de Riesz}.
\end{proof}

We skip the proof of the following result as it is similar to that
 of Proposition \ref{caracterisation dual de H^p_0(div)}.

\begin{proposition}\label{caraterisation dual de H^p_0(curl)}
A distribution $\mathbf{f}$ belongs to
$[\mathring{\mathbf{H}}_{k}^2(\operatorname{\bf curl},\Omega)]'$
if and only if there exist functions
$\boldsymbol{\psi}\in\mathbf{W}_{-k}^{0,2}(\Omega)$ and
$\boldsymbol{\xi}\in \mathbf{W}_{-k-1}^{0,2}(\Omega)$, such that
$\mathbf{f}=\boldsymbol{\psi}+\operatorname{\bf curl}\boldsymbol{\xi}$.
Moreover
\begin{equation*}%\label{estimation d'un element dans le dual de H^2_0(curl)}
\|\mathbf{f}\|_{[\mathring{\mathbf{H}}_{k}^2(\operatorname{\bf curl},\Omega)]'}
=\max\{  \|\boldsymbol{\psi}\|_{\mathbf{W}_{-k}^{0,2}(\Omega)},
\|\boldsymbol{\xi}\|_{\mathbf{W}_{-k-1}^{0,2}(\Omega)}\}  .
\end{equation*}
\end{proposition}

\begin{definition} \label{def1.5} \rm
Let $\mathbf{X}^2_{k,N}(\Omega)$, $\mathbf{X}^2_{k,T}(\Omega)$ and
$\mathring{\mathbf{X}}^2_{k}(\Omega)$ be the following subspaces of
$\mathbf{X}^2_{k}(\Omega)$:
\begin{gather*}
 \mathbf{X}^2_{k,N}(\Omega)=\{ \mathbf{v}\in
 \mathbf{X}^2_{k}(\Omega);\,\mathbf{v}\times \mathbf{n}=\mathbf{0} \text{ on }
 \Gamma\}, \\
 \mathbf{X}^2_{k,T}(\Omega)=\{ \mathbf{v}\in
 \mathbf{X}^2_{k}(\Omega);\mathbf{v}\cdot\mathbf{n}=0\text{ on }\Gamma\},
\\
\mathring{\mathbf{X}}^2_{k}(\Omega)=\mathbf{X}^2_{k,N}(\Omega)
\cap\mathbf{X}^2_{k,T}(\Omega).
\end{gather*}
\end{definition}

\section{Preliminary results}

Now, we give some results related to the Dirichlet problem and Neumann problem
which are essential to ensure the existence and the uniqueness of some
vectors potentials and one usually forces either the normal component
to vanish or the tangential components to vanish. We start by giving
the definition of the kernel of the Laplace operator for any integer $k\in \mathbb{Z}$:
\begin{equation*}
\mathcal{A}_{k-1}^{\Delta}=\{ \chi \in W_{-k}^{1,2}(\Omega):
\Delta \chi=0\text{ in $\Omega$ and $\chi= 0$ on } \Gamma\}.
\end{equation*}
In contrast to a bounded domain, the Dirichlet problem for the Laplace
operator with zero data can have nontrivial solutions in an exterior domain;
it depends upon the exponent of the weight. The result that we state
below is established by Giroire in \cite{these Giroire}.

\begin{proposition}\label{noyau de dirichlet vivet}
For any integer $k\geq 1$, the space $\mathcal{A}_{k-1}^{\Delta}$
is a subspace of all functions  in $W_{-k}^{1,2}(\Omega)$ of the form
$v(p)-p$, where $p$ runs over all polynomials of $\mathcal{P}_{k-1}^{\Delta}$
and $v(p)$ is the unique solution in $W_{0}^{1,2}(\Omega)$ of the Dirichlet
problem
\begin{equation}\label{problem de Dirichlet liee au noyeau}
\Delta v(p)=0\quad\text{in}\quad \Omega\quad\text{and}\quad v(p)
= p\quad\text{on } \Gamma.
\end{equation}
The space $\mathcal{A}_{k-1}^{\Delta}$ is a finite-dimentional space of the same
dimension as $\mathcal{P}_{k-1}^{\Delta}$ and
$\mathcal{A}_{k-1}^{\Delta}=\{ 0\} $ when $k\leq0$.
\end{proposition}

Our second proposition is established also by Giroire in \cite{these Giroire},
it characterizes the kernel of the Laplace operator with Neumann boundary
condition. For any integer $k\in \mathbb{Z}$,
\begin{equation*}
\mathcal{N}_{k-1}^{\Delta}=\{ \chi \in W_{-k}^{1,2}(\Omega):
\Delta \chi=0\text{ in $\Omega$ and $\frac{\partial \chi}{\partial \mathbf{n}}= 0$
on } \Gamma\}.
\end{equation*}

\begin{proposition}\label{noyau de neumann vivet}
For any integer $k\geq1$, $\mathcal{N}_{k-1}^{\Delta}$ the subspace of all
functions in $W_{-k}^{1,2}(\Omega)$ of the form $w(p)-p$, where $p$ runs
over all polynomials of $\mathcal{P}_{k-1}^{\Delta}$ and $w(p)$ is
 the unique solution in $W_{0}^{1,2}(\Omega)$ of the Neumann problem
\begin{equation}\label{problem de Neumann liee au noyeau}
\Delta w(p)=0\quad\text{in } \Omega\quad\text{and}\quad
\frac{\partial w(p)}{\partial \mathbf{n}}=\frac{\partial p}{\partial \mathbf{n}}
\quad\text{on } \Gamma.
\end{equation}
Here also, we set $\mathcal{N}_{k-1}^{\Delta}=\{ 0\} $ when $k\leq0$;
$\mathcal{N}_{k-1}^{\Delta}$ is a finite-dimentional space of the same
dimension as $\mathcal{P}_{k-1}^{\Delta}$ and in particular,
$\mathcal{N}_{0}^{\Delta}=\mathbb{R}$.
\end{proposition}

Next, the uniqueness of the solutions of Problem \eqref{eST} and
Problem \eqref{eSN} will follow from the characterization of the kernel.
 For all integers $k$ in $\mathbb{Z}$, we define
\begin{gather*}
\mathbf{Y}_{k,N}^2(\Omega)=\{ \mathbf{w} \in \mathbf{X}^2_{-k,N}(\Omega):
\operatorname{div}\mathbf{w}=0 \text{ and }
 \operatorname{\bf curl}\mathbf{w}=\mathbf{0} \text{ in }\Omega\}
\\
\mathbf{Y}_{k,T}^2(\Omega)=\{ \mathbf{w} \in \mathbf{X}^2_{-k,T}(\Omega):
\operatorname{div}\mathbf{w}=0 \text{ and } \operatorname{\bf curl}
\mathbf{w}=\mathbf{0}\text{ in }\Omega\} .
\end{gather*}

The proof of the following propositions can be easily deduced
from \cite{vecteur potentiel vivet}.

\begin{proposition}\label{Caracterisation du kernel vectoriel n=0}
Let $k\in \mathbb{Z}$ and suppose that  $\Omega'$ is of class $C^{1,1}$,
simply-connected and with a Lipshitz-continuous and connected boundary
$\Gamma$.
\begin{itemize}
\item If $k< 1$, then $\mathbf{Y}_{k,N}^2(\Omega)=\{ \mathbf{0}\} $.
\item If $k\geq 1$, then $\mathbf{Y}_{k,N}^2(\Omega)= \{ \nabla(v(p)-p),\,
p \in \mathcal{P}^{\Delta}_{k-1}\} $, where $v(p)$ is the unique solution
in $W_{0}^{1,2}(\Omega)$ of the Dirichlet problem
\eqref{problem de Dirichlet liee au noyeau}.
\end{itemize}
\end{proposition}

\begin{proof}
Let $k\in \mathbb{Z}$ and let $\mathbf{w} \in \mathbf{X}^2_{-k,N}(\Omega)$ such that
 $\operatorname{div}\mathbf{w}=0$ and
$ \operatorname{\bf curl}\mathbf{w}=\mathbf{0}$ in $\Omega$.
Then since $\Omega'$ is simply-connected, there exists
$\chi \in W_{-k}^{1,2}(\Omega)$, unique up to an additive constant,
such that $\mathbf{w}=\nabla \chi$. But $\mathbf{w}\times \mathbf{n}=\mathbf{0}$,
hence, $\chi$ is constant on $\Gamma$ ($\Gamma $ is a connected boundary)
and we choose the additive constant in $\chi$ so that $\chi=0$ on $\Gamma$.
Thus $\chi$ belongs to  $\mathcal{A}_{k-1}^{\Delta}(\Omega)$ .
Due to Proposition \ref{noyau de dirichlet vivet}, we deduce that if
 $k<1$, $\chi$ is equal to zero and if $k\geq 1$, $\chi=v(p)-p$, where
$p$ runs over all polynomials of $\mathcal{P}^{\Delta}_{k-1}$ and $v(p)$
is the unique solution in $W_{0}^{1,2}(\Omega)$ of problem
\eqref{problem de Dirichlet liee au noyeau} and thus
$\mathbf{w}=\nabla(v(p)-p)$. Now, to finish the proof we shall prove that
$\nabla(v(p)-p)$ belongs to $\mathbf{Y}_{k,N}^2(\Omega)$ and this is a simple
consequence of the definition of  $p$ and $v(p)$.
\end{proof}

We skip the proof of the following result as it is entirely similar
to that of Proposition \ref{Caracterisation du kernel vectoriel n=0}.

\begin{proposition}\label{Caracterisation du kernel scalaire n=0}
Let the assumptions of Proposition \ref{Caracterisation du kernel scalaire n=0}
hold.
\begin{itemize}
\item If $k< 1$, then $\mathbf{Y}_{k,T}^2(\Omega)=\{ \mathbf{0}\}$.
\item If $k\geq 1$, then $\mathbf{Y}_{k,T}^2(\Omega)= \{ \nabla(w(p)-p),\,
p \in \mathcal{P}^{\Delta}_{k-1}\} $, where $w(p)$ is the unique solution in
$W_{0}^{1,2}(\Omega)$ of the Neumann problem
\eqref{problem de Neumann liee au noyeau}
\end{itemize}
\end{proposition}

The imbedding results that we state below are established by
 Girault in \cite{vecteur potentiel vivet}. The first imbedding result is
given by the following theorem.

\begin{theorem}\label{injection in X_T  homogene}
Let $k\leqslant 2$ and assume that $\Omega'$ is of class $C^{1,1}$.
Then the space $\mathbf{X}^2_{k-1,T}(\Omega)$ is continuously imbedded
in $\mathbf{W}_{k}^{1,2}(\Omega)$. In addition there exists a constant $C$
such that for any $\boldsymbol{\varphi} \in \mathbf{X}^2_{k-1,T}(\Omega)$,
\begin{equation}\label{inegalite injection dans X_T 2}
 \|\boldsymbol{\varphi}\|_{\mathbf{W}_{k}^{1,2}(\Omega)}
\leq C \Big(  \|\boldsymbol{\varphi}\|_{\mathbf{W}_{k-1}^{0,2}(\Omega)}
+\|\operatorname{div}\boldsymbol{\varphi}\|_{\mathbf{W}_{k}^{0,2}(\Omega)}
+\|\operatorname{\bf curl}\boldsymbol{\varphi} \|_{\mathbf{W}_{k}^{0,2}(\Omega)}
\Big) .
\end{equation}
If in addition, $\Omega'$ is simply-connected, there exists a constant $C$
such that for all $\boldsymbol{\varphi} \in \mathbf{X}^2_{k-1,T}(\Omega)$
we have
\begin{equation}\label{equivalence de norme avec curl X_T 2}
\begin{aligned}
\|\boldsymbol{\varphi}\|_{\mathbf{W}_{k}^{1,2}(\Omega)}
&\leq C (\|\operatorname{div}\boldsymbol{\varphi}\|_{\mathbf{W}_{k}^{0,2}(\Omega)}
+\|\operatorname{\bf curl}\boldsymbol{\varphi} \|_{\mathbf{W}_{k}^{0,2}(\Omega)}
\\
&\quad + \sum_{j=2}^{N(-k)}|\int_{\Gamma}\boldsymbol{\varphi} \cdot \nabla
w(q_{j})\,d\sigma|),
\end{aligned}
\end{equation}
where $\{ q_{j}\} _{j=2}^{N(-k)}$ denotes a basis of
$\{ q \in \mathcal{P}_{-k}^{\Delta}:\,\,q(\mathbf{0})=0\} $,
$N(-k)$ denotes the dimension of $\mathcal{P}_{-k}^{\Delta}$ and $w(q_{j})$
is the corresponding function of  $\mathcal{N}_{-k}^{\Delta}$.
Thus, the seminorm in the right-hand side of
\eqref{equivalence de norme avec curl X_T 2} is a norm on
$\mathbf{X}^2_{k-1,T}(\Omega)$ equivalent to the norm
$\|\boldsymbol{\varphi}\|_{\mathbf{W}_{k}^{1,2}(\Omega)}$.
\end{theorem}

The second imbedding result is given by the following theorem.

\begin{theorem}\label{injection in X_N  homogene}
Let $k\leqslant 2$ and assume that $\Omega'$ is of class $C^{1,1}$.
Then the space $\mathbf{X}^2_{k-1,N}(\Omega)$ is continuously imbedded
in $\mathbf{W}_{k}^{1,2}(\Omega)$. In addition there exists a constant
 $C$ such that for any $\boldsymbol{\varphi} \in \mathbf{X}^2_{k-1,N}(\Omega)$,
\begin{equation}\label{inegalite  injection X_N}
 \|\boldsymbol{\varphi}\|_{\mathbf{W}_{k}^{1,2}(\Omega)}
\leq C \Big(  \|\boldsymbol{\varphi}\|_{\mathbf{W}_{k-1}^{0,2}(\Omega)}
+\|\operatorname{div}\boldsymbol{\varphi}\|_{\mathbf{W}_{k}^{0,2}(\Omega)}
+\|\operatorname{\bf curl}\boldsymbol{\varphi} \|_{\mathbf{W}_{k}^{0,2}(\Omega)}
\Big) .
\end{equation}
If in addition, $\Omega'$ is simply-connected and its boundary $\Gamma$
is connected, there exists a constant $C$ such that for all
$\boldsymbol{\varphi} \in \mathbf{X}^2_{k-1,N}(\Omega)$ we have
\begin{equation}\label{equivalence de norme avec curl X_N 2}
\begin{aligned}
\|\boldsymbol{\varphi}\|_{\mathbf{W}_{k}^{1,2}(\Omega)}
&\leq  C (\|\operatorname{div}\boldsymbol{\varphi}\|_{\mathbf{W}_{k}^{0,2}(\Omega)}
+\|\operatorname{\bf curl}\boldsymbol{\varphi} \|_{\mathbf{W}_{k}^{0,2}(\Omega)}
\\
&\quad + |\int_{\Gamma}(\boldsymbol{\varphi} \cdot \mathbf{n})d\sigma|
+\sum_{j=1}^{N(-k)}|\int_{\Gamma}(\boldsymbol{\varphi}
\cdot \mathbf{n})q_{j}\,d\sigma|),
\end{aligned}
\end{equation}
where the term $|\int_{\Gamma}(\boldsymbol{\varphi} \cdot \mathbf{n})d\sigma|$
can be dropped if $k\neq1$ and where $\{ q_{j}\} _{j=1}^{N(-k)}$ denotes
a basis of $\mathcal{P}_{-k}^{\Delta}$. In other words, the seminorm in
the right-hand side of \eqref{equivalence de norme avec curl X_N 2} is a norm
on $\mathbf{X}^2_{k-1,N}(\Omega)$ equivalent to the norm
$\|\boldsymbol{\varphi}\|_{\mathbf{W}_{k}^{1,2}(\Omega)}$.
\end{theorem}

Finally, let us recall the abstract setting of Babu\v{s}ka-Brezzi's
Theorem (see  Babu\v{s}ka \cite{Babuska}, Brezzi \cite{Brezzi}
and Amrouche-Selloula \cite{Nour1}).

\begin{theorem}\label{Babuchca brezi}
 Let $X$ and $M$ be two reflexive Banach spaces and $X'$ and $M'$ their
dual spaces. Let $a$ be the continuous bilinear form defined on
$X\times M$, let $A\in \mathcal{L}(X;\ M')$ and  $A'\in \mathcal{L}(M;\ X')$
be the operators defined by
$$
\forall v\in X,\; \forall w\in M, \quad a(v, w) = \langle Av, w\rangle
=\langle v, A'w\rangle
$$
and $V=\ker A$.
The following statements are equivalent:
\begin{itemize}
\item[(i)] There exist $\beta > 0$ such that
\begin{equation}\label{CISabstraite}
\inf_{w\in M,\,w\neq 0} \,\sup_{v\in X,\, v\neq 0}
\frac{a(v, w)}{\| v\|_X\|  w\|_M}\geq\beta.
\end{equation}

\item[(ii)] The operator $A: X/V\mapsto M'$ is an isomophism and
$1/\beta$ is the continuity constant of $A^{-1}$.

\item[(iii)]  The operator $A': M\mapsto X'\bot V$ is an isomophism and
$1/\beta$ is the continuity constant of $(A')^{-1}$.
\end{itemize}
\end{theorem}

\begin{remark}\label{remark sur c inf sup}{\rm
As consequence, if the inf-sup condition \eqref{CISabstraite}
is satisfied, then we have the following properties:
\begin{itemize}
\item[(i)] If $V = \{0\}$, then for any $f\in X'$, there exists a unique
$ w\in M$ such that
\begin{equation}\label{inegalite util ds inf-sup}
\forall v\in X,\;  a(v, w) = \langle f, v\rangle \quad \text{and}\quad
\| w\|_{M} \leq \frac{1}{\beta} \| f\|_{X'}.
\end{equation}

\item[(ii)] If $V \neq  \{0\}$, then for any $f\in X'$, satisfying the
compatibility condition:
$\forall v\in V,\;\langle  f, v\rangle  = 0$, there exists a unique
$ w\in M$ such that \eqref{inegalite util ds inf-sup}.

\item[(iii)] For any $g\in M'$, there exists $v\in X$, unique up an
additive element of $V$, such that:
$$
\forall w\in M,\;  a(v, w) =  \langle g, w\rangle \quad \text{and}\quad
\| v\|_{X/V} \leq \frac{1}{\beta} \| g\|_{M'}.
$$
\end{itemize}
}\end{remark}



\section{Inequalities and inf-sup conditions}

In this sequel, we prove some imbedding results. More precisely,
we show that the results of Theorem \ref{injection in X_T  homogene}
and the result of  Theorem \ref{injection in X_N  homogene} can be
extended to the case where the boundary conditions
$\mathbf{v}\cdot\mathbf{n}=0$ or $\mathbf{v}\times\mathbf{n}=\mathbf{0}$ on
$\Gamma$ are replaced by inhomogeneous one. Next, we study some problems
posed in an exterior domain which are essentials to prove the regularity
of solutions for  Problem \eqref{eST} and Problem \eqref{eSN}.

 For any integer $k$ in $\mathbb{Z}$, we introduce the following spaces:
\begin{gather*}
\mathbf{Z}_{k,T}^2(\Omega)
=\{ \mathbf{v}\in \mathbf{X}^2_{k}(\Omega)\text{ and }
\mathbf{v}\cdot\mathbf{n}\in H^{1/2}(\Gamma)\} ,\\
\mathbf{Z}_{k,N}^2(\Omega)=\{ \mathbf{v}\in \mathbf{X}_{k}^2(\Omega)\text{ and }
\mathbf{v}\times\mathbf{n}\in \mathbf{H}^{1/2}(\Gamma)\}
\end{gather*}
and
\begin{align*}
\mathbf{M}_{k,T}^2(\Omega)
=\{&\mathbf{v}\in \mathbf{W}_{k+1}^{1,2}(\Omega),\;
  \operatorname{div}\mathbf{v}\in W_{k+2}^{1,2}(\Omega),
\;\operatorname{\bf curl}\mathbf{v}\in \mathbf{W}_{k+2}^{1,2}(\Omega)\\
&\text{and } \mathbf{v}\cdot \mathbf{n}\in H^{3/2}(\Gamma)\} .
\end{align*}

\begin{proposition}\label{injection in X_T non homogene}
Let $k=-1$ or $k=0$, then the space $\mathbf{Z}_{k,T}^2(\Omega)$
is continuously imbedded in $\mathbf{W}_{k+1}^{1,2}(\Omega)$
and we have the following estimate for any $\mathbf{v}$ in
$\mathbf{Z}_{k,T}^2(\Omega)$:
\begin{equation}\label{inequality Z_T}
\|{\mathbf{v}}\|_{\mathbf{W}_{k+1}^{1,2}(\Omega)}
\leq  C\big(\|\mathbf{v}\|_{\mathbf{W}_{k}^{0,2}(\Omega)}
 + \|\operatorname{\bf curl}{\mathbf{v}}\|_{\mathbf{W}_{k+1}^{0,2}(\Omega)}
 +\| \operatorname{div}{\mathbf{v}}\|_{{W}_{k+1}^{0,2}(\Omega)}
 +\|\mathbf{v}\cdot\mathbf{n}\|_{ H^{1/2}(\Gamma)}\big).
\end{equation}
\end{proposition}

\begin{proof}
 Let $k=-1$ or $k=0$ and let $\mathbf{v}$ any function of
 $\mathbf{Z}_{k,T}^2(\Omega)$. Let us study the  Neumann problem
\begin{equation}\label{problem de neumann avec v.n}
\Delta{\chi}=\operatorname{div} {\mathbf{v}} \text{ in }\Omega
\quad\text{and}\quad
\partial_{n}{\chi}=\mathbf{v}\cdot\mathbf{n}\text{ on }\Gamma.
\end{equation}
It is shown in \cite[Theorems 3.7 and  3.9]{vecteur potentiel vivet},
that Problem \eqref{problem de neumann avec v.n} has a unique solution
$\chi$ in ${W}_{k+1}^{2,2}(\Omega)/\mathbb{R}$ if $k=-1$ and $\chi$ is unique
in ${W}_{k+1}^{2,2}(\Omega)$ if $k=0$. With the estimate
\begin{equation}\label{estimation chi solution du probleme de Neumann}
  \| \nabla \chi\|_{W_{k+1}^{1,2}(\Omega)}
\leq C\big( \| \operatorname{div}\mathbf{v}\|_{W_{k+1}^{0,2}(\Omega)}
+\| \mathbf{v}\cdot\mathbf{n}\|_{H^{1/2}(\Gamma)}\big).
\end{equation}
Let $\mathbf{w}=\mathbf{v}-\operatorname{\bf grad} \chi$, then
$\mathbf{w}$ is a divergence-free function.
Since $\mathbf{W}_{k+1}^{1,2}(\Omega)\hookrightarrow
\mathbf{W}_{k}^{0,2}(\Omega)$, then $\mathbf{w} \in \mathbf{X}^2_{k,T}(\Omega)$.
 Applying Theorem \ref{injection in X_T  homogene}, we have $\mathbf{w}$
belongs to $\mathbf{W}_{k+1}^{1,2}(\Omega)$ and then $\mathbf{v}$ is in
 $\mathbf{W}_{k+1}^{1,2}(\Omega)$.
According to Inequality \eqref{inegalite injection dans X_T 2}, we obtain
\begin{equation*}
\| \mathbf{w}\|_{\mathbf{W}_{k+1}^{1,2}(\Omega)}
\leq C\big( \| \mathbf{w}\|_{\mathbf{W}_{k}^{0,2}(\Omega)}
+\| \operatorname{\bf curl}\mathbf{w}\|_{\mathbf{W}_{k+1}^{0,2}(\Omega)} \big).
\end{equation*}
Then,  inequality \eqref{inequality Z_T} follows directly from
\eqref{estimation chi solution du probleme de Neumann}.
\end{proof}

Similarly, we can prove the following imbedding result.

\begin{proposition}\label{injection in X_T non homogene fort}
Suppose that $\Omega'$ is of class $C^{2,1}$.
Then the space $\mathbf{M}_{-1,T}^2(\Omega)$ is continuously imbedded in
$\mathbf{W}_1^{2,2}(\Omega)$ and we have the following estimate for any
$\mathbf{v}$ in $\mathbf{M}_{-1,T}^2(\Omega)$:
\begin{equation}\label{inequality M_T}
\|{\mathbf{v}}\|_{\mathbf{W}_1^{2,2}(\Omega)}
\leq  C\big(\|\mathbf{v}\|_{\mathbf{W}_{0}^{1,2}(\Omega)}
+ \|\operatorname{\bf curl}{\mathbf{v}}\|_{\mathbf{W}_1^{1,2}(\Omega)}
+\| \operatorname{div}{\mathbf{v}}\|_{{W}_1^{1,2}(\Omega)}
+\|\mathbf{v}\cdot\mathbf{n}\|_{ H^{\,3/2}(\Gamma)}\big).
\end{equation}
\end{proposition}

\begin{proof}
Proceeding as in Proposition \ref{injection in X_T non homogene}.
Let $\mathbf{v}$ in $\mathbf{M}_{-1,T}^2(\Omega)$.
Since $\Omega'$ is of class $C^{2,1}$, then according to
\cite[Theorem 3.9]{vecteur potentiel vivet}, there exists a unique
solution $\chi$ in $W_1^{3,2}(\Omega)/\mathbb{R}$ of Problem
\eqref{problem de neumann avec v.n}. Setting
$\mathbf{w}=\mathbf{v}-\operatorname{\bf grad} \,\chi$.
Since $W_1^{2,2}(\Omega)$ is imbedded in $W_{0}^{1,2}(\Omega)$,
it follows from \cite[Corollary 3.16]{vecteur potentiel vivet},
that $\mathbf{w}$ belongs to $\mathbf{W}_1^{2,2}(\Omega)$ and moreover
we have the  estimate
\begin{equation*}
\| \mathbf{w}\|_{\mathbf{W}_1^{2,2}(\Omega)}
\leq C\big( \| \mathbf{w}\|_{\mathbf{W}_{0}^{1,2}(\Omega)}
+\| \operatorname{\bf curl}\mathbf{w}\|_{\mathbf{W}_1^{1,2}(\Omega)} \big).
\end{equation*}
Then $\mathbf{v}=\mathbf{w}+\operatorname{\bf grad} \chi$ belongs to
$\mathbf{W}_1^{2,2}(\Omega)$ and we have the estimate \eqref{inequality M_T}.
\end{proof}


Although we are under the Hilbertian case but the Lax-Milgram lemma is not
always valid to ensure the existence of solutions. Thus, we shall establish
two ``inf-sup'' conditions in order to apply Theorem \ref{Babuchca brezi}.
First recall the following spaces for all integers $k\in \mathbb{Z}$:
\begin{align*}
 \mathbf{V}_{k,T}^2(\Omega)
=\Big\{& \mathbf{z} \in \mathbf{X}^2_{k,T}(\Omega):
\operatorname{div}\mathbf{z}=0\text{ in }\Omega \quad\text{and}\\
&\int_{\Gamma}\mathbf{z} \cdot \nabla (w(q)-q)\,d\sigma=0, \,\forall
(w(q)-q)\in \mathcal{N}_{-k-1}^{\Delta} \Big\}
\end{align*}
and
\begin{equation*}
 \mathbf{V}_{k,N}^2(\Omega)=\{ \mathbf{z} \in \mathbf{X}^2_{k,N}(\Omega):
\operatorname{div}\mathbf{z}=0\text{ in }\Omega\text{ and }
\int_{\Gamma}(\mathbf{z} \cdot \mathbf{n})q\,d\sigma=0,\,\forall
 q \in\mathcal{P}_{-k-1}^{\Delta} \}.
\end{equation*}
The first  ``inf-sup'' condition is given by the following lemma.

\begin{lemma}\label{lemme condition inf sup 2}
The following  inf-sup Condition holds: there exists a constant $\beta>0$,
such that
\begin{equation}\label{condition inf sup 2}
 \inf_{\boldsymbol{\varphi}\in\mathbf{V}^2_{0,T}(\Omega),\,
\boldsymbol{\varphi}\neq 0}
\sup_{\boldsymbol{\psi}\in\mathbf{V}^2_{-2,T}(\Omega),\, \boldsymbol{\psi}\neq 0}
\frac{\int_{\Omega}\operatorname{\bf curl}\boldsymbol{\psi}
\cdot\operatorname{\bf curl}\boldsymbol{\varphi}\,\text{d}\mathbf{x}}
{\| \boldsymbol{\psi}\|_{\mathbf{X}^2_{-2,T}(\Omega)}\|
 \boldsymbol{\varphi}\|_{\mathbf{X}^2_{0,T}(\Omega)}}\geq\beta.
\end{equation}
\end{lemma}


\begin{proof}
Let $\mathbf{g}\in\mathbf{W}_{-1}^{0,2}(\Omega)$ and let us introduce the
 Dirichlet problem
\begin{equation*}
 -\Delta \chi=\operatorname{div}\mathbf{g}\quad\text{in } \Omega,\quad
\chi=0\quad\text{on } \Gamma.
\end{equation*}
It is shown in \cite[Theorem 3.5]{vecteur potentiel vivet},
that this problem has a solution $\chi \in \mathring{W}_{-1}^{1,2}(\Omega) $
unique up to an element of $\mathcal{A}^{\Delta}_{0}$ and we can choose $\chi$
such that
\begin{equation*}
 \|\nabla \chi \|_{\mathbf{W}_{-1}^{0,2}(\Omega)}
\leq C\|\mathbf{g}\|_{\mathbf{W}_{-1}^{0,2}(\Omega)}.
\end{equation*}
Set $\mathbf{z}= \mathbf{g}-\nabla \chi$. Then we have
$\mathbf{z} \in \mathbf{W}_{-1}^{0,2}(\Omega)$,
$\operatorname{div}\mathbf{z}=0$ and  we have
\begin{equation} \label{inegalite decomposition de helmholt 2}
 \|\mathbf{z} \|_{\mathbf{W}_{-1}^{0,2}(\Omega)}
\leq C\|\mathbf{g}\|_{\mathbf{W}_{-1}^{0,2}(\Omega)}.
\end{equation}
Let $\boldsymbol{\varphi}$ any function of $\mathbf{V}^2_{0,T}(\Omega)$,
by Theorem \ref{injection in X_T  homogene} we have
$\boldsymbol{\varphi} \in \mathbf{X}^2_{0,T}(\Omega)
\hookrightarrow \mathbf{W}_1^{1,2}(\Omega)$. Then due to
\eqref{equivalence de norme avec curl X_T 2} we can write
\begin{equation}\label{dualite 2}
 \|\boldsymbol{\varphi}\|_{\mathbf{X}^2_{0,T}(\Omega)}
\leq C\|\operatorname{\bf curl}\boldsymbol{\varphi}\|_{\mathbf{W}_1^{0,2}(\Omega)}
= C\,\sup_{\mathbf{g}\in\mathbf{W}_{-1}^{0,2}(\Omega),\,
\mathbf{g}\neq 0}
\frac{\big{\vert}\int_{\Omega}\operatorname{\bf curl}
\boldsymbol{\varphi}\cdot\mathbf{g}\,\text{d}\mathbf{x}\big{\vert}}
{\| \mathbf{g}\|_{\mathbf{W}_{-1}^{0,2}(\Omega)}}.
\end{equation}
Using the fact that
$\operatorname{\bf curl}\boldsymbol{\varphi}
\in \mathbf{H}_1^2(\operatorname{div},\Omega)$
and applying \eqref{formule de green v.n}, we obtain
\begin{equation}\label{calcul integ zero 2}
 \int_{\Omega}\operatorname{\bf curl}\boldsymbol{\varphi}\cdot\nabla \chi\,
\text{d}\mathbf{x}=0.
\end{equation}
Now, let $\lambda \in W_{0}^{1,2}(\Omega)$ the unique solution of the
 problem
\begin{equation*}
\Delta \lambda=0\quad\text{in } \Omega\quad\text{and}\quad
\lambda=1\quad\text{on } \Gamma.
\end{equation*}
It follows from \cite[Lemma 3.11]{vecteur potentiel vivet} that
\[
\int_{\Gamma}\frac{\partial \lambda}{\partial \mathbf{n}}\,d\sigma=C_1>0.
\]
 Now, setting
\begin{equation*}
\widetilde{\mathbf{z}}=\mathbf{z}-\frac{1}{C_1}\langle \mathbf{z}
\cdot\mathbf{n},1 \rangle _{\Gamma}\nabla \lambda.
\end{equation*}
It is clear that $\widetilde{\mathbf{z}}\in \mathbf{W}_{-1}^{0,2}(\Omega)$,
$\operatorname{div}\widetilde{\mathbf{z}}=0$ in $\Omega$ and that
 $\langle \widetilde{\textbf{\textit{•z}}}\cdot\mathbf{n},1 \rangle _{\Gamma}=0$.
Due to \cite[Theorem 3.15]{vecteur potentiel vivet},
there exists a potential vector
$\boldsymbol{\psi}\in \mathbf{W}_{-1}^{1,2}(\Omega)$ such that
\begin{equation}\label{conséquence vecteur potentiel 2}
 \widetilde{\mathbf{z}}=\operatorname{\bf curl}\boldsymbol{\psi},\quad
\operatorname{div}\boldsymbol{\psi}=0\quad\text{in }\Omega\quad\text{and}\quad
\boldsymbol{\psi}\cdot\mathbf{n}=0\quad\text{on }\Gamma.
\end{equation}
and we have
\begin{equation}\label{integrale nul de base de polynomes}
\forall \mathbf{v}(q)  \in \mathcal{N}_1^{\Delta},\quad
\int_{\Gamma}\boldsymbol{\psi} \cdot \nabla \mathbf{v}(q)\,d\sigma=0.
\end{equation}
In addition, we have the estimate
\begin{equation}\label{relation de norme entre z tilde et z}
\| \boldsymbol{\psi}\|_{\mathbf{W}_{-1}^{1,2}(\Omega)}
\leq C \|\widetilde{\mathbf{z}}\|_{ \mathbf{W}_{-1}^{0,2}(\Omega)}
\leq C  \|\mathbf{z}\|_{ \mathbf{W}_{-1}^{0,2}(\Omega)}.
\end{equation}
Using \eqref{integrale nul de base de polynomes}, we obtain that
$\boldsymbol{\psi}$ belongs to $\mathbf{V}^2_{-2,T}(\Omega)$.
Since $\boldsymbol{\varphi}$ is $\mathbf{H}^{1}$ in a neighborhood of $\Gamma$,
then $\boldsymbol{\varphi}$ has an $\mathbf{H}^{1}$ extension in $\Omega'$
denoted by $\widetilde{\boldsymbol{\varphi}}$. Applying Green's formula in
 $\Omega'$, we obtain
\begin{equation*}
 0=\int_{\Omega'}\operatorname{div}(\operatorname{\bf curl}
\widetilde{\boldsymbol{\varphi}})\,\text{d}\mathbf{x}
=\langle \operatorname{\bf curl}\widetilde{\boldsymbol{\varphi}}\cdot\mathbf{n},
1\rangle _{\Gamma}=\langle \operatorname{\bf curl}\boldsymbol{\varphi}\cdot
\mathbf{n},1\rangle _{\Gamma}.
\end{equation*}
Using the fact that $\operatorname{\bf curl}\boldsymbol{\varphi}$ in
$\mathbf{H}_1^2(\operatorname{div},\Omega)$ and $\lambda$ in
$W_{-1}^{1,2}(\Omega)$ and applying \eqref{formule de green v.n}, we obtain
\begin{equation}\label{calcul integrale de nabla lambda nul}
0=\langle \operatorname{\bf curl}\boldsymbol{\varphi}\cdot
\mathbf{n},1\rangle _{\Gamma}
=\langle \operatorname{\bf curl}\boldsymbol{\varphi}\cdot
\mathbf{n},\lambda\rangle _{\Gamma}=\int_{\Omega}
\operatorname{\bf curl}\boldsymbol{\varphi}\cdot\nabla \lambda\,\text{d}
\mathbf{x}.
\end{equation}
Using  \eqref{calcul integ zero 2} and
\eqref{calcul integrale de nabla lambda nul}, we deduce that
\begin{equation}\label{egalite des integrale total}
\int_{\Omega}\operatorname{\bf curl}\boldsymbol{\varphi}\cdot\mathbf{g}\,\text{d}\mathbf{x}=\int_{\Omega}\operatorname{\bf curl}\boldsymbol{\varphi}\cdot\mathbf{z}\,\text{d}\mathbf{x}=\int_{\Omega}\operatorname{\bf curl}\boldsymbol{\varphi}\cdot \widetilde{\mathbf{z}}\,\text{d}\mathbf{x}.
\end{equation}
From \eqref{relation de norme entre z tilde et z},
\eqref{inegalite decomposition de helmholt 2} and
\eqref{egalite des integrale total}, we deduce that
\begin{equation*}
\frac{\big{\vert}\int_{\Omega}\operatorname{\bf curl}\boldsymbol{\varphi}
\cdot\mathbf{g}\,\text{d}\mathbf{x}\big{\vert}}
{\| \mathbf{g}\|_{\mathbf{W}_{-1}^{0,2}(\Omega)}}
\leq  C\frac{\big{\vert}\int_{\Omega}\operatorname{\bf curl}
\boldsymbol{\varphi}\cdot \widetilde{\mathbf{z}}\,\text{d}
\mathbf{x}\big{\vert}}
{\| \widetilde{\mathbf{z}}\|_{\mathbf{W}_{-1}^{0,2}(\Omega)}}
=C\frac{\big{\vert}\int_{\Omega}\operatorname{\bf curl}
\boldsymbol{\varphi}\cdot\operatorname{\bf curl}\boldsymbol{\psi}\,
\text{d}\mathbf{x}\big{\vert}}{\| \operatorname{\bf curl}
\boldsymbol{\psi}\|_{\mathbf{W}_{-1}^{0,2}(\Omega)}}.
\end{equation*}
Applying again \eqref{equivalence de norme avec curl X_T 2} and using
\eqref{integrale nul de base de polynomes}, we obtain
\begin{equation*}
 \frac{\big{\vert}\int_{\Omega}\operatorname{\bf curl}
 \boldsymbol{\varphi}\cdot\mathbf{g}\,\text{d}\mathbf{x}\big{\vert}}{\|
 \mathbf{g}\|_{\mathbf{W}_{-1}^{0,2}(\Omega)}}
\leq  C\frac{\big{\vert}\int_{\Omega}\operatorname{\bf curl}
 \boldsymbol{\varphi}\cdot\operatorname{\bf curl}\boldsymbol{\psi}\,\text{d}
\mathbf{x}\big{\vert}}{\| \boldsymbol{\psi}\|_{\mathbf{X}_{-2,T}^2(\Omega)}},
\end{equation*}
 and the inf-sup Condition \eqref{condition inf sup 2} follows immediately
from \eqref{dualite 2}.
\end{proof}

The second "inf sup" condition is given by the following lemma:

\begin{lemma} \label{lem3.4}
The following  inf-sup Condition holds: there exists a constant $\beta>0$,
such that
\begin{equation}\label{condition inf sup 3}
\inf_{\boldsymbol{\varphi}\in\mathbf{V}^2_{-2,N}(\Omega),\,
\boldsymbol{\varphi}\neq 0}
\sup_{\boldsymbol{\psi}\in\mathbf{V}^2_{0,N}(\Omega),\,
\boldsymbol{\psi}\neq 0}
\frac{\int_{\Omega}\operatorname{\bf curl}\boldsymbol{\psi}
\cdot\operatorname{\bf curl}\boldsymbol{\varphi}\,\text{d}\mathbf{x}}
{\| \boldsymbol{\psi}\|_{\mathbf{X}^2_{0,N}(\Omega)}\|
\boldsymbol{\varphi}\|_{\mathbf{X}^2_{-2,N}(\Omega)}}\geq\beta.
 \end{equation}
 \end{lemma}

\begin{proof}
The proof is similar to that of Lemma \ref{lemme condition inf sup 2}.
 Let $\mathbf{g} \in \mathbf{W}_1^{0,2}(\Omega)$ and let us introduce the
 generalized Neumann problem
\begin{equation}\label{problem jdid}
 \operatorname{div}(\nabla \chi-\mathbf{g})=0\quad \text{in }
 \Omega\quad\text{and}\quad
(\nabla \chi-\mathbf{g})\cdot\mathbf{n}=0\quad\text{on } \Gamma.
\end{equation}
It follows from \cite{these Giroire} that Problem \eqref{problem jdid}
has a solution $\chi \in W_1^{1,2}(\Omega)$ and we have
\begin{equation*}
\|\nabla \chi\|_{ W_1^{0,2}(\Omega)}\leq C \| \mathbf{g}\|_{\mathbf{W}_1^{0,2}
(\Omega)}.
\end{equation*}
Setting $\mathbf{z}=\mathbf{g}-\nabla \chi$, then we have
$\mathbf{z} \in \mathring{\mathbf{H}}_1^2(\operatorname{div},\Omega)$ and
$\operatorname{div}\mathbf{z}=0$ with the estimate
\begin{equation}\label{estimation important}
\|\mathbf{z}\|_{\mathbf{W}_1^{0,2}(\Omega)}\leqslant
C\|\mathbf{g}\|_{\mathbf{W}_1^{0,2}(\Omega)}.
\end{equation}
Let $\boldsymbol{\varphi}$ be any function of $\mathbf{V}^2_{-2,N}(\Omega)$.
 Due to Theorem \ref{injection in X_N  homogene}, we have
$\mathbf{X}^2_{-2,N}(\Omega)\hookrightarrow \mathbf{W}_{-1}^{1,2}(\Omega)$
and by \eqref{equivalence de norme avec curl X_N 2} we can write
\begin{equation}\label{dualite 3}
 \|\boldsymbol{\varphi}\|_{\mathbf{X}^2_{-2,N}(\Omega)}
\leq C\|\operatorname{\bf curl}\boldsymbol{\varphi}\|_{\mathbf{W}_{-1}^{0,2}
(\Omega)}
= C\sup_{\mathbf{g}\in\mathbf{W}_1^{0,2}(\Omega),\, \mathbf{g}\neq 0}
\frac{\big{\vert}\int_{\Omega}\operatorname{\bf curl}\boldsymbol{\varphi}
\cdot\mathbf{g}\,\text{d}\mathbf{x}\big{\vert}}
{\| \mathbf{g}\|_{\mathbf{W}_1^{0,2}(\Omega)}}.
\end{equation}
Observe that  $\operatorname{\bf curl}\boldsymbol{\varphi}$ belongs to
$\mathbf{H}_{-1}^2(\operatorname{div},\Omega)$ with
$ \boldsymbol{\varphi} \times \mathbf{n}=\mathbf{0}$ on $\Gamma$ and
$\chi \in W_1^{1,2}(\Omega)$. Then using \eqref{formule de green v.n},
 we obtain
\begin{equation}\label{calcul integ zero 3}
 \int_{\Omega}\,\operatorname{\bf curl}\boldsymbol{\varphi}\cdot\,\nabla \chi\,dx
=\langle \operatorname{\bf curl}\boldsymbol{\varphi}\cdot\mathbf{n},
\chi \rangle _{\Gamma}=0.
\end{equation}
Due to \cite[Proposition 3.12]{vecteur potentiel vivet},
there exists a potential vector $\boldsymbol{\psi}\in \mathbf{W}_1^{1,2}(\Omega)$
such that
\begin{gather}\label{conséquence vecteur potentiel 3}
 \mathbf{z}=\operatorname{\bf curl}\boldsymbol{\psi},
\quad\operatorname{div}\boldsymbol{\psi}=0\quad\text{in }\Omega\quad\text{and}\quad
 \boldsymbol{\psi}\times\mathbf{n}=\mathbf{0}\quad\text{on }\Gamma,
\\
\label{consequence du vecteur curl}
\int_{\Gamma}\boldsymbol{\psi}\cdot \mathbf{n}\,d\sigma=0.
\end{gather}
In addition, we have
\begin{equation}
\|\boldsymbol{\psi}\|_{\mathbf{W}_1^{1,2}(\Omega)}
\leq C \|\mathbf{z}\|_{\mathbf{W}_1^{0,2}(\Omega)}.
\end{equation}
Then, we deduce that $\boldsymbol{\psi}$ belongs to $\mathbf{V}^2_{0,N}(\Omega)$.
Using \eqref{estimation important}, \eqref{calcul integ zero 3} and
\eqref{conséquence vecteur potentiel 3}, we deduce that
\begin{equation*}
\frac{\big{\vert}\int_{\Omega}\operatorname{\bf curl}\boldsymbol{\varphi}
\cdot\mathbf{g}\,\text{d}\mathbf{x}\big{\vert}}
{\| \mathbf{g}\|_{\mathbf{W}_1^{0,2}(\Omega)}}
\leq  C\frac{\big{\vert}\int_{\Omega}\operatorname{\bf curl}
\boldsymbol{\varphi}\cdot\mathbf{z}\,\text{d}\mathbf{x}\big{\vert}}
{\| \mathbf{z}\|_{\mathbf{W}_1^{0,2}(\Omega)}}
=C\frac{\big{\vert}\int_{\Omega}\operatorname{\bf curl}
\boldsymbol{\varphi}\cdot\operatorname{\bf curl}\boldsymbol{\psi}\,\text{d}
\mathbf{x}\big{\vert}}{\| \operatorname{\bf curl}
\boldsymbol{\psi}\|_{\mathbf{W}_1^{0,2}(\Omega)}}.
\end{equation*}
Applying again \eqref{equivalence de norme avec curl X_N 2} and using
\eqref{consequence du vecteur curl}, we obtain
\begin{equation*}
 \frac{\big\vert \int_{\Omega}\operatorname{\bf curl}\boldsymbol{\varphi}
\cdot\mathbf{g}\,\text{d}\mathbf{x}\big\vert}{\|
\mathbf{g}\|_{\mathbf{W}_1^{0,2}(\Omega)}}
\leq  C\frac{\big{\vert}\int_{\Omega}\operatorname{\bf curl}
\boldsymbol{\varphi}\cdot\operatorname{\bf curl}\boldsymbol{\psi}
\,\text{d}\mathbf{x}\big{\vert}}{\| \boldsymbol{\psi}
\|_{\mathbf{X}_{0,N}^2(\Omega)}},
\end{equation*}
and the inf-sup condition \eqref{condition inf sup 3} follows immediately
from \eqref{dualite 3}.
\end{proof}


\section{Elliptic problems with different boundary conditions}

Next, we  study the problem
\begin{equation}\label{problem de EN} %(E_{\,N})
\begin{gathered}
-\Delta \boldsymbol{\xi} =\mathbf{f}\quad \text{and}\quad\operatorname{div}
\boldsymbol{\xi}  =0 \quad \text{in }\Omega,\\
 \boldsymbol{\xi}\times\mathbf{n}=\mathbf{g}\times\mathbf{n}\quad\text{on }
 \Gamma\quad \text{and}\quad \int_{\Gamma}(\boldsymbol{\xi}\cdot \mathbf{n})q
\,d\sigma=0,\quad\forall  q \in\mathcal{P}_{k}^{\Delta}.
 \end{gathered}
\end{equation}


\begin{proposition}\label{existence E_N homogene 2}
Let $k=-1$ or $k=0$ and suppose that $\mathbf{g}\times\mathbf{n}=\mathbf{0}$
and let $\mathbf{f} \in [\mathring{\mathbf{H}}_{k-1}^2(\operatorname{\bf curl},
\Omega)]'$ with $\operatorname{div}\mathbf{f}=0$ in $\Omega$
and satisfying the compatibility condition
\begin{equation}\label{condition de compatibilite E_N Homogene 2}
\forall\mathbf{v}\in\mathbf{Y}^2_{1-k,N}(\Omega),\quad
 \langle\mathbf{f}, \mathbf{v}\rangle_{[\mathring{\mathbf{H}}_{k-1}^2
(\operatorname{\bf curl}\Omega)]'\times \mathring{\mathbf{H}}_{k-1}^2
(\operatorname{\bf curl}\Omega)}=0.
 \end{equation}
Then, Problem \eqref{problem de EN} has a unique solution in
$\mathbf{W}_{-k}^{1,2}(\Omega)$ and we have
\begin{equation}\label{estimation E_N homogene 2}
 \|\boldsymbol{\xi}\|_{\mathbf{W}_{-k}^{1,2}(\Omega)}\leqslant
 C \| \mathbf{f}\|_{[\mathring{\mathbf{H}}_{k-1}^2
(\operatorname{\bf curl},\Omega)]'}.
\end{equation}
Moreover, if $\mathbf{f}$ in $\mathbf{W}_{-k+1}^{0,2}(\Omega)$ and $\Omega'$
is of class $C^{2,1}$, then the solution $\boldsymbol{\xi}$ is in
$\mathbf{W}_{-k+1}^{2,2}(\Omega)$ and satisfies the estimate
\begin{equation}\label{estimation E_N homogene 2 réguliere}
 \|\boldsymbol{\xi}\|_{\mathbf{W}_{-k+1}^{2,2}(\Omega)}
\leqslant C \| \mathbf{f}\|_{\mathbf{W}_{-k+1}^{0,2}(\Omega)}.
\end{equation}
\end{proposition}

\begin{proof}
(i) On the one hand, observe that  Problem \eqref{problem de EN} is
reduced to the variational problem:
Find $\boldsymbol{\xi} \in \mathbf{V}_{-k-1,N}^2(\Omega)$ such that
\begin{equation}\label{formul vario E_N 2}
 \forall  \boldsymbol{\varphi} \in \mathbf{X}_{k-1,N}^2(\Omega), \quad
 \int_{\Omega}\operatorname{\bf curl}\boldsymbol{\xi}\cdot\operatorname{\bf curl}
\boldsymbol{\varphi}\,\text{d}\mathbf{x}=\langle\mathbf{f},\,
\boldsymbol{\varphi}\rangle_{\Omega},
\end{equation}
where the duality on $\Omega$ is
$$
\langle\cdot,\,\cdot\rangle_{\Omega}=\langle\cdot,\,\cdot
\rangle_{[\mathring{\mathbf{H}}_{k-1}^2(\operatorname{\bf curl},\,\Omega)]'
\times\mathring{\mathbf{H}}_{k-1}^2(\operatorname{\bf curl},\,\Omega)}.
$$
On the other hand,  \eqref{formul vario E_N 2} is equivalent to the
problem:
Find $\boldsymbol{\xi} \in \mathbf{V}_{-k-1,N}^2(\Omega)$ such that
\begin{equation}\label{formul vario 2 E_N 2}
 \forall \boldsymbol{\varphi} \in \mathbf{V}_{k-1,N}^2(\Omega),\quad
 \int_{\Omega}\operatorname{\bf curl}\boldsymbol{\xi}\cdot\operatorname{\bf curl}
\boldsymbol{\varphi}\,\text{d}\mathbf{x}=\langle\mathbf{f},
\boldsymbol{\varphi}\rangle_{\Omega}.
\end{equation}
Indeed, every solution of \eqref{formul vario E_N 2} also solves
\eqref{formul vario 2 E_N 2}. Conversely, assume that
\eqref{formul vario 2 E_N 2} holds, and let
$\boldsymbol{\varphi} \in \mathbf{X}_{k-1,N}^2(\Omega)$.
Let us solve the exterior Dirichlet problem
\begin{equation}\label{Dirichlet 2}
 -\Delta \chi=\operatorname{div}\boldsymbol{\varphi} \quad\text{in }
 \Omega\quad\text{and}\quad\chi=0\quad\text{on } \Gamma.
\end{equation}
It is shown in \cite[Theorem 3.5]{vecteur potentiel vivet}
 that problem \eqref{Dirichlet 2} has a unique solution
 $\chi \in W_{k}^{2,2}(\Omega)/\mathcal{A}^{\Delta}_{-k}$.

\subsection*{First case} if $k=0$, we set
 \begin{equation*}
 \widetilde{\boldsymbol{\varphi}}=\boldsymbol{\varphi}-\nabla \chi
-\frac{1}{C_1}\langle \boldsymbol{\varphi}-\nabla \chi,1\rangle_{\Gamma}
\nabla(v(1)-1),
\end{equation*}
where $v(1)$ is the unique solution in $W_{0}^{1,2}(\Omega)$ of the Dirichlet
problem \eqref{problem de Dirichlet liee au noyeau} and
 \[
C_1=\int_{\Gamma}\frac{\partial v(1)}{\partial \mathbf{n}}\,d\sigma.
\]
 It follows from \cite[Lemma 3.11]{vecteur potentiel vivet}
 that $C_1>0$ and since $\nabla(v(1)-1)$ belongs to $\mathbf{Y}^2_{1,N}(\Omega)$,
we deduce that $\widetilde{\boldsymbol{\varphi}}$ belongs to
$\mathbf{V}_{-1,N}^2(\Omega)$.

\subsection*{Second case}
 if $k=-1$, for each polynomial $p$ in $\mathcal{P}^{\Delta}_1$, we take
$\widetilde{\boldsymbol{\varphi}}$ of the form
\begin{equation*}
 \widetilde{\boldsymbol{\varphi}}=\boldsymbol{\varphi}-\nabla \chi-\nabla(v(p)-p)
\end{equation*}
where $v(p)$ is the unique solution in $W_{0}^{1,2}(\Omega)$ of the Dirichlet
problem \eqref{problem de Dirichlet liee au noyeau}. The polynomial $p$
is chosen to satisfy the  condition
\begin{equation}\label{condition pour appartenir  dans V}
\int_{\Gamma}(\widetilde{\boldsymbol{\varphi}}\cdot \mathbf{n})
\,q\,d\sigma=0\quad \quad\forall q \in \mathcal{P}^{\Delta}_1.
\end{equation}
To show that this is possible, let $T$ be a linear form defined by
$T: \mathcal{P}^{\Delta}_1  \to \mathbb{R}^4$,
\begin{align*}
T(p)&=\Big( \int_{\Gamma}\frac{\partial  (v(p)-p)}{\partial \mathbf{n}}d\sigma,
 \int_{\Gamma}\frac{\partial  (v(p)-p) }{\partial \mathbf{n}}\,x_1d\sigma,\\
&\quad \int_{\Gamma}\frac{\partial (v(p)-p) }{\partial \mathbf{n}}\,x_2d\sigma,
 \int_{\Gamma}\frac{\partial  (v(p)-p)}{\partial \mathbf{n}}\,x_3d\sigma\Big),
\end{align*}
where $\{ 1,x_1,x_2,x_3\} $ denotes a basis of $\mathcal{P}^{\Delta}_1$.
 It is shown in the proof of \cite[Theorem 7]{Sequeira 2}, that if
\[
\int_{\Gamma}\frac{\partial  (v(p)-p)}{\partial \mathbf{n}}\,q\,d\sigma=0
\quad   \forall q \in \mathcal{P}^{\Delta}_1,
\]
then $p=0$.
 This implies that $T$ is injective and so bijective. And so, there exists
a unique $p$ in $\mathcal{P}^{\Delta}_1$ so that condition
\eqref{condition pour appartenir  dans V} is satisfied and since
 $\nabla(v(p)-p)$ belongs to $\mathbf{Y}^2_{\,2,N}(\Omega)$,
we prove that $\widetilde{\boldsymbol{\varphi}} \in
\mathbf{V}_{-2,N}^2(\Omega)$.

Finally, using \eqref{condition de compatibilite E_N Homogene 2},
we obtain for $k=0$ and $k=-1$ that
\begin{equation*}
\langle\mathbf{f},\nabla(v(p)-p)\rangle_{\Omega}=0 \quad\text{and}\quad
\langle\mathbf{f},\nabla(v(1)-1)\rangle_{\Omega}=0
\end{equation*}
and as $\boldsymbol{\mathcal{D}}(\Omega)$ is dense in
$\mathring{\mathbf{H}}_{k-1}^2(\operatorname{\bf curl},\Omega)$,
we obtain that
\begin{equation*}
\langle\mathbf{f},\nabla \chi\rangle_{\Omega}=0 .
\end{equation*}
Then we have
\begin{equation*}
\int_{\Omega}\operatorname{\bf curl}\boldsymbol{\xi}\cdot\operatorname{\bf curl}
\boldsymbol{\varphi}\,\text{d}\mathbf{x} = \int_{\Omega}\operatorname{\bf curl}
\boldsymbol{\xi}\cdot\operatorname{\bf curl}\widetilde{\boldsymbol{\varphi}}
\,\text{d}\mathbf{x}=\langle\mathbf{f},\boldsymbol{\varphi}\rangle_{\Omega}.
\end{equation*}
Then Problem \eqref{formul vario E_N 2} and Problem \eqref{formul vario 2 E_N 2}
 are equivalent. Now, to solve Problem \eqref{formul vario 2 E_N 2}, we use
Lax-Milgram lemma for $k=0$ and the inf-sup condition \eqref{condition inf sup 3}
for $k=-1$. Let us start by $k=0$. We consider the bilinear form
$\boldsymbol{a}:\mathbf{V}^2_{-1,N}(\Omega)\times \mathbf{V}^2_{-1,N}(\Omega)
\to \mathbb{R}$ such that
\begin{equation*}
\boldsymbol{a}(\boldsymbol{\xi},\boldsymbol{\varphi})
={\int}_{\Omega}\operatorname{\bf curl}\boldsymbol{\xi}\cdot
\operatorname{\bf curl}\boldsymbol{\varphi}\,\,\text{d}\mathbf{x}.
\end{equation*}
According to Theorem \ref{injection in X_N  homogene},
$\boldsymbol{a}$ is continuous and coercive on  $\mathbf{V}^2_{-1,N}(\Omega)$.
Due to Lax-Milgram lemma,  there exists a unique solution
$\boldsymbol{\xi}\in \mathbf{V}^2_{-1,N}(\Omega)$ of
Problem \eqref{formul vario 2 E_N 2}. Using again Theorem \ref{injection in X_N  homogene}, we prove that this solution $\boldsymbol{\xi}$ belongs to $\mathbf{W}_{0}^{1,2}(\Omega)$ and the following estimate follows immediately
\begin{equation}\label{consequence inf sup condition}
\| \boldsymbol{\xi} \|_{ \mathbf{W}_{0}^{1,2}(\Omega)}\leqslant
 C \| \mathbf{f} \|_{[\mathring{\mathbf{H}}_{-1}^2
(\operatorname{\bf curl},\Omega)]'}.
\end{equation}
When $k=-1$, we have that Problem \eqref{formul vario 2 E_N 2} satisfies
the inf-sup condition \eqref{condition inf sup 3}.
Let us consider the  mapping
$\boldsymbol{\ell}:\mathbf{V}^2_{-2,N}(\Omega)\to \mathbb{R}$ such that
$\boldsymbol{\ell}(\boldsymbol{\varphi})=\langle \mathbf{f},
\boldsymbol{\varphi}\rangle _{\Omega}$. It is clear that $\boldsymbol{\ell}$
 belongs to $(\mathbf{V}^2_{-2,N}(\Omega))'$ and according to
Remark \ref{remark sur c inf sup}, there exists a unique solution
$\boldsymbol{\xi}\in \mathbf{V}^2_{0,N}(\Omega)$ of
Problem \eqref{formul vario 2 E_N 2}. Due to Theorem
\ref{injection in X_N  homogene}, we prove that this solution
$\boldsymbol{\xi}$ belongs to $\mathbf{W}_1^{1,2}(\Omega)$.
It follows from Remark \ref{remark sur c inf sup} i) that
\begin{equation}\label{consequence inf sup condition2}
\|\boldsymbol{\xi}\|_{ \mathbf{W}_1^{1,2}(\Omega)}\leqslant
 C\|\mathbf{f}\|_{[\mathring{\mathbf{H}}_{-2}^2(\operatorname{\bf curl},\Omega)]'}.
\end{equation}

(ii) We suppose in addition that $\mathbf{f}$ is in
$\mathbf{W}_{-k+1}^{0,2}(\Omega)$ for $k=-1$ or $k=0$ and $\Omega'$ is of class
 $C^{2,1}$ and  we set $\mathbf{z}=\operatorname{\bf curl}\boldsymbol{\xi}$,
 where $\boldsymbol{\xi} \in \mathbf{W}_{-k}^{1,2}(\Omega)$ is the unique
solution of  Problem \eqref{problem de EN}. Then we have
\begin{equation*}
\mathbf{z} \in \mathbf{W}_{-k}^{0,2}(\Omega),\quad \operatorname{\bf curl}
\mathbf{z}=\mathbf{f}\in \mathbf{W}_{-k+1}^{0,2}(\Omega),\quad
\operatorname{div}\mathbf{z}=0\quad\text{and}\quad \mathbf{z}\cdot\mathbf{n}=0
\quad\text{on } \Gamma
\end{equation*}
and thus $\mathbf{z}$ belongs to $\mathbf{X}^2_{-k,T}(\Omega)$.
By Theorem \ref{injection in X_T  homogene}, we prove that $\mathbf{z}$
belongs to $\mathbf{W}_{-k+1}^{1,2}(\Omega)$ and using
\eqref{equivalence de norme avec curl X_T 2}, we prove that $\mathbf{z}$
 satisfies
 \begin{equation}\label{estimation de z en fct f}
\|\mathbf{z}\|_{\mathbf{W}_{-k+1}^{1,2}(\Omega)}\leqslant
C\|\mathbf{f}\|_{\mathbf{W}_{-k+1}^{0,2}(\Omega)}.
 \end{equation}
As a consequence $\boldsymbol{\xi}$ satisfies
\begin{equation*}
\boldsymbol{\xi} \in \mathbf{W}_{-k}^{1,2}(\Omega),\quad
\operatorname{\bf curl}\boldsymbol{\xi} \in \mathbf{W}_{-k+1}^{1,2}(\Omega),\quad
\operatorname{div}\boldsymbol{\xi}=0\quad\text{and}\quad
\boldsymbol{\xi}\times\mathbf{n}=\mathbf{0}\quad\text{on }\, \Gamma.
\end{equation*}
Applying \cite[Corollary 3.14]{vecteur potentiel vivet}, we deduce that
$\boldsymbol{\xi}$ belongs to $\mathbf{W}_{-k+1}^{2,2}(\Omega)$ and
using in addition the boundary condition of \eqref{problem de EN}
we prove that
\begin{equation}\label{estimation de xi réguliere}
\|\boldsymbol{\xi}\|_{\mathbf{W}_{-k+1}^{2,2}(\Omega)}
\leqslant C\|\operatorname{\bf curl}
\boldsymbol{\xi}\|_{\mathbf{W}_{-k+1}^{1,2}(\Omega)}.
\end{equation}
Finally, estimate \eqref{estimation E_N homogene 2 réguliere}
follows from \eqref{estimation de z en fct f} and
\eqref{estimation de xi réguliere}.
\end{proof}

\begin{corollary}\label{existence E_N non homogene 2}
Let $k=-1$ or $k=0$ and let $\mathbf{f} \in
 [\mathring{\mathbf{H}}_{k-1}^2(\operatorname{\bf curl},\Omega)]'$ with
$\operatorname{div}\mathbf{f}=0$ in $\Omega$ and
$\mathbf{g} \in \mathbf{H}^{1/2}(\Gamma)$ and satisfying the compatibility
condition \eqref{condition de compatibilite E_N Homogene 2}. Then,
 Problem \eqref{problem de EN} has a unique solution $\boldsymbol{\xi}$ in
$\mathbf{W}_{-k}^{1,2}(\Omega)$ and we have:
\begin{equation}\label{estimation E_N non homogene 2}
 \| \boldsymbol{\xi}\| _{\mathbf{W}_{-k}^{1,2}(\Omega)}\leqslant
C \Big( \| \mathbf{f} \|_{[\mathring{\mathbf{H}}_{k-1}^2
(\operatorname{\bf curl},\Omega)]'}+\|\mathbf{g}\times
\mathbf{n}\|_{\mathbf{H}^{1/2}(\Gamma)}\Big) .
\end{equation}
Moreover, if $\mathbf{f}$ in $\mathbf{W}_{-k+1}^{0,2}(\Omega)$,
 $\mathbf{g}$ in $\mathbf{H}^{\,3/2}(\Gamma)$ and $\Omega'$ is of class
$C^{2,1}$, then the solution $\boldsymbol{\xi}$ is in
$\mathbf{W}_{-k+1}^{2,2}(\Omega)$ and satisfies
\begin{equation}\label{estimation E_N non homogene 2 fort}
 \|\boldsymbol{\xi}\|_{\mathbf{W}_{-k+1}^{2,2}(\Omega)}
\leqslant C \Big( \| \mathbf{f} \|_{\mathbf{W}_{-k+1}^{0,2}(\Omega)}
+\|\mathbf{g}\times\mathbf{n}\|_{\mathbf{H}^{3/2}(\Gamma)}\Big) .
\end{equation}
\end{corollary}

\begin{proof}
Let $k=0$ or $k=-1$ and let $\mathbf{g} \in \mathbf{H}^{1/2}(\Gamma)$.
We know that there exists  $\boldsymbol{\xi}_{0}$ in $\mathbf{H}^{1}(\Omega)$
with compact support satisfying
\begin{equation*}
\boldsymbol{\xi}_{0}=\mathbf{g}_{\tau}\quad\text{on }
 \Gamma\quad\text{and}\quad\operatorname{div}\boldsymbol{\xi}_{0}=0\quad
\text{in } \Omega,
\end{equation*}
where $\mathbf{g}_{\tau}$ is the tangential component of $\mathbf{g}$ on $\Gamma$.
Since support of $\boldsymbol{\xi}_{0}$ is compact, we deduce that
$\boldsymbol{\xi}_{0}$ belongs to $ \mathbf{W}_{-k}^{1,2}(\Omega)$
for $k=-1$ or $k=0$ and satisfies
 \begin{equation}\label{estimation de lifting}
\|\boldsymbol{\xi}_{0}\|_{\mathbf{W}_{-k}^{1,2}(\Omega)}
\leq C\| \mathbf{g}_{\tau}\| _{\mathbf{H}^{1/2}(\Gamma)}.
\end{equation}
Setting $\mathbf{z}=\boldsymbol{\xi}-\boldsymbol{\xi}_{0}$, then Problem
\eqref{problem de EN} is equivalent to: find
$\mathbf{z}\in\mathbf{W}_{-k}^{1,2}(\Omega) $ such that
\begin{equation}\label{problem E_N Z Homogene 2}
 \begin{gathered}
  -\Delta \mathbf{z}=\mathbf{f}+\Delta \boldsymbol{\xi}_{0} \quad \text{and} \quad
 \operatorname{div}\mathbf{z}=0 \quad\text{in } \Omega,\\
\mathbf{z}\times\mathbf{n}=\mathbf{0}\quad \text{on }
 \Gamma\quad \text{and}\quad {\int}_{\Gamma}(\mathbf{z}\cdot \mathbf{n})\,q\,
d\sigma=0,\quad \forall  q \in\mathcal{P}_{k}^{\Delta}.
\end{gathered}
\end{equation}

Observe that $\mathbf{F}=\mathbf{f}-\operatorname{\bf curl}
\operatorname{\bf curl}\boldsymbol{\xi}_{0}$ belongs to
$[\mathring{\mathbf{H}}_{k-1}^2(\operatorname{\bf curl},\Omega)]'$.
Since $\boldsymbol{\mathcal{D}}(\Omega)$ is dense in
$\mathring{\mathbf{H}}_{k-1}^2(\operatorname{\bf curl},\,\Omega)$,
we have for any $\mathbf{v}\in\mathbf{Y}^2_{1-k,N}(\Omega)$:
\begin{equation*}
 \langle \operatorname{\bf curl}\operatorname{\bf curl}\boldsymbol{\xi}_{0},
\mathbf{v}\rangle _{\Omega}=\int_{\Omega}\,\operatorname{\bf curl}
\boldsymbol{\xi}_{0}\cdot\operatorname{\bf curl}\mathbf{v}\,dx=0.
\end{equation*}
Thus $\mathbf{F}$ satisfies the compatibility condition
\eqref{condition de compatibilite E_N Homogene 2}.
Due to Proposition \ref{existence E_N homogene 2}, there exists a unique
$\mathbf{z}\in\mathbf{W}_{-k}^{1,2}(\Omega)$ solution of problem
\eqref{problem E_N Z Homogene 2} such that
\begin{equation}\label{estimation du cas homogen}
\|\mathbf{z}\|_{\mathbf{W}_{-k}^{1,2}(\Omega)}
\leqslant C \| \mathbf{F}\| _{[\mathring{\mathbf{H}}_{k-1}^2
(\operatorname{\bf curl},\Omega)]'}
\leq C \Big( \| \mathbf{f}\| _{[\mathring{\mathbf{H}}_{k-1}^2
(\operatorname{\bf curl},\Omega)]'}+\|\operatorname{\bf curl}\boldsymbol{\xi}_{0}
\|_{\mathbf{W}_{-k}^{0,2}(\Omega)}\Big) .
\end{equation}
Then $\boldsymbol{\xi}=\mathbf{z}+\boldsymbol{\xi}_{0}$ belongs to
$\mathbf{W}_{-k}^{1,2}(\Omega)$ is the unique solution of \eqref{eN}
and estimate \eqref{estimation E_N non homogene 2} follows immediately
from \eqref{estimation de lifting} and \eqref{estimation du cas homogen}.

\noindent \textbf{Regularity of the solution:}
 Suppose in addition that  $\Omega'$ is of class $C^{2,1}$, $\mathbf{f}$
in $\mathbf{W}_{-k+1}^{0,2}(\Omega)$ and $\mathbf{g}$ in
$\mathbf{H}^{\,3/2}(\Gamma)$. Then the function  $\boldsymbol{\xi}_{0}$
defined above belongs to $\mathbf{H}^2(\Omega)$ with compact support and
thus $\boldsymbol{\xi}_{0}$ belongs to $\mathbf{W}_{-k+1}^{2,2}(\Omega)$
and we have
 \begin{equation}\label{estimation de lifting plus regulier}
\| \boldsymbol{\xi}_{0}\| _{\mathbf{W}_{-k+1}^{2,2}(\Omega)}
\leq C\| \mathbf{g}_{\tau}\| _{\mathbf{H}^{\,3/2}(\Gamma)}.
\end{equation}
Using again Proposition \ref{existence E_N homogene 2}, we prove that
$\mathbf{z}$ belongs to $\mathbf{W}_{-k+1}^{2,2}(\Omega)$ and satisfies
\begin{equation*}
\|\mathbf{z}\|_{\mathbf{W}_{-k+1}^{2,2}(\Omega)}
\leqslant C \| \mathbf{F}\| _{\mathbf{W}_{-k+1}^{0,2}(\Omega)}.
\end{equation*}
Then $\boldsymbol{\xi}$ is in $\mathbf{W}_{-k+1}^{2,2}(\Omega)$ and
estimate \eqref{estimation E_N non homogene 2 fort} follows from
\eqref{estimation de lifting plus regulier}.
\end{proof}

The next theorem solves an other type of exterior problem.

\begin{theorem}\label{resolution du probleme de stokes sans pression}
Let $k=-1$ or $k=0$ and let $\mathbf{v}$ belongs to $\mathbf{W}_{k}^{0,2}(\Omega)$. Then, the following problem
\begin{equation}\label{problem en xi}
\begin{gathered}
 -\Delta \boldsymbol{\xi} =\operatorname{\bf curl}\mathbf{v}\quad\text{and}\quad
\operatorname{div}\boldsymbol{\xi}  =0\quad \text{in } \Omega,
\\
  \boldsymbol{\xi}\cdot\mathbf{n} =0\quad\text{and}\quad
(\operatorname{\bf curl} \boldsymbol{\xi}-\mathbf{v})\times\mathbf{n}
=\mathbf{0} \quad \text{on }\, \Gamma,
\\
{\int}_{\Gamma}\boldsymbol{\xi}\cdot  \nabla (w(q)-q)\,d\sigma=0,\quad
\forall \,(w(q)-q)\in \mathcal{N}_{-k}^{\Delta}
  \end{gathered}
\end{equation}
has a unique solution $\boldsymbol{\xi}$ in $\mathbf{W}_{k}^{1,2}(\Omega)$
and we have
\begin{equation}\label{estimation du problem bien definie}
 \| \boldsymbol{\xi}\|_{\mathbf{W}_{k}^{1,2}(\Omega)}
\leq C\| \mathbf{v}\|_{\mathbf{W}_{k}^{0,2}(\Omega)}.
\end{equation}
 Moreover, if $\mathbf{v}\in\mathbf{W}_{k+1}^{1,2}(\Omega)$ and $\Omega'$ is
of class $\mathcal{C}^{2,1}$, then the solution $\boldsymbol{\xi}$ is in
$\mathbf{W}_{k+1}^{2,2}(\Omega)$ and satisfies the estimate
\begin{equation}
 \| \boldsymbol{\xi}\|_{\mathbf{W}_{k+1}^{2,2}(\Omega)}
\leq C\| \mathbf{v}\|_{\mathbf{W}_{k+1}^{1,2}(\Omega)}.
\end{equation}
\end{theorem}

\begin{proof}
At first observe that if $\boldsymbol{\xi} \in \mathbf{W}_{k}^{1,2}(\Omega)$
is a solution of Problem \eqref{problem en xi} for $k=-1$ or $k=0$, then
 $\operatorname{\bf curl} \boldsymbol{\xi}-\mathbf{v}$ belongs to
 $\mathbf{H}_{k}^2(\operatorname{\bf curl},\Omega)$ and thus
$(\operatorname{\bf curl} \boldsymbol{\xi}-\mathbf{v})\times\mathbf{n}$
is well defined in $\Gamma$ and belongs to $\mathbf{H}^{-1/2}(\Gamma)$.

On the other hand, note that \eqref{problem en xi} can be reduced to the
 variational problem:
Find $\boldsymbol{\xi} \in \mathbf{V}_{k-1,T}^2(\Omega)$ such that
\begin{equation}\label{formulation variationnelle pour prob en xi}
 \forall\boldsymbol{\varphi}\in \mathbf{X}^2_{-k-1,T}(\Omega)\quad\int_{\Omega}\operatorname{\bf curl}\boldsymbol{\xi}\cdot\operatorname{\bf curl}\boldsymbol{\varphi}\,\text{d}\mathbf{x}=\int_{\Omega}\mathbf{v}\cdot\operatorname{\bf curl}\boldsymbol{\varphi}\,\text{d}\mathbf{x}.
\end{equation}
Indeed, every solution of \eqref{problem en xi}  also solves \eqref{formulation variationnelle pour prob en xi}.  Conversely, let $\boldsymbol{\xi} \in \mathbf{V}_{k-1,T}^2(\Omega)$ a solution of the problem \eqref{formulation variationnelle pour prob en xi}. Then,
\begin{equation*}
 \forall \boldsymbol{\varphi} \in\boldsymbol{\mathcal{D}}(\Omega),\quad
\langle\operatorname{\bf curl}\operatorname{\bf curl}\boldsymbol{\xi}
-\operatorname{\bf curl}\mathbf{v},\,\boldsymbol{\varphi}
\rangle_{\boldsymbol{\mathcal{D}}'(\Omega)\times\boldsymbol{\mathcal{D}}
(\Omega)}=0.
\end{equation*}
Then
\begin{equation}\label{premiere equation}
 -\Delta \boldsymbol{\xi}=\operatorname{\bf curl}\mathbf{v}\quad
\text{in } \Omega.
\end{equation}
Moreover, by the fact that $\boldsymbol{\xi}$ belongs to the space
$\mathbf{V}^2_{k-1,T}(\Omega)$ we have $\operatorname{div}\boldsymbol{\xi}=0$
in $\Omega$ and $\boldsymbol{\xi}\cdot\mathbf{n}=0$ on $\Gamma$.
Then, it remains to verify the boundary condition
 $(\operatorname{\bf curl} \boldsymbol{\xi}-\mathbf{v})\times\mathbf{n}
=\mathbf{0}\,\,\text{on }\Gamma$. Now setting
$\mathbf{z}=\operatorname{\bf curl}\boldsymbol{\xi}-\mathbf{v}$, then
$\mathbf{z}$ belongs to $\mathbf{H}_{k}^2(\operatorname{\bf curl},\Omega)$.
Therefore, \eqref{premiere equation} becomes
\begin{equation}\label{curl z =0}
\operatorname{\bf curl}\mathbf{z}=\mathbf{0}\quad  \text{in }\, \Omega.
\end{equation}
Let $\boldsymbol{\varphi} \in\mathbf{X}^2_{-k-1,T}(\Omega)$, by Theorem
\ref{injection in X_T  homogene} we have
$\mathbf{X}^2_{-k-1,T}(\Omega) \hookrightarrow \mathbf{W}_{-k}^{1,2}(\Omega)$.
Thank's to \eqref{formule de green v*n} we obtain
\begin{equation}\label{nulite de zxn}
\int_{\Omega}\mathbf{z}\cdot\operatorname{\bf curl}\boldsymbol{\varphi}\,
\text{d}\mathbf{x}=\langle\mathbf{z}\times\mathbf{n},
\boldsymbol{\varphi}\rangle_{\mathbf{H}^{-1/2}(\Gamma)
\times\mathbf{H}^{1/2}(\Gamma)}+\int_{\Omega}\,\operatorname{\bf curl}
\mathbf{z}\cdot\boldsymbol{\varphi}\,\text{d}\mathbf{x}.
\end{equation}
Compare \eqref{nulite de zxn} with
\eqref{formulation variationnelle pour prob en xi} and using \eqref{curl z =0},
we deduce that
$$
\forall\boldsymbol{\varphi}\in\mathbf{X}^2_{-k-1,T}(\Omega),\quad
\langle\mathbf{z}\times\mathbf{n},\boldsymbol{\varphi}\rangle_{\Gamma}=0.
$$
Let now $\boldsymbol{\mu}$ any element of the space $\mathbf{H}^{1/2}(\Gamma)$.
As $\Omega'$ is bounded, we can fix once for all a ball $B_{R}$, centered at
the origin and with radius $R$, such that $\overline{\Omega'}\subset B_{R}$.
Setting $\Omega_{R}=\Omega \cap B_{R}$, then we have the existence of
$\boldsymbol{\varphi}$ in $ \mathbf{H}^{1}(\Omega_{R})$ such that
$\boldsymbol{\varphi}=\mathbf{0}$ on $ \partial B_{R}$ and
$\boldsymbol{\varphi}=\boldsymbol{\mu}_{t}$ on $\Gamma$, where
$\boldsymbol{\mu}_{t}$ is the tangential component of $\boldsymbol{\mu}$
on $\Gamma$. The function $\boldsymbol{\varphi}$ can be extended by zero
outside $B_R$ and the extended function, still denoted by $\boldsymbol{\varphi}$,
 belongs to $\mathbf{W}_{\alpha}^{1,p}(\Omega)$, for any $\alpha$ since its
support is bounded. Thus $\boldsymbol{\varphi}$, belongs to
$\mathbf{W}_{-k}^{1,2}(\Omega)$. It is clear that $\boldsymbol{\varphi}$
belongs to $\mathbf{X}^2_{-k-1,T}(\Omega)$ and
\begin{equation}
 \langle\mathbf{z}\times\mathbf{n},\boldsymbol{\mu}\rangle_{\Gamma}
=\langle\mathbf{z}\times\mathbf{n}, \boldsymbol{\mu}_{t}\rangle_{\Gamma}
=\langle\mathbf{z}\times\mathbf{n},\boldsymbol{\varphi}\rangle_{\Gamma}=0.
\end{equation}
This implies that $\mathbf{z}\times\mathbf{n}=\mathbf{0}$ on $\Gamma$ which
is the last boundary condition in \eqref{problem en xi}.

On the other hand, let us introduce the problem:
Find $\boldsymbol{\xi} \in \mathbf{V}_{k-1,T}^2(\Omega)$ such that
\begin{equation}\label{deuxieme form pour xi}
\forall\boldsymbol{\varphi}\in \mathbf{V}^2_{-k-1,T}(\Omega)\quad
\int_{\Omega}\operatorname{\bf curl}\boldsymbol{\xi}\cdot
\operatorname{\bf curl}\boldsymbol{\varphi}\,\text{d}\mathbf{x}
=\int_{\Omega}\mathbf{v}\cdot\operatorname{\bf curl}\boldsymbol{\varphi}\,
\text{d}\mathbf{x}.
\end{equation}
Problem \eqref{deuxieme form pour xi} can be solved by Lax-Milgram lemma
if $k=0$ and by Lemma \ref{lemme condition inf sup 2} if $k=-1$.

We start by the case $k=-1$. Observe that Problem \eqref{deuxieme form pour xi}
satisfies the inf-sup condition \eqref{condition inf sup 2}.
Let consider the  mapping $\boldsymbol{\ell}:\mathbf{V}^2_{0,T}(\Omega)\to \mathbb{R}$
such that $\boldsymbol{\ell}(\boldsymbol{\varphi})
=\int_{\Omega}\mathbf{v}\cdot\operatorname{\bf curl}\boldsymbol{\varphi}
\,\text{d}\mathbf{x}$. It is clear that $\boldsymbol{\ell}$ belongs to
$(\mathbf{V}^2_{0,T}(\Omega))'$ and according to Remark
\ref{remark sur c inf sup}, there exists a unique solution
 $\boldsymbol{\xi} \in \mathbf{V}^2_{-2,T}(\Omega)$.
Applying Theorem \ref{injection in X_T  homogene}, we deduce that this
solution $\boldsymbol{\xi}$ belongs to $\mathbf{W}_{-1}^{1,2}(\Omega)$.
It follows from Remark \ref{remark sur c inf sup} i) and
Theorem \ref{injection in X_N  homogene} that
 \begin{equation}\label{estimation avec inf sup et vivet}
\| \boldsymbol{\xi}\| _{\mathbf{W}_{-1}^{1,2}(\Omega)}
\leq C \| \boldsymbol{\ell}\| _{(\mathbf{V}^2_{0,T}(\Omega))'}
\leq C \| \mathbf{v}\|_{\mathbf{W}_{-1}^{0,2}(\Omega)}.
\end{equation}
For $k=0$, let us consider the bilinear form
$\boldsymbol{b}:\mathbf{V}^2_{-1,T}(\Omega)\times \mathbf{V}^2_{-1,T}(\Omega)
\to \mathbb{R}$ such that
\begin{equation*}
\boldsymbol{b}(\boldsymbol{\xi},\boldsymbol{\varphi})
=\int_{\Omega}\operatorname{\bf curl}\boldsymbol{\xi}\cdot
\operatorname{\bf curl}\boldsymbol{\varphi}\,\text{d}\mathbf{x}.
\end{equation*}
According to Theorem \ref{injection in X_T  homogene}, $\boldsymbol{b}$
is continuous and coercive on  $\mathbf{V}^2_{-1,T}(\Omega)$.
Due to Lax-Milgram lemma,  there exists a unique solution
$\boldsymbol{\xi}\in \mathbf{V}^2_{-1,T}(\Omega)$ of Problem
\eqref{deuxieme form pour xi}. Using again
Theorem \ref{injection in X_T homogene}, we prove that this solution
$\boldsymbol{\xi}$ belongs to $\mathbf{W}_{0}^{1,2}(\Omega)$ and
estimate \eqref{estimation du problem bien definie} follows immediately.

Next, we  extend \eqref{deuxieme form pour xi} to any test function in
 $\mathbf{X}^2_{-k-1,T}(\Omega)$.
Let $\boldsymbol{\varphi} \in \mathbf{X}^2_{-k-1,T}(\Omega)$ and let us solve
the exterior Neumann problem
\begin{equation}\label{prob 2 de Neumann}
\Delta \chi =\operatorname{div}\boldsymbol{\varphi}\quad \text{in } \Omega\quad
\text{and}\quad
\frac{\partial \,\chi}{\partial \mathbf{n}}=0 \quad\text{on } \Gamma.
 \end{equation}
It is shown in \cite[Lemma 3.7 and Theorem 3.9]{vecteur potentiel vivet}
 that  this problem has a unique solution $\chi$ in $W_{-k-1}^{1,2}(\Omega)$
if $k=-1$ and unique up to a constant if $k=0$. Set
\begin{equation}
\widetilde{ \boldsymbol{\varphi}}=\boldsymbol{\varphi}-\nabla \chi.
\end{equation}
It is clear that for $k=0$ and $k=-1$,
${\int}_{\Gamma}\widetilde{ \boldsymbol{\varphi}}\cdot \nabla (w(q)-q)\,d\sigma=0$
 for any $(w(q)-q)\in \mathcal{N}_{k}^{\Delta}$.
Then $\widetilde{\boldsymbol{\varphi}}$ belongs to
$\mathbf{V}^2_{-k-1,T}(\Omega)$. Now, if \eqref{deuxieme form pour xi} holds,
we have
\begin{align*}
\int_{\Omega}\operatorname{\bf curl}\boldsymbol{\xi}\cdot\operatorname{\bf curl}
\boldsymbol{\varphi}\,\text{d}\mathbf{x} =\int_{\Omega}\operatorname{\bf curl}
\boldsymbol{\xi}\cdot\operatorname{\bf curl}\widetilde{\boldsymbol{\varphi}}
\,\text{d}\mathbf{x}&= \int_{\Omega}\mathbf{v} \cdot\operatorname{\bf curl}
\boldsymbol{\varphi}\,\text{d}\mathbf{x}.
\end{align*}
Hence, problem \eqref{formulation variationnelle pour prob en xi}
and problem \eqref{deuxieme form pour xi} are equivalent.
This implies that problem \eqref{problem en xi} has a unique solution
$\boldsymbol{\xi}$ in $\mathbf{W}_{k}^{1,2}(\Omega)$ for $k=0$ or $k=-1$.

\subsection*{Regularity} Now, we suppose that
$\mathbf{v}\in\mathbf{W}_{k+1}^{1,2}(\Omega)\hookrightarrow
\mathbf{W}_{k}^{0,2}(\Omega)$ and $\Omega'$ is of class
$\mathcal{C}^{2,1}$. Let $\boldsymbol{\xi}\in\mathbf{W}_{k}^{1,2}(\Omega)$
the weak solution of \eqref{problem en xi} and we set
$\mathbf{z}=\operatorname{\bf curl}\boldsymbol{\xi}-\mathbf{v}$.
It is clear that $\mathbf{z}$ belongs to $\mathbf{X}^2_{k,N}(\Omega)$.
Applying Theorem \ref{injection in X_N  homogene}, we obtain that
$\mathbf{z}\in\mathbf{W}_{k+1}^{1,2}(\Omega)$ and using
\eqref{inegalite  injection X_N} and \eqref{estimation du problem bien definie}
we obtain that
\begin{equation}\label{estimation de z et de v}
\begin{aligned}
\| \mathbf{z}\| _{\mathbf{W}_{k+1}^{1,2}(\Omega)}
&\leq  C \Big( \| \mathbf{z}\| _{\mathbf{W}_{k}^{0,2}(\Omega)}
+\|\operatorname{div}\mathbf{z}\|_{W_{k+1}^{0,2}(\Omega)}\Big) \\
&\leq  C  \Big( \| \operatorname{\bf curl}\boldsymbol{\xi}
\| _{\mathbf{W}_{k}^{0,2}(\Omega)}
+\| \mathbf{v}\| _{ \mathbf{W}_{k}^{0,2}(\Omega)}
+\|\operatorname{div}\mathbf{v}\|_{W_{k+1}^{0,2}(\Omega)}\Big) \\
&\leq  C\| \mathbf{v}\| _{ \mathbf{W}_{k+1}^{1,2}(\Omega)}  .
\end{aligned}
\end{equation}
This implies that $\boldsymbol{\xi}$ satisfies
\begin{equation*}
\boldsymbol{\xi}\in \mathbf{W}_{k}^{1,2}(\Omega),\quad
\operatorname{div}\boldsymbol{\xi}=0\in\mathbf{W}_{k+1}^{1,2}(\Omega),\quad
\operatorname{\bf curl}\boldsymbol{\xi}\in\mathbf{W}_{k+1}^{1,2}(\Omega),\quad
\boldsymbol{\xi}\cdot\mathbf{n}=0\quad\text{on } \Gamma.
\end{equation*}
Applying \cite[Corollary 3.16]{vecteur potentiel vivet},
we deduce that $\boldsymbol{\xi}$ belongs to $\mathbf{W}_{k+1}^{2,2}(\Omega)$
and using \eqref{estimation de z et de v}, we obtain
\begin{align*}
\| \boldsymbol{\xi}\| _{\mathbf{W}_{k+1}^{2,2}(\Omega)}
& \leq  C  \Big( \| \boldsymbol{\xi}\| _{\mathbf{W}_{k}^{1,2}(\Omega)}
+\|\operatorname{\bf curl}\boldsymbol{\xi}\|_{\mathbf{W}_{k+1}^{1,2}(\Omega)}\Big)
\\
& \leq  C \Big( \| \mathbf{v}\| _{\mathbf{W}_{k}^{0,2}(\Omega)}
+\|\mathbf{z}\|_{\mathbf{W}_{k+1}^{1,2}(\Omega)}
+ \| \mathbf{v}\| _{\mathbf{W}_{k+1}^{1,2}(\Omega)} \Big)\\
& \leq  \| \mathbf{v}\| _{ \mathbf{W}_{k+1}^{1,2}(\Omega)}.
\end{align*}
This completes the proof of the theorem.
\end{proof}

As consequence, we can prove other imbedding results.
We start by the following theorem.


\begin{theorem}\label{injection in X_N non homogene}
Let $k=-1$ or $k=0$. Then the space $\mathbf{Z}_{k,N}^2(\Omega)$
is continuously imbedded in $\mathbf{W}_{k+1}^{1,2}(\Omega)$ and we have
the following estimate for any $\mathbf{v}$ in $\mathbf{Z}_{k,N}^2(\Omega)$:
\begin{equation}\label{inequality Z_N}
\begin{aligned}
\|{\mathbf{v}}\|_{\mathbf{W}_{k+1}^{1,2}(\Omega)}
&\leq  C\big(\|\mathbf{v}\|_{\mathbf{W}_{k}^{0,2}(\Omega)}
  + \|\operatorname{\bf curl}{\mathbf{v}}\|_{\mathbf{W}_{k+1}^{0,2}(\Omega)}\\
&\quad +\| \operatorname{div}{\mathbf{v}}\|_{{W}_{k+1}^{0,2}(\Omega)}
 +\|\mathbf{v}\times\mathbf{n}\|_{\mathbf{H}^{1/2}(\Gamma)}\big).
\end{aligned}
\end{equation}
\end{theorem}

\begin{proof}
Let $k=-1$ or $k=0$ and let $\mathbf{v}$ be any function of
$\mathbf{Z}^2_{k,N}(\Omega)$. We set
$\mathbf{z}=\operatorname{\bf curl}\boldsymbol{\xi}-\mathbf{v}$ where
 $\boldsymbol{\xi} \in \mathbf{W}_{k}^{1,2}(\Omega)$ is the solution of
the problem \eqref{problem en xi}. Hence, $\mathbf{z}$ belongs to the space
$\mathbf{X}^2_{k,N}(\Omega)$. By Theorem \ref{injection in X_N  homogene}
and \eqref{inegalite  injection X_N}, $\mathbf{z}$ even belongs to
 $\mathbf{W}_{k+1}^{1,2}(\Omega)$ with the estimate
\begin{equation}\label{estim z}
 \|\mathbf{z}\|_{\mathbf{W}_{k+1}^{1,2}(\Omega)}
\leq C\big(\|\mathbf{z}\|_{\mathbf{W}_{k}^{0,2}(\Omega)}
+\|\operatorname{div}\mathbf{z}\|_{W_{k+1}^{0,2}(\Omega)}
+\|\operatorname{\bf curl}\mathbf{z}\|_{\mathbf{W}_{k+1}^{0,2}(\Omega)}\big).
\end{equation}
Then, it suffices to prove that
$\operatorname{\bf curl}\boldsymbol{\xi}\in\mathbf{W}_{k+1}^{1,2}(\Omega)$
 to obtain $\mathbf{v}\in\mathbf{W}_{k+1}^{1,2}(\Omega)$.
Setting $\boldsymbol{\omega}=\operatorname{\bf curl}\boldsymbol{\xi}$.
It is clear that
\begin{equation}\label{condition du bord du problem EN}
{\int}_{\Gamma}\boldsymbol{\omega}\cdot \mathbf{n}\,d\sigma=0
\end{equation}
and then $\boldsymbol{\omega}$ satisfies
\begin{equation}\label{problem de omega}
\begin{gathered}
-\Delta \boldsymbol{\omega} = \operatorname{\bf curl}\operatorname{\bf curl}
\mathbf{v}\quad \text{and}\quad\operatorname{div}\boldsymbol{\omega}  =0
\quad \text{in }\Omega\\
\boldsymbol{\omega}\times\mathbf{n}=\mathbf{v}\times\mathbf{n}\quad\text{on }
 \Gamma\quad \text{and}\quad {\int}_{\Gamma}(\boldsymbol{\omega}\cdot
 \mathbf{n})q\,d\sigma=0,\,\forall \, q \in\mathcal{P}_{-k-1}^{\Delta}.
 \end{gathered}
\end{equation}
Note that $\operatorname{\bf curl}\mathbf{v}\in\mathbf{W}_{k+1}^{0,2}(\Omega)$
then $\operatorname{\bf curl}\operatorname{\bf curl}\mathbf{v}$ is in
 $[\mathring{\mathbf{H}}_{-k-2}^2(\operatorname{\bf curl},\Omega)]'$ and we
 have $\mathbf{v}\times\mathbf{n}\in\mathbf{H}^{1/2}(\Gamma)$.
Since $\boldsymbol{\mathcal{D}}(\Omega)$ is dense in
$\mathring{\mathbf{H}}_{-k-2}^2(\operatorname{\bf curl},\Omega)$, we prove that
\begin{equation*}
\forall\boldsymbol{\phi}\in\mathbf{Y}^2_{k+2,N}(\Omega),\quad
\langle \operatorname{\bf curl}\operatorname{\bf curl}\mathbf{v},
 \boldsymbol{\phi}\rangle_{[\mathring{\mathbf{H}}_{-k-2}^2
(\operatorname{\bf curl},\Omega)]'\times \mathring{\mathbf{H}}_{-k-2}^2
(\operatorname{\bf curl},\Omega)}=0.
 \end{equation*}
Due to Corollary \ref{existence E_N non homogene 2}, the function
$\boldsymbol{\omega}$ belongs to $\mathbf{W}_{k+1}^{1,2}(\Omega)$
and satisfies the estimate
\begin{equation}\label{estim en w}
\begin{aligned}
\|\boldsymbol{\omega}\|_{\mathbf{W}_{k+1}^{1,2}(\Omega)}
&\leq  C \big( \|\operatorname{\bf curl}\operatorname{\bf curl}
\mathbf{v}\|_{[\mathring{\mathbf{H}}_{-k-2}^2(\operatorname{\bf curl},\Omega)]'}
+ \|\mathbf{v}\times\mathbf{n}\|_{\mathbf{H}^{1/2}(\Gamma)}\big)\\
& \leq  C \big( \|\operatorname{\bf curl}\mathbf{v}\|_{\mathbf{W}_{k+1}^{0,2}
(\Omega)}+ \|\mathbf{v}\times\mathbf{n}\|_{\mathbf{H}^{1/2}(\Gamma)}\big).
\end{aligned}
\end{equation}
Finally, estimate \eqref{inequality Z_N} can be deduced by using inequalities
 \eqref{estim z} and \eqref{estim en w}.
\end{proof}

Before giving the second imbedding result, we need to introduce the following
space for any integer $k$ in $\mathbb{Z}$,
\begin{align*}
\mathbf{M}_{k,N}^2(\Omega)
=\Big\{ &\mathbf{v}\in \mathbf{W}_{k+1}^{1,2}(\Omega),\;
\operatorname{div}\mathbf{v}\in W_{k+2}^{1,2}(\Omega),\;
\operatorname{\bf curl}\mathbf{v}\in \mathbf{W}_{k+2}^{1,2}(\Omega),\\
&\mathbf{v}\times \mathbf{n}\in \mathbf{H}^{3/2}(\Gamma)\Big\}\,.
\end{align*}

\begin{proposition}\label{Injection de M_k,N}
Suppose that $\Omega'$ is of class $C^{2,1}$. Then the space
$\mathbf{M}_{-1,N}^2(\Omega)$ is continuously imbedded in
$\mathbf{W}_1^{2,2}(\Omega)$ and we have the following estimate for any
 $\mathbf{v}$ in $\mathbf{M}_{-1,N}^2(\Omega)$,
\begin{equation}\label{inequality M_N}
 \|{\mathbf{v}}\|_{\mathbf{W}_1^{2,2}(\Omega)}
\leq  C\big(\|\mathbf{v}\|_{\mathbf{W}_{0}^{1,2}(\Omega)}
 + \|\operatorname{\bf curl}{\mathbf{v}}\|_{\mathbf{W}_1^{1,2}(\Omega)}
 +\| \operatorname{div}{\mathbf{v}}\|_{{W}_1^{1,2}(\Omega)}
 +\|\mathbf{v}\times\mathbf{n}\|_{\mathbf{H}^{\,3/2}(\Gamma)}\big).
\end{equation}
\end{proposition}

\begin{proof}
The proof is very similar to that of Theorem \ref{injection in X_N non homogene}.
 Let $\mathbf{v}$ be any function of $\mathbf{M}^2_{-1,N}(\Omega)$ and set
$\mathbf{z}=\operatorname{\bf curl}\boldsymbol{\xi}-\mathbf{v}$ where
$\boldsymbol{\xi} \in \mathbf{W}_{0}^{2,2}(\Omega)$ is the solution of the
problem \eqref{problem en xi}. According to
\cite[Corollary 3.14]{vecteur potentiel vivet}, we prove that
$\mathbf{z}$ belongs to $\mathbf{W}_1^{2,2}(\Omega)$ with the estimate
\begin{equation}\label{estim z 2}
 \|\mathbf{z}\|_{\mathbf{W}_1^{2,2}(\Omega)}
\leq C\big(\|\mathbf{z}\|_{\mathbf{W}_{0}^{1,2}(\Omega)}
+\|\operatorname{div}\mathbf{z}\|_{W_1^{1,2}(\Omega)}
+\|\operatorname{\bf curl}\mathbf{z}\|_{\mathbf{W}_1^{1,2}(\Omega)}\big).
\end{equation}
Then, it suffices to prove that
$\operatorname{\bf curl}\boldsymbol{\xi}\in\mathbf{W}_1^{2,2}(\Omega)$
to obtain $\mathbf{v}\in\mathbf{W}_1^{2,2}(\Omega)$.
We set $\boldsymbol{\omega}=\operatorname{\bf curl}\boldsymbol{\xi}$.
Using \cite[Theorem 3.1]{vecteur potentiel vivet}, we prove that
$\boldsymbol{\omega}$ satisfies Problem \eqref{problem de omega}.
Using the regularity of Corollary \ref{existence E_N non homogene 2},
we prove that $\boldsymbol{\omega}$ belongs to $\mathbf{W}_1^{2,2}(\Omega)$
and satisfies
\begin{equation*}
 \| \boldsymbol{\omega}\| _{\mathbf{W}_{-k}^{1,2}(\Omega)}
\leqslant C \Big( \| \operatorname{\bf curl}\operatorname{\bf curl}\mathbf{v}
\|_{\mathbf{W}_1^{0,2}(\Omega)}+\|\mathbf{v}\times\mathbf{n}\|_{\mathbf{H}^{3/2}
(\Gamma)}\Big)
\end{equation*}
and then estimate \eqref{inequality M_N} follows from \eqref{estim z 2}.
\end{proof}

\section{Generalized solutions for \eqref{eST} and \eqref{eSN}}

We start this sequel by introducing the  space
$$
\mathbf{E}^2(\Omega)=\{\mathbf{v}\in\mathbf{W}_{0}^{1,2}(\Omega):
\Delta \mathbf{v}\in[\mathring{\mathbf{H}}_{-1}^2(\operatorname{div},\Omega)]'\}.
$$
This is a Banach space with the norm
$$
\|\mathbf{v}\|_{\mathbf{E}^2(\Omega)}=\|\mathbf{v}\|_{\mathbf{W}_{0}^{1,2}
(\Omega)}+\|\Delta \mathbf{v}\|_{[\mathring{\mathbf{H}}_{-1}^2
(\operatorname{div},\,\Omega)]'}.
$$
We have the following preliminary result.

\begin{lemma}\label{desite dans W^1,p avec Delta dans (H^p'_0(curl))'}
The space $\boldsymbol{\mathcal{D}}(\overline{\Omega})$ is dense in
$\mathbf{E}^2(\Omega)$.
\end{lemma}

\begin{proof}
Let $P$ be a continuous linear mapping from $\mathbf{W}_{0}^{1,2}(\Omega)$
to $\mathbf{W}_{0}^{1,2}(\mathbb{R}^{3})$, such that
$P\mathbf{v} |_{\Omega}=\mathbf{v}$ and let
$\boldsymbol{\ell} \in (\mathbf{E}^2(\Omega))'$, such that for any
$\mathbf{v} \in \boldsymbol{\mathcal{D}}(\overline{\Omega})$,
 we have $\langle \boldsymbol{\ell},\mathbf{v}\rangle =0$.
We want to prove that $\boldsymbol{\ell}=\mathbf{0}$ on $\mathbf{E}^2(\Omega)$.
Then there exists $(\mathbf{f},\mathbf{g}) \in
\mathbf{W}_{0}^{-1,2}(\mathbb{R}^{3}) \times \mathring{\mathbf{H}}_{-1}^2
(\operatorname{div},\,\Omega)$ such that: for any
$\mathbf{v}\in \mathbf{E}^2(\Omega)$,
\begin{equation*}
\langle \boldsymbol{\ell},\mathbf{v}\rangle
=\langle  \mathbf{f},P\, \mathbf{v} \rangle _{\mathbf{W}_{0}^{-1,2}
(\mathbb{R}^{3})\times\mathbf{W}_{0}^{1,2}(\mathbb{R}^{3})}
+\langle \Delta \mathbf{v},\mathbf{g}\rangle _{ [\mathring{\mathbf{H}}_{-1}^2
(\operatorname{div},\,\Omega)]'\times \mathring{\mathbf{H}}_{-1}^2
(\operatorname{div},\,\Omega)}.
\end{equation*}
Observe that we can easily extend by zero the function  $\mathbf{g}$ in such a
 way that $\widetilde{\mathbf{g}} \in \mathbf{H}_{-1}^2(\operatorname{div},
\mathbb{R}^{3})$. Now we take
$\boldsymbol{\varphi} \in \boldsymbol{\mathcal{D}}(\mathbb{R}^{3})$.
Then we have by assumption that
\begin{equation*}
\langle \mathbf{f},\boldsymbol{\varphi}\rangle_{\mathbf{W}_{0}^{-1,2}
(\mathbb{R}^{3})\times\mathbf{W}_{0}^{1,2}(\mathbb{R}^{3})}
+  \int_{\mathbb{R}^{3}} \widetilde{\mathbf{g}} \cdot
\Delta \boldsymbol{\varphi}dx =0,
\end{equation*}
because $\langle \mathbf{f},\boldsymbol{\varphi}\rangle =\langle  \mathbf{f},
P \mathbf{v} \rangle $ where $\mathbf{v}=\boldsymbol{\varphi}|_{\Omega}$.
Thus we have $\mathbf{f}+\Delta \widetilde{\mathbf{g}}=\mathbf{0}$ in
$\boldsymbol{\mathcal{D}}'(\mathbb{R}^{3})$. Then we can deduce that
$\Delta \widetilde{\mathbf{g}}=-\mathbf{f} \in
\mathbf{W}_{0}^{-1,2}(\mathbb{R}^{3})$ and due to \cite[Theorem 1.3]{AGG1},
there exists a unique $\boldsymbol{\lambda} \in
\mathbf{W}_{0}^{1,2}(\mathbb{R}^{3})$ such that
$\Delta \boldsymbol{\lambda}=\Delta \widetilde{\mathbf{g}}$.
Thus the harmonic function $\boldsymbol{\lambda}-\widetilde{\mathbf{g}}$
belonging to $\mathbf{W}_{-1}^{0,2}(\mathbb{R}^{3})$ is necessarily equal
to zero. Since $\mathbf{g} \in \mathbf{W}_{0}^{1,2}(\Omega)$ and
$\widetilde{\mathbf{g}} \in \mathbf{W}_{0}^{1,2}(\mathbb{R}^{3})$,
we deduce that $\mathbf{g} \in \mathring{\mathbf{W}}_{0}^{1,2}(\Omega)$.
As $\boldsymbol{\mathcal{D}}(\Omega)$ is dense in
$\mathring{\mathbf{W}}_{0}^{1,2}(\Omega)$, there exists a sequence
$\mathbf{g}_{k} \in \boldsymbol{\mathcal{D}}(\Omega)$ such that
$\mathbf{g}_{k}\to\mathbf{g}$ in $\mathbf{W}_{0}^{1,2}(\Omega)$,
when $k\to\infty$. Then $\nabla \cdot \mathbf{g}_{k}\to \nabla \cdot \mathbf{g}$
in $L^2(\Omega)$. Since $\mathbf{W}_{0}^{1,2}(\Omega)$ is imbedded in
$\mathbf{W}_{-1}^{0,2}(\Omega)$, we deduce that $\mathbf{g}_{k}\to\mathbf{g}$
in $\mathbf{H}_{-1}^2(\operatorname{div},\,\Omega)$.
Now, we consider $\mathbf{v} \in \mathbf{E}^2(\Omega)$ and we want to prove
that $\langle  \boldsymbol{\ell},\mathbf{v}\rangle=0$.
Observe that:
\begin{align*}
 \langle \boldsymbol{\ell},\mathbf{v}\rangle
&=-\langle \Delta \widetilde{\mathbf{g}},P \mathbf{v}
\rangle _{\mathbf{W}_{0}^{-1,2}(\mathbb{R}^{3})\times
\mathbf{W}_{0}^{1,2}(\mathbb{R}^{3})} +\langle \Delta \mathbf{v},
\mathbf{g}\rangle _{[\mathring{\mathbf{H}}_{-1}^2(\operatorname{div},
\Omega)]'\times \mathring{\mathbf{H}}_{-1}^2(\operatorname{div},\Omega)}
\\
&=\lim _{k\to\infty}(- \int_{\Omega} \Delta \mathbf{g}_{k}\cdot
\mathbf{v} dx+\langle \Delta \mathbf{v},\mathbf{g}_{k}
 \rangle _{[\mathring{\mathbf{H}}_{-1}^2(\operatorname{div},\,\Omega)]'
 \times \mathring{\mathbf{H}}_{-1}^2(\operatorname{div},\,\Omega)}
\\
&=\lim _{k\to\infty}(- \int_{\Omega} \Delta \mathbf{g}_{k}\cdot \mathbf{v} dx
+ \int_{\Omega} \mathbf{v} \cdot \Delta \mathbf{g}_{k} dx )=0.
\end{align*}
\end{proof}

As a consequence, we have the following result.

\begin{corollary}\label{lemme formule de grenn dans E^2}
 The linear mapping $\gamma :\mathbf{v}\to\operatorname{\bf curl}\mathbf{v}
\vert_{\Gamma}\times\mathbf{n}$ defined on
$\boldsymbol{\mathcal{D}}(\overline{\Omega})$ can be extended to a linear
continuous mapping
$$
\gamma:\mathbf{E}^2(\Omega)\to \mathbf{H}^{-1/2}(\Gamma).
$$
Moreover, we have the Green formula:for any
$\mathbf{v}\in\mathbf{E}^2(\Omega)$ and any
$\boldsymbol{\varphi}\in \mathbf{W}_{0}^{1,2}(\Omega)$ such that
$\operatorname{div}\boldsymbol{\varphi}=0$ in $\Omega$ and
$\boldsymbol{\varphi}\cdot \mathbf{n}=0$ on $\Gamma$,
\begin{equation}\label{formule de grenn dans E^p}
-\langle\Delta \mathbf{v},\,\boldsymbol{\varphi}
\rangle _{[\mathring{\mathbf{H}}_{-1}^2(\mathrm {div},\Omega)]'
\times\mathring{\mathbf{H}}_{-1}^2(\mathrm {div},\,\Omega)}
=\int_{\Omega}\,\operatorname{\bf curl}\mathbf{v}\cdot\operatorname{\bf curl}
\boldsymbol{\varphi}\,\text{d}\mathbf{x}-\langle\operatorname{\bf curl}
\mathbf{v}\times\mathbf{n},\,\boldsymbol{\varphi}\rangle_{\Gamma},
\end{equation}
where the duality on $\Gamma$ is defined by
$\langle\cdot,\cdot\rangle_{\Gamma}
=\langle\cdot,\cdot\rangle_{\mathbf{H}^{-1/2}(\Gamma)
\times\mathbf{H}^{1/2}(\Gamma)}$.
\end{corollary}

\begin{proof}
Let $\mathbf{v} \in \boldsymbol{\mathcal{D}}(\overline{\Omega})$.
 Observe that if  $\boldsymbol{\varphi}\in \mathbf{W}_{0}^{1,2}(\Omega)$
such that $\boldsymbol{\varphi}\cdot \mathbf{n}=0$ on $\Gamma$ we deduce
that $\boldsymbol{\varphi}\in \mathbf{X}_{-1,T}^2(\Omega)$,
then \eqref{formule de grenn dans E^p} holds for such $\boldsymbol{\varphi} $.
Now, let $\boldsymbol{\mu} \in \mathbf{H}^{1/2}(\Gamma)$, then there exists
$\boldsymbol{\varphi}\in \mathbf{W}_{0}^{1,2}(\Omega)$ such that
$\boldsymbol{\varphi}=\boldsymbol{\mu}_{t}$ on $\Gamma$ and that
$\operatorname{div}\boldsymbol{\varphi}=0$ with
\begin{equation}\label{estimation de relevement n1}
\|\boldsymbol{\varphi}\|_{\mathbf{W}_{0}^{1,2}(\Omega)}
\leqslant C \|\boldsymbol{\mu}_{t}\|_{\mathbf{H}^{1/2}(\Gamma)}
\leqslant C \|\boldsymbol{\mu}\|_{\mathbf{H}^{1/2}(\Gamma)}.
\end{equation}
As a consequence, using \eqref{formule de grenn dans E^p}, we have
\begin{equation*}
|\langle \operatorname{\bf curl}\mathbf{v}\times\mathbf{n},
\boldsymbol{\mu}\rangle_{\Gamma}|\leqslant C\|\mathbf{v}
\|_{\mathbf{E}^2(\Omega)}\|\boldsymbol{\mu}\|_{\mathbf{H}^{1/2}(\Gamma)}.
\end{equation*}
Thus,
\begin{equation*}
\| \operatorname{\bf curl}\mathbf{v}\times\mathbf{n}\|_{\mathbf{H}^{-1/2}
(\Gamma)}|\leqslant C\|\mathbf{v}\|_{\mathbf{E}^2(\Omega)}.
\end{equation*}
We deduce that the linear mapping $\gamma$ is continuous for the norm
$\mathbf{E}^2(\Omega)$. Since $\boldsymbol{\mathcal{D}}(\overline{\Omega})$
is dense in $\mathbf{E}^2(\Omega)$, $\gamma$ can be extended to by continuity
to $\gamma \in \mathcal{L}(\mathbf{E}^2(\Omega),\mathbf{H}^{-1/2}(\Gamma))$
and formula \eqref{formule de grenn dans E^p} holds for all
$\mathbf{v}\in\mathbf{E}^2(\Omega)$ and
$\boldsymbol{\varphi}\in \mathbf{W}_{0}^{1,2}(\Omega)$ such that
$\operatorname{div}\boldsymbol{\varphi}=0$ in $\Omega$ and
$\boldsymbol{\varphi}\cdot \mathbf{n}=0$ on $\Gamma$.
\end{proof}

\begin{proposition}
\label{proposition probleme faible et fort sans pression non homogene cas K^p_T}
 Let $\mathbf{f}$ belongs to $\mathbf{W}_1^{0,2}(\Omega)$ with
$\operatorname{div}\mathbf{f}=0$ in $\Omega$, $g\in H^{1/2}(\Gamma)$ and
$\mathbf{h}\in\mathbf{H}^{-1/2}(\Gamma)$ verify the  compatibility
conditions: for any $\mathbf{v}\in\mathbf{Y}^2_{1,T}(\Omega)$,
\begin{gather}
\int_{\Omega}\mathbf{f}\cdot\mathbf{v}\,\text{d}\mathbf{x}
 +\langle\mathbf{h}\times\mathbf{n},\,\mathbf{v}\rangle_{\mathbf{H}^{-1/2}
 (\Gamma)\times\mathbf{H}^{1/2}(\Gamma)}=0,
\label{condition de compatibilite cas X^p_T}\\
\mathbf{f}\cdot\mathbf{n}+\operatorname{div}_{\Gamma}
(\mathbf{h}\times\mathbf{n})=0\quad \text{on }\Gamma,
\label{condition de compatibilite avec divergence surfacique}
\end{gather}
where $\operatorname{div}_{\Gamma}$ is the surface divergence on $\Gamma$.
Then, the problem
\begin{equation} \label{eET}
\begin{gathered}
  -\Delta \mathbf{z}=\mathbf{f}\quad\text{and}\quad
\operatorname{div}\mathbf{z}=0 \quad\text{in }\Omega,\\
\mathbf{z}\cdot\mathbf{n}=g\quad\text{and}\quad
\operatorname{\bf curl}\mathbf{z}\times\mathbf{n}=\mathbf{h}\times\mathbf{n}
\quad\text{on }\Gamma ,
   \end{gathered}
\end{equation}
has a unique solution $\mathbf{z}$ in $\mathbf{W}_{0}^{1,2}(\Omega)$
satisfying the estimate
\begin{equation}
\label{estimation solution probleme sans pression non homogene cas X^p_T faible}
\| \mathbf{z}\| _{\mathbf{W}_{0}^{1,2}(\Omega)}
\leq C\Big( \| \mathbf{f}\| _{\mathbf{W}_1^{0,2}(\Omega)}
+\|  g\| _{H^{1/2}(\Gamma)}+\| \mathbf{h}
\times\mathbf{n}\| _{\mathbf{H}^{-1/2}(\Gamma)}\Big).
\end{equation}
Moreover, if $\mathbf{h}$ in $\mathbf{H}^{1/2}(\Gamma)$, $g$ in
$H^{3/2}(\Gamma)$ and $\Omega'$ is of class $C^{2,1}$, then the solution
$\mathbf{z}$ is in $\mathbf{W}_1^{2,2}(\Omega)$ and satisfies the estimate
 \begin{equation}
\label{estimation solution probleme sans pression non homogene cas X^p_T fort}
 \| \mathbf{z}\| _{\mathbf{W}_1^{2,2}(\Omega)}
\leq C\Big( \| \mathbf{f}\| _{\mathbf{W}_1^{0,2}(\Omega)}
+\|  g\| _{H^{\,3/2}(\Gamma)}
+\| \mathbf{h}\times\mathbf{n}\| _{\mathbf{H}^{1/2}(\Gamma)}\Big).
\end{equation}
\end{proposition}

\begin{proof}
First, note that if $\mathbf{h}\in\mathbf{H}^{-1/2}(\Gamma)$, then
$\mathbf{h}\times \mathbf{n}$ also belongs to $\mathbf{H}^{-1/2}(\Gamma)$.
On the other hand, let us consider the Neumann problem:
\begin{equation} \label{eN}
\Delta \theta=0\quad \text{in }\Omega\quad\text{and}\quad
\frac{\partial\,\theta}{\partial\,\mathbf{n}}=g\quad\text{on }\Gamma.
\end{equation}
It is shown in \cite[Theorem 3.9]{vecteur potentiel vivet},
that this problem has a unique solution $\theta\in W_{0}^{2,2}(\Omega)/\mathbb{R}$
satisfying the estimate
\begin{equation}\label{estimation solution neumann}
 \| \theta\| _{W_{0}^{2,2}(\Omega)}\leq C\|  g\| _{H^{1/2}(\Gamma)}.
\end{equation}
Setting $\boldsymbol{\xi}=\mathbf{z}-\nabla \theta$, then problem \eqref{eET}
 becomes: find $\boldsymbol{\xi} \in \mathbf{W}_{0}^{1,2}(\Omega)$ such that
\begin{equation}\label{problem ET avec g=0}
 \begin{gathered}
  -\Delta \boldsymbol{\xi}=\mathbf{f}\quad\text{and}\quad
\operatorname{div}\boldsymbol{\xi}=0 \quad\text{in }\Omega,\\
\boldsymbol{\xi}\cdot\mathbf{n}=0\quad\text{and}\quad\operatorname{\bf curl}
\boldsymbol{\xi}\times\mathbf{n}=\mathbf{h}\times\mathbf{n}\quad\text{on }\Gamma .
\end{gathered}
\end{equation}
Now, observe that problem \eqref{problem ET avec g=0} is reduced to the
variational problem: Find $\boldsymbol{\xi} \in \mathbf{V}_{-1,T}^2(\Omega)$
such that
\begin{equation}\label{premiere formulation variationnelle}
  \forall\boldsymbol{\varphi}\in \mathbf{X}^2_{-1,T}(\Omega)\quad
\int_{\Omega}\operatorname{\bf curl}\boldsymbol{\xi}\cdot\operatorname{\bf curl}
\boldsymbol{\varphi}\,\text{d}\mathbf{x}=\int_{\Omega}\mathbf{f}
\cdot\boldsymbol{\varphi}\,\text{d}\mathbf{x}
+\langle \mathbf{h}\times\mathbf{n},\boldsymbol{\varphi}\rangle_{\Gamma}.
\end{equation}
Indeed, every solution of \eqref{problem ET avec g=0}  also solves
\eqref{premiere formulation variationnelle}.  Conversely,
let $\boldsymbol{\xi}$ a solution of the problem
\eqref{premiere formulation variationnelle}. Then,
\begin{equation*}
 \forall \boldsymbol{\varphi} \in\boldsymbol{\mathcal{D}}(\Omega),\quad
\langle\operatorname{\bf curl}\operatorname{\bf curl}\boldsymbol{\xi}
-\mathbf{f},\boldsymbol{\varphi}\rangle_{\boldsymbol{\mathcal{D}}'
(\Omega)\times\boldsymbol{\mathcal{D}}(\Omega)}=0.
\end{equation*}
So $-\Delta \boldsymbol{\xi} =\mathbf{f}$ in $\Omega$. Moreover, by the
fact that $\boldsymbol{\xi}$ belongs to the space $\mathbf{V}^2_{-1,T}(\Omega)$,
we have $\operatorname{div}\boldsymbol{\xi}=0$ in $\Omega$ and
$\boldsymbol{\xi}\cdot\mathbf{n}=0$ on $\Gamma$. Then, it remains to verify
the boundary condition
$\operatorname{\bf curl} \boldsymbol{\xi}\times\mathbf{n}
=\mathbf{h}\times\mathbf{n}\,\,\text{on }\Gamma$. Observe that
$ \boldsymbol{\xi}$ belongs to $\mathbf{E}^2(\Omega)$ so by
\eqref{formule de grenn dans E^p} and comparing with
\eqref{premiere formulation variationnelle} we deduce that for any
$\boldsymbol{\varphi}\in\mathbf{X}^2_{-1,T}(\Omega)$, we have
\begin{align*}
\langle \operatorname{\bf curl}\boldsymbol{\xi}\times\mathbf{n},
\boldsymbol{\varphi}\rangle_{\Gamma}
=\langle \mathbf{h}\times\mathbf{n},\,\boldsymbol{\varphi}\rangle_{\Gamma}.
\end{align*}
Proceeding as in the proof of Theorem
\ref{resolution du probleme de stokes sans pression}, we prove that
$\operatorname{\bf curl}\boldsymbol{\xi}\times\mathbf{n}
=\mathbf{h}\times\mathbf{n}$ on $\Gamma$.

On the other hand, let us introduce the following problem:
Find $\boldsymbol{\xi} \in \mathbf{V}_{-1,T}^2(\Omega)$ such that
\begin{equation}\label{deuxieme formulation vart}
 \forall\boldsymbol{\varphi}\in \mathbf{V}^2_{-1,T}(\Omega)
\quad\int_{\Omega}\operatorname{\bf curl}\boldsymbol{\xi}
\cdot\operatorname{\bf curl}\boldsymbol{\varphi}\,\text{d}\mathbf{x}
=\int_{\Omega}\mathbf{f}\cdot\boldsymbol{\varphi}\,\text{d}\mathbf{x}
+\langle \mathbf{h}\times\mathbf{n},\boldsymbol{\varphi}\rangle_{\Gamma}.
\end{equation}
As in the proof of Theorem \ref{resolution du probleme de stokes sans pression},
we use Lax-Milgram lemma to prove the existence of a unique solution
$\boldsymbol{\xi}$ in $\mathbf{V}^2_{-1,T}(\Omega)$ of Problem
\eqref{deuxieme formulation vart}. Using Theorem
\ref{injection in X_T  homogene}, we prove that this solution $\boldsymbol{\xi}$
belongs to $\mathbf{W}_{0}^{1,2}(\Omega)$ and the following estimate
follows immediately
\begin{equation}\label{estimation du prob homegene sans pression}
\|\boldsymbol{\xi}\|_{\mathbf{W}_{0}^{1,2}(\Omega)}
\leqslant C\Big( \|\mathbf{f}\|_{\mathbf{W}_1^{0,2}(\Omega)}
+\| \mathbf{h}\times\mathbf{n}\| _{\mathbf{H}^{-1/2}(\Gamma)}\Big).
\end{equation}

Next, we  extend \eqref{deuxieme formulation vart} to any test function
$\boldsymbol{\varphi}$ in $\mathbf{X}^2_{-1,T}(\Omega)$.
 Let $\widetilde{\boldsymbol{\varphi}} \in \mathbf{X}^2_{-1,T}(\Omega)$
and let us solve  the exterior Neumann problem:
\begin{equation}\label{probleme de Neumann en xi}
 \Delta \chi =\operatorname{div}\widetilde{\boldsymbol{\varphi}}\quad
\text{in }\Omega\quad\text{and}\quad
  \frac{\partial \chi}{\partial \mathbf{n}}=0 \quad\text{ on }\Gamma.
 \end{equation}
It is shown in \cite[Theorem 4.12]{these Giroire}
 that  this problem has a unique solution $\chi$ in $W_{0}^{2,2}(\Omega)$
up to an additive constant. Then, we set
\begin{equation}
 \boldsymbol{\varphi}=\widetilde{\boldsymbol{\varphi}}-\nabla \chi.
\end{equation}
Since $W_{0}^{2,2}(\Omega)$ is imbedded in $W_{-1}^{1,2}(\Omega)$, then $\boldsymbol{\varphi}$ belongs to $\mathbf{V}^2_{-1,T}(\Omega)$. Now, if \eqref{deuxieme formulation vart} holds, we have
\[
 \int_{\Omega}\operatorname{\bf curl}\boldsymbol{\xi}
\cdot\operatorname{\bf curl}\widetilde{\boldsymbol{\varphi}}
\,\text{d}\mathbf{x}
= \int_{\Omega}\mathbf{f}\cdot\widetilde{\boldsymbol{\varphi}}\,\text{d}
\mathbf{x}+\langle \mathbf{h}\times\mathbf{n},
\widetilde{\boldsymbol{\varphi}}\rangle_{\Gamma}
- \int_{\Omega}\mathbf{f}\cdot\,\nabla \chi\text{d}\mathbf{x}
-\langle \mathbf{h}\times\mathbf{n},\,\nabla \chi\rangle_{\Gamma}.
\]
Using \eqref{formule de green v.n} and
\eqref{condition de compatibilite avec divergence surfacique}, we obtain
\begin{equation*} %\label{e5.14}
 \int_{\Omega}\operatorname{\bf curl}\boldsymbol{\xi}\cdot\operatorname{\bf curl}
\widetilde{\boldsymbol{\varphi}}\,\text{d}\mathbf{x}
=\int_{\Omega}\mathbf{f}\cdot\widetilde{\boldsymbol{\varphi}}\,\text{d}
\mathbf{x}+\langle \mathbf{h}\times\mathbf{n},
\widetilde{\boldsymbol{\varphi}}\rangle_{\Gamma}.
\end{equation*}
This implies that problem \eqref{premiere formulation variationnelle}
and problem \eqref{deuxieme formulation vart} are equivalent and thus
problem \eqref{problem ET avec g=0}   has a unique solution $\boldsymbol{\xi}$
in $\mathbf{W}_{0}^{1,2}(\Omega)$. Finally, we set
$\mathbf{z}=\boldsymbol{\xi}+\nabla \theta \in \mathbf{W}_{0}^{1,2}(\Omega)$
the unique solution of \eqref{eET}. Finally,
\eqref{estimation solution probleme sans pression non homogene cas X^p_T faible}
follows immediately from \eqref{estimation du prob homegene sans pression}
and \eqref{estimation solution neumann}.

\subsection*{Regularity of the solution}
We suppose in addition that $\mathbf{h}$ is in $\mathbf{H}^{1/2}(\Gamma)$,
 $g$ in $H^{3/2}(\Gamma)$ and $\Omega'$ is of class $C^{2,1}$ and let
 $\mathbf{z}$ in $\mathbf{W}_{0}^{1,2}(\Omega)$ be the weak solution of Problem
 \eqref{eET}. Setting $\boldsymbol{\omega}=\operatorname{\bf curl}\mathbf{z}$,
then $\boldsymbol{\omega}$ satisfies
\begin{gather*}
\boldsymbol{\omega}\in \mathbf{L}^2(\Omega),\quad
\operatorname{div}\boldsymbol{\omega}=0\in W_1^{0,2}(\Omega),\\\
\operatorname{\bf curl}\boldsymbol{\omega}=\mathbf{f}\in
 \mathbf{W}_1^{0,2}(\Omega),\quad
\boldsymbol{\omega}\times  \mathbf{n}=\operatorname{\bf curl}\mathbf{z}
\times\mathbf{n}\in \mathbf{H}^{1/2}(\Gamma).
\end{gather*}
Applying Theorem \ref{injection in X_N non homogene} (with $k=0$),
we prove that $\boldsymbol{\omega}$ belongs to $\mathbf{W}_1^{1,2}(\Omega)$.
This implies that $\mathbf{z}$ satisfies
\begin{equation*}
\mathbf{z}\in \mathbf{W}_{0}^{1,2}(\Omega),\quad
\operatorname{div}\mathbf{z}=0\in W_1^{1,2}(\Omega),\quad
\operatorname{\bf curl}\mathbf{z}\in \mathbf{W}_1^{1,2}(\Omega)\quad
\mathbf{z}\cdot \mathbf{n}\in H^{3/2}(\Gamma).
\end{equation*}
Applying Proposition \ref{injection in X_T non homogene fort},
we prove that $\mathbf{z}$ belongs to $\mathbf{W}_1^{2,2}(\Omega)$
and we have the estimate
\eqref{estimation solution probleme sans pression non homogene cas X^p_T fort}.
\end{proof}

Next, we solve the Stokes problem \eqref{eST}.

\begin{theorem}[Weak solutions for \eqref{eST}]
\label{theoreme stokes faible homogene condition type1}
Suppose that $g=0$ and $\chi=0$. For $\mathbf{f}$ given in
$[ \mathring{\mathbf{H}}_{-1}^2(\operatorname{div},\Omega)]'$ and
$\mathbf{h}$ given in $\mathbf{H}^{-1/2}(\Gamma)$ satisfying
\eqref{condition de compatibilite cas X^p_T}.
The Stokes problem \eqref{eST} has a unique solution
 $(\mathbf{u},\pi)\in\mathbf{W}_{0}^{1,2}(\Omega)\times L^2(\Omega)$
and we have the estimate
\begin{equation}\label{estimation Stokes  homogene}
 \|\mathbf{u}\| _{\mathbf{W}_{0}^{1,2}(\Omega)} + \|\pi\|_{L^2(\Omega)}
\leq C \big( \| \mathbf{f}\| _{[\mathring{\mathbf{H}}_{-1}^2\operatorname{div},
\Omega)]'}
+ \|\mathbf{h}\times\mathbf{n}\|_{\mathbf{H}^{-1/2}(\Gamma)}\big).
\end{equation}
\end{theorem}

\begin{proof}
 At first, observe that problem \eqref{eST} is reduced to the
variational problem:
Find  $\mathbf{u} \in \mathbf{V}_{-1,T}^2(\Omega)$ such that
\begin{equation}\label{formulation var pour S_T}
\begin{gathered}
  \forall\boldsymbol{\varphi}\in \mathbf{V}^2_{-1,T}(\Omega),\\
\int_{\Omega}\operatorname{\bf curl}\mathbf{u}\cdot\operatorname{\bf curl}
\boldsymbol{\varphi}\,\text{d}\mathbf{x}
=\langle \mathbf{f},\boldsymbol{\varphi}\rangle _{[\mathring{\mathbf{H}}_{-1}^2
(\operatorname{div},\,\Omega)]'\times \mathring{\mathbf{H}}_{-1}^2
(\operatorname{div},\,\Omega)}
+\langle \mathbf{h}\times\mathbf{n},\,\boldsymbol{\varphi}\rangle_{\Gamma}.
\end{gathered}
\end{equation}
Indeed, every solution of \eqref{eST} also solves
\eqref{formulation var pour S_T}.  Conversely, let $\mathbf{u}$ a solution
of problem \eqref{formulation var pour S_T}. Then
\begin{equation*}
\text{for all $\boldsymbol{\varphi} \in\boldsymbol{\mathcal{D}}(\Omega)$
such that $\operatorname{div}\boldsymbol{\varphi}=0$},\quad
\langle-\Delta \mathbf{u}-\mathbf{f},\boldsymbol{\varphi}
\rangle_{\boldsymbol{\mathcal{D}}'(\Omega)\times\boldsymbol{\mathcal{D}}(\Omega)}
=0.
\end{equation*}
By De Rham theorem, there exists  $ q \in \mathcal{D}'(\Omega)$ such that
$$
-\Delta \mathbf{u}-\mathbf{f}=\nabla q\quad\text{in }\,\Omega.
$$
Note that $[\mathring{\mathbf{H}}_{-1}^2(\operatorname{div},\Omega)]'$
is imbedded in $\mathbf{W}_{0}^{-1,2}(\Omega)$ and thus
$-\Delta \mathbf{u}-\mathbf{f} \in \mathbf{W}_{0}^{-1,2}(\Omega)$.
It follows from \cite[Theorem 2.7]{vecteur potentiel vivet},
that there exists a unique real constant $C$ and a unique
$\pi \in L^2(\Omega)$ such that $\pi $ has the decomposition $q=\pi+C$.

Observe that since $\mathbf{f}$ and $\nabla \pi$ are two elements of
$[\mathring{\mathbf{H}}_{-1}^2(\operatorname{div},\,\Omega)]'$,
it is the same for $\Delta \mathbf{u}$. Since $\boldsymbol{\mathcal{D}}(\Omega)$
is dense in $\mathring{\mathbf{H}}_{-1}^2(\operatorname{div},\,\Omega)$,
we obtain for any
$\boldsymbol{\varphi}\in \mathring{\mathbf{H}}_{-1}^2(\operatorname{div},
\Omega)$ such that $\operatorname{div}\boldsymbol{\varphi}=0$:
\begin{align*}
 \langle\nabla \pi,\boldsymbol{\varphi}
\rangle_{[\mathring{\mathbf{H}}_{-1}(\operatorname{div},\,\Omega)]'
\times\mathring{\mathbf{H}}_{-1}(\operatorname{div},\,\Omega)}=0.
\end{align*}
Moreover, if $\boldsymbol{\varphi}\in\mathbf{V}^2_{-1,T}(\Omega)$,
using Corollary \ref{lemme formule de grenn dans E^2} we have
\begin{align*}
&\langle-\Delta \mathbf{u},\,\boldsymbol{\varphi}
\rangle_{[\mathring{\mathbf{H}}_{-1}^2(\operatorname{div},\,\Omega)]'
\times{\mathbf{H}}_{-1}^2(\operatorname{div},\,\Omega)}\\
&= \int_{\Omega}\operatorname{\bf curl}\mathbf{u}\cdot\operatorname{\bf curl}
\boldsymbol{\varphi}\,\text{d}\mathbf{x}
-\langle\operatorname{\bf curl}\mathbf{u}\times\mathbf{n},
\boldsymbol{\varphi}\rangle_{\mathbf{H}^{-1/2}(\Gamma)
\times\mathbf{H}^{1/2}(\Gamma)}.
\end{align*}
We  deduce that for all $\boldsymbol{\varphi}\in\mathbf{V}^2_{-1,T}(\Omega)$
\begin{equation*}
 \langle\operatorname{\bf curl}\mathbf{u}\times\mathbf{n},
\boldsymbol{\varphi}\rangle_{\mathbf{H}^{-1/2}(\Gamma)\times\mathbf{H}^{1/2}
(\Gamma)}=\langle \mathbf{h}\times\mathbf{n},
\boldsymbol{\varphi}\rangle_{\mathbf{H}^{-1/2}(\Gamma)\times\mathbf{H}^{1/2}
(\Gamma)}.
\end{equation*}
Let now $\boldsymbol{\mu}$ any element of the space $\mathbf{H}^{1/2}(\Gamma)$.
 So, there exists an element $\boldsymbol{\varphi}\in\mathbf{W}_{0}^{1,2}(\Omega)$
 such that $\operatorname{div}\boldsymbol{\varphi}=0$ in $\Omega$ and
$\boldsymbol{\varphi}=\boldsymbol{\mu}_{t}$ on $\Gamma$. It is clear that
$\boldsymbol{\varphi}\in\mathbf{V}^2_{-1,T}(\Omega)$ and
\begin{align*}
 \langle\operatorname{\bf curl}\mathbf{u}\times\mathbf{n},
\boldsymbol{\mu}\rangle_{\Gamma}-\langle\mathbf{h}\times\mathbf{n},
\boldsymbol{\mu}\rangle_{\Gamma}
&= \langle\operatorname{\bf curl}\mathbf{u}
\times\mathbf{n},\boldsymbol{\mu}_{t}\rangle_{\Gamma}
-\langle\mathbf{h}\times\mathbf{n},\,\boldsymbol{\mu}_{t}\rangle_{\Gamma}\\
&= \langle\operatorname{\bf curl}\mathbf{u}\times\mathbf{n},
\boldsymbol{\varphi}\rangle_{\Gamma}-\langle\mathbf{h}\times\mathbf{n},
\boldsymbol{\varphi}\rangle_{\Gamma}=0.
\end{align*}
This implies that
$\mathbf{cur}\,\mathbf{u}\times\mathbf{n}=\mathbf{h}\times\mathbf{n}$ on
$\Gamma$. As a consequence, Problem \eqref{formulation var pour S_T}
and \eqref{eST} are equivalent. As in the proof of Theorem
\ref{resolution du probleme de stokes sans pression},
 we use Lax-Milgram lemma to prove the existence of a unique solution
$\mathbf{u}$ in $\mathbf{V}^2_{-1,T}(\Omega)$ of Problem
\eqref{formulation var pour S_T}. Using Theorem \ref{injection in X_T  homogene},
 we prove that this solution $\mathbf{u}$ belongs to
$\mathbf{W}_{0}^{1,2}(\Omega)$.  Then the pair
$(\mathbf{u},\,\pi)\in\mathbf{W}_{0}^{1,2}(\Omega)\times\mathbf{L}^2(\Omega)$
is the unique solution of the problem \eqref{eST}.
The estimate \eqref{estimation Stokes  homogene} follows from
\eqref{equivalence de norme avec curl X_T 2}.
\end{proof}

\begin{corollary}\label{solution faible chi diferent de zero}
Let $\mathbf{f}$, $\chi$, $g$, $\mathbf{h}$ such that
\begin{equation*}
\mathbf{f}\in[ \mathring{\mathbf{H}}_{-1}^2(\operatorname{div},\Omega)]',\quad
\chi \in L^2(\Omega),\quad
 g \in H^{1/2}(\Gamma),\quad
\mathbf{h}\in\mathbf{H}^{-1/2}(\Gamma),
\end{equation*}
and that \eqref{condition de compatibilite cas X^p_T} holds.
Then, the Stokes problem \eqref{eST} has a unique solution
$(\mathbf{u},\pi)\in\mathbf{W}_{0}^{1,2}(\Omega)\times L^2(\Omega)$
and we have:
\begin{equation}\label{estimation Stokes non homogene chi}
\begin{aligned}
&\|\mathbf{u}\| _{\mathbf{W}_{0}^{1,2}(\Omega)}+ \| \pi\| _{L^2(\Omega)} \\
&\leqslant  C\left(  \| \mathbf{f}\| _{[\mathring{\mathbf{H}}_{-1}^2
(\operatorname{div},\,\Omega)]'}+\| \,\chi\| _{L^2(\Omega)}
+\| g\|_{H^{1/2}(\Gamma)}+ \|\mathbf{h}\times\mathbf{n}\|_{\mathbf{H}^{-1/2}
(\Gamma)}\right) .
\end{aligned}
\end{equation}
\end{corollary}

\begin{proof}
\textbf{First case:} We suppose that $\chi=0$.
Let  $\theta \in W_{0}^{2,2}(\Omega)$ be a solution of the exterior
 Neumann problem \eqref{eN}. Setting $\mathbf{z}=\mathbf{u}-\nabla \theta$,
then, problem \eqref{eST} becomes:
Find $(\mathbf{z},\pi) \in \mathbf{W}_{0}^{1,2}(\Omega)\times L^2(\Omega)$
 such that
\begin{equation}
\begin{gathered}
-\Delta \mathbf{z}+\nabla \pi=\mathbf{f}\quad\text{and}\quad
\operatorname{div}\mathbf{z}=0 \quad\text{in }\Omega,\\
\mathbf{z}\cdot\mathbf{n}=0\quad\text{and}\quad
\operatorname{\bf curl}\mathbf{z}\times\mathbf{n}
=\mathbf{h}\times\mathbf{n}\quad\text{on }\Gamma \,.
\end{gathered}
\end{equation}
Due to Theorem \ref{theoreme stokes faible homogene condition type1},
this problem has a unique solution
$(\mathbf{z},\pi)\in\mathbf{W}_{0}^{1,2}(\Omega)\times L^2(\Omega)$.
Thus $\mathbf{u}=\mathbf{z}+\nabla \theta$ belongs to
$\mathbf{W}_{0}^{1,2}(\Omega)$ and using \eqref{estimation solution neumann}
and \eqref{estimation Stokes homogene},  we deduce that
\begin{equation}\label{estimation Stokes non homogene}
 \|\mathbf{u}\| _{\mathbf{W}_{0}^{1,2}(\Omega)}+ \|\pi\|_{L^2(\Omega)}\leq C \big( \| \mathbf{f}\| _{[\mathring{\mathbf{H}}_{-1}^2(\operatorname{div},\,\Omega)]'}+ \| g\|_{H^{1/2}(\Gamma)}+ \|\mathbf{h}\times\mathbf{n}\|_{\mathbf{H}^{-1/2}(\Gamma)}\big).
\end{equation}
\textbf{Second case:} We suppose that $\chi \in L^2(\Omega)$.
We solve the following Neumann problem in $\Omega$:
\begin{equation}\label{Neumann avec done chi m=1}
\Delta \theta=\chi\quad\text{in } \Omega,\quad
\frac{\partial \theta}{\partial \mathbf{n}}=g\quad\text{on } \Gamma.
\end{equation}
It follows from \cite[Theorem 3.9]{vecteur potentiel vivet}
 that Problem \eqref{Neumann avec done chi m=1} has a unique solution
 $\theta$ in $W_{0}^{2,2}(\Omega)/\mathbb{R}$ and we have
\begin{equation}\label{estimation de vivet}
\|\theta\|_{W_{0}^{2,2}(\Omega)/\mathbb{R}}
\leqslant C \Big( \|\chi\|_{L^2(\Omega)}+\|g\|_{H^{1/2}(\Gamma)}\Big) .
\end{equation}
Setting $\mathbf{z}=\mathbf{u}-\nabla \theta$, then Problem problem \eqref{eST}
becomes: Find
$(\mathbf{z},\pi) \in \mathbf{W}_{0}^{1,2}(\Omega)\times L^2(\Omega)$ such that
\begin{equation}
\begin{gathered}
-\Delta \mathbf{z}+\nabla \pi=\mathbf{f}+\nabla \chi\quad\text{and}\quad
\operatorname{div}\mathbf{z}=0 \quad\text{in }\Omega,\\
\mathbf{z}\cdot\mathbf{n}=0\quad\text{and}\quad
\operatorname{\bf curl}\mathbf{z}\times\mathbf{n}
=\mathbf{h}\times\mathbf{n}\quad\text{on }\Gamma \,.
\end{gathered}
\end{equation}
Observe that $\mathbf{f}+\nabla \chi$ belongs to
$[ \mathring{\mathbf{H}}_{-1}^2(\operatorname{div},\,\Omega)]'$ and
$\langle \nabla \chi,\mathbf{v} \rangle _{[ \mathring{\mathbf{H}}_{-1}^2
(\operatorname{div},\,\Omega)]'\times\mathring{\mathbf{H}}_{-1}^2
(\operatorname{div},\,\Omega)}=0$ for all $\mathbf{v}$  in
$\mathbf{Y}^2_{1,T}(\Omega)$. According to the first step, this
problem has a unique solution $(\mathbf{z},\pi)
\in\mathbf{W}_{0}^{1,2}(\Omega)\times L^2(\Omega)$.
Thus $\mathbf{u}=\mathbf{z}+\nabla \theta$ belongs to
$\mathbf{W}_{0}^{1,2}(\Omega)$ and estimate
\eqref{estimation Stokes non homogene chi} follows from
\eqref{estimation Stokes non homogene} and \eqref{estimation de vivet}.
\end{proof}

Now, we study the problem \eqref{eSN}.

\begin{theorem}[Weak solutions for \eqref{eSN}]
\label{theoreme stokes faible homogene condition type2}
Assume that $\chi=0$,  $\mathbf{f}$ is  in the space
$[\mathring{\mathbf{H}}_{-1}^2(\operatorname{\bf curl},\Omega)]'$,
$\mathbf{g}$ is  in $\mathbf{H}^{1/2}(\Gamma)$ and
$\pi_{0}$ is  in $H^{1/2}(\Gamma)$, satisfying the  compatibility
condition
\begin{equation}\label{condition de compatibility de SN}
\forall\boldsymbol{\lambda}\in\mathbf{Y}^2_{1,N}(\Omega),\quad
\langle \mathbf{f},\boldsymbol{\lambda}
\rangle_{[\mathring{\mathbf{H}}_{-1}^2(\operatorname{\bf curl},\Omega)]'
\times \mathring{\mathbf{H}}_{-1}^2(\operatorname{\bf curl},\Omega)}
=\langle \boldsymbol{\lambda}\cdot\mathbf{n},\pi_{0}\rangle _{\Gamma}.
 \end{equation}
Then the Stokes problem \eqref{eSN} has a unique solution
$(\mathbf{u},\pi) \in \mathbf{W}_{0}^{1,2}(\Omega)\times W_1^{1,2}(\Omega)$
and we have
\begin{equation}\label{estimation du problem SN homogene12}
\| \mathbf{u}\| _{\mathbf{W}_{0}^{1,2}(\Omega)}
+\| \pi\| _ {W_1^{1,2}(\Omega)}
\leqslant  C ( \| \mathbf{f}\| _{[\mathring{\mathbf{H}}_{-1}^2
(\operatorname{\bf curl},\,\Omega)]'}
+\| \mathbf{g}\times\mathbf{n}\| _{\mathbf{H}^{1/2}(\Gamma)}
+\| \pi_{0}\| _{H^{1/2}(\Gamma)})  .
\end{equation}
\end{theorem}

\begin{proof}
First, we consider the  problem
\begin{equation}\label{problem neumann avec pression}
\Delta \pi=\operatorname{div}\mathbf{f}\quad\text{in } \Omega,\quad
\pi=\pi_{0}\quad\text{on } \Gamma.
\end{equation}
Since $\mathbf{f} \in [\mathring{\mathbf{H}}_{-1}^2(\operatorname{\bf curl},
\Omega)]'$, we deduce from Proposition \ref{caraterisation dual de H^p_0(curl)}
that $\operatorname{div}\mathbf{f}$ belongs to $\mathbf{W}_1^{-1,2}(\Omega)$.
Now, let $(v(1)-1)$ an element of $\mathcal{A}_0^{\Delta}$,
it is clear that $\nabla(v(1)-1)$ belongs to $\mathbf{Y}^2_{1,N}(\Omega)$.
Then using the density of $\boldsymbol{\mathcal{D}}(\Omega)$ in
$\mathring{\mathbf{W}}_{-1}^{1,2}(\Omega)$ and
\eqref{condition de compatibility de SN}, we prove that
\begin{equation}\label{condition de compatibilite prouver par vivet}
\begin{aligned}
\langle \operatorname{div}\mathbf{f},(v(1)-1)
\rangle _{\mathbf{W}_1^{-1,2}(\Omega)\times
\mathring{\mathbf{W}}_{-1}^{1,2}(\Omega)}
&= -\langle \mathbf{f},  \nabla (v(1)-1)
 \rangle_{[\mathring{\mathbf{H}}_{-1}^2(\operatorname{\bf curl},\Omega)]'
 \times \mathring{\mathbf{H}}_{-1}^2(\operatorname{\bf curl},\Omega)}  \\
&= -\langle\nabla (v(1)-1)\cdot\mathbf{n}, \pi_{0} \rangle _{\Gamma}.
\end{aligned}
\end{equation}
Since \eqref{condition de compatibilite prouver par vivet} is satisfied,
we apply \cite[Theorem 3.6]{vecteur potentiel vivet}
 to prove that Problem \eqref{problem neumann avec pression} has a unique
solution $\pi \in W_1^{1,2}(\Omega)$ and we have the following estimate:
\begin{equation}\label{estimation sans condition comp}
\| \pi\| _{W_1^{1,2}(\Omega)}
\leqslant C\Big( \|  \operatorname{div}\mathbf{f}\| _{\mathbf{W}_1^{-1,2}(\Omega)}
+\| \pi_{0}\| _{H^{1/2}(\Gamma)}\Big) .
\end{equation}
Setting $\mathbf{F}=\mathbf{f}-\nabla \pi$, then $\mathbf{F}$ belongs to
$[\mathring{\mathbf{H}}_{-1}^2(\operatorname{\bf curl},\,\Omega)]'$.
Thus Problem \eqref{eSN} becomes:
Find $\mathbf{u} \in \mathbf{W}_{0}^{1,2}(\Omega)$ such that:
\begin{equation}\label{equation avec grand F}
\begin{gathered}
-\Delta \mathbf{u}=\mathbf{F}\quad\text{and}\quad
\operatorname{div}\mathbf{u}=0 \quad\text{in }\Omega,\\
\mathbf{u}\times\mathbf{n}=\mathbf{g}\times\mathbf{n}\quad\text{on }
\Gamma\quad\text{and}\quad{\int}_{\Gamma}\mathbf{u}\cdot \mathbf{n}\,d\sigma=0.
\end{gathered}
\end{equation}
Using \eqref{condition de compatibility de SN} and the fact that
$\boldsymbol{\mathcal{D}}(\Omega)$ is dense in
$\mathring{\mathbf{H}}_{-1}^2(\operatorname{\bf curl},\Omega)$,
we prove that
\begin{equation}
\forall\boldsymbol{\lambda}\in\mathbf{Y}^2_{1,N}(\Omega),\quad
\langle \mathbf{F}, \boldsymbol{\lambda}
\rangle_{[\mathring{\mathbf{H}}_{-1}^2(\operatorname{\bf curl},\Omega)]'\times
\mathring{\mathbf{H}}_{-1}^2(\operatorname{\bf curl},\Omega)}=0.
 \end{equation}
Therefore, $\mathbf{F}$ satisfies the assumptions of Corollary
\ref{existence E_N non homogene 2} and thus Problem
\eqref{equation avec grand F} has a unique solution
$\mathbf{u} \in \mathbf{W}_{0}^{1,2}(\Omega)$ with
\begin{equation}\label{estimation du corollaire de SN}
\|\mathbf{u}\|_{\mathbf{W}_{0}^{1,2}(\Omega)}
\leqslant C
\Big( \|\mathbf{F}\|_{[\mathring{\mathbf{H}}_{-1}^2(\operatorname{\bf curl},
\Omega)]'}+\|\mathbf{g}\times\mathbf{n}\|_{\mathbf{H}^{1/2}(\Gamma)}\Big).
\end{equation}
Thus estimate \eqref{estimation du problem SN homogene12} follows
from \eqref{estimation du corollaire de SN} and from
\eqref{estimation sans condition comp}.
\end{proof}

\begin{corollary}\label{problem SN avec div non nul}
Let $\mathbf{f}$, $\chi$, $\mathbf{g}$, $\pi_{0}$ such that
\begin{equation*}
\mathbf{f} \in[\mathring{\mathbf{H}}_{-1}^2
(\operatorname{\bf curl},\Omega)]',\quad
\chi \in W_1^{1,2}(\Omega),\quad
\mathbf{g}\in \mathbf{H}^{1/2}(\Gamma),\quad
\pi_{0}\in H^{1/2}(\Gamma),
\end{equation*}
and satisfying the compatibility condition:
\begin{equation}\label{condition de compatibility de SN non homogen}
\forall \boldsymbol{\lambda}\in\mathbf{Y}^2_{1,N}(\Omega),\quad
 \langle \mathbf{f},  \boldsymbol{\lambda}
\rangle_{[\mathring{\mathbf{H}}_{-1}^2(\operatorname{\bf curl},\Omega)]'\times
 \mathring{\mathbf{H}}_{-1}^2(\operatorname{\bf curl},\Omega)}
=\langle\boldsymbol{\lambda}\cdot\mathbf{n},\pi_{0}-\chi\rangle _{\Gamma}.
\end{equation}
Then the Stokes problem \eqref{eSN} has a unique solution
$(\mathbf{u},\pi) \in \mathbf{W}_{0}^{1,2}(\Omega)\times W_1^{1,2}(\Omega)$.
Moreover, we have the estimate
\begin{equation}\label{estimation de SN non homogen 13}
\begin{aligned}
&\| \mathbf{u}\| _{\mathbf{W}_{0}^{1,2}(\Omega)}
+\| \pi\| _ {W_1^{1,2}(\Omega)}\\
&\leqslant  C ( \| \mathbf{f}\|_{[\mathring{\mathbf{H}}_{-1}^2
(\operatorname{\bf curl},\,\Omega)]'}
+\|\mathbf{g}\times\mathbf{n}\| _{\mathbf{H}^{1/2}(\Gamma)}
+\| \pi_{0}\| _{H^{1/2}(\Gamma)}+\| \chi\| _ {W_1^{1,2}(\Omega)})  .
\end{aligned}
\end{equation}
\end{corollary}

\begin{proof}
First, we consider the  problem
\begin{equation}\label{equation avec pression et xi}
\Delta \pi=\operatorname{div}\mathbf{f}+\Delta \chi\quad\text{in } \Omega,\quad
\pi=\pi_{0}\quad\text{on }\Gamma.
\end{equation}
Since $\mathbf{f} \in [\mathring{\mathbf{H}}_{-1}^2(\operatorname{\bf curl},
\Omega)]'$, we deduce from Proposition \ref{caraterisation dual de H^p_0(curl)}
that $\operatorname{div}\mathbf{f}+\Delta \chi$ belongs to
$\mathbf{W}_1^{-1,2}(\Omega)$. Proceeding as in the proof of Theorem
\ref{theoreme stokes faible homogene condition type2}, we prove that
\begin{align*}
\langle \operatorname{div}\mathbf{f}+\Delta \chi,(v(1)-1)
\rangle _{\mathbf{W}_1^{-1,2}(\Omega)\times
\mathring{\mathbf{W}}_{-1}^{1,2}(\Omega)}
=- \langle   \nabla (v(1)-1)\cdot\mathbf{n},\pi_{0}\rangle_{\Gamma}
\end{align*}
and then we apply \cite[Theorem 3.6]{vecteur potentiel vivet}
 to prove that Problem \eqref{equation avec pression et xi} has a unique
solution $\pi \in W_1^{1,2}(\Omega)$ and we have the following estimate:
\begin{equation}\label{estimation sans condition comp pour Sn non homogen}
\| \pi\| _{W_1^{1,2}(\Omega)}
\leqslant C\Big( \|  \operatorname{div}\mathbf{f}
+\Delta \chi\| _{\mathbf{W}_1^{-1,2}(\Omega)}+\| \pi_{0}\| _{H^{1/2}(\Gamma)}
\Big).
\end{equation}
Thus Problem \eqref{eSN} becomes:
Find $\mathbf{u} \in \mathbf{W}_{0}^{1,2}(\Omega)$ such that
\begin{equation}\label{equation avec grand F et chi}
\begin{gathered}
-\Delta \mathbf{u}=\mathbf{f}-\nabla \pi\quad\text{and}\quad
\operatorname{div}\mathbf{u}=\chi \quad\text{in }\Omega,\\
\mathbf{u}\times\mathbf{n}=\mathbf{g}\times\mathbf{n}\quad\text{on }\Gamma
\quad\text{and}\quad
{\int}_{\Gamma}\mathbf{u}\cdot \mathbf{n}\,d\sigma=0.
\end{gathered}
\end{equation}
On the other hand, let us solve the  Dirichlet problem
\begin{equation*}
\Delta \theta=\chi\quad\text{in } \Omega,\quad
\theta=0\quad\text{on }\Gamma.
\end{equation*}
Since $W_1^{1,2}(\Omega)$ is imbedded in $L^2(\Omega)$, it follows
from \cite[Theorem 3.5]{vecteur potentiel vivet}, that this problem
has a unique solution $\theta \in W_{0}^{2,2}(\Omega)$
(there is no compatibility condition) and we have the  estimate
\begin{equation}\label{estimation dirichlet modifier}
\| \theta\| _{W_{0}^{2,2}(\Omega)}\leqslant C \|  \chi\| _{L^2(\Omega)}.
\end{equation}
Setting
\begin{equation*}
 \mathbf{z}=\mathbf{u}
-\Big( \nabla \theta-\frac{1}{C_1}\langle \nabla
\theta\cdot\mathbf{n},1\rangle_{\Gamma}\nabla(v(1)-1)\Big) ,
\end{equation*}
where $v(1)$ is the unique solution in $W_{0}^{1,2}(\Omega)$
of the Dirichlet problem
\eqref{problem de Dirichlet liee au noyeau} and
$C_1=\int_{\Gamma}\frac{\partial v(1)}{\partial \mathbf{n}}\,d\sigma$.
We know from \cite[Lemma 3.11]{vecteur potentiel vivet}
 that $C_1>0$ and that $\nabla(v(1)-1)$ belongs to $\mathbf{Y}^2_{1,N}(\Omega)$.
Then Problem \eqref{equation avec grand F et chi} becomes:
Find $\mathbf{z} \in \mathbf{W}_{0}^{1,2}(\Omega)$ such that
\begin{equation}\label{equation avec grand F et chi=0}
\begin{gathered}
-\Delta \mathbf{z}=\mathbf{f}-\nabla \pi+\nabla \chi\quad\text{and}\quad
\operatorname{div}\mathbf{z}=0 \quad\text{in } \Omega,\\
\mathbf{z}\times\mathbf{n}=\mathbf{g}\times\mathbf{n}\quad\text{on }
\Gamma\quad\text{and}\quad{\int}_{\Gamma}\mathbf{z}\cdot \mathbf{n}\,d\sigma=0.
\end{gathered}
\end{equation}
Now, we will solve the Problem \eqref{equation avec grand F et chi=0}.
Setting $\mathbf{F}=\mathbf{f}-\nabla \pi+\nabla \chi$, then
$\mathbf{F}$ belongs to
$[\mathring{\mathbf{H}}_{-1}^2(\operatorname{\bf curl},\,\Omega)]'$.
Using \eqref{condition de compatibility de SN non homogen} and the fact
that $\boldsymbol{\mathcal{D}}(\Omega)$ is dense in
$\mathring{\mathbf{H}}_{-1}^2(\operatorname{\bf curl},\Omega)$, we prove that
\begin{equation}
\forall\boldsymbol{\lambda}\in\mathbf{Y}^2_{1,N}(\Omega),\quad
\langle \mathbf{F},
\boldsymbol{\lambda}\rangle_{[\mathring{\mathbf{H}}_{-1}^2
(\operatorname{\bf curl},\Omega)]'\times \mathring{\mathbf{H}}_{-1}^2
(\operatorname{\bf curl},\Omega)}=0.
 \end{equation}
Therefore, $\mathbf{F}$ satisfies the assumptions of Corollary
\ref{existence E_N non homogene 2} and thus Problem
\eqref{equation avec grand F et chi=0} has a unique solution
$\mathbf{z} \in \mathbf{W}_{0}^{1,2}(\Omega)$ with
\begin{equation}\label{estimation du corollaire de SN2}
\|\mathbf{z}\|_{\mathbf{W}_{0}^{1,2}(\Omega)}
\leqslant C \Big( \|\mathbf{F}\|_{[\mathring{\mathbf{H}}_{-1}^2
(\operatorname{\bf curl},\Omega)]'}+\|\mathbf{g}
\times\mathbf{n}\|_{\mathbf{H}^{1/2}(\Gamma)}\Big)
\end{equation}
and estimate \eqref{estimation de SN non homogen 13} holds.
\end{proof}

\section{Strong solutions for \eqref{eST} and \eqref{eSN}}

We prove in this sequel the existence and the uniqueness of strong
solutions for Problem \eqref{eST} and \eqref{eSN}, we start with
Problem \eqref{eST}.

\begin{theorem}\label{solution forte de ST}
Suppose that $\Omega'$ is of class $C^{2,1}$. Let $\mathbf{f}$, $\chi$,
$g$, $\mathbf{h}$ be such that
\begin{equation*}
\mathbf{f}\in \mathbf{W}_1^{0,2}(\Omega),\quad
\chi \in W_1^{1,2}(\Omega),\,\, g \in H^{\,3/2}(\Gamma),\quad
\mathbf{h}\in\mathbf{H}^{1/2}(\Gamma),
\end{equation*}
and that \eqref{condition de compatibilite cas X^p_T} holds.
Then, the Stokes problem \eqref{eST} has a unique solution
$(\mathbf{u}, \pi)\in\mathbf{W}_1^{2,2}(\Omega)\times W_1^{1,2}(\Omega)$
and we have
\begin{equation}\label{estimation Stokes non homogene chi fort}
 \|\mathbf{u}\| _{\mathbf{W}_1^{2,2}(\Omega)}
+ \| \pi\| _{W_1^{1,2}(\Omega)}
\leqslant  \| \mathbf{f}\| _{\mathbf{W}_1^{0,2}(\Omega)}
+\| \chi\| _{W_1^{1,2}(\Omega)}+\| g\|_{H^{\,3/2}(\Gamma)}
+ \|\mathbf{h}\times\mathbf{n}\|_{\mathbf{H}^{1/2}(\Gamma)}.
\end{equation}
\end{theorem}

\begin{proof}
\textbf{First case:} $\chi=0$.
Since $\mathbf{W}_1^{0,2}(\Omega)$ is included in
$[ \mathring{\mathbf{H}}_{-1}^2(\operatorname{div},\,\Omega)]'$,
we deduce that we are under the hypothesis of Corollary
\ref{solution faible chi diferent de zero} and so Problem \eqref{eST}
has a unique solution $(\mathbf{u}, \pi)\in\mathbf{W}_{0}^{1,2}(\Omega)
\times L^2(\Omega)$. Setting $\mathbf{z}=\operatorname{\bf curl}\mathbf{u}$,
 then $\mathbf{z}$ satisfies
\begin{gather*}
\mathbf{z} \in \mathbf{L}^2(\Omega),\quad
\operatorname{div}\mathbf{z}=0\in W_1^{0,2}(\Omega),\\
-\Delta\mathbf{z}=\operatorname{\bf curl}\mathbf{f},
\quad
\mathbf{z}\times \mathbf{n}=\mathbf{h}\times\mathbf{n}
\in \mathbf{H}^{1/2}(\Gamma).
\end{gather*}
Applying Corollary \ref{existence E_N non homogene 2}
(with $k = -1$) and the uniqueness argument, we prove 
that $\mathbf{z}$ belongs to $\mathbf{W}_1^{1,2}(\Omega)$.
This implies that $\mathbf{u}$ satisfies
\begin{equation*}
\mathbf{u}\in \mathbf{W}_{0}^{1,2}(\Omega),\quad
\operatorname{div}\mathbf{u}=0\in W_1^{1,2}(\Omega),\quad
\operatorname{\bf curl}\mathbf{u}\in \mathbf{W}_1^{1,2}(\Omega),\quad
\mathbf{u}\cdot \mathbf{n}=g\in H^{3/2}(\Gamma).
\end{equation*}
Applying Proposition \ref{injection in X_T non homogene fort},
we prove that $\mathbf{u}$ belongs to $\mathbf{W}_1^{2,2}(\Omega)$ and
thus $\nabla \pi=\mathbf{f}+\Delta \mathbf{u} \in \mathbf{W}_1^{0,2}(\Omega)$.
Since $\pi$ is in $L^2(\Omega)$ then $\pi$ is in $W_1^{1,2}(\Omega)$.

\textbf{Second case:} $\chi$ is in $W_1^{1,2}(\Omega)$.
Since $\Omega'$ is of class $C^{2,1}$,  it follows from
\cite[Theorem 3.9]{vecteur potentiel vivet} that there exists a unique
solution $\theta$ in $W_1^{3,2}(\Omega)/\mathbb{R}$ satisfies Problem
\eqref{Neumann avec done chi m=1} and
\begin{equation}\label{estimation de vivet fort}
\|\theta\|_{W_1^{3,2}(\Omega)/\mathbb{R}}
\leqslant C \Big( \|\chi\|_{W_1^{1,2}(\Omega)}+\|g\|_{H^{3/2}(\Gamma)}\Big).
\end{equation}
 The rest of the proof is similar to that  Corollary
\ref{solution faible chi diferent de zero}.
\end{proof}

\begin{remark} \label{rmk6.2} \rm
Assume that the hypothesis of Theorem \ref{solution forte de ST}
hold and suppose in addition that $\chi=0$.
Let $(\mathbf{u}, \pi)\in\mathbf{W}_{0}^{1,2}(\Omega)\times L^2(\Omega)$
the unique solution of Problem  \eqref{eST} then $\pi$ satisfies the
 problem
\begin{equation}\label{problme generalis de Neumann avec la pression}
\operatorname{div}(\nabla \pi-\mathbf{f})=0\quad\text{in } \Omega
\quad\text{and}\quad
(\nabla \pi-\mathbf{f})\cdot\mathbf{n}=-\operatorname{div}_{\Gamma}
(\mathbf{h}\times\mathbf{n})\quad\text{on } \Gamma.
\end{equation}
It follows from \cite{these Giroire} that Problem
\eqref{problme generalis de Neumann avec la pression}
has a solution $\pi$ in $W_1^{1,2}(\Omega)$. Setting
$\mathbf{F}=\nabla \pi-\mathbf{f} \in  \mathbf{W}_1^{0,2}(\Omega)$.
Then problem \eqref{eST} becomes
\begin{gather*}
  -\Delta \mathbf{u}=\mathbf{F}\quad\text{and}\quad
\operatorname{div}\mathbf{u}=0 \quad\text{in }\Omega,\\
\mathbf{u}\cdot\mathbf{n}=g\quad\text{and}\quad\operatorname{\bf curl}
\mathbf{u}\times\mathbf{n}=\mathbf{h}\times\mathbf{n}\quad\text{on }\Gamma .
\end{gather*}
Therefore, $\mathbf{F}$, $g$ and $\mathbf{h}$ satisfy the assumptions
of Proposition
\ref{proposition probleme faible et fort sans pression non homogene cas K^p_T}
and thus $\mathbf{u}$ belongs to $\mathbf{W}_1^{2,2}(\Omega)$.
\end{remark}



Next, we study the regularity of the solution for Problem \eqref{eSN}.

\begin{theorem}\label{solution fortes de SN}
Suppose that $\Omega'$ is of class $C^{2,1}$. Let $\mathbf{f}$, $\chi$,
 $\mathbf{g}$, $\pi_{0}$ be such that
\begin{equation*}
\mathbf{f} \in \mathbf{W}_1^{0,2}(\Omega),\quad
\chi \in W_1^{1,2}(\Omega),\quad
\mathbf{g}\in \mathbf{H}^{\,3/2}(\Gamma),\quad
\pi_{0}\in H^{1/2}(\Gamma),
\end{equation*}
and satisfying the compatibility condition
\eqref{condition de compatibility de SN non homogen}.
Then the Stokes problem \eqref{eSN} has a unique solution
$(\mathbf{u},\pi) \in \mathbf{W}_1^{2,2}(\Omega)\times W_1^{1,2}(\Omega)$.
Moreover, we have the  estimate
\begin{equation}\label{estimation de SN non homogen 13 fort}
\|\mathbf{u}\| _{\mathbf{W}_1^{2,2}(\Omega)}
+\| \pi\| _ {W_1^{1,2}(\Omega)}
\leqslant  C ( \| \mathbf{f}\| _{\mathbf{W}_1^{0,2}(\Omega)}
 +\| \mathbf{g}\times\mathbf{n}\| _{\mathbf{H}^{3/2}(\Gamma)}
 +\| \pi_{0}\| _{H^{1/2}(\Gamma)}+\| \chi\| _ {W_1^{1,2}(\Omega)})  .
\end{equation}
\end{theorem}

\begin{proof}
\textbf{First case:} We suppose that $\chi=0$.
Since $\mathbf{W}_1^{0,2}(\Omega)$ is  in
$[ \mathring{\mathbf{H}}_{-1}^2(\operatorname{\bf curl},\Omega)]'$,
we deduce that we are under the hypothesis of Corollary
\ref{theoreme stokes faible homogene condition type2} and so Problem
\eqref{eSN} has a unique solution
$(\mathbf{u}, \pi)\in\mathbf{W}_{0}^{1,2}(\Omega)\times W_1^{1,2}(\Omega)$.
Setting $\mathbf{z}=\operatorname{\bf curl}\mathbf{u}$. Observe that
$\mathbf{u}\times \mathbf{n}=\mathbf{g}\times\mathbf{n}$ belongs to
 $\mathbf{H}^{\,3/2}(\Gamma)$ and thus
$\operatorname{\bf curl}\mathbf{u}\cdot\mathbf{n}$ belongs to
$H^{1/2}(\Gamma)$ and so $\mathbf{z}$ satisfies
\begin{gather*}
\mathbf{z} \in \mathbf{L}^2(\Omega),\quad
\operatorname{div}\mathbf{z}=0\in W_1^{0,2}(\Omega),\\
\operatorname{\bf curl}\mathbf{z}=\mathbf{f} 
- \nabla \pi \in \mathbf{W}_1^{0,2}(\Omega), \quad 
\mathbf{z}\cdot \mathbf{n}=\operatorname{\bf curl}\mathbf{u}\cdot\mathbf{n}\in
\mathbf{H}^{1/2}(\Gamma).
\end{gather*}
Applying Proposition \ref{injection in X_T non homogene} (with $k=0$),
we prove that $\mathbf{z}$ belongs to $\mathbf{W}_1^{1,2}(\Omega)$.
This implies that $\mathbf{u}$ satisfies
\begin{gather*}
\mathbf{u}\in \mathbf{W}_{0}^{1,2}(\Omega),\quad
\operatorname{div}\mathbf{u}=0\in W_1^{1,2}(\Omega),\\\
\operatorname{\bf curl}\mathbf{u}\in \mathbf{W}_1^{1,2}(\Omega),\quad
\mathbf{u}\times \mathbf{n}=\mathbf{g}\times\mathbf{n}\in H^{3/2}(\Gamma).
\end{gather*}
Applying Proposition \ref{Injection de M_k,N}, we prove that $\mathbf{u}$
belongs to $\mathbf{W}_1^{2,2}(\Omega)$.

\textbf{Second case:} $\chi$ is in $W_1^{1,2}(\Omega)$.
The proof of this case is very similar to that Corollary
\ref{problem SN avec div non nul}.
\end{proof}


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