#  $L^{p}$-theory for the exterior Stokes problem with Navier’s type slip-
without-friction boundary conditions

Anis Dhifaoui

( UR Analysis and Control of PDEs, UR13ES64,
Department of Mathematics
Faculty of Sciences of Monastir
University of Monastir
5019 Monastir, Tunisia

Email adresses<EMAIL_ADDRESS>
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###### Abstract

In this paper, we consider the stationary Stokes equations in an exterior
domain three-dimensional under a slip boundary condition without friction. We
set the problem in weighted Sobolev spaces in order to control the behavior at
infinity of the solutions. In this work, we try to investigate the existence
and uniqueness of the weak solutions related to this problem in $L^{P}-$theory
when $p>3$. Our proof are based on obtaining inf-sup conditions that play a
fundamental role.

Keywords: Fluid mechanics, Stokes equations, slip boundary condition, exterior
domain, Inf-Sup conditions, weighted Sobolev spaces.

##  1 Introduction

In this paper, we are interested in the Stationary Stokes equations, this
problem describes the flow of a viscous and incompressible fluid past an
obstacle:

$\begin{split}-\Delta\textit{{u}}+\nabla\pi&=\textit{{f}}\quad\text{in}\quad\Omega,\\\
\mathrm{div}\,\textit{{u}}&=0\quad\text{in}\quad\Omega.\end{split}$ (1.1)

In (1.1), the flow domain $\Omega$ is a three-dimensional external domain,
i.e. the complement of a bounded domain which represents the obstacle, the
unknowns are the velocity of the fluid u and the pressure $\pi$ and f  the
external forces acting on the fluid. For many years, the standard boundary
conditions are the Dirichlet or no-slip boundary condition, has been the most
widely used, given its success in reproducing the standard velocity profiles
for incompressible viscous fluid. These latter conditions prescribe that the
fluid adheres to the boundary of the domain. More generally, in the case where
the obstacles have an approximate limit, the standard conditions are no longer
valid (see for example [30]). Therefore, another approach has been introduced,
which assumes that due to the roughness of the boundary and the viscosity of
the fluid, there is a stagnant fluid layer near the boundary, which allows the
fluid to slip (see for example [23]). To overcome the complicated description
of the problem, the Navier slip boundary conditions are commonly used.

These conditions, proposed by Navier [28], can be written as

$\textbf{{u}}\cdot\textbf{{n}}=0,\quad\big{(}(\mathrm{\textbf{D}}\textbf{{u}})\textbf{{n}}+\alpha\textbf{{u}}\big{)}_{\tau}=\boldsymbol{0}\quad\text{on}\quad\Gamma,$
(1.2)

where

$\textbf{D}\textbf{{u}}=\dfrac{1}{2}\left(\nabla\,\textbf{{u}}+\nabla\,\textbf{{u}}^{T}\right)$

denotes the rate-of-strain tensor field, $\alpha$ is a scalar friction
function, n is the unit normal vector to $\Gamma$ and the notation
$(\cdot)_{\tau}$ denotes the tangential component of a vector on $\Gamma$. The
first condition in (1.2) is the no-penetration condition and the second
condition expresses the fact that the tangential velocity is proportional to
the tangential stress.

Now, let us recall some notations related to the boundary condition. First,
for any vector field u on $\Gamma$, we can write

$\textbf{{u}}=\textbf{{u}}_{\tau}+(\textbf{{u}}\cdot\textbf{{n}})\textbf{{n}},$

where $\textbf{{u}}_{\tau}$ is the projection of u on the tangent hyper-plan
to $\Gamma$. Next, for any point x on $\Gamma$, one may choose an open
neighbourhood $\mathcal{W}$ of x in $\Gamma$ small enough to allow the
existence of two families of $\mathcal{C}^{2}$ curves on $\mathcal{W}$ and
where the lengths $s_{1}$ and $s_{2}$ along each family of curves are possible
system of coordinates. Denoting by $\boldsymbol{\tau}_{1}$,
$\boldsymbol{\tau}_{2}$ the unit tangent vectors to each family of curves, we
have

$\textbf{{u}}_{\tau}=(\textbf{{u}}\cdot\boldsymbol{\tau}_{1})\boldsymbol{\tau}_{1}+(\textbf{{u}}\cdot\boldsymbol{\tau}_{2})\boldsymbol{\tau}_{2}.$

For a mathematical analysis of the Stokes system satisfying (1.2), the first
pionnering paper is due to Solonnikov and Scadilov [31] with $\alpha=0$. More
recently, Beirao da Veiga [11] proved existence results for weak and strong
solutions in the $L^{2}$ setting. However, we can prove that

$\displaystyle
2\big{(}(\mathrm{\textbf{D}}\textbf{{u}})\textbf{{n}}\big{)}_{\tau}=\nabla_{\tau}(\textbf{{u}}\cdot\textbf{{n}})+\left(\frac{\partial\textbf{{u}}}{\partial\textbf{{n}}}\right)_{\tau}-\sum_{k=1}^{2}\left(\textbf{{u}}_{\tau}\cdot\frac{\partial\textbf{{n}}}{\partial
s_{k}}\right)\boldsymbol{\tau}_{k}.$

On the other hand, we have the following relation:

$\displaystyle\mathrm{curl}\,\textbf{{u}}\times\textbf{{n}}=\nabla_{\tau}(\textbf{{u}}\cdot\textbf{{n}})-\left(\frac{\partial\textbf{{u}}}{\partial\textbf{{n}}}\right)_{\tau}-\sum_{k=1}^{2}\left(\textbf{{u}}_{\tau}\cdot\frac{\partial\textbf{{n}}}{\partial
s_{k}}\right)\boldsymbol{\tau}_{k}.$

In the particular case $\textbf{{u}}\cdot\textbf{{n}}=0$ on $\Gamma$, which
implies that

$2\big{(}(\mathrm{\textbf{D}}\textbf{{u}})\textbf{{n}}\big{)}_{\tau}=-\mathrm{curl}\,\textbf{{u}}\times\textbf{{n}}-\sum_{k=1}^{2}\left(\textbf{{u}}_{\tau}\cdot\frac{\partial\textbf{{n}}}{\partial
s_{k}}\right)\boldsymbol{\tau}_{k},$

Comparing with (1.2), the following boundary condition

$\displaystyle\textbf{{u}}\cdot\textbf{{n}}=0\,\,\,\,\text{and}\quad\mathbf{curl}\,\textbf{{u}}\times\textbf{{n}}=\textbf{{0}}\quad\text{
on }\,\Gamma,$ (1.3)

is in fact a slip-without-friction Navier boundary condition type for more
details the interested reader can also refer to [17, 12, 27] and references
therein.

Stokes problem with the boundary conditions (1.3) was studied by many authors.
In the one hand, in the bounded domains, one can refer for instance to [26, 8,
13, 15, 16]. In the other hand, the case for the exterior domains, we can just
mention [4, 25]. The exterior problem (1.1)–(1.2), where as far as know, we
can mention [33, 29, 19]. Although the Stokes problem set in bounded domains
with conditions (1.2) has been well studied by various authors (see for
instance [6, 10] or [7] for the case $\alpha=0$ and [1, 5] for the case
$\alpha\neq 0$).

The purpose of this work is to study the existence and the uniqueness of the
weak solution for the problem (1.1)–(1.3). Since the flow domain is unbounded,
we set the problem in weighted Sobolev spaces in order to control the behavior
of functions at infinity. This functional framework allows us to find
solutions with different behaviours to infinity (decay or polynomial growth).
The study is based on a $L^{p}$-theory, $p>3$. Our proof of solvability the
problem (1.1)–(1.3) is based on a variational formulations obtained and the
Inf-Sup conditions.

The outline of this paper is as follows. In Section 2, we introduce the
notations and the functional framework based on weighted Sobolev spaces. We
shall precise in which sense, the Navier’s type slip-without-friction boundary
conditions (1.3), we finish this section by giving a result concerning the
Laplace problem with Neumann boundary conditions with data $f$ belongs to
$L^{p}(\Omega)$, we got the existence of solutions in $W^{1,p}_{-1}(\Omega)$.
In Section 3, we recall one result about the vector potential problem (see
[24]) and we prove the Inf-Sup conditions, which plays a crucial role in the
existence and uniqueness of solutions. Finally, in Section 4, we conclude with
the main result of this paper related to well-posedness of the Stokes problem
(1.1)–(1.3). We prove the existence and the uniqueness of weak solutions, when
$p>3$.

##  2 Notations and Preliminaries

###  2.1 Notations

We recall the main notations and results, concerning the weighted Sobolev
spaces, which we shall use later on. In what follows, $p$ is a real number in
the interval $]1,\infty[$. The dual exponent of $p$ denoted by $p^{\prime}$ is
given by the following relation:

$\frac{1}{p}+\frac{1}{p^{\prime}}=1.$

We will use bold characters for vector and matrix fields. A point in
$\mathbb{R}^{3}$ is denoted by $\textbf{{x}}=(x_{1},x_{2},x_{3})$ and its
distance to the origin by

$r=|\textbf{{x}}|=\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\right)^{1/2}.$

For any multi-index $\boldsymbol{\lambda}\in\mathbb{N}^{3}$, we denote by
$\partial^{\boldsymbol{\lambda}}$ the differential operator of order
$\boldsymbol{\lambda}$,

$\partial^{\boldsymbol{\lambda}}={\partial^{|\lambda|}\over\partial_{1}^{\lambda_{1}}\partial_{2}^{\lambda_{2}}\partial_{3}^{\lambda_{3}}},\quad|\lambda|=\lambda_{1}+\lambda_{2}+\lambda_{3}.$

Let $\Omega^{\prime}$ be a bounded connected open set in $\mathbb{R}^{3}$ with
boundary $\partial\Omega^{\prime}=\Gamma$ of class $\mathcal{C}^{1,1}$ and let
$\Omega$ its complement i.e
$\Omega=\mathbb{R}^{3}\diagdown\overline{\Omega^{\prime}}$. In this work, we
shall also denote by $B_{R}$ the open ball of radius $R>0$ centred at the
origin with boundary $\partial B_{R}$. In particular, since $\Omega^{\prime}$
is bounded, we can find some $R_{0}$, such that $\Omega^{\prime}\subset
B_{R_{0}}$ and we introduce, for any $R\geq R_{0}$, the set

$\Omega_{R}=\Omega\cap B_{R}.$

We denote by $\mathcal{D}(\Omega)$ the space of $\mathcal{C}^{\infty}$
functions with compact support in $\Omega$, $\mathcal{D}(\overline{\Omega})$
the restriction to $\Omega$ of functions belonging to
$\mathcal{D}(\mathbb{R}^{3})$. We recall that $\mathcal{D}^{\prime}(\Omega)$
is the well-known space of distributions defined on $\Omega$. We recall that
$L^{p}(\Omega)$ is the well-known Lebesgue real space and for $m\geq 1$, we
recall that $W^{m,p}(\Omega)$ is the classical Sobolev space. We shall write
$u\in W_{loc}^{m,p}(\Omega)$ to mean that $u\in W^{m,p}(\mathcal{O})$, for any
bounded domain $\mathcal{O}$, with $\overline{\mathcal{O}}\subset\Omega$. We
denote by $[s]$ the integer part of $s$. For any $k\in\mathbb{Z}$,
$\mathcal{P}_{k}$ stands for the space of polynomials of degree less than or
equal to $k$ and $\mathcal{P}^{\Delta}_{k}$ the harmonic polynomials of
$\mathcal{P}_{k}$. If ${k}$ is a negative integer, we set by convention
$\mathcal{P}_{k}=\\{0\\}$.

Given a Banach space $X$, with dual space $X^{\prime}$ and a closed subspace
$Y$ of $X$, we denote by $X^{\prime}\perp Y$ the subspace of $X^{\prime}$
orthogonal to $Y$, i.e.

$X^{\prime}\perp Y=\\{f\in X^{\prime};<f,v>=0\,\,\forall\,v\in
Y\\}=(X/Y)^{\prime}.$

The space $X^{\prime}\perp Y$ is also called the polar space of $Y$ in
$X^{\prime}$. Finally, as usual, $C>0$ denotes a generic constant the value of
which may change from line to line and even at the same line.

###  2.2 Weighted Sobolev spaces

In order to control the behavior at infinity of our functions and
distributions we use for basic weights the quantity
$\rho(\textbf{{x}})=(1+r^{2})^{1/2}$ which is equivalent to $r$ at infinity,
and to one on any bounded subset of $\mathbb{R}^{3}$.
For $\alpha\in\mathbb{Z}$, we introduce

$W_{\alpha}^{0,p}(\Omega)=\Big{\\{}u\in\mathcal{D}^{\prime}(\Omega),\,\rho^{\alpha}u\in
L^{p}(\Omega)\Big{\\}},$

which is a Banach space equipped with the norm:

$\|u\|_{W_{\alpha}^{0,p}(\Omega)}=\|\rho^{\alpha}u\|_{L^{p}(\Omega)}.$

For any non-negative integers $m$, real numbers $p>1$ and
$\alpha\in\mathbb{Z}$. We define the weighted Sobolev space for
$3/p+\alpha\notin\\{1,\cdots,m\\}$:

$W_{\alpha}^{m,p}(\Omega)=\Big{\\{}u\in\mathcal{D}^{\prime}(\Omega);\,\forall\boldsymbol{\lambda}\in\mathbb{N}^{3}:\,0\leq|\boldsymbol{\lambda}|\leq
m,\,\rho^{\alpha-m+|\boldsymbol{\lambda}|}\partial^{\boldsymbol{\lambda}}u\in
L^{p}(\Omega)\Big{\\}}.$

It is a reflexive Banach space equipped with the norm:

$\|u\|_{W_{\alpha}^{m,p}(\Omega)}=\left(\sum_{0\leqslant|\boldsymbol{\lambda}|\leqslant
m}\|\rho^{\alpha-m+|\boldsymbol{\lambda}|}\partial^{\boldsymbol{\lambda}}u\|^{p}_{L^{p}(\Omega)}\right)^{1/p}.$

We define the semi-norm

$|u|_{W_{\alpha}^{m,p}(\Omega)}=\left(\sum_{|\boldsymbol{\lambda}|=m}\|\rho^{\alpha}\partial^{\boldsymbol{\lambda}}u\|_{L^{p}(\Omega)}\right)^{1/p}.$

Let us give some examples of such space that will be often used in the
remaining of the thesis.

  1. 1.

For $m=1$, we have

$\displaystyle{W_{\alpha}^{1,p}(\Omega)}:={\\{}$ $\displaystyle
u\in\mathcal{D}^{\prime}(\Omega);\,\rho^{\alpha-1}u\in
L^{p}(\Omega),\,\rho^{\alpha}\,\nabla\,u\in L^{p}(\Omega)\\}$

  2. 2.

For $m=2$, we have

$W_{\alpha+1}^{2,p}(\Omega):=\left\\{u\in{W_{\alpha}^{1,p}(\Omega)},\rho^{k+1}\nabla^{2}u\in
L^{p}(\Omega)\right\\},$

######  Remark 2.1.

Note that if $\dfrac{3}{p}+\alpha\in\left\\{1,...,m\right\\}$, we need to add
the logarithmic weight in the definition of weight Sobolev spaces introduced
above. The logarithmic weight is defined by $ln(2+|x|^{2})$ see [21] for more
details. For example if $3/p+\alpha=1$, we have

$\displaystyle{{W}_{\alpha}^{1,p}(\Omega)}:={\\{}$
$\displaystyle\textbf{{u}}\in\mathcal{D}^{\prime}(\Omega);\,\rho^{\alpha-1}(ln(2+r^{2}))^{-1}\textbf{{u}}\in
L^{p}(\Omega),\,\rho^{\alpha}\,\nabla\,\textbf{{u}}\in L^{p}(\Omega)\\}.$

Now, we present some basic properties on weighted Sobolev spaces. For more
details, the reader can refer to [3, 2, 22].

######  Properties 2.2.

  1. 1.

The space $\mathcal{D}(\overline{\Omega})$ is dense in
$W^{m,p}_{\alpha}(\Omega)$.

  2. 2.

For any $m\in\mathbb{N}^{*}$ and $3/p+\alpha\neq 1$, we have the following
continuous embedding:

$W_{\alpha}^{m,p}(\Omega)\hookrightarrow W_{\alpha-1}^{m-1,p}(\Omega).$ (2.1)

  3. 3.

For any $\alpha$, $m\in\mathbb{Z}$ and for any $\lambda\in\mathbb{N}^{3}$, the
mapping

$u\in W_{\alpha}^{m,p}(\Omega)\longrightarrow\,\,\partial^{\lambda}u\in
W_{\alpha}^{m-|\lambda|,p}(\Omega)$ (2.2)

is continuous.

The space $W_{\alpha}^{m,p}(\Omega)$ sometimes contains some polynomial
functions. Let $j$ be defined as follow:

$j=\begin{cases}[m-(3/p+\alpha)]\quad\quad\mathrm{if}\quad
3/p+\alpha\notin\mathbb{Z}^{-},\\\
m-3/p-\alpha-1\qquad\qquad\mathrm{otherwise}.\end{cases}$ (2.3)

Then $\mathcal{P}_{j}$ is the space of all polynomials included in
$W_{\alpha}^{m,p}(\Omega)$.

The norm of the quotient space $W_{\alpha}^{m,p}(\Omega)/\mathcal{P}_{j}$ is
given by:

$\displaystyle||u||_{W_{\alpha}^{m,p}(\Omega)/\mathcal{P}_{j}}=\inf_{\mu\in\mathcal{P}_{j}}||u+\mu||_{W_{\alpha}^{m,p}(\Omega)}.$

All the local properties of $W_{\alpha}^{m,p}(\Omega)$ coincide with those of
the corresponding classical Sobolev spaces $W^{m,p}(\Omega)$. Hence, it also
satisfies the usual trace theorems on the boundary $\Gamma$. Therefore, we can
define the space

$\mathring{W}_{\alpha}^{m,p}(\Omega)=\\{u\in
W_{\alpha}^{m,p}(\Omega),\,\,\,\gamma_{0}u=0,\,\gamma_{1}u=0,\,\cdots,\gamma_{m-1}u=0\\}.$

Note that when $\Omega=\mathbb{R}^{3}$, we have
$\mathring{W}_{\alpha}^{m,p}(\mathbb{R}^{3})=W_{\alpha}^{m,p}(\mathbb{R}^{3})$.
The space $\mathcal{D}(\Omega)$ is dense in
$\mathring{W}^{m,p}_{\alpha}(\Omega)$. Therefore, the dual space of
$\mathring{W}_{\alpha}^{m,p}(\Omega)$, denoting by
$W_{-\alpha}^{-m,p^{\prime}}(\Omega)$, is a space of distributions with the
norm

$||\,u\,||_{W_{-\alpha}^{-m,p^{\prime}}(\Omega)}=\sup_{v\in\mathring{W}_{\alpha}^{m,p}(\Omega)}\dfrac{\left\langle
u,v\right\rangle_{W_{-\alpha}^{-m,p^{\prime}}(\Omega)\times\mathring{W}_{\alpha}^{m,p}(\Omega)}}{||\,v\,||_{W_{\alpha}^{m,p}(\Omega)}}.$

We state the Hardy’s inequalities which play a key role in solving elliptic
problems.

######  Theorem 2.3.

Let $\Omega$ be an lipschitzian exterior domain. Let $m\geqslant 1$,
$\alpha\in\mathbb{Z}$ and $1<p<\infty$. There exists a constant
$C=C(p,\alpha,\Omega)>0$ such that

  1. 1.

$\displaystyle\forall u\in
W^{m,p}_{\alpha}(\Omega),\quad||u||_{W_{\alpha}^{m,p}(\Omega)/\mathcal{P}_{j^{\prime}}}\leqslant|u|_{W_{\alpha}^{m,p}(\Omega)},$
(2.4)

where $j^{\prime}=\min(j,0)$ and j is the highest degree of polynomials
belonging to $W^{m,p}_{\alpha}(\Omega)$.

  2. 2.

$\displaystyle\forall
u\in\mathring{W}^{m,p}_{\alpha}(\Omega),\quad||u||_{W_{\alpha}^{m,p}(\Omega)}\leqslant|u|_{W_{\alpha}^{m,p}(\Omega)}.$
(2.5)

The inequalities (2.4) and (2.5) are the reason of choosing the weight
functions in the definition of $W^{m,p}_{\alpha}(\Omega)$. The proof of this
theorem can be found in the case $\Omega=\mathbb{R}^{3}$ in [2] and its
extension to an exterior domain $\Omega$ in [3] and in [21] for $p=2$. From
Theorem 2.3 and the Sobolev embeddings, we have the following continuous and
dense embedding:

$\displaystyle W^{1,p}_{0}(\Omega)\hookrightarrow
L^{\frac{3p}{3-p}}(\Omega),\quad\text{ if }\quad 1<p<3.$ (2.6)

By duality, we have

$\displaystyle L^{\frac{3p^{\prime}}{3+p^{\prime}}}(\Omega)\hookrightarrow
W^{-1,p^{\prime}}_{0}(\Omega),\quad\text{ if }\quad 3/2<p^{\prime}<\infty.$
(2.7)

The proof of the following theorem can be found in [24, Proposition 2.1].

######  Theorem 2.4.

Let $k$, $k^{\prime}$ be real numbers. Let $\lambda$ be a polynomial that
belongs to $W^{1,p}_{k}(\Omega)+W^{1,q}_{k^{\prime}}(\Omega)$. Then $\lambda$
belongs to $\mathcal{P}_{\gamma}$ where

$\gamma=\max\Big{(}[1-\frac{3}{p}-k,[1-\frac{3}{q}-k^{\prime}\Big{)}.$

###  2.3 Functional spaces and Trace Results

The purpose of this section is to introduce some weighted Sobolev spaces that
are specific for the study of the Stokes problem (1.1) with the Navier
boundary conditions (1.3). Let us first introduce some notations. For any
vector fields u and v of $\mathbb{R}^{3}$, we define

$\textit{{u}}\times\textit{{v}}=(u_{2}v_{3}-v_{3}u_{2},\,u_{3}v_{1}-u_{1}v_{3},\,u_{1}v_{2}-u_{2}v_{1})^{T}$

and

$\textbf{curl }\textit{{u}}=\nabla\times\textit{{u}}.$

We note that, the vector-valued Laplace operator of a vector field v is
equivalently defined by

$\Delta\,\textbf{{v}}=\nabla\,\mathrm{div}\,\textbf{{v}}-\mathbf{curl}\,\mathbf{curl}\,\textbf{{v}}.$
(2.8)

We start by introducing the following spaces for $\alpha\in\mathbb{Z}$ and
$1<p<\infty$.

######  Definition 2.5.

The space $H_{\alpha}^{p}(\mathbf{curl},\Omega)$ is defined by:

$H_{\alpha}^{p}(\mathbf{curl},\Omega)=\left\\{\textbf{{v}}\in
W_{\alpha}^{0,p}(\Omega);\mathbf{curl}\,\,\textbf{{v}}\in
W_{\alpha+1}^{0,p}(\Omega)\right\\}\,,$

and is provided with the norm:

$\|\textbf{{v}}\|_{H_{\alpha}^{p}(\mathbf{curl},\Omega)}=\left({\|\textbf{{v}}\|^{p}}_{W_{\alpha}^{0,p}(\Omega)}+{\|\mathbf{curl}\,\,\textbf{{v}}\|^{p}}_{W_{\alpha+1}^{0,p}(\Omega)}\right)^{\frac{1}{p}}.$

The space $H_{\alpha}^{p}(\mathrm{div},\Omega)$ is defined by:

$H_{\alpha}^{p}(\mathrm{div},\Omega)=\left\\{\textbf{{v}}\in
W_{\alpha}^{0,p}(\Omega);\mathrm{div}\,\,\textbf{{v}}\in
W_{\alpha+1}^{0,p}(\Omega)\right\\}\,,$

and is provided with the norm:

$\|\textbf{{v}}\|_{H_{\alpha}^{p}(\mathrm{div},\Omega)}=\left(\|\textbf{{v}}\|^{p}_{W_{\alpha}^{0,p}(\Omega)}+\|\mathrm{div}\,\,\textbf{{v}}\|^{p}_{{W}_{\alpha+1}^{0,p}(\Omega)}\right)^{\frac{1}{p}}.$

Finally, we set

$\textbf{{X}}_{\alpha}^{p}(\Omega)=H_{\alpha}^{p}(\mathbf{curl},\Omega)\cap
H_{\alpha}^{p}(\mathrm{div},\Omega).$

It is provided with the norm

$\textbf{{X}}_{\alpha}^{p}(\Omega)=\left(\|\textbf{{v}}\|^{p}_{W_{\alpha}^{0,p}(\Omega)}+\|\mathrm{div}\,\,\textbf{{v}}\|^{p}_{{W}_{\alpha+1}^{0,p}(\Omega)}+\|\mathbf{curl}\,\,\textbf{{v}}\|^{p}_{W_{\alpha+1}^{0,p}(\Omega)}\right)^{\frac{1}{p}}.$

These definitions will be also used when $\Omega$ is replaced by
${\mathbb{R}}^{3}$.

Note that $\mathcal{D}(\overline{\Omega})$ is dense in
$H_{\alpha}^{p}(\mathrm{div},\Omega)$ and in
$H_{\alpha}^{p}(\mathbf{curl},\Omega)$ and so in
$\textbf{{X}}_{\alpha}^{p}(\Omega)$. For the proof, one can use the same
arguments than for the proof of the density of
$\mathcal{D}(\overline{\Omega})$ in $W_{\alpha}^{m,p}(\Omega)$ (see [21, 22]).
If v belongs to $H_{\alpha}^{p}(\mathrm{div},\Omega)$, then v has normal trace
$\textbf{{v}}\cdot\textbf{{n}}$ in $W^{-1/p,p}(\Gamma)$, where
$W^{-1/p,p}(\Gamma)$ denotes the dual space of $W^{1/p,p^{\prime}}(\Gamma)$.
By the same way, if v belongs to $H_{\alpha}^{p}(\mathbf{curl},\Omega)$, then
v has a tangential trace $\textbf{{v}}\times\textbf{{n}}$ that belongs to
$W^{-1/p,p}(\Gamma)$. Similarly as in bounded domain, we have the trace
theorems i.e, for $\alpha\in\mathbb{Z}$ there exists a constant $C>0$, such
that

$\forall\textbf{{v}}\in
H_{\alpha}^{p}(\mathrm{div},\Omega),\,\,||\textbf{{v}}\cdot\textbf{{n}}||_{W^{-1/p,p}(\Gamma)}\leq
C||\textbf{{v}}||_{H_{\alpha}^{p}(\mathrm{div},\Omega)},$ (2.9)
$\forall\textbf{{v}}\in
H_{\alpha}^{p}(\mathbf{curl},\Omega),\,\,||\textbf{{v}}\times\textbf{{n}}||_{W^{-1/p,p}(\Gamma)}\leq
C||\textbf{{v}}||_{H_{\alpha}^{p}(\mathbf{curl},\Omega)}$ (2.10)

and the following Green’s formulas holds: For any $\textbf{{v}}\in
H_{\alpha}^{p}(\mathrm{div},\Omega)$ and $\varphi\in
W_{-\alpha}^{1,p^{\prime}}(\Omega)$, we have

$\left\langle\textbf{{v}}\cdot\textbf{{n}},\varphi\right\rangle_{\Gamma}=\int_{\Omega}\textbf{{v}}\cdot\nabla\,\varphi\,dx+\int_{\Omega}\varphi\,\mathrm{div}\,\textbf{{v}}\,dx.$
(2.11)

For any $\textbf{{v}}\in H_{\alpha}^{p}(\mathbf{curl},\Omega)$ and
$\boldsymbol{\varphi}\in W_{-\alpha}^{1,p^{\prime}}(\Omega)$, we have

$\left\langle\textbf{{v}}\times\textbf{{n}},\boldsymbol{\varphi}\right\rangle_{\Gamma}=\int_{\Omega}\textbf{{v}}\cdot\mathbf{curl}\,\boldsymbol{\varphi}\,dx-\int_{\Omega}\mathbf{curl}\,\textbf{{v}}\cdot\boldsymbol{\varphi}dx,$
(2.12)

where $\left\langle.,.\right\rangle_{\Gamma}$ denotes the duality pairing
between $W^{-1/p,p}(\Gamma)$ and $W^{1/p,p^{\prime}}(\Gamma)$.

The closures of $\boldsymbol{\mathcal{D}}(\Omega)$ in
$H_{\alpha}^{\,p}(\mathrm{div},\Omega)$ and in
$H_{\alpha}^{\,p}(\mathbf{curl},\Omega)$ are denoted respectively by
$\mathring{H}_{\alpha}^{\,p}(\mathbf{curl},\Omega)$ and
$\mathring{H}_{\alpha}^{\,p}(\mbox{div},\Omega)$ and can be characterized
respectively by:

$\mathring{H}_{\alpha}^{\,p}(\mathbf{curl},\Omega)=\left\\{\textbf{{v}}\in
H_{\alpha}^{p}(\mathbf{curl},\Omega);\,\,\textbf{{v}}\times\textbf{{n}}=\textbf{0}\,\,\text{on}\,\,\Gamma\right\\},$
$\mathring{H}_{\alpha}^{\,p}(\mbox{div},\Omega)=\left\\{\textbf{{v}}\in
H_{\alpha}^{p}(\mbox{div},\Omega);\,\,\textbf{{v}}\cdot\textbf{{n}}=0\,\,\text{on}\,\,\Gamma\right\\}.$

For $1<p<\infty$ and $\alpha\in\mathbb{Z}$, we denote by
$[\mathring{H}_{\alpha}^{p}(\mathrm{div};\Omega)]^{\prime}$ and
$[\mathring{H}_{\alpha}^{p}(\mathbf{curl};\Omega)]^{\prime}$ the dual spaces
of $\mathring{H}_{\alpha}^{\,p}(\mathbf{curl},\Omega)$ and
$\mathring{H}_{\alpha}^{\,p}(\mbox{div},\Omega)$ respectively. We can
characterize theses spaces as it is stated in the following proposition.

######  Proposition 2.6.

  1. 1.

A distribution f belongs to
$[\mathring{H}_{\alpha}^{p}(\mathrm{div};\Omega)]^{\prime}$ if and only if
there exist functions $\boldsymbol{\psi}\in
W^{0,p^{\prime}}_{-\alpha}(\Omega)$ and $\chi\in
W^{0,p^{\prime}}_{-\alpha-1}(\Omega)$, such that
$\textbf{{f}}=\boldsymbol{\psi}+\nabla\chi$. Moreover

$\displaystyle\|\boldsymbol{\psi}\|_{W^{0,p^{\prime}}_{-\alpha}(\Omega)}+\|\chi\|_{W^{0,p^{\prime}}_{-\alpha-1}(\Omega)}\leq
C\|\textbf{{f
}}\|_{[\mathring{H}_{\alpha}^{p}(\mathrm{div};\Omega)]^{\prime}}.$ (2.13)

  2. 2.

A distribution f belongs to
$[\mathring{H}_{\alpha}^{p}(\mathbf{curl},\Omega)]^{\prime}$ if and only if
there exist functions $\psi\in W^{0,p^{\prime}}_{-\alpha}(\Omega)$ and
$\chi\in W^{0,p^{\prime}}_{-\alpha-1}(\Omega)$, such that
$\textbf{{f}}=\psi+\mathbf{curl}\,\chi$. Moreover

$\displaystyle\|\psi\|_{W^{0,p^{\prime}}_{-\alpha}(\Omega)}+\|\chi\|_{W^{0,p^{\prime}}_{-\alpha-1}(\Omega)}\leq
C\|\,\textbf{{f}}\,\|_{[\mathring{H}_{\alpha}^{p}(\mathbf{curl},\Omega)]^{\prime}}.$
(2.14)

The proof of Proposition 2.6 can be found in [18, Proposition II.2]. As a
consequence of Proposition 2.6 and the imbedding (2.1) we have, for any
$\alpha\in\mathbb{Z}$ and $1<p<\infty$, the following imbeddings

$[\mathring{H}_{\alpha}^{p}(\mathrm{div};\Omega)]^{\prime}\subset
W_{-\alpha-1}^{-1,p^{\prime}}(\Omega)$ (2.15)

and

$[\mathring{H}_{\alpha}^{p}(\mathrm{curl};\Omega)]^{\prime}\subset
W_{-\alpha-1}^{-1,p^{\prime}}(\Omega).$ (2.16)

Let us consider the following space:

$\displaystyle\textbf{E}^{p}(\Omega)=\\{\textbf{{v}}\in
W^{1,p}_{0}(\Omega);\Delta\,\textbf{{v}}\in[\mathring{H}_{-1}^{p^{\prime}}(\mathrm{div},\Omega)]^{\prime}\\}.$

equipped with the following norm

$\displaystyle\|\textbf{{v}}\|_{\textbf{E}^{p}(\Omega)}=\|\textbf{{v}}\|_{W^{1,p}_{0}(\Omega)}+\|\Delta\textbf{{v}}\|_{[\mathring{H}_{-1}^{p^{\prime}}(\mathrm{div},\Omega)]^{\prime}}.$

We have the following preliminary result.

######  Lemma 2.7.

The space $\mathcal{D}(\overline{\Omega})$ is dense in
$\textbf{E}^{p}(\Omega)$.

###### Proof.

The proof is quite similar to the proof when p=2 (see [4]). Let $P$ be a
continuous linear mapping from $W_{0}^{1,p}(\Omega)$ to
$W_{0}^{1,p}(\mathbb{R}^{3})$, such that
$P\,\textbf{{v}}|_{\Omega}=\textbf{{v}}$ and let
$\boldsymbol{\ell}\in(\textbf{{E}}^{\,p}(\Omega))^{\prime}$, such that for any
$\textbf{{v}}\in\boldsymbol{\mathcal{D}}(\overline{\Omega})$, we have
$\left\langle\boldsymbol{\ell},\textbf{{v}}\right\rangle=0$. We want to prove
that $\boldsymbol{\ell}=\boldsymbol{0}$ on $\textbf{{E}}^{\,p}(\Omega)$. Then
there exists $(\textbf{{f}},\textbf{{g}})\in
W_{0}^{-1,p^{\prime}}(\mathbb{R})\times\mathring{H}_{-1}^{\,p^{\prime}}(\mathrm{div},\,\Omega)$
such that : for any $\textbf{{v}}\in\textbf{{E}}^{\,p}(\Omega)$,

$\left\langle\boldsymbol{\ell},\textbf{{v}}\right\rangle=\left\langle\textbf{{f}},P\,\textbf{{v}}\right\rangle_{W_{0}^{-1,p^{\prime}}(\mathbb{R}^{3})\times
W_{0}^{1,p}(\mathbb{R}^{3})}+\left\langle\Delta\textbf{{v}},\textbf{{g}}\right\rangle_{[\mathring{H}_{-1}^{\,p^{\prime}}(\mathrm{div},\,\Omega)]^{\prime}\times\mathring{H}_{-1}^{\,p^{\prime}}(\mathrm{div},\,\Omega)}.$

Observe that we can easily extend by zero the function g in such a way that
$\widetilde{\textbf{{g}}}\in
H_{-1}^{\,p^{\prime}}(\mathrm{div},\,\mathbb{R}^{3})$. Now we take
$\boldsymbol{\varphi}\in\boldsymbol{\mathcal{D}}(\mathbb{R}^{3})$. Then we
have by assumption that:

$\left\langle\textbf{{f}},\boldsymbol{\varphi}\right\rangle_{W_{0}^{-1,p^{\prime}}(\mathbb{R}^{3})\times
W_{0}^{1,p}(\mathbb{R}^{3})}+\displaystyle\int_{\mathbb{R}^{3}}\widetilde{\textbf{{g}}}\cdot\Delta\boldsymbol{\varphi}dx=0,$

because
$\left\langle\textbf{{f}},\boldsymbol{\varphi}\right\rangle=\left\langle\textbf{{f}},P\,\textbf{{v}}\right\rangle$
where $\textbf{{v}}=\boldsymbol{\varphi}|_{\Omega}$. Thus we have
$\textbf{{f}}+\Delta\widetilde{\textbf{{g}}}=\boldsymbol{0}$ in
$\boldsymbol{\mathcal{D}}^{\prime}(\mathbb{R}^{3})$. Then we can deduce that
$\Delta\widetilde{\textbf{{g}}}=-\textbf{{f}}\in
W_{0}^{-1,p^{\prime}}(\mathbb{R}^{3})$ and due to [3, Theorem 1.3], there
exists a unique $\boldsymbol{\lambda}\in W_{0}^{1,p^{\prime}}(\mathbb{R}^{3})$
such that $\Delta\boldsymbol{\lambda}=\Delta\widetilde{\textbf{{g}}}$. Thus
the harmonic function $\boldsymbol{\lambda}-\widetilde{\textbf{{g}}}$
belonging to $W_{-1}^{0,p^{\prime}}(\mathbb{R}^{3})$ is necessarily equal to
zero. Since $\textbf{{g}}\in W_{0}^{1,p^{\prime}}(\Omega)$ and
$\widetilde{\textbf{{g}}}\in W_{0}^{1,p^{\prime}}(\mathbb{R}^{3})$, we deduce
that $\textbf{{g}}\in\mathring{W}_{0}^{1,p^{\prime}}(\Omega)$. As
$\boldsymbol{\mathcal{D}}(\Omega)$ is dense in
$\mathring{W}_{0}^{1,p^{\prime}}(\Omega)$, there exists a sequence
$\textbf{{g}}_{k}\in\boldsymbol{\mathcal{D}}(\Omega)$ such that
$\textbf{{g}}_{k}\rightarrow\textbf{{g}}$ in $W_{0}^{1,p^{\prime}}(\Omega)$,
when $k\rightarrow\infty$. Then
$\nabla\cdot\textbf{{g}}_{k}\rightarrow\nabla\cdot\textbf{{g}}$ in
$L^{\,p^{\prime}}(\Omega)$. Since $W_{0}^{1,p^{\prime}}(\Omega)$ is imbedded
in $W_{-1}^{0,p^{\prime}}(\Omega)$, we deduce that
$\textbf{{g}}_{k}\rightarrow\textbf{{g}}$ in
$H_{-1}^{\,p^{\prime}}(\mathrm{div},\,\Omega)$. Now, we consider
$\textbf{{v}}\in\textbf{{E}}^{\,p}(\Omega)$ and we want to prove that
$\left\langle\boldsymbol{\ell},\textbf{{v}}\right\rangle=0$. Observe that:

$\left\langle\boldsymbol{\ell},\textbf{{v}}\right\rangle=-\left\langle\Delta\widetilde{\textbf{{g}}},P\textbf{{v}}\right\rangle_{W_{0}^{-1,p^{\prime}}(\mathbb{R}^{3})\times
W_{0}^{1,p}(\mathbb{R}^{3})}+\left\langle\Delta\textbf{{v}},\textbf{{g}}\right\rangle_{[\mathring{H}_{-1}^{\,p^{\prime}}(\mathrm{div},\,\Omega)]^{\prime}\times\mathring{H}_{-1}^{\,p^{\prime}}(\mathrm{div},\,\Omega)}$
$=\lim\limits_{k\rightarrow\infty}(-\displaystyle\int_{\Omega}\Delta\textbf{{g}}_{k}\cdot\textbf{{v}}dx+\left\langle\Delta\textbf{{v}},\textbf{{g}}_{k}\right\rangle_{[\mathring{H}_{-1}^{\,p^{\prime}}(\mathrm{div},\,\Omega)]^{\prime}\times\mathring{H}_{-1}^{\,p^{\prime}}(\mathrm{div},\,\Omega)}$
$=\lim\limits_{k\rightarrow\infty}(-\displaystyle\int_{\Omega}\Delta\textbf{{g}}_{k}\cdot\textbf{{v}}dx+\displaystyle\int_{\Omega}\textbf{{v}}\cdot\Delta\textbf{{g}}_{k}dx)=0.$

∎

Next, we introduce the following space:

$\displaystyle\textbf{{V}}_{0,T}^{\,p}(\Omega)=\left\\{\textbf{{z}}\in\textbf{{X}}^{\,p}_{0,T}(\Omega);\,\,\mathrm{div}\,\textbf{{z}}=0\,\,\mathrm{in}\,\,\,\Omega\right\\},$

where

$\textbf{{X}}^{\,p}_{0,T}(\Omega)=\\{\textbf{{z}}\in\textbf{{X}}^{\,p}_{0}(\Omega)\,\,;\,\,\textbf{{z}}\cdot\textbf{{n}}=0\,\,\,on\,\,\,\Gamma\\}.$

As a consequence, we have the following result.

######  Lemma 2.8.

The linear mapping
$\gamma:\textbf{{v}}\rightarrow\mathbf{curl}\,\textbf{{v}}\mid_{\Gamma}\times\textbf{{n}}$
defined on $\mathcal{D}(\overline{\Omega})$ can be extended to a linear
continuous mapping

$\displaystyle\gamma:\textbf{E}^{p}(\Omega)\longrightarrow
W^{-1/p,p}(\Gamma).$

Moreover, we have the Green formula: for any
$\textbf{{v}}\in\textbf{E}^{p}(\Omega)$ and any
$\varphi\in\textbf{{V}}^{p^{\prime}}_{0,T}(\Omega)$,

$\displaystyle\langle-\Delta\,\textbf{{v}},\varphi\rangle_{\Omega}=\int_{\Omega}\mathbf{curl}\,\textbf{{v}}\,.\,\mathbf{curl}\,\varphi\,d\textbf{x}+\langle\mathbf{curl}\,\textbf{{v}}\times\textbf{{n}},\varphi\rangle_{\Gamma}$
(2.17)

Where $\left\langle.\,,\,.\right\rangle_{\Gamma}$ denotes the duality between
$W^{-1/p,p}(\Gamma)$ and $W^{1/p,p^{\prime}}(\Gamma)$ and
$\left\langle.\,,\,.\right\rangle_{\Omega}$ denotes the duality between
$[\mathring{H}_{-1}^{\,\,p^{\prime}}(\mathrm{div},\Omega)]^{\prime}$ and
$\mathring{H}^{\,\,p^{\prime}}_{-1}(\mathrm{div},\Omega)$.

###### Proof.

Let $\textbf{{v}}\in\boldsymbol{\mathcal{D}}(\overline{\Omega})$. Observe that
if $\boldsymbol{\varphi}\in W_{0}^{1,p^{\prime}}(\Omega)$ such that
$\boldsymbol{\varphi}\cdot\textbf{{n}}=0$ on $\Gamma$ we deduce that
$\boldsymbol{\varphi}\in\textbf{{X}}_{T}^{\,p^{\prime}}(\Omega)$, then (2.17)
holds for such $\boldsymbol{\varphi}$.

For every $\boldsymbol{\mu}\in W^{\,1/p,p^{\prime}}(\Gamma)$, then there
exists $\boldsymbol{\varphi}\in W_{0}^{1,p^{\prime}}(\Omega)$ such that
$\boldsymbol{\varphi}=\boldsymbol{\mu}_{t}$ on $\Gamma$ and that
$\mathrm{div}\,\boldsymbol{\varphi}=0$ with

$||\boldsymbol{\varphi}||_{W_{0}^{1,p^{\prime}}(\Omega)}\leqslant
C||\boldsymbol{\mu}_{t}||_{W^{\,1/p,p^{\prime}}(\Gamma)}\leqslant
C||\boldsymbol{\mu}||_{W^{\,1/p,p^{\prime}}(\Gamma)}.$ (2.18)

As a consequence, using (2.17) we have

$|\langle\mathbf{curl}\,\textbf{{v}}\times\textbf{{n}},\,\boldsymbol{\mu}\rangle_{\Gamma}|\leqslant
C||\textbf{{v}}||_{\textbf{{E}}^{\,p}(\Omega)}||\boldsymbol{\mu}||_{W^{\,1/p,p^{\prime}}(\Gamma)}.$

Thus,

$||\mathbf{curl}\,\textbf{{v}}\times\textbf{{n}}||_{W^{\,-1/p,p}(\Gamma)}|\leqslant
C||\textbf{{v}}||_{\textbf{{E}}^{\,p}(\Omega)}.$

We deduce that the linear mapping $\gamma$ is continuous for the norm
$\textbf{{E}}^{\,p}(\Omega)$. Since
$\boldsymbol{\mathcal{D}}(\overline{\Omega})$ is dense in
$\textbf{{E}}^{\,p}(\Omega)$, $\gamma$ can be extended to by continuity to
$\gamma\in\mathcal{L}(\textbf{{E}}^{\,p}(\Omega),W^{\,-1/p,p}(\Gamma))$ and
formula (2.17) holds for all $\textbf{{v}}\in\textbf{{E}}^{\,p}(\Omega)$ and
$\boldsymbol{\varphi}\in W_{0}^{1,p^{\prime}}(\Omega)$ such that
$\mathrm{div}\,\boldsymbol{\varphi}=0$ in $\Omega$ and
$\boldsymbol{\varphi}\cdot\textbf{{n}}=0$ on $\Gamma$. ∎

###  2.4 The Laplace problem

This section is devoted to recall the solution of the Laplace equations in
$\mathbb{R}^{3}$ and we give a result concerning the Laplace problem with
Neumann boundary conditions. The result that we state below was proved by
Amrouche et all in [2].

######  Theorem 2.9.

Assume that $3/p+\alpha\neq 1$ and $3/p^{\prime}-\alpha\neq 1$, then for any
integer $m\geqslant 1$, the following Laplace operator is an isomorphism:

$\displaystyle\Delta:W^{1+m,p}_{\alpha+m}(\mathbb{R}^{3})/P_{[1-3/p-\alpha]}^{\bigtriangleup}$
$\displaystyle\longrightarrow$ $\displaystyle
W^{-1+m,p}_{\alpha+m}(\mathbb{R}^{3})\perp
P_{[1-3/p^{\prime}+\alpha]}^{\bigtriangleup}$ (2.19)

Now, we are interested into the following Neumann problem:

$\displaystyle-\Delta\,u=f\quad\text{ in}\,\,\,\Omega\quad\text{ and
}\quad\dfrac{\partial u}{\partial n}=g\quad\text{ on}\,\,\,\Gamma.$ (2.20)

J. Giroire in [21], studied the problem (2.20) in the Hilbert framework. In
[24], the authors investigated the harmonic Neumann problem in $L^{p}$ theory,
for the exterior domain with boundary of class $\mathcal{C}^{1,1}$, note that
these results have been also proved by Specovius-Neugebauer [32] with boundary
of class at leas $\mathcal{C}^{2}$. C. Amrouche, V. Girault and J. Giroire in
[3], studied the problem (2.20) with data f belongs to
$W^{-1,p}_{0}(\Omega)\cap L^{p}(\Omega)$, they got the existence of solutions
in $W^{1,p}_{0}(\Omega)$. In this work, we give a result concerning the
Laplace problem with Neumann boundary conditions with data $f$ belongs to
$L^{p}(\Omega)$, we got the existence of solutions in $W^{1,p}_{-1}(\Omega)$.

Our first proposition is established also in [24], it characterizes the kernel
of the Laplace operator with Neumann boundary condition. For any integer
$k\in\mathbb{Z}$ and $1<p<\infty$,

$\mathcal{N}_{p,\alpha}^{\Delta}=\left\\{w\in
W_{\alpha}^{1,p}(\Omega);\,\,\,\,\Delta\,w=0\quad\mathrm{in}\,\,\Omega\quad\mathrm{and}\quad\dfrac{\partial
w}{\partial n}=0\quad\mathrm{on}\,\,\Gamma\right\\}.$

######  Proposition 2.10.

For any integer $k\geq 1$, $\mathcal{N}_{p,\alpha}^{\Delta}$ the subspace of
all functions in $W_{\alpha}^{1,p}(\Omega)$ of the form $w(p)-p$, where $p$
runs over all polynomials of $\mathcal{P}_{[1-3/p-\alpha]}^{\Delta}$ and
$w(p)$ is the unique solution in $W_{0}^{1,2}(\Omega)\cap
W_{\alpha}^{1,p}(\Omega)$ of the Neumann problem

$\Delta\,w(p)=0\quad\mathrm{in}\,\,\Omega\quad\mathrm{and}\quad\dfrac{\partial
w(p)}{\partial n}=\dfrac{\partial p}{\partial n}\quad\mathrm{on}\,\,\Gamma.$
(2.21)

Here also, we set $\mathcal{N}_{p,\alpha}^{\Delta}=\left\\{0\right\\}$ when
$\alpha\leq 0$; $\mathcal{N}_{p,\alpha}^{\Delta}$ is a finite-dimentional
space of the same dimension as $\mathcal{P}_{[1-3/p-\alpha]}^{\Delta}$ and in
particular, $\mathcal{N}_{0}^{\Delta}=\mathbb{R}$.

The next theorem states an existence, uniqueness and regularity result for
problem (2.20).

######  Theorem 2.11.

Let $1<p<\infty$ be any real number and assume that $\Gamma$ is of class
$\mathcal{C}^{1,1}$ if $p\neq 2$. For any $f$ in $L^{p}(\Omega)$ and $g$ in
$W^{-1/p,p}(\Gamma)$. Then, the problem (2.20) has a solution $u\in
W^{1,p}_{-1}(\Omega)$ unique up to element of
$\mathcal{N}^{\Delta}_{p,-1}(\Omega)$ and we have the following estimate:

$\displaystyle\|u\|_{W^{1,p}_{-1}(\Omega)/\mathcal{N}^{\Delta}_{p,-1}(\Omega)}\leqslant
C\big{(}\|f\|_{L^{p}(\Omega)}+\|g\|_{W^{-1/p,p}(\Omega)}\big{)}$ (2.22)

If in addition,$f\in W^{1,p}_{1}(\Omega)$ and $g$ in
$W^{1/p^{\prime},p}(\Gamma)$, the solution $u$ of problem (2.20) belongs to
$W^{2,p}_{0}(\Omega)$ and satisfies

$\displaystyle\|u\|_{W^{2,p}_{0}(\Omega)}\leqslant
C\big{(}\|f\|_{L^{p}(\Omega)}+\|g\|_{W^{1/p^{\prime},p}(\Omega)}\big{)}$
(2.23)

###### Proof.

Let us extend $f$ by zero in $\Omega^{\prime}$ and let $\widetilde{f}$ denote
the extended function. Then $\widetilde{f}$ belongs to
$L^{p}(\mathbb{R}^{3})$. Using Theorem 2.9, there exists a unique function
$\widetilde{v}\in W^{2,p}_{0}(\mathbb{R}^{3})/\mathcal{P}^{\Delta}_{[2-3/p]}$
such that

$\displaystyle-\Delta\,\widetilde{v}=\widetilde{f}\quad\text{ in
}\,\,\mathbb{R}^{3}.$ (2.24)

Then $\nabla\,\widetilde{v}\cdot n$ belongs to
$W^{1-1/p,p}(\Gamma)\hookrightarrow W^{-1/p,p}(\Gamma)$. It follows from [24,
Theorem 3.12], that the following problem:

$\displaystyle\Delta\,w=0\quad\text{ in }\,\,\,\Omega\quad\text{ and
}\quad\nabla\,w\cdot n=g-\nabla\,\widetilde{v}\cdot n\quad\text{ on
}\,\,\,\Gamma,$ (2.25)

has a solution $w\in W^{1,p}_{-1}(\Omega)$ unique up to element of
$\mathcal{N}^{\Delta}_{p,-1}(\Omega)$. Thus
$u=\widetilde{v}_{\mid_{\Omega}}+w\in W^{1,p}_{-1}(\Omega)$ is the required
solution of (2.20). The uniqueness follows immediately from Proposition 2.10 .

Now, suppose that $g$ belongs to $W^{1/p^{\prime},p}(\Gamma).$ The aim is to
prove that $u$ belongs to $W^{2,p}_{0}(\Omega)$. To that end, let us introduce
the following partition of unity:

$\begin{split}&\varphi,\,\psi\in\mathcal{C}^{\infty}(\mathbb{R}^{3}),\quad
0\leq\varphi,\,\psi\leq
1,\quad\varphi+\psi=1\quad\text{in}\quad\mathbb{R}^{3},\\\
&\varphi=1\quad\text{in}\quad B_{R},\quad\text{supp }\varphi\subset
B_{R+1}.\end{split}$

Let $P$ be a continuous linear mapping from $W^{1,p}_{-1}(\Omega)$ to
$W^{1,p}_{-1}(\mathbb{R}^{3})$, such that $P\,u=\widetilde{u}$. Then
$\widetilde{u}$ belongs to $W^{1,p}_{-1}(\mathbb{R}^{3})$ and can be written
as:

$\displaystyle\widetilde{u}=\varphi\,{\widetilde{u}}+\psi\,\widetilde{u}.$

Next, one can easily observe that $\widetilde{u}$ satisfies the following
problem:

$\displaystyle-\Delta\,\psi\,\widetilde{u}=f_{1}\quad\text{ in
}\,\,\,\mathbb{R}^{3},$ (2.26)

with

$\displaystyle
f_{1}=\widetilde{f}{\psi}-(2\nabla{\widetilde{u}}\nabla{\psi}+\widetilde{u}\Delta\,{\psi}).$

Owing to the support of $\psi$, $f_{1}$ has the same regularity as $f$ and so
belongs to $L^{p}(\mathbb{R}^{3})$. It follows from Theorem 2.9, that there
exists $z$ in $W^{2,p}_{0}(\mathbb{R}^{3})$ such that $-\Delta\,z=f_{1}$ in
$\mathbb{R}^{3}.$ This implies that $\psi\,\widetilde{f}-z$ is a harmonic
tempered distribution and therefore a harmonic polynomial that belongs to
$\mathcal{P}^{\Delta}_{[2-3/p]}$. The fact that
$\mathcal{P}^{\Delta}_{[2-3/p]}\subset W^{2,p}_{0}(\mathbb{R}^{3})$ yields
that ${\psi\,\widetilde{u}}$ belongs to $W^{2,p}_{0}(\mathbb{R}^{3})$. In
particular, we have ${\psi\,\widetilde{u}}=u$ outside $B_{R+1}$, so the
restriction of $u$ to $\partial\,B_{R+1}$ belongs to
$W^{2-1/p,p}(\partial\,B_{R+1})$. Therefore, $\varphi\,u\in
W^{1,p}(\Omega_{R+1})$ satisfies the following problem:

$\displaystyle\begin{cases}-\Delta\,\varphi\,u=f_{2}\quad\text{ in
}\,\,\,\Omega_{R+1},\\\ \nabla\,\varphi\,u\cdot n=g\quad\text{ on
}\,\,\,\Gamma,\\\ \varphi\,u={\psi\,\widetilde{u}}\quad\text{ on
}\,\,\,\partial\,B_{R+1},\end{cases}$ (2.27)

where $f_{2}\in L^{p}(\Omega_{R+1})$ have similar expression as $f_{1}$ with
$\psi$ remplaced by $\varphi$. According [Remark 3.2, [3]], $\varphi\,u\in
W^{2,p}(\Omega_{R+1})$, which in turn shows that $\varphi\,\widetilde{u}$ also
belongs to $W^{2,p}(\Omega_{R+1})$. This implies that $u\in
W^{2,p}_{0}(\Omega).$
∎

##  3 Potential vector and inf-sup condition

In this section, we recall a result concerning the existence a vector
potential, and we give the proof of the Inf-sup condition, which plays a
crucial role in the existence and uniqueness of solutions. Let us recall the
abstract setting of Babu$\check{s}$ka-Brezzi’s Theorem (see Babu$\check{s}$ka
[9] and Brezzi [14]).

######  Theorem 3.1.

Let $X$ and $M$ be two reflexive Banach spaces and $X^{\prime}$ and
$M^{\prime}$ their dual spaces. Let $a$ be the continuous bilinear form
defined on $X\times M$, let $A\in\mathcal{L}(X;\ M^{\prime})$ and
$A^{\prime}\in\mathcal{L}(M;\ X^{\prime})$ be the operators defined by

$\forall v\in X,\ \forall w\in M,\ a(v,w)=\ <Av,w>\ =\ <v,A^{\prime}w>$

and $V=\mathrm{Ker}\,A$. The following statements are equivalent:

i) There exist $\beta>0$ such that

$\inf_{\begin{subarray}{c}w\in M\\\
w\not=0\end{subarray}}\,\sup_{\begin{subarray}{c}v\in X\\\
v\not=0\end{subarray}}\dfrac{a(v,w)}{\|v\|_{X}\,\|w\|_{M}}\geq\beta.$ (3.1)

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

iii) The operator $A^{\prime}:\ M\mapsto X^{\prime}\bot\,V$ is an isomophism
and $1/\beta$ is the continuity constant of $(A^{\prime})^{-1}$.

######  Remark 3.2.

As consequence, if the Inf-Sup condition (3.1) is satisfied, then we have the
following properties:

i) If $V=\\{0\\}$, then for any $f\in X^{\prime}$, there exists a unique $w\in
M$ such that

$\forall v\in X,\ a(v,w)=\
<f,v>\quad\mathrm{and}\quad\|w\|_{M}\leq\dfrac{1}{\beta}\|f\|_{X^{\prime}}.$
(3.2)

ii) If $V\not=\\{0\\}$, then for any $f\in X^{\prime},\ $ satisfying the
compatibility condition:

$\forall v\in V,\ <f,v>\ =0$, there exists a unique $w\in M$ such that (3.2).

iii) For any $g\in M^{\prime},\ \exists v\in X$, unique up an additive element
of $V$, such that:

$\forall w\in M,\ a(v,w)=\
<g,w>\quad\mathrm{and}\quad\|v\|_{X/V}\leq\dfrac{1}{\beta}\|g\|_{M^{\prime}}.$

Now, we recall a result concerning the potential vector provided in [24].

######  Theorem 3.3.

Assume that $1<p<3/2$. Let z in $H^{p}_{0}(\mathrm{div},\Omega)$ satisfying

$\mathrm{div}\,\textbf{{z}}=0$ in  $\Omega$ and
$\langle\textbf{{z}}\cdot\textbf{{n}},1\rangle_{\Gamma}=0$.

Then there exists a unique vector potential $\boldsymbol{\psi}$ in
$W^{1,p}_{0}(\Omega)$ such that:

$\displaystyle\textbf{{z}}=\textbf{curl}\,\boldsymbol{\psi}\text{,}\quad\mathrm{div}\,\boldsymbol{\psi}=0\quad\text{
in }\Omega\quad\text{ and
}\quad\boldsymbol{\psi}\cdot\textbf{{n}}=0\quad\text{ on }\,\,\Gamma.$ (3.3)

In addition we have the following estimate:

$\displaystyle\|\boldsymbol{\psi}\|_{W^{1,p}_{0}(\Omega)}\leqslant
C\|\textbf{{z}}\|_{L^{p}(\Omega)}.$ (3.4)

We introduce the following space:

$\displaystyle\textbf{{Z}}^{\,p}_{T}(\Omega)=\Big{\\{}\textbf{{v}}\in
W^{0,p}_{-1}(\Omega):\,\mathrm{div}\,\textbf{{v}}\in
L^{p}(\Omega),\,\,\mathbf{curl}\,\textbf{{v}}\in
L^{p}(\Omega)\,\,,\,\,\textbf{{v}}\cdot\textbf{{n}}=0\,\,\text{ on
}\,\,\Gamma\Big{\\}}.$

######  Theorem 3.4.

Let $1<p<3/2$. There exists a constant $C>0$ such that for all
$\boldsymbol{\varphi}\in\textbf{{Z}}^{\,p}_{T}(\Omega),$ we have

$\displaystyle||\boldsymbol{\varphi}||_{W_{0}^{1,p}(\Omega)}$
$\displaystyle\leq$ $\displaystyle
C\Big{(}||\mathrm{div}\,\boldsymbol{\varphi}||_{L^{p}(\Omega)}+||\mathbf{curl}\,\boldsymbol{\varphi}||_{L^{p}(\Omega)}\Big{)}.$
(3.5)

###### Proof.

Let $\boldsymbol{\varphi}$ in $\textbf{{Z}}^{\,p}_{T}(\Omega)$. It follows
from [[24], Theorem 5.1], that $\textbf{{Z}}^{\,p}_{T}(\Omega)$ is
continuously imbedded in $W^{1,p}_{0}(\Omega)$ and so $\boldsymbol{\varphi}$
belongs to $W^{1,p}_{0}(\Omega)$. Observe that for any $\chi\in
W_{0}^{1,p^{\prime}}(\Omega)$, the following Green formula holds:

$\displaystyle\langle\mathbf{curl}\,\varphi\cdot\textbf{{n}},\chi\rangle_{\Gamma}=\displaystyle\int_{\Omega}\mathbf{curl}\,\varphi\cdot\nabla\,\chi\,d\textbf{x}.$
(3.6)

Because $1<p<3/2$, the constants belong to $W_{0}^{1,p^{\prime}}(\Omega)$,
then (3.6) is still valid for $\chi=1$ and thus we have:

$\langle\mathbf{curl}\,\varphi\cdot\textbf{{n}},1\rangle_{\Gamma}=0.$

Using Theorem 3.3, we prove that there exists a unique vector potential
$\boldsymbol{\psi}\in W_{0}^{1,p}(\Omega)$ such that:

$\displaystyle\mathbf{curl}\,\boldsymbol{\varphi}=\textbf{curl}\,\boldsymbol{\psi}\text{,}\quad\mathrm{div}\,\boldsymbol{\psi}=0\quad\text{
in }\Omega\quad\text{ and
}\quad\boldsymbol{\psi}\cdot\textbf{{n}}=0\quad\text{ on }\,\,\Gamma.$

In addition, we have

$\displaystyle\|\boldsymbol{\psi}\|_{W^{1,p}_{0}(\Omega)}\leqslant
C\|\mathbf{curl}\,\boldsymbol{\varphi}\|_{L^{p}(\Omega)}.$

On the other hand, let us solve the following problem,

$\displaystyle\Delta\,\textit{{w}}=\mathrm{div}\,\boldsymbol{\varphi}\,\,\text{
in }\,\,\Omega\quad\text{ and
}\,\,\dfrac{\partial\textit{{w}}}{\partial\textbf{{n}}}=0\,\,\text{ on
}\,\,\Gamma.$ (3.7)

Thanks to Theorem 2.11, problem (3.7) has a solution $\textit{{w}}\in
W^{2,p}_{0}(\Omega)$ that satisfies

$\displaystyle\|\nabla\,\textit{{w}}\|_{W^{1,p}_{0}(\Omega)}\leqslant
C\|\mathrm{div}\,\boldsymbol{\varphi}\|_{L^{p}(\Omega)}$ (3.8)

Setting ${\textbf{{z}}}=\boldsymbol{\varphi}-\nabla\,\textit{{w}}$. It is
clear that z belongs to $W^{1,p}_{0}(\Omega)$, it is divergence-free,
${\textbf{{z}}}\cdot\textbf{{n}}=0$ on $\Gamma$ and
$\mathbf{curl}\,\varphi=\mathbf{curl}\,{\textbf{{z}}}$ in $\Omega$. The
uniqueness of $\boldsymbol{\psi}$ implies that
${\textbf{{z}}}=\boldsymbol{\psi}$ and thus we obtain

$\displaystyle\|{\textbf{{z}}}\|_{W^{1,p}_{0}(\Omega)}\leqslant
C\|\mathbf{curl}\,\boldsymbol{\varphi}\|_{L^{p}(\Omega)}.$ (3.9)

Finally, the inequality (3.5) follows immediately from (3.9) and (3.8). ∎

######  Remark 3.5.

As a consequence of Theorem 3.4, the seminorm in the right-hand side of (3.5)
is a norm on $\textbf{{Z}}^{\,p}_{T}(\Omega)$ equivalent to the norm
$||\boldsymbol{\varphi}||_{W_{0}^{1,p}(\Omega)}$.

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

######  Lemma 3.6.

Assume that $p>3$. The following Inf-Sup Condition holds: there exists a
constant $\beta>0$, such that

$\inf_{\begin{subarray}{c}\boldsymbol{\varphi}\in\textbf{{V}}^{\,p^{\prime}}_{0,T}(\Omega)\\\
\boldsymbol{\varphi}\neq
0\end{subarray}}\sup_{\begin{subarray}{c}\boldsymbol{\psi}\in\textbf{{V}}^{\,p}_{0,T}(\Omega)\\\
\boldsymbol{\psi}\neq
0\end{subarray}}\dfrac{\int_{\Omega}\mathbf{curl}\,\boldsymbol{\psi}\cdot\mathbf{curl}\,\boldsymbol{\varphi}\,\mathrm{d}\textbf{{x}}}{\|\boldsymbol{\psi}\|_{\textbf{{Z}}^{\,p}_{T}(\Omega)}\|\boldsymbol{\varphi}\|_{\textbf{{Z}}^{\,p^{\prime}}_{T}(\Omega)}}\geq\beta.$
(3.10)

###### Proof.

Let $\textbf{{g}}\in\textbf{{L}}^{p}(\Omega)$ and let us introduce the
following Dirichlet problem:

$-\Delta\,\chi=\mathrm{div}\,\textbf{{g}}\quad\mathrm{in}\,\,\Omega,\quad\chi=0\quad\mathrm{on}\,\,\Gamma.$

It is shown in Theorem $3.7$ of [24], that this problem has a solution
$\chi\in\mathring{W}_{0}^{1,p}(\Omega)$ and we have

$\|\nabla\,\chi\|_{\textbf{{L}}^{p}(\Omega)}\leq
C\|\textbf{{g}}\|_{\textbf{{L}}^{p}(\Omega)}.$

Set $\textbf{{z}}=\textbf{{g}}-\nabla\,\chi$. Then we have
$\textbf{{z}}\in\textbf{{L}}^{p}(\Omega)$, $\mathrm{div}\,\textbf{{z}}=0$ and
we have

$\|\textbf{{z}}\|_{\textbf{{L}}^{p}(\Omega)}\leq
C\|\textbf{{g}}\|_{\textbf{{L}}^{p}(\Omega)}.$ (3.11)

Let $\boldsymbol{\varphi}$ any function of
$\textbf{{V}}^{\,p^{\prime}}_{0,T}(\Omega)$, by Theorem 3.4 we have
$\boldsymbol{\varphi}\in\textbf{{Z}}^{\,p^{\prime}}_{T}(\Omega)\hookrightarrow
W_{0}^{1,p^{\prime}}(\Omega)$. Then due to (3.5) we can write

$\|\boldsymbol{\varphi}\|_{W^{\,1,p^{\prime}}_{0}(\Omega)}\leq
C\|\mathbf{curl}\,\boldsymbol{\varphi}\|_{\textbf{{L}}^{p^{\prime}}(\Omega)}=C\,\sup_{\begin{subarray}{c}\textbf{{g}}\in\textbf{{L}}^{p}(\Omega)\\\
\textbf{{g}}\neq
0\end{subarray}}\dfrac{\big{|}\int_{\Omega}\mathbf{curl}\,\boldsymbol{\varphi}\cdot\textbf{{g}}\,\mathrm{d}\textbf{{x}}\big{|}}{\|\textbf{{g}}\|_{\textbf{{L}}^{p}(\Omega)}}.$
(3.12)

Using the fact that $\mathbf{curl}\,\boldsymbol{\varphi}\in
H_{0}^{\,p^{\prime}}(\mathrm{div},\Omega)$ and applying (2.11), we obtain

$\int_{\Omega}\mathbf{curl}\,\boldsymbol{\varphi}\cdot\nabla\,\chi\,\mathrm{d}\textbf{{x}}=0.$
(3.13)

Now, let $\lambda\in W_{1}^{2,p}(\Omega)\cap W_{1}^{2,2}(\Omega)$ the solution
of the following problem:

$\Delta\,\lambda=0\quad\mathrm{in}\,\,\Omega\quad\mathrm{and}\quad\lambda=1\quad\mathrm{on}\,\,\Gamma.$

It follows from Lemma $3.11$ of [20] that

$\Big{\langle}\dfrac{\partial\lambda}{\partial\textbf{{n}}},1\Big{\rangle}_{\Gamma}=\int_{\Gamma}\dfrac{\partial\lambda}{\partial\textbf{{n}}}d\sigma=C_{1}>0.$

Now, setting

$\widetilde{\textbf{{z}}}=\textbf{{z}}-\dfrac{1}{C_{1}}\left\langle\textbf{{z}}\cdot\textbf{{n}},1\right\rangle_{\Gamma}\nabla\,\lambda.$

It is clear that $\widetilde{\textbf{{z}}}\in\textbf{{L}}^{p}(\Omega)$,
$\mathrm{div}\,\widetilde{\textbf{{z}}}=0$ in $\Omega$ and that
$\left\langle\widetilde{\textbf{{z}}}\cdot\textbf{{n}},1\right\rangle_{\Gamma}=0$.
Due to Theorem 3.3, there exists a potential vector $\boldsymbol{\psi}\in
W_{0}^{1,p}(\Omega)$ such that

$\widetilde{\textbf{{z}}}=\mathbf{curl}\,\boldsymbol{\psi},\quad\mathrm{div}\,\boldsymbol{\psi}=0\quad\mathrm{in}\,\Omega\quad\mathrm{and}\quad\boldsymbol{\psi}\cdot\textbf{{n}}=0\quad\mathrm{on}\,\Gamma.$
(3.14)

In addition, we have the estimate

$||\boldsymbol{\psi}||_{W_{0}^{1,p}(\Omega)}\leq
C||\widetilde{\textbf{{z}}}||_{\textbf{{L}}^{p}(\Omega)}\leq
C||\textbf{{z}}||_{\textbf{{L}}^{p}(\Omega)}.$ (3.15)

Then, we obtain that $\boldsymbol{\psi}$ belongs to
$\textbf{{V}}^{\,p}_{0,T}(\Omega)$. Since $\boldsymbol{\varphi}$ is
$W^{1,p^{\prime}}_{loc}$ in a neighborhood of $\Gamma$, then
$\boldsymbol{\varphi}$ has an $W^{1,p^{\prime}}_{loc}$ extension in
$\Omega^{\prime}$ denoted by $\widetilde{\boldsymbol{\varphi}}$. Applying
Green’s formula in $\Omega^{\prime}$, we obtain

$0=\int_{\Omega^{\prime}}\mathrm{div}(\mathbf{curl}\,\widetilde{\boldsymbol{\varphi}})\,\mathrm{d}\textbf{{x}}=\left\langle\mathbf{curl}\,\widetilde{\boldsymbol{\varphi}}\cdot\textbf{{n}},1\right\rangle_{\Gamma}=\left\langle\mathbf{curl}\,\boldsymbol{\varphi}\cdot\textbf{{n}},1\right\rangle_{\Gamma}.$

Using the fact that $\mathbf{curl}\,\boldsymbol{\varphi}$ in
$H_{0}^{\,p^{\prime}}(\mathrm{div},\Omega)$ and $\lambda$ in
$W_{1}^{2,p}(\Omega)\hookrightarrow W_{0}^{1,p}(\Omega)$ and applying (2.11),
we obtain

$0=\left\langle\mathbf{curl}\,\boldsymbol{\varphi}\cdot\textbf{{n}},1\right\rangle_{\Gamma}=\left\langle\mathbf{curl}\,\boldsymbol{\varphi}\cdot\textbf{{n}},\lambda\right\rangle_{\Gamma}=\int_{\Omega}\mathbf{curl}\,\boldsymbol{\varphi}\cdot\nabla\,\lambda\,\mathrm{d}\textbf{{x}}.$
(3.16)

Using (3.13) and (3.16), we deduce that

$\int_{\Omega}\mathbf{curl}\,\boldsymbol{\varphi}\cdot\textbf{{g}}\,\mathrm{d}\textbf{{x}}=\int_{\Omega}\mathbf{curl}\,\boldsymbol{\varphi}\cdot\textbf{{z}}\,\mathrm{d}\textbf{{x}}=\int_{\Omega}\mathbf{curl}\,\boldsymbol{\varphi}\cdot\widetilde{\textbf{{z}}}\,\mathrm{d}\textbf{{x}}.$
(3.17)

From (3.11), (3.15) and (3.17), we deduce that

$\dfrac{\big{|}\int_{\Omega}\mathbf{curl}\,\boldsymbol{\varphi}\cdot\textbf{{g}}\,\mathrm{d}\textbf{{x}}\big{|}}{\|\textbf{{g}}\|_{\textbf{{L}}^{p}(\Omega)}}\leq
C\dfrac{\big{|}\int_{\Omega}\mathbf{curl}\,\boldsymbol{\varphi}\cdot\widetilde{\textbf{{z}}}\,\mathrm{d}\textbf{{x}}\big{|}}{\|\widetilde{\textbf{{z}}}\|_{\textbf{{L}}^{p}(\Omega)}}=C\dfrac{\big{|}\int_{\Omega}\mathbf{curl}\,\boldsymbol{\varphi}\cdot\mathbf{curl}\,\boldsymbol{\psi}\,\mathrm{d}\textbf{{x}}\big{|}}{\|\mathbf{curl}\,\boldsymbol{\psi}\|_{\textbf{{L}}^{p}(\Omega)}}.$

Applying again (3.5), we obtain

$\dfrac{\big{|}\int_{\Omega}\mathbf{curl}\,\boldsymbol{\varphi}\cdot\textbf{{g}}\,\mathrm{d}\textbf{{x}}\big{|}}{\|\textbf{{g}}\|_{W^{p}(\Omega)}}\leq
C\dfrac{\big{|}\int_{\Omega}\mathbf{curl}\,\boldsymbol{\varphi}\cdot\mathbf{curl}\,\boldsymbol{\psi}\,\mathrm{d}\textbf{{x}}\big{|}}{\|\boldsymbol{\psi}\|_{W_{0}^{\,1,p}(\Omega)}},$

and the Inf-Sup Condition (3.10) follows immediately from (3.12). ∎

##  4  Existence of weak solution for $p>3$

We prove in this sequel the existence and the uniqueness of weak solutions for
the following Problem:

$\displaystyle(\mathcal{S}_{T})\begin{cases}-\Delta\,\textbf{{u}}+\nabla\,{\pi}=\textbf{{f}}\quad
and\quad\mathrm{div}\,\textbf{{u}}={\chi}\quad\text{ in }\,\Omega,\\\
\quad\textbf{{u}}\cdot\textbf{{n}}=g\,\,\,\,and\quad\mathbf{curl}\,\textbf{{u}}\times\textbf{{n}}=\textbf{{h}}\quad\text{
on }\,\Gamma.\end{cases}$

The following result concerns the existence and uniqueness of the weak
solution of the problem $(\mathcal{S}_{T})$ where $\chi=g=0$.

######  Theorem 4.1.

Assume that $p>3$, suppose that $\chi=g=0$. Then, for any
$\textbf{{f}}\in[\mathring{H}_{-1}^{\,\,p^{\prime}}(\mathrm{div},\Omega)]^{\prime}$
and $\textbf{{h}}\in W^{-1/p,p}(\Gamma)$, the problem $(\mathcal{S}_{T})$ has
a unique solution
$(\textbf{{u}},{\pi})\in(W^{1,p}_{0}(\Omega)\times{{L}}^{p}(\Omega))$. In
addition, we have the following estimate:

$\displaystyle\|\textbf{{u}}\|_{W^{1,p}_{0}(\Omega)}+\|{\pi}\|_{{{L}}^{p}(\Omega)}\leqslant
C(\|\textbf{{f}}\|_{[\mathring{H}_{-1}^{\,\,p^{\prime}}(\mathrm{div},\Omega)]^{\prime}}+\|\textbf{{h}}\|_{W^{-1/p,p}(\Gamma)}).$
(4.1)

###### Proof.

On the first hand, observe that Problem $(\mathcal{S}_{T})$ is reduced to the
following variational problem:

$\displaystyle\begin{cases}Find\quad\textbf{{u}}\in\textbf{V}_{0,T}^{p}(\Omega)\quad
such\quad that,\\\
\quad\displaystyle{\int}_{\Omega}\textbf{curl}\textbf{{u}}.\textbf{curl}\boldsymbol{\varphi}d\textbf{x}=\langle\textbf{{f}},\boldsymbol{\varphi}\rangle_{\Omega}+\left\langle\textbf{{h}},\boldsymbol{\varphi}\right\rangle_{\Gamma},\forall\boldsymbol{\varphi}\in\textbf{{V}}^{p^{\prime}}_{0,T}(\Omega)\end{cases}$
(4.2)

Let us introduce the space

$\mathcal{D}_{\sigma}(\Omega)=\Big{\\{}\boldsymbol{\varphi}\in\mathcal{D}(\Omega),\,\mathrm{div}\,\boldsymbol{\varphi}=0\Big{\\}}.$

Indeed, every solution of $(\mathcal{S}_{T})$ also solves (4.2). Conversely,
let us take $\boldsymbol{\varphi}\in\mathcal{D}_{\sigma}(\Omega)$ as the test
function in (4.2), then we get

$\displaystyle\langle-\Delta\,\textbf{{u}}-\textbf{{f}},\boldsymbol{\varphi}\rangle_{\mathcal{D}_{\sigma}^{\prime}(\Omega)\times\mathcal{D}_{\sigma}(\Omega)}=0.$
(4.3)

By De Rham theorem, there exist a function $\pi\in L^{p}(\Omega)$ such that

$-\Delta\,\textbf{{u}}+\nabla\,\pi=\textbf{{f}}\quad\text{ in }\Omega.$

Moreover, by the fact that u belongs to the space
$\textbf{{V}}^{p}_{0,T}(\Omega)$, we have $\mathrm{div}\,\textbf{{u}}=0$ in
$\Omega$, $\textbf{{u}}\cdot\textbf{{n}}=0$ on $\Gamma$. The remainder
boundary condition
$\mathbf{curl}\,\textbf{{u}}\times\textbf{{n}}=\textbf{{H}}$ on $\Gamma$ is
implicitly contained in (4.2). Observe that since $\nabla\,\pi$ is elements of
$[\mathring{H}^{p^{\prime}}_{-1}(\mathrm{div},\Omega)]^{\prime}$, it is the
same for $\Delta\,\textbf{{u}}$. Since, $\mathcal{D}(\Omega)$ is dense in
$\mathring{H}^{p^{\prime}}_{-1}(\mathrm{div},\Omega)$, it is clear then that
for any
$\boldsymbol{\varphi}\in\mathring{H}^{p^{\prime}}_{-1}(\mathrm{div},\Omega)$

$\displaystyle\langle\nabla\,\pi,\boldsymbol{\varphi}\rangle_{\Omega}=-\displaystyle\int_{\Omega}\pi\mathrm{div}\,\boldsymbol{\varphi}\,d\textbf{{x}},\quad\forall\boldsymbol{\varphi}\in\mathring{H}^{p^{\prime}}_{-1}(\mathrm{div};\Omega).$

Particularly

$\langle\nabla\,\pi,\boldsymbol{\varphi}\rangle_{\Omega}=0\quad\text{if}\quad\boldsymbol{\varphi}\in{\textbf{V}}^{p^{\prime}}_{0,\sigma}(\Omega).$

Moreover, if $\boldsymbol{\varphi}\in\textbf{{V}}^{p^{\prime}}_{0,T}(\Omega)$,
using the Green formula (2.17) we have

$\displaystyle\langle-\Delta\,\textbf{{u}},\boldsymbol{\varphi}\rangle_{[\mathring{H}_{-1}^{p^{\prime}}(\mathrm{div},\Omega)]^{\prime}\times\mathring{H}_{-1}^{p^{\prime}}(\mathrm{div},\Omega)}=\int_{\Omega}\mathbf{curl}\,\textbf{{u}}\,.\,\mathbf{curl}\,\varphi\,d\textbf{x}+\langle\mathbf{curl}\,\textbf{{u}}\times\textbf{{n}},\varphi\rangle_{\Gamma}$
(4.4)

Therefore, from (4.2) and (4.4) we deduce that for all
$\boldsymbol{\varphi}\in\textbf{{V}}^{p^{\prime}}_{0,T}(\Omega)$

$\displaystyle\langle\mathbf{curl}\,\textbf{{u}}\times\textbf{{n}},\boldsymbol{\varphi}\rangle_{\Gamma}=\langle\textbf{{H}},\boldsymbol{\varphi}\rangle_{\Gamma}$

Let now $\boldsymbol{\mu}$ any element of the space
$W^{1-1/p^{\prime},p^{\prime}}(\Gamma)$. So, there exists an element
$\boldsymbol{\varphi}\in W^{1,p^{\prime}}_{0}(\Omega)$ such that
$\mathrm{div}\,\boldsymbol{\varphi}=0$ in $\Omega$ and
$\boldsymbol{\varphi}=\boldsymbol{\mu}_{\tau}$ on $\Gamma$. Then, we have

$\displaystyle\langle\mathbf{curl}\,\textbf{{u}}\times\textbf{{n}},\boldsymbol{\mu}\rangle_{\Gamma}-\langle\textbf{{H}},\boldsymbol{\mu}\rangle_{\Gamma}$
$\displaystyle=$
$\displaystyle\langle\mathbf{curl}\,\textbf{{u}}\times\textbf{{n}},\boldsymbol{\mu}_{\tau}\rangle_{\Gamma}-\langle\textbf{{H}},\boldsymbol{\mu}_{\tau}\rangle_{\Gamma}$
$\displaystyle=$
$\displaystyle\langle\mathbf{curl}\,\textbf{{u}}\times\textbf{{n}},\boldsymbol{\varphi}\rangle_{\Gamma}-\langle\textbf{{H}},\boldsymbol{\varphi}\rangle_{\Gamma}$

This implies that $\mathbf{curl}\,\textbf{{u}}\times\textbf{{n}}=\textbf{{H}}$
on $\Gamma$.

This implies that problem $(\mathcal{S}_{T})$ and problem (4.2) are
equivalent. Now, to solve Problem (4.2), we use the Inf-Sup condition (3.10).
We consider the bilinear form $\textbf{a}:$
$\textbf{{V}}_{0,T}^{\,p}(\Omega)\times\textbf{{V}}_{0,T}^{\,p^{\prime}}(\Omega)\longrightarrow\mathbb{R}$
such that

$\displaystyle\textbf{a}(\textbf{{u}},\varphi)=\displaystyle{\int}_{\Omega}\mathbf{curl}\,{\textbf{{u}}}\cdot\mathbf{curl}\,\boldsymbol{\varphi}\,\,\mathrm{d}\textbf{{x}}.$

Let consider the following mapping
$\boldsymbol{\ell}:\,\textbf{{V}}^{\,p^{\prime}}_{0,T}(\Omega)\longrightarrow\,\mathbb{R}$
such that
$\boldsymbol{\ell}(\boldsymbol{\varphi})=\left\langle\textbf{{f}},\boldsymbol{\varphi}\right\rangle_{\Omega}+\langle\textbf{{h}},\boldsymbol{\varphi}\rangle_{\Gamma}$.
It is clear that $\boldsymbol{\ell}$ belongs to
$(\textbf{{V}}^{\,p^{\prime}}_{0,T}(\Omega))^{\prime}$ and according to Remark
3.2, there exists a unique solution
$\textbf{{u}}\in\textbf{{V}}^{\,p}_{0,T}(\Omega)$ of Problem (4.2). Due to
Theorem 3.4, we prove that this solution u belongs ro $W^{1,p}_{0}(\Omega)$.
It follows from Remark 3.2 $i)$ that

$\displaystyle\|\textbf{{u}}\|_{W^{1,p}_{0}(\Omega)}\leqslant
C(\|\textbf{{f}}\|_{[\mathring{H}_{-1}^{\,\,p^{\prime}}(\mathrm{div},\Omega)]^{\prime}}+\|\textbf{{h}}\|_{W^{-1/p,p}(\Gamma)}).$
(4.5)

Since $\nabla\,\pi=\textbf{{f}}+\Delta\,\textbf{{u}}$, then we have the
following estimate

$\displaystyle{\|\pi\|_{L^{p}(\Omega)}\leqslant
C\|\nabla\,\pi\|_{W^{-1,p}_{0}(\Omega)}}\leqslant
C\big{(}\|\textbf{{f}}\|_{[\mathring{H}_{-1}^{p^{\prime}}(\mathrm{div},\Omega)]^{\prime}}+\|\textbf{{u}}\|_{W^{1,p}_{0}(\Omega)}\big{)}$
(4.6)

The estimate (4.1) follows from (4.5) and (4.6). ∎

We can also solve the Stokes problem when the divergence operator does not
vanish and $g\neq 0$.

######  Corollary 4.2.

Assume that $p>3$. Let f, $\chi$, $g$, h such that

$\textbf{{f}}\in[\mathring{H}_{-1}^{\,p^{\prime}}(\mathrm{div},\,\Omega)]^{\prime},\,\chi\in
L^{\,p}(\Omega),\,\,g\in
W^{\,1-1/p,p}(\Gamma)\,\,\,\mathrm{and}\,\,\,\,\textbf{{h}}\in
W^{\,-1/p,p}(\Gamma).$

Then, the Stokes problem $(\mathcal{S}_{T})$ has a unique solution
$(\textbf{{u}}\,,\,\pi)\in W_{0}^{\,1,p}(\Omega)\times L^{\,p}(\Omega)$ and we
have:

$\|\textbf{{u}}\,\|_{W_{0}^{\,1,p}(\Omega)}+\|\,\pi\,\|_{L^{\,p}(\Omega)}\leqslant
C\left(\|\,\textbf{{f}}\,\|_{[\mathring{H}_{-1}^{\,p^{\prime}}(\mathrm{div},\,\Omega)]^{\prime}}+\|\,\chi\,\|_{L^{\,p}(\Omega)}+\|g\|_{W^{\,1-1/p,p}(\Gamma)}+\|\textbf{{h}}\|_{W^{\,-1/p,p}(\Gamma)}\right).$
(4.7)

###### Proof.

Let $\chi\in L^{p}(\Omega)$ and $g\in W^{1-1/p,p}(\Gamma)$. We solve the
following Neumann problem in $\Omega$:

$-\Delta\,\theta=\chi\quad\mathrm{in}\,\,\Omega,\quad\frac{\partial\theta}{\partial\textbf{{n}}}=g\quad\mathrm{on}\,\,\Gamma.$
(4.8)

It follows from Theorem 2.11, that the Problem (4.8) has a solution $\theta$
in $W_{0}^{\,2,p}(\Omega)$ and we have:

$||\theta||_{W_{0}^{\,2,p}(\Omega)}\leqslant
C\left(||\chi||_{L^{\,p}(\Omega)}+||g||_{W^{1-1/p,p}(\Gamma)}\right).$ (4.9)

Setting $\textbf{{z}}=\textbf{{u}}-\nabla\,\theta$, then Problem
$(\mathcal{S}_{T})$ becomes: Find $(\textbf{{z}},\pi)\in
W_{0}^{1,p}(\Omega)\times L^{\,p}(\Omega)$ such that

$\begin{cases}-\Delta\,\textbf{{z}}+\nabla\,\pi=\textbf{{f}}+\nabla\,\chi\quad\mathrm{and}\quad\mathrm{div}\,\textbf{{z}}=0&\quad\mathrm{in}\,\Omega,\\\
\textbf{{z}}\cdot\textbf{{n}}=0\quad\mathrm{and}\quad\mathbf{curl}\,\textbf{{z}}\times\textbf{{n}}=\textbf{{h}}&\quad\mathrm{on}\,\mathrm{\Gamma},\\\
\end{cases}$ (4.10)

Observe that $\textbf{{f}}+\nabla\,\chi$ belongs to
$[\mathring{H}_{-1}^{\,p^{\prime}}(\mathrm{div},\,\Omega)]^{\prime}$.
According to Theorem 4.1, this problem has a unique solution
$(\textbf{{z}}\,,\,\pi)\in W_{0}^{\,1,p}(\Omega)\times L^{\,p}(\Omega)$. Thus
$\textbf{{u}}=\textbf{{z}}+\nabla\,\theta$ belongs to $W_{0}^{\,1,p}(\Omega)$
and estimate (4.7) follows from (4.1) and (4.9). ∎

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# A Closed Universe: de Sitter Cosmic Gate

S. Danial Forghani<EMAIL_ADDRESS>Faculty of Engineering, Final
International University, Kyrenia, North Cyprus via Mersin 10, Turkey  S.
Habib Mazharimousavi<EMAIL_ADDRESS>Department of Physics, Faculty
of Arts and Sciences, Eastern Mediterranean University, Famagusta, North
Cyprus via Mersin 10, Turkey

###### Abstract

A new cosmological object in analogy with the concept of a wormhole in general
relativity is introduced. As wormholes connect two distant points through a
tunnel in a spacetime, this new object connects two spacetime through a large
mouth which is referred to as a ”Cosmic Gate”. In this context, two identical
copies of the regular part of the de Sitter spacetime are cut through a
timelike hyperplane, and are glued at their identical boundaries (the only
boundary) to form a complete spacetime. The stability of the cosmic gate
against a linear radial perturbation is studied as well. Finally, the radial
geodesics of the spacetime for a timelike and a null particle are presented.

Wormhole; Thin-shell; de Sitter; Closed universe; Cosmic gate

##  I Introduction

In his 1985 hard sci-fi novel ”Contact”, Carl Sagan portraits a spacetime
journey via a wormhole in the most scientific way possible for its time. In
his endeavor to build up such an accurate presentation of wormholes for a
novel, he benefited from Kip Thorne’s consultant who at the time was a physics
professor at Caltech. Later in 1987, the communication between the two
scientists initiated the idea of two scientific masterworks, published in 1988
by Morris and Thorne 1 ; 2 . The significance of these articles lies in the
fact that, although wormholes and their application for space and time travel
were known by 1988, they were not considered to be practically traversable.
Therefore, the ever-fainting trend of wormhole physics since the 1950s was
considerably revived, which in turn led to many other great papers in the
followed by years 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10 . However, it was already
shown by Morris and Thorne 2  that in order to keep the throat of a wormhole
open, matter with negative energy density had to be distributed over the
spacetime. In 1989, Matt Visser, inspired by thin-shell formalism, constructed
a new class of traversable wormholes - known as thin-shell wormholes - which
had this exotic matter limited only to the location of the throat 3 . Six
years later, Poisson and Visser proposed a method by which the stability of
such wormholes could be somehow determined under a linear radial perturbation
11 .

In this study, we would like to employ the idea of thin-shell wormholes and
apply it to construct a new entity that represents a closed and complete
universe. In thin-shell wormhole studies, it is usual to cut two exterior
solutions in a way that the existed singularities and their associated event
horizons are excised out. Then the two remaining manifolds (that are
geodesically incomplete) are identified at the cut hypersurface to form a
geodesically complete manifold. Of course, the identification of the two
manifolds requires certain conditions to be satisfied which results in
determining the surface energy density and pressure of the matter that
supports the identification location, known as the throat of the wormhole. In
this paper, we instead excise out the exterior solutions and keep the interior
ones to construct a closed and complete universe by gluing them. In this
regard, we must choose a solution that is free of any singularities and/or
horizons at or towards the center at $r=0$. Evidently, the de Sitter solution
12 ; 13 ; 14  perfectly fits the situation. The de Sitter solution is a
maximally symmetric vacuum solution in general relativity that describes a
$3+1$-dimensional expanding universe with a positive cosmological constant.
The main significance of the de Sitter solution is that our universe is
believed to be asymptotically de Sitter, based on cosmological observations 15
.

The paper is arranged as follows. In section II, we briefly discuss wormholes
in Morris and Thorne’s sense to review some of their important features
related to our discussion. Section III goes into the details of our
construction. We show that the resultant thin-shell is supported by ordinary
matter and is stable under a linear radial perturbation. In the continuation,
we show in section IV that our constructed close universe is also geodesically
complete. We conclude our paper in section V. All over the paper we have used
the conventional units $c=G=1$.

##  II Wormhole vs. Gate

The so-called Morris-Thorne wormhole spacetime is described by

$ds^{2}=-e^{2\Phi\left(r\right)}dt^{2}+\frac{dr^{2}}{1-\frac{b\left(r\right)}{r}}+r^{2}d\Omega^{2},$
(1)

in which the red-shift function $e^{2\Phi\left(r\right)}$ admits no root in
the domain of $r$ and $\lim_{r\rightarrow\infty}\Phi\rightarrow 0$.
Furthermore, the shape function $b\left(r\right)$ satisfies the so-called
flare-out conditions 16 . These conditions are obtained through applying the
embedding process to (1), which leads to an embedding diagram for the
wormhole. The embedding diagram is to visualize a $2$-dimensional version of a
$3+1$-dimensional universe by considering an instant of time ($t=t_{0}$) on
the equator ($\theta=\pi/2$), that reduces the original line element in (1) to

$ds^{2}=\frac{dr^{2}}{1-\frac{b\left(r\right)}{r}}+r^{2}d\phi^{2}.$ (2)

Then, we embed this $2$-dimensional surface onto a $3$-dimensional flat
cylindrical space by comparing it to the line element

$ds^{2}=dr^{2}+dz^{2}+r^{2}d\phi^{2}.$ (3)

With the cylindrical symmetry configuration, we assume $z=z\left(r\right)$,
which takes us to

$\frac{1}{1-\frac{b\left(r\right)}{r}}=1+z^{\prime}\left(r\right)^{2},$ (4)

where $z^{\prime}\left(r\right)=\frac{dz\left(r\right)}{dr}$. Consequently, we
obtain

$z\left(r\right)=\pm\int_{R_{0}}^{r}\frac{1}{\sqrt{\frac{r}{b\left(r\right)}-1}}dr,$
(5)

in which at $r=R_{0}$ we assume $z\left(r\right)=z\left(R_{0}\right)=0.$
Accordingly, $z\left(R_{0}\right)$ is called the throat of the wormhole
connecting the region $z\geq 0$ to the region $z\leq 0$ provided that
$r\left(z\right)$ is the absolute minimum at $r\left(0\right)=R_{0}.$
Technically speaking, this condition means that $b\left(R_{0}\right)=R_{0}$
and $b-rb^{\prime}\geq 0$ for $r\geq R_{0}$. In order to keep the signature of
the spacetime unaltered for $r\geq R_{0}$, one should also add the condition
$b\left(r\right)<r$ to the previous conditions. Therefore, while in the
embedding diagram we have $z\in\mathbb{R}$, the condition $r\geq R_{0}$ always
holds. One can mathematically prove that the requirement of these conditions,
i.e. the flare-out conditions, implies that the energy-momentum tensor
supporting the wormhole spacetime does not satisfy the null energy condition,
hence the corresponding matter is exotic 2 .

Next, let us assume, there exists a spacetime described by the following line
element

$ds^{2}=-e^{2\Phi\left(r\right)}dt^{2}+\frac{dr^{2}}{\frac{b\left(r\right)}{r}-1}+r^{2}d\Omega^{2},$
(6)

such that $b\left(r\right)\geq r$ for $r\leq R_{0}$ and the equality occurs at
$r=R_{0}.$ Here, $r=R_{0}$, is not a throat but in the sequel we shall refer
to it as a “Gate”. Looking at its embedding diagram, we follow the steps as in
the case for the Morris-Thorne wormhole. Hence, we obtain

$\frac{1}{\frac{b\left(r\right)}{r}-1}=1+z^{\prime}\left(r\right)^{2},$ (7)

and as a result

$z^{\prime}\left(r\right)=\pm\sqrt{\frac{2-\frac{b\left(r\right)}{r}}{\frac{b\left(r\right)}{r}-1}.}$
(8)

Having $z^{\prime}\left(r\right)$ real, one has to impose the condition

$1\leq\frac{b}{r}\leq 2.$ (9)

We observe that, at $r=R_{0}$, $\frac{dr}{dz}=\frac{1}{z^{\prime}}$ is still
zero, however, its second derivative is negative, i.e.

$\frac{d^{2}r}{dz^{2}}<0.$ (10)

This implies that $r\left(z\right)$ at $z=0$ becomes an absolute maximum. The
function $z\left(r\right)$ is obtained to be

$z\left(r\right)=\pm\int_{R_{0}}^{r}\sqrt{\frac{2-\frac{b\left(r\right)}{r}}{\frac{b\left(r\right)}{r}-1},}$
(11)

where we again considered $z\left(R_{0}\right)=0$ and $z\in[-R_{0},R_{0}].$ On
this account, unlike the wormhole spacetime, here the gate connects two
bounded/closed spacetimes at their maximum surface. The resultant spacetime is
complete and consists of two parts connected by what we are calling a “Cosmic
Gate” from here onwards, in the hope that the terminology pleases the reader
both scientifically and phonaesthetically.

##  III de Sitter thin-shell cosmic gate

Let us start by setting the line element of a $3+1$-dimensional spherically
symmetric universe

$ds^{2}=-f\left(r\right)dt^{2}+\frac{1}{f\left(r\right)}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right),$ (12)

which represents the maximally symmetric de Sitter spacetime with a positive
scalar curvature for

$f\left(r\right)=1-\frac{r^{2}}{\ell^{2}},$ (13)

where $\ell^{2}=\frac{3}{\Lambda}$, with $\Lambda>0$ being the cosmological
constant. The spacetime is nonsingular, although it has a cosmological horizon
at $r_{c}=\ell$. In what follows we construct a closed and geodesically
complete universe by virtue of Visser’s cut-and-paste method originally used
for constructing thin-shell wormholes 3 . However, what eventually results in
our consideration is essentially a thin-shell cosmic gate, as we will show
here. Hence, in the bulk metric (12) we make two identical copies of the
spacetime inside a timelike hypersurface defined by $\Sigma:r=R<r_{c}$, and
paste them at this common hypersurface $\Sigma$ to construct a closed
universe. To show the closeness of this universe, we rely on its embedding
diagram, which is shown in Fig. 1 for arbitrary values of $r_{c}$ and
$R<r_{c}$.

Figure 1: Embedding diagram of two identical de-Sitter universes joined at the
cosmic gate. The result is a non-singular closed universe that is geodesically
complete.

The function regarding the diagram displayed in Fig. 1 is given by

$z=\pm\ell\,\left[\sqrt{1-\frac{r^{2}}{\ell^{2}}}-\sqrt{1-\frac{R^{2}}{\ell^{2}}}\right],$
(14)

following the method described in section II. Once again, we emphasize that
the flare-out conditions for thin-shell wormholes 17  behave absolutely
inversely in the case of thin-shell cosmic gates to satisfy
$\frac{d^{2}r}{dz^{2}}=-\frac{\ell^{2}}{r^{3}}<0$, in our case.

It is known from Darmois-Israel formalism that the boundary hypersurface
$\Sigma$ should satisfy certain junction conditions 18 . The details of the
procedure are explained in many references 19 ; 20 ; 21 ; 22 , so here we
refrain from excessive repetition. In short, the first fundamental form (the
induced metric tensor) has to be continuous across the thin-shell whereas the
second fundamental form (the induced curvature tensor) is discontinuous due to
the presence of matter at the shell. Satisfying these conditions leads to
mathematical expressions for the energy density $\sigma$ and angular pressure
$p$ of the matter present on the shell. The symmetry of the bulk and the
hyperplane $\Sigma$ yields the matter on the shell to be a perfect fluid with
the surface energy-momentum tensor given by
$S_{a}^{b}=diag\left(-\sigma,p,p\right)$. We should comment that there is a
fundamental difference between a thin-shell wormhole and a thin-shell cosmic
gate in this context, that lies in the direction of the $4$-normal of the
hypersurface $\Sigma$ \- which are actually in opposite directions.
Eventually, the expressions for the energy density and angular pressure read

$\sigma=\frac{1}{2\pi R}\,\sqrt{f\left(R\right)+\dot{R}^{2}},$ (15)

and

$p=-\frac{1}{4\pi}\,\frac{\ddot{R}+f^{\prime}\left(R\right)/2}{\sqrt{f\left(R\right)+\dot{R}^{2}}}-\frac{\sigma}{2},$
(16)

respectively. Here, $\left.\dot{R}\equiv\frac{dr}{d\tau}\right|_{r=R}$ is the
speed of the shell measured by an observer on the shell, and
$\left.f^{\prime}\left(R\right)\equiv\frac{df\left(r\right)}{dr}\right|_{r=R}$
is the total radial derivative of $f\left(r\right)$ at the shell. Note that
due to the opposite direction of the $4$-normals explained earlier, the signs
of the energy density $\sigma$ and angular pressure $p$ are opposite the signs
of their counterparts in thin-shell wormhole theory. It is evident from (15)
that in static equilibrium configuration where $\ddot{R}=\dot{R}=0$, the weak
energy condition 23  is satisfied for the perfect fluid at the gate i.e.,
$\sigma\geq 0$ and $\sigma+p\geq 0$. Therefore unlike for a wormhole the
matter on the shell/gate is ordinary. Besides, it can be shown by direct
substitution that the relation

$\frac{d\sigma}{dR}=-\frac{2}{R}\,\left(\sigma+p\right),$ (17)

illustrative of conservation of energy, holds for the matter at the gate.
Performing a second derivative on the latter equation results in

$\frac{d^{2}\sigma}{dR^{2}}=\frac{2}{R^{2}}\,\left(\sigma+p\right)\left(3+2\beta^{2}\right)+\frac{2\gamma}{R},$
(18)

in which $\beta^{2}\equiv\frac{\partial p}{\partial\sigma}$, and
$\gamma\equiv-\frac{\partial p}{\partial R}$, provided that a variable
equation of state $p=p\left(\sigma,R\right)$ 24  governs. Eq. (15) could be
written in an alternative form to recollect a classical equation of motion

$\dot{R}^{2}+V_{e}\left(R\right)=0,$ (19)

where

$V_{e}\left(R\right)=f\left(R\right)-\left(2\pi R\sigma\right)^{2},$ (20)

is the effective potential. In static equilibrium of thin-shell wormholes 11 ,
the first derivative of the effective potential i.e.,
$V_{e}^{\prime}\left(R_{0}\right)$ becomes zero at the supposed equilibrium
radius $R_{0}$, where the sign of the second derivative of the potential
$V_{e}^{\prime\prime}\left(R_{0}\right)$ determines whether the equilibrium is
stable or not. Although in this context what we are confronting is not a thin-
shell wormhole, technically speaking, the same method could be applied
consistently. To this, one should benefit from Eqs. (17) and (18) to arrive at
an expression for $V_{e}^{\prime\prime}\left(R_{0}\right)$. Then, by setting
$V_{e}^{\prime\prime}\left(R_{0}\right)=0$ and solving for $\beta^{2}$ we
obtain

$\beta^{2}=-\frac{1+4\pi\gamma
R_{0}^{2}\,f^{3/2}\left(R_{0}\right)}{2f\left(R_{0}\right)},$ (21)

where $f\left(R_{0}\right)=1-\frac{R_{0}^{2}}{\ell^{2}}$. The purpose here is
to graph $\beta^{2}$ versus the equilibrium radius $R_{0}$ to spot the
stability regions for which $V_{e}^{\prime\prime}\left(R_{0}\right)>0$. The
parameter $\beta^{2}$ is physically interpreted as the speed of sound in
matter 11 . In this regard, we are particularly interested in the regions
where $0<\beta^{2}<1$ since we have set $c=1$. As can also be observed from
Fig. 2, $\beta^{2}$ is negative-definite for a barotropic equation of state
where $\gamma=0$. On the other hand, for negative values of $\gamma$, the
value of $\beta^{2}$ may fall into the interval $\left(0,1\right)$ for stable
situations. In Fig. 2, $\beta^{2}$ is graphed versus $R_{0}$ for $\ell=1$ and
given values of $\gamma$.

Figure 2: The graph for speed of sound $\beta^{2}$ versus the equilibrium
radius $R_{0}$ for $\alpha=1$ and four given values of $\gamma$. The stable
regions are shown in colors.

##  IV The Geodesics

In this section, we study the geodesics of the constructed closed universe. In
this regard, we are interested in investigating the radial timelike and null
geodesics for which the angular momentum of the test particles is zero. The
Lagrangian of a particle moving on its equatorial plane along its geodesics in
a static spherically symmetric space-time is given by 25

$\mathcal{L}=\frac{1}{2}\left[-f\left(r\right)\dot{t}^{2}+\frac{1}{f\left(r\right)}\dot{r}^{2}+r^{2}\dot{\phi}^{2}\right]$
(22)

where an overhead dot represents differentiation with respect to an arbitrary
affine parameter. Furthermore the 4-velocity satisfies
$g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=\epsilon$ or explicitly

$-f\left(r\right)\dot{t}^{2}+\frac{1}{f\left(r\right)}\dot{r}^{2}+r^{2}\dot{\phi}^{2}=\epsilon$
(23)

with $\epsilon=-1$ ($\epsilon=0$) for a timelike (null) geodesics. In the
first observation, since $\mathcal{L}$ is independent of $t$ and $\phi$ the
energy

$E\equiv-\frac{\partial\mathcal{L}}{\partial\dot{t}}=f\left(r\right)\dot{t}$
(24)

and the angular momentum

$L\equiv\frac{\partial\mathcal{L}}{\partial\dot{\phi}}=r^{2}\dot{\phi}$ (25)

are conserved. By substituting the energy and angular momentum into Eq. (23)
one obtains the differential equation governing the radial coordinate of the
geodesics expressed by

$\dot{r}^{2}=E^{2}+f\left(r\right)\left(\epsilon-\frac{L^{2}}{r^{2}}\right).$
(26)

In what follows, we consider both timelike ($\epsilon=-1$) and null
($\epsilon=0$) cases for a radial motion with no angular momentum ($L=0$).

###  IV.1 The Timelike Geodesics

For $\epsilon=-1$, once we set $L=0$ in Eq. (26) it reduces to

$\frac{dr}{d\tau}=\pm\sqrt{E^{2}-1+\frac{r^{2}}{\ell^{2}}},$ (27)

where we considered the affine parameter to be the proper time. The latter
equation of motion is exactly solvable with the following solution

$r=r_{0}\cosh\left(\pm\frac{\left(\tau-\tau_{0}\right)}{r_{c}}\right),$ (28)

where $r_{0}=\ell\sqrt{1-E^{2}}$ ($E^{2}<1$) is the initial radial position
provided that the particle starts from the rest at $\tau=\tau_{0}$. Naturally,
we assume that $r_{0}<R_{0}$. Solving for $r=R_{0}$ shows that the particle
reaches the cosmic gate at ($\tau_{0}=0$)

$\tau_{R}=r_{c}\cosh^{-1}\left(\frac{R_{0}}{r_{0}}\right).$ (29)

This, however, is measured by the commoving observer, not by a distant
observer. For the distant observer, one needs to solve

$\frac{dr}{dt}=\pm\left(1-\frac{r^{2}}{r_{c}^{2}}\right)\sqrt{\frac{r^{2}-r_{0}^{2}}{r_{c}^{2}-r_{0}^{2}}},$
(30)

which directly results from the Lagrangian in Eq. (22) when
$\dot{r}^{2}/\dot{t}^{2}=\left(dr/dt\right)^{2}$. It can be shown that the
solution to this differential equation is

$r=\frac{r_{0}}{\sqrt{1-\left(1-\frac{r_{0}^{2}}{r_{c}^{2}}\right)\tanh^{2}\left(\pm\frac{t}{r_{c}}\right)}}$
(31)

where we set $t_{0}=0.$ Note that as $t\rightarrow\infty$, the radial
coordinate approaches $r_{c}$, so the distant observer never sees the particle
crossing the cosmic horizon in the de Sitter spacetime. However, in the
presence of a cosmic gate at $r=R_{0}$, the distant observer witnesses the
transition of the particle through the gate at

$t_{R}=r_{c}\tanh^{-1}\left(\sqrt{\frac{1-r_{0}^{2}/R_{0}^{2}}{1-r_{0}^{2}/r_{c}^{2}}}\right).$
(32)

Figure 3: Radial timelike-geodesics for a massive particle as seen by a
distant observer.

In Fig. 3, we plotted the time $t$ versus radial coordinates $r$ for a massive
particle as observed by a distant observer in the presence (solid line) and
absence (dash line) of the cosmic gate. As it is seen, the particle’s
geodesics is continuous across the cosmic gate, although it undergoes an
impulse in colliding with the matter at the gate.

###  IV.2 The Null Geodesics

From (24) and (26), with $L=0$ and $\epsilon=0$ for a photon (null particle),
one finds

$\dot{r}^{2}=E^{2}=\left(1-\frac{r^{2}}{\ell^{2}}\right)^{2}\left(\frac{dr}{dt}\right)^{2}$
(33)

for a radial motion, which admits the integral ($t_{0}=0$)

$r=r_{c}\left(\frac{r_{0}+r_{c}\tanh\left(\pm\frac{t}{r_{c}}\right)}{r_{c}+r_{0}\tanh\left(\pm\frac{t}{r_{c}}\right)}\right)$
(34)

between the radial coordinate and the time $t$ measured by a distant observer.
It could be easily checked that the distant observer never detects the light
reaching the cosmic horizon, although it could pass over the cosmic gate at a
finite time given by

$t=r_{c}\tanh^{-1}\left(\frac{r_{c}\left(R_{0}-r_{0}\right)}{r_{c}^{2}-R_{0}r_{0}}\right)$
(35)

if it existed. The geodesics of the massless particle are plotted in Fig. 4.

Figure 4: Radial null-geodesics for a massless particle as seen by a distant
observer.

##  V Conclusion

In contrast with the concept of a wormhole that connects two different distant
points of the same or different universe(s) through a hyperplane/throat whose
surface area is a local minimum, we introduced a cosmic gate that connects two
closed universes through a hyperplane/gate whose surface area is a local
maximum. In particular, we applied the junction formalism and glued the
regular part of two de-Sitter spacetimes at a hyperplane located before the
cosmological horizon. The resultant universe in its embedded diagram displayed
in Fig. 1 is a closed universe in the sense that the radial coordinate is
confined to a finite interval. We investigated the mechanical stability of the
newly built cosmological object (we called it Cosmic Gate) and showed that it
is stable against a radial linear perturbation. Furthermore, we studied the
timelike and null geodesics of the massive and massless particles in this
spacetime. Specifically, we focused our attention on radial geodesics to show
that the spacetime is geodesically complete. Our results have been presented
analytically and as have been depicted in Fig. 3 and 4, the particle reaches
the gate in a finite time and travels to the other side. As a final remark,
due to the direction of the 4-vector on the gate which is in the opposite
direction of its counterpart on a thin-shell wormhole, the matter presented at
the gate satisfies the physical energy conditions and therefore is normal
matter. This can also be seen from the equal signs of the principal curvature
of the gate (see Fig. 1). We recall that at the throat of a wormhole, the
signs of the two principal curvatures are opposite of each other indicating
the violation of the energy conditions.

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