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\begin{document}
\title{\sc Hodge decomposition of variable exponent spaces of Clifford-valued functions and applications to Dirac and Stokes equations}
%\title{Solutions for $p(x)$-Laplacian equations without AR-condition via Morse theory}
\author{Yongqiang Fu$^{\mathrm{a}}$, \ Vicen\c{t}iu D. R\u{a}dulescu$^{\mathrm{b},\mathrm{c}}$\footnote{Corresponding author.
E-mail address:\ {\tt vicentiu.radulescu@imar.ro}},\ Binlin Zhang$^{\mathrm{a}}$\\\
\small $^{\mathrm{a}}$ Department of Mathematics, Harbin Institute of Technology, 150001 Harbin , China\\
\small $^{\mathrm{b}}$ Institute of Mathematics ``Simion Stoilow" of the Romanian Academy, 014700 Bucharest, Romania\\
\small $^{\mathrm{c}}$ Department of Mathematics, University of Craiova,  200585 Craiova, Romania\\
} \date{ }
 \maketitle


\begin{abstract}
{In this paper, we establish a Hodge-type decomposition of
variable exponent Lebesgue spaces of Clifford-valued functions, where one of the
subspaces is the space of all monogenic  $L^{p(x)}$-functions. Using
this decomposition, we obtain the existence and uniqueness of
solutions to the homogeneous $A$-Dirac equations with variable growth under certain appropriate conditions
and to the Stokes equations in the setting of variable exponent spaces of Clifford-valued functions.}
\\
\emph{\bf Keywords:} Clifford analysis;  $L^{p(x)}$-decomposition;
$A$-Dirac equation;  Stokes equation; variable exponent.\\
\emph{\bf 2010 Mathematics Subject Classification:} 30G35; 35J60; 35D05; 35Q30; 46E30.
\end{abstract}

%{\section*{{\bf 1. Introduction }}} \setcounter{section}{1}
 %\setcounter{equation}{0}
\section{Introduction}
The central purpose of this paper is the study of the Hodge decomposition of the variable exponent Lebesgue space of Clifford-valued functions. In the compact case, the Hodge theory is a central tool for characterizing the topology of the underlying manifold. In the second part of this paper, we apply this decomposition
in order to solve the homogeneous $A$-Dirac equations with variable growth and the  Stokes equations in the setting of variable exponent spaces of Clifford-valued functions. The
approach we develop is in the spirit of Helmholtz [1858]. He first formulated a result on the
splitting of vector fields into vortices and gradients, which can be understood
as a rudimentary form of what is now called the ``Hodge decomposition". We also point out the pioneering papers by  Hodge \cite{hodge1, hodge2}, de Rham \cite{rham1,rham2}, and Weyl \cite{weyl}.

Variable exponent Lebesgue spaces $L^{p(x)}$  appeared in the literature for the first time already in an article by W. Orlicz in 1931, who considered the variable exponent function space  $L^{p(x)}$ on the real line, see \cite{5}, \cite{radnla} and \cite{radrep}.
   The next major step in the research of variable exponent spaces was the paper by O. Kov\'{a}\v{c}ik and J. R\'{a}kosn\'{i}k \cite{25}, which established many of the basic properties of Lebesgue spaces $L^{p(x)}$ and the corresponding Sobolev spaces $W^{k, p(x)}$.
   In the last twenty years, these spaces have attracted more and more attention. The study of these spaces has been stimulated by problems in elastic mechanics, fluid dynamics, calculus of variations and differential equations with variable growth conditions, see \cite{5, 13, 14, 25, 31}.
   In particular, one of the reasons that forced the rapid expansion of the theory of variable exponent spaces have been the models of electrorheological fluids introduced by Rajagopal and Ru\v{z}i\v{c}ka \cite{30}, which can be described by the boundary value problem for the generalized Navier-Stokes equations, where the extra stress tensor is coercive and satisfies appropriate growth assumptions.
   For the detailed accounts we refer to the monograph of Ru\v{z}i\v{c}ka \cite{31}.

   As a powerful tool for solving elliptic boundary value problems in the plane, the methods of complex functions theory play an important role. One way to extend these ideas to higher dimension is to begin with a generalization of algebraic and geometrical properties of the complex numbers.
   In this way, W.R. Hamilton studied the algebra of quaternions in 1843. Further generalizations were investigated by W.K. Clifford in 1878.
   He introduced the so-called {\it geometric algebras} or {\it Clifford algebras}, which are generalizations of the complex numbers, the quaternions, and the exterior algebras, see \cite{21}.
   Initially developed in the 1930--1940 and very intensively since 1970, theories for functions with values in Clifford algebras were developed. Many advantages of the complex function theory were preserved in this process.
   Clifford analysis usually studies the solutions of the Dirac equation or of a generalized Cauchy-Riemann system for functions defined on domains in Euclidean space and taken value in Clifford algebras,
   see \cite{1, 32}. Of course, the motivation for this development is closely related with systems of partial differential equations in physics, see \cite{7}.

 In \cite{27, 28}, C.A. Nolder first introduced $A$-Dirac equations
$DA(x,Du) = 0$ and developed tools for the study of weak solutions
to nonlinear $A$-Dirac equations in space $W^{1,p}_{0}(\Omega,\mathrm{C}\ell _{n})$.
Note that under appropriate identifications, the scalar part of
 $DA(x,Du)=0$ is actually the {\it $A$-harmonic equation} $\mathrm{div}\,A(x, \nabla u)=0$.
These equations have been extensively studied with many applications, see for instance \cite{19}.
Recently,
L. Diening and P. Kaplicky \cite{7} studied the interior regularity of the local weak solutions
$u \in W^{1, \varphi}(\Omega)$ and $\omega \in L^{\varphi^{\ast}}(\Omega)$ of the following stationary generalized Stoked system:
\begin{equation*}
\begin{cases}
-\mathrm{div}A(Du) + \nabla \omega = -\mathrm{div}\,G\ \ \ \mathrm{in}\ \Omega\\
\mathrm{div}\  u = 0\ \ \ \mathrm{in}\ \Omega,
\end{cases}
\end{equation*}
where $D$ is the symmetric part of the gradient, the extra stress tensor $A$ determines properties of the fluid.
For more detailed discussions, we refer to \cite{7, 8, 9} and the references therein.
Obviously, $A$-Dirac equations correspond to the extra stress tensor in the real-valued case.

However, the existence
of weak solutions to the $A$-Dirac equations has not been proved.
Y. Fu and B. Zhang \cite{16, 17, 35} proved
the existence of weak solutions for $A$-Dirac equations with variable growth. For this purpose, they also established a
theory of variable exponent spaces of Clifford-valued functions. But so far they have proved the existence of weak solutions
to the scalar part of the $A$-Dirac equations in space
$W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$, see \cite{30}. With the help of the orthogonal decomposition of the space $L^{2}(\Omega)$, K. G\"{u}rlebeck and W. Spr\"{o}{\ss}ig \cite{18, 19} obtained the existence of uniqueness of solutions to the Stokes equations.
Therefore, it is natural to consider an extension of the orthogonal decomposition of the spaces $L^{2}(\Omega, \mathrm{C}\ell _{n})$.


One of the most interesting results of complex and hyper-complex function theory is the orthogonal decomposition of the space $L^{2}(\Omega, \mathrm{C}\ell _{n})$, namely
$$L^{2}(\Omega, \mathrm{C}\ell _{n}) = (\mathrm{ker}D \cap L^{2}(\Omega, \mathrm{C}\ell _{n})) \oplus DW^{1,2}_{0}(\Omega, \mathrm{C}\ell _{n}),\eqno(1.1)$$
where $\mathrm{ker}D$  denotes the set of all monogenic functions on
$\Omega$. This decomposition has a number of applications,
especially to the theory of partial differential equations, see \cite{11}
for the complex case and \cite{19} for the hypercomplex case. In \cite{26}, U.
K\"{a}hler extended the orthogonal decomposition (1.1) to the spaces
$L^{p}(\Omega)$ in form of a direct decomposition in the case of
Clifford analysis. In \cite{12}, J. Dubinski and M. Reissig considered
decompositions for the spaces $W^{m,p}(\Omega)$, which can be
applied to study nonlinear variational problems.


The goal of this paper is to generalize decomposition (1.1) in the framework of
the variable exponent spaces and to give  applications to the $A$-Dirac equations and the Stokes equations.
Next, using the Hodge-type decomposition that we obtain in this paper, we prove the existence and
uniqueness of weak solutions to $A$-Dirac equations with variable growth under certain suitable assumptions.
At the same time, the existence and uniqueness of solutions to Stokes equations are showed in variable exponent
product spaces of Clifford-valued functions.  The whole treatment applies to a much larger
class of elliptic problems.


This paper is organized as follows. In Section 2, we start with a
brief summary of basic knowledge of Clifford algebras and variable
exponent spaces of Clifford-valued functions, then discuss
properties of some operators, which will be needed in the sequel. In
Section 3, we establish a Hodge-type decomposition for the space
$L^{p(x)}(\Omega, \mathrm{C}\ell _{n})$. In Section 4, appealing to this
decomposition in combination with the Minty-Browder theory, we prove the
existence and uniqueness of a solution to the $A$-Dirac equations with
variable growth in $W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n})$. On the other
hand, with the help of this decomposition, we prove the existence
and uniqueness of a solution in $W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n})
\times L^{p(x)}(\Omega, \mathbb{R})$
  of the Stokes problem, provided that the right-hand side is in  $W^{-1,p(x)}(\Omega, \mathrm{C}\ell _{n})$.

We refer to the excellent book by Ciarlet \cite{ciarlet} for necessary abstract notions and useful related examples.

\section{Preliminaries}

\subsection{Clifford algebra}
 We first recall some related notions and results from Clifford
algebra. For a detailed account we refer to \cite{1, 10, 19} and the references therein.

Let $\mathrm{C}\ell _{n}$ be
the real universal Clifford algebra over $\mathbb{R}^{n}$. Then
$$\mathrm{C}\ell _{n} =\mathrm{ span} \{\mathrm{e}_{0},\mathrm{e}_{1},\mathrm{e}_{2},\ldots ,\mathrm{e}_{n},\mathrm{e}_{1}\mathrm{e}_{2},\ldots ,\mathrm{e}_{n-1}\mathrm{e}_{n},\ldots ,\mathrm{e}_{1}\mathrm{e}_{2}\cdots \mathrm{e}_{n}\}$$
where $\mathrm{e}_{0}=1$ (the identity element in $\mathbb{R}^{n}$),
$\{\mathrm{e}_{1},\mathrm{e}_{2},\ldots ,\mathrm{e}_{n}\}$ is an
orthonormal basis of $\mathbb{R}^{n}$ with the relation
$\mathrm{e}_{i}\mathrm{e}_{j}+\mathrm{e}_{j}\mathrm{e}_{i}=-2\delta_{ij}\mathrm{e}_{0}$.
Thus, the dimension of $\mathrm{C}\ell _{n}$ is $2^{n}$.
In particular,  we denote by $\mathrm{C}\ell _{2}=\mathbb{H}$ the algebra of real quaternions.
For
$I=\{i_{1},\ldots,i_{r}\}\subset \{1,\ldots ,n\}$
    with $1\leq i_{1}<i_{2}<\cdots <i_{n}\leq n$, put $\mathrm{e}_{I}=\mathrm{e}_{i_{1}}\mathrm{e}_{i_{2}}\ldots \mathrm{e}_{i_{r}}$, while for $I=\emptyset$, $\mathrm{e}_{\emptyset }=\mathrm{e}_{0}$.
     For $0\leq r\leq n$ fixed, the space $\mathrm{C}\ell _{n}^{r}$ is defined by
$$\mathrm{C}\ell _{n}^{r}=\mathrm{span}\{\mathrm{e}_{I}:|I|:=\mathrm{card}(I)=r\}.$$
The Clifford algebra $\mathrm{C}\ell _{n}$ is a graded algebra as
$$\mathrm{C}\ell _{n}=\bigoplus_{1\leq r\leq n}\mathrm{C}\ell _{n}^{r}.$$
    Any element $a\in \mathrm{C}\ell _{n}$ may thus be written in a unique way as
$$a=[a]_{0}+[a]_{1}+\ldots +[a]_{n},$$
    where $[\ ]_{r}:\mathrm{C}\ell _{n}\rightarrow \mathrm{C}\ell _{n}^{r}$ denotes the projection of $\mathrm{C}\ell _{n}$ onto $\mathrm{C}\ell _{n}^{r}$. It is customary to identify $\mathbb{R}$ with $\mathrm{C}\ell _{n}^{0}$ and identify $\mathbb{R}^{n}$ with $\mathrm{C}\ell _{n}^{1}$ respectively.
     This means that each element $x$ of $\mathbb{R}^{n}$ may be represented by $$x=\sum_{i=1}^{n}x_{i}\mathrm{e}_{i}.$$ For $u\in \mathrm{C}\ell _{n}$, we denotes by $[u]_{0}$ the scalar part of $u$, that is, the coefficient of the element $\mathrm{e}_{0}$. We define the Clifford conjugation as follows:
$$\overline{\mathrm{e}_{i_{1}}\mathrm{e}_{i_{2}}\ldots \mathrm{e}_{i_{r}}}=(-1)^{\frac{r(r+1)}{2}}\mathrm{e}_{i_{1}}\mathrm{e}_{i_{2}}\ldots \mathrm{e}_{i_{r}}.$$
For $A\in \mathrm{C}\ell _{n}$, $B\in \mathrm{C}\ell _{n}$, we have
$$\overline{AB}=\overline{B}\ \overline{A},\ \
\overline{\overline{A}}=A.$$
 We denote
$$(A,B)=[\overline{A}B]_{0}.$$
Then an inner product is thus obtained, which defines the norm $|\cdot|$ on $\mathrm{C}\ell _{n}$  by
$$|A|^{2}=[\overline{A}A]_{0}.$$
From \cite{20} we know that this norm is submultiplicative, namely
$$|AB|\leq C|A||B|,$$ where $C$ is a positive constant depending only on ${n}$ and smaller than $2^{n/2}$.



Let $\Omega \subset \mathbb{R}^{n}$ be a
bounded domain with  smooth boundary $\partial\Omega$.
A Clifford-valued function $u:\Omega\rightarrow \mathrm{C}\ell _{n}$ can be
written as $u=\displaystyle\Sigma_{I}u_{I}\mathrm{e}_{I}$, where the coefficients
$u_{I}:\Omega\rightarrow \mathbb{R}$ are real-valued functions.


    The Dirac operator on the Euclidean space used here is introduced by
$$D=\sum_{j=1}^{n}e_{j}\frac{\partial}{\partial x_{j}}.$$
This is a special case of the Atiyah-Singer-Dirac operator acting on sections of a spinor bundle.
We also point out that the most famous Dirac operator describes the propagation of a free fermion in three dimensions.


    If $u$ is a real-valued function defined on  $\Omega$, then $Du=\nabla u$.
    Moreover, $D^{2}=-\Delta$, where $\Delta$ is the
Laplace operator which operates only on coefficients. A function is
left monogenic if it satisfies the equation $Du(x)=0$ for each
$x\in\Omega$. A similar definition can be given for right monogenic
function. An important example of a left monogenic function is the
generalized Cauchy kernel
$$G(x)=\frac{1}{\omega_{n}}\frac{\overline{x}}{|x|^{n}},$$
where $\omega_{n}$ denotes the surface area of the unit ball in $\mathbb{R}^{n}$. This function is a fundamental solution of the Dirac operator. We refer the readers to \cite{19, 21, 26, 32} for basic properties of left monogenic functions.

\subsection{Variable exponent spaces of Clifford-valued functions}

Next, we investigate some basic properties of variable exponent
spaces of Clifford-valued functions. Note that in what follows, we
use the short notation $L^{p(x)}(\Omega)$, $W^{1,p(x)}(\Omega)$,
etc., instead of $L^{p(x)}(\Omega, \mathbb{R})$,
$W^{1,p(x)}(\Omega,\mathbb{R})$, etc.

 Throughout this paper we assume that
$$p \in \mathrm{P^{log}}(\Omega) \ \mathrm{and} \ 1 < p_{-}:= \inf_{x \in \overline{\Omega}}p(x) \leq p(x) \leq \sup_{x \in \overline{\Omega}}p(x) =:p_{+} < \infty,\eqno(2.1)$$
where  $\mathrm{P^{log}}(\Omega)$ is the set of exponents $p$
satisfying the so-called $\log$--H\"older continuity,
that is,
$$|p(x) - p(y)| \leq \frac{C}{\mathrm{log}(\mathrm{e} + |x-y|^{-1})}$$
holds for all $x,y \in \Omega$, see \cite{5}. Let $\mathcal{P}(\Omega)$  be the
set of all Lebesgue measurable functions $p: \Omega \rightarrow
(1,\infty)$. Given $p \in \mathcal{P}(\Omega)$ we define the conjugate
function $p'(x)\in \mathcal{P}(\Omega)$ by
$$p'(x)=\frac{p(x)}{p(x)-1},\ \  \mathrm{for}\ \mathrm{each} \
x\in\Omega.$$

The variable exponent Lebesgue space $L^{p(x)}(\Omega)$ is defined
by
$$L^{p(x)}(\Omega)=\Big\{ u\in \mathcal{P}(\Omega):\int_{\Omega}|u|^{p(x)}dx <\infty \Big\},$$
with the norm
$$||u||_{L^{p(x)}(\Omega)}=\mathrm{inf} \Big\{t>0:\int_{\Omega}|\frac{u(x)}{t}|^{p(x)}dx \leq 1\Big\}.$$
The variable exponent Sobolev space $W^{1,p(x)}(\Omega)$ is defined
by
$$W^{1,p(x)}(\Omega)=\Big\{u\in L^{p(x)}(\Omega): |\nabla  u| \in L^{p(x)}(\Omega)\Big\},$$
with the norm
$$||u||_{W^{1,p(x)}(\Omega)}=||\nabla  u||_{L^{p(x)}(\Omega)} + ||u||_{L^{p(x)}(\Omega)}.\eqno(2.2)$$
Denote by $W_{0}^{1,p(x)}(\Omega)$ the completion of
$C_{0}^{\infty}(\Omega)$ in $W^{1,p(x)}(\Omega)$ with respect to the
norm (2.2). The space $W^{-1,p(x)}(\Omega)$ is defined as the dual
of the space $W_{0}^{1,p'(x)}(\Omega)$.
For more details we refer to \cite{5, 15, 25} and the references therein.

In what follows, we understand $u\in L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$ coordinatewisely.
 For example, $u\in L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$ means
 that $\{u_{I}\} \subset L^{p(x)}(\Omega)$ for $u=\Sigma_{I}u_{I}e_{I}\in \mathrm{C}\ell _{n}$
 with the norm $||u||_{L^{p(x)}(\Omega, \mathrm{C}\ell _{n})}= \sum_{I}||u_{I}||_{L^{p(x)}(\Omega)}$.
 In this way, the spaces $W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})$,
 $W_{0}^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})$, $C_{0}^{\infty}(\Omega,\mathrm{C}\ell _{n})$, etc., can be understood similarly.
 In particular, the space $L^{2}(\Omega,\mathrm{C}\ell _{n})$ can be converted into a right Hilbert $\mathrm{C}\ell _{n}$-module by defining
 the following Clifford-valued inner product (see \cite[Definition 3.74]{19})
 $$\big(f,g\big)_{\mathrm{C}\ell _{n}}=\int_{\Omega}{\overline{f(x)}g(x)}dx.  \eqno(2.3)$$


\noindent\textbf{Remark 2.1.} A simple computation shows that
$||u||_{L^{p(x)}(\Omega, \mathrm{C}\ell _{n})}$ is equivalent to
$\big|\big||u|\big|\big|_{L^{p(x)}(\Omega)}$. Furthermore, we also
have that for every $u \in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$,
$||Du||_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}$ is an equivalent norm of
$||u||_{W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})}$ (see \cite{16}).\\

\noindent\textbf{Lemma 2.2.} (\cite{13}) \begin{itshape}Let $\rho (u) =
\displaystyle\int_{\Omega} |u(x)|^{p(x)}dx$. For $u\in L^{p(x)}(\Omega)$, we
have

$(1)$ If  $||u||_{L^{p(x)}(\Omega)} \geq 1$, then
$||u||^{p_{-}}_{L^{p(x)}(\Omega)} \leq \rho (u) \leq
||u||^{p_{+}}_{L^{p(x)}(\Omega)}$.

$(2)$ If  $||u||_{L^{p(x)}(\Omega)} \leq 1$, then $||u||^{p_{+}}_{L^{p(x)}(\Omega)} \leq \rho (u) \leq ||u||^{p_{-}}_{L^{p(x)}(\Omega)}$.\end{itshape}\\

\noindent\textbf{Lemma 2.3.} (\cite{16}) \begin{itshape} Assume that $p(x) \in \mathcal{P}(\Omega)$.
Then the inequality
$$\int_{\Omega}|uv|dx \leq C(n,p)||u||_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}||v||_{L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})}$$
holds for every $u \in L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$ and $ v \in L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})$.\end{itshape}\\

\noindent\textbf{Lemma 2.4.} (\cite{17}) \begin{itshape} Assume that $p(x) \in
\mathcal{P}(\Omega)$. Then the following properties are true.

$(1)$ The dual of the space $L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$ is
$L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})$, i.e.,
$\big(L^{p(x)}(\Omega,\mathrm{C}\ell _{n})\big)^{\ast}$ $=
L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})$. Thus, the space $L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$
is a reflexive Banach space.

$(2)$ The space $W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})$ is a reflexive Banach
space.\end{itshape}\\

\noindent\textbf{Definition 2.5.} (\cite{19}) Let $u \in C(\Omega, \mathrm{C}\ell _{n})$.
The Teodorescu operator is defined by
$$Tu(x) = \int_{\Omega}G(x-y)u(y)dy,$$
where $G(x)$ is the generalized Cauchy kernel mentioned above.\\

\noindent\textbf{Definition 2.6.} Let $u \in
L^{1}_{loc}(\mathbb{R}^{n})$. The Hardy-Littlewood maximal operator is defined by
$$Mu\big(x\big) = \sup_{r>0}\frac{1}{\mathrm{meas}\big(B(x,r)\big)}
\int_{B(x,r)}\big|u(y)\big|dy,$$
where $B(x,r)=\{y\in \mathbb{R}^{n}: |y-x|<r \}$.\\

\noindent \textbf{Lemma 2.7.} (\cite{5})
\begin{itshape}Let  $x \in \Omega$ and $u \in
L^{1}_{loc}(\mathbb{R}^{n})$. Then
$$\int_{\Omega}\frac{1}{|x-y|^{n-1}}|u(y)|dy \leq C(n)\,(\mathrm{diam}
\Omega)\,Mu(x).$$\end{itshape}

\noindent\textbf{Lemma 2.8.} (\cite{5})
\begin{itshape}If $p(x)$ satisfies $(2.1)$, then $M$ is bounded in
$L^{p(x)}(\Omega)$. Thus, there exists a constant $C =
C(n,p)$ such that
$$||Mu||_{L^{p(x)}(\Omega)} \leq
C(n,p)||u||_{L^{p(x)}(\Omega)}.$$\end{itshape}

\noindent\textbf{Lemma 2.9.} (\cite{5}) \begin{itshape}Let $\Phi$ be a
Calder\'{o}n-Zygmund operator with Calder\'{o}n-Zygmund kernel $K$
on $\mathbb{R}^{n} \times \mathbb{R}^{n}$. Then $\Phi$ is bounded on
$L^{p(x)}(\mathbb{R}^{n})$.\end{itshape}\\

\noindent\textbf{Lemma 2.10.} (\cite{17}) \begin{itshape} The operator $D: W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n}) \rightarrow L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$ is continuous.\end{itshape}\\

\noindent\textbf{Lemma 2.11.} (\cite{17}) \begin{itshape}The operator $T: L^{p(x)}(\Omega,\mathrm{C}\ell _{n}) \rightarrow W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})$ is continuous.\end{itshape}\\

\noindent\textbf{Lemma 2.12.}\begin{itshape} Let $p(x) \in \mathcal{P}(\Omega)$. Then
for $u \in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$, the Borel-Pompeiu formula
$TDu(x) = u(x)$ holds for all $x \in \Omega$. Moreover, for  $u \in
L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$, the equation $DTu(x) = u(x)$ holds for all
$x \in \Omega$.\end{itshape}

{\it Proof.}  According to Remark 4.21 in \cite{19}, the conclusions are
implied by $W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})$ $\hookrightarrow
W^{1,p_{-}}(\Omega,\mathrm{C}\ell _{n})$ and
$L^{p(x)}(\Omega,\mathrm{C}\ell _{n})\hookrightarrow
L^{p_{-}}(\Omega,\mathrm{C}\ell _{n})$.
\qed

\noindent\textbf{Lemma 2.13.}\begin{itshape} There exists a unique linear
extension $\widetilde{T}$ of the operator $T$ such that the operator
$\widetilde{T}: W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n}) \rightarrow
L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$ is continuous. \end{itshape}

{\it Proof.}  In view of Proposition 12.3.2 in \cite{5}, we know that
 for each $f \in W^{-1,p(x)}(\Omega)$, there exists $f_{k} \in L^{p(x)}(\Omega),k=0,1,\ldots,n,$ such that
$$\langle f,\varphi\rangle = \sum^{n}_{k=0}\int_{\Omega}
f_{k}\frac{\partial \varphi}{\partial x_{k}} dx,\eqno(2.4)$$ for all
$\varphi \in W^{1,p'(x)}_{0}(\Omega)$. Moreover,
$\|f\|_{W^{-1,p(x)}(\Omega)}$ is equivalent to
$\sum^{n}_{k=0}\|f_{k}\|_{L^{p(x)}(\Omega)}$. Obviously, for every
$f \in W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})$ the equality (2.4) still holds
for $f_{k} \in L^{p(x)}(\Omega,\mathrm{C}\ell _{n}),$ $k=0,1,\ldots,n$. Moreover,
$\|f\|_{W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})}$ is equivalent to
$\sum^{n}_{k=0}\|f_{k}\|_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}$. On the other
hand, by Proposition 12.3.4 in \cite{5},  the
space $C^{\infty}_{0}(\Omega,\mathrm{C}\ell _{n})$ is dense in
$W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})$. Thus, we may choose
$$u^{j} = u_{0}^{j} + \sum^{n}_{k=1}\frac{\partial u_{k}^{j}}{\partial x_{k}},$$
where $u_{0}^{j}, u_{k}^{j} \in C^{\infty}_{0}(\Omega,\mathrm{C}\ell _{n})$, such
that $\|u^{j}-f\|_{W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})}\rightarrow 0$ and
$\|u_{k}^{j}-f_{k}\|_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}$ $\rightarrow 0$ as
$j\rightarrow\infty$, where $k=0,1,\ldots,n.$ Here, we are using the
fact that $C^{\infty}_{0}(\Omega,\mathrm{C}\ell _{n})$ is dense in
$L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$(see \cite{17}). Set
$$Tu^{j} = \int_{ \Omega}G(x-y)u^{j}(y)dy,$$
where $G(x)$ is the above-mentioned generalized Cauchy kernel. Therefore
$$\aligned
Tu^{j} &= \int_{\Omega}G(x-y)\big(u_{0}^{j}(y) + \sum^{n}_{k=1}\frac{\partial}{\partial y_{k}}u_{k}^{j}(y)\big)dy\\
&=\int_{ \Omega}G(x-y)u_{0}^{j}(y)dy + \sum^{n}_{k=1} \int_{
\Omega}\frac{\partial}{\partial x_{k}}G(x-y)u_{k}^{j}(y)dy.
\endaligned$$
Since
$$\Big|\int_{ \Omega}G(x-y)u_{0}^{j}(y)dy\Big|\leq \int_{ \Omega}\frac{1}{|x-y|^{n-1}}\big|u_{0}^{j}(y)\big|dy,$$
 Remark 2.1, Lemma 2.7 and Lemma 2.8 imply that there exists a constant
$C_{0}
> 0$ such that
$$\Big|\Big|\int_{ \Omega}G(x-y)u_{0}^{j}(y)dy \Big|\Big|_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})} \leq C_{0}||u_{0}^{j}||_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}.\eqno(2.5)$$
Now, let us extend $u_{k}^{j}(x)$ by zero to $\mathbb{R}^{n}\setminus
\Omega$. Then  $\displaystyle\frac{\partial}{\partial
x_{k}}G(x-y)$ satisfies the conditions of Calder\'{o}n-Zygmund
kernel on $\mathbb{R}^{n} \times \mathbb{R}^{n}$ (see \cite{17}). In view
of Lemma 2.9, there exist positive constant $C_{k}\ (k=1,\ldots,n)$
such that
$$\Big|\Big|\int_{ \Omega}\frac{\partial}{\partial x_{k}}G(x-y)u_{k}^{j}(y) \Big|\Big|_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})} \leq C_{k}||u_{k}^{j}||_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}.\eqno(2.6)$$
Combining (2.5) with (2.6), we have
$$\aligned
\big|\big|&Tu^{j}\big|\big|_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}& \\
\leq &\Big|\Big|\int_{ \Omega}G(x-y)u_{0}^{j}(y)dy \Big|\Big|_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})} + \sum^{n}_{k=1}\Big|\Big|\int_{ \Omega}\frac{\partial}{\partial x_{k}}G(x-y)u_{k}^{j}(y) \Big|\Big|_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}\\
\leq & C_{0}\big|\big|u_{0}^{j}\big|\big|_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}
+ \sum^{n}_{k=1}
C_{k}\big|\big|u_{k}^{j}\big|\big|_{L^{p(x)}(\Omega,Cl_{n})}.
\endaligned$$
Letting $j\rightarrow \infty$, by means of the continuous linear
extension theorem, the operator $T$ can be uniquely extended to a
bounded linear operator $\widetilde{T}$ such that for all $f \in
W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})$, there exists a constant
$\widetilde{C}>0$ such that
$$||\widetilde{T}f||_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})} \leq C\Big(||f_{0}||_{L^{p(x)}(\Omega,Cl_{n})} + \sum^{n}_{k=1}||f_{k}||_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}\Big) \leq
\widetilde{C}||f||_{W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})}.$$
This completes the proof of Lemma 2.13.\qed

In \cite{6}, L. Diening, D. Lengeler and M. Ru\v{z}i\v{c}ka showed that
the Dirichlet problem of the Poisson equation with homogeneous
boundary data
$$\left\{
  \begin{array}{ll}
    -\Delta u = f, &\mathrm{in} \ \Omega \\
    u = 0, &\mathrm{on} \ \partial \Omega
  \end{array}
\right.\eqno(2.7)$$
possesses a unique weak solution $u\in
W^{1,p(x)}(\Omega)$ for each $f \in W^{-1,p(x)}(\Omega)$. Moreover, the following
estimate holds
 $$||u||_{W^{1,p(x)}(\Omega)} \leq C(n,p,\Omega)||f||_{W^{-1,p(x)}(\Omega)}.$$

We say that $u$ is a {\it weak solution} of problem (2.7) provided that
$$\langle f,\varphi\rangle=\int_{\Omega}\nabla u\cdot \nabla \varphi dx ,\ \ \forall\  \varphi \in W^{1,p'(x)}_{0}(\Omega).$$ Then it is easy to see
that for all $f \in W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})$ the problem (2.7)
still has a unique weak solution $u\in W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})$.
We denote by $\Delta^{-1}_{0}$ the solution operator. On the other
hand,  the operator
$$\Delta : W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n}) \rightarrow W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})$$
is continuous, so we obtain that the operator $\widetilde{D}=-\Delta
T:L^{p(x)}(\Omega,\mathrm{C}\ell _{n}) \rightarrow W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})$ is
continuous from Lemma 2.11, where the operator $\widetilde{D}$ can
be considered as a unique continuous linear extension of the Dirac
operator. Hence we can derive two useful results which will be needed later.\\

\noindent\textbf{Lemma 2.14.}\begin{itshape} Assume that $p(x)$ satisfies relation $(2.1)$.

$(i)$ If $u \in L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$, then the equation
$\widetilde{T}\widetilde{D}u(x) = u(x)$ holds for all $x \in
\Omega$.

$(ii)$ If $u \in W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})$, then the equation
$\widetilde{D}\widetilde{T}u(x) = u(x)$ holds
for all $x \in \Omega$.\end{itshape}


{\it Proof.}  (i) follows from Lemma 2.12 and the density of
$W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n})$ in $L^{p(x)}(\Omega, \mathrm{C}\ell _{n})$.

(ii)
follows from Lemma 2.12 and the density of
$C^{\infty}_{0}(\Omega,\mathrm{C}\ell _{n})$ in $W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})$.\qed


\noindent\textbf{Lemma 2.15.}\begin{itshape} Assume that $f\in
W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ and $g\in L^{p'(x)}(\Omega, \mathrm{C}\ell _{n})$.
Then the following equality holds
$$\big(Df, g\big)_{\mathrm{C}\ell _{n}}=\big(f, \widetilde{D}g \big)_{\mathrm{C}\ell _{n}},$$ where $(\cdot,\cdot)_{\mathrm{C}\ell _{n}}$ is Clifford-valued product $(2.3)$ above mentioned. \end{itshape}


{\it Proof.}  Let $g_{k} \in W^{1,p'(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ with
$g_{k}\rightarrow g$ in $L^{p'(x)}(\Omega, \mathrm{C}\ell _{n})$. Then we have
$$\big(Df,g_{k}\big)_{\mathrm{C}\ell _{n}}=\big(f,Dg_{k}\big)_{\mathrm{C}\ell _{n}}.$$
By the density of $W^{1,p'(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ in
$L^{p'(x)}(\Omega, \mathrm{C}\ell _{n})$, together with the continuity of
$\widetilde{D}$ and Lemma 2.3, the desired conclusion follows.
\qed





\section{The Hodge decomposition of variable exponent Lebesgue spaces}


Now we are ready to generalize (1.1) to the case of variable exponent Lebesgue spaces.\\



\noindent\textbf{Theorem 3.1.}\begin{itshape} The space
$L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$ allows the Hodge-type decomposition
$$L^{p(x)}(\Omega,\mathrm{C}\ell _{n}) =(\mathrm{ker}\widetilde{D} \cap L^{p(x)}(\Omega,\mathrm{C}\ell _{n})) \oplus DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})\eqno(3.1)$$
with respect to the Clifford-valued product $(2.3)$.\end{itshape}

{\it Proof.}  Similar to the proof of Theorem 6 in \cite{26}, we first
prove that
$$(\mathrm{ker}\widetilde{D} \cap L^{p(x)}(\Omega,\mathrm{C}\ell _{n})) \cap DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n}) = \{0\}.$$
Suppose $f \in (\mathrm{ker}\widetilde{D} \cap
L^{p(x)}(\Omega,\mathrm{C}\ell _{n})) \cap DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$, then
$\widetilde{D}f = 0$ . Because of $f \in
DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ there exists a function  $v \in
W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ such that $Dv =f$. Hence, we get that
$-\Delta v = 0$ and $v = 0$ on $\partial \Omega$. From the
uniqueness of $\Delta^{-1}_{0}$ we obtain $v = 0$. Consequently, $f
= 0$. Therefore, the sum of the two subspaces is a direct one.

Now let $u \in L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$. Then we have $u_{2} =
D\Delta^{-1}_{0}\widetilde{D}u \in DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$.
Let $u_{1} = u - u_{2}$. Then $u_{1} \in L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$.
Furthermore, we take $u_{k}\in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ such
that $u_{k} \rightarrow u $ in $L^{p(x)}(\Omega, \mathrm{C}\ell _{n})$, then by
the density of $W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ in $L^{p(x)}(\Omega,
\mathrm{C}\ell _{n})$ and Lemma 2.3, we have for any $\varphi \in
W^{1,p'(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$
$$\aligned
\big(u_{1},D\varphi \big)_{\mathrm{C}\ell _{n}}=\big(u - u_{2},D\varphi
\big)_{\mathrm{C}\ell _{n}}&=\lim_{k \rightarrow \infty}\big(Du_{k} -
DD\Delta^{-1}_{0}Du_{k},\varphi \big)_{\mathrm{C}\ell _{n}}\\
&=\lim_{k \rightarrow \infty}\big(Du_{k} - Du_{k},\varphi
\big)_{\mathrm{C}\ell _{n}}=0.
\endaligned$$
Thus, we get $u_{1} \in \mathrm{ker}\widetilde{D}$. Since $u \in
L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$ is arbitrary, the desired result
follows immediately. \qed

Beginning with this decomposition we  get the following
projections
$$P: L^{p(x)}(\Omega,\mathrm{C}\ell _{n}) \rightarrow \mathrm{ker}\widetilde{D}\cap L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$$
$$Q: L^{p(x)}(\Omega,\mathrm{C}\ell _{n}) \rightarrow DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n}).$$
For $p(x) \equiv 2$, these are ortho-projections. Notice that directly from the proof of Theorem 3.1 we obtain
$$Q = D\Delta^{-1}_{0}\widetilde{D}, P = I - Q.\eqno(3.2)$$
It follows from (3.2) that the operator $Q$ as well as $P$ maps the
space $L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$ into itself.

In the following we discuss the properties of the operator $Q$, which will be used later.\\


\noindent\textbf{Theorem 3.2.} \begin{itshape}The space
$L^{p(x)}(\Omega,\mathrm{C}\ell _{n}) \cap \mathrm{im}Q$ is a closed subspace of
$L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$, that is, the space
$DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$
is closed in $L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$.\end{itshape}

{\it Proof.}  Let $u \in
\overline{DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})}$. Then there exists $u_{k}
\in DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ such that
 $||Du_{k} - u||_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})} \rightarrow 0$ as $k \rightarrow \infty$. Since $W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ is a reflexive Banach space,
we can extract a subsequence of $\{u_{k}\}$( still denote by
$\{u_{k}\}$ ), such that $u_{k} \rightharpoonup v$ weakly in
$W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$. Since the norm in a Banach space is
weakly lower semicontinuous and the operator $D:
W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n}) \rightarrow L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$
is continuous, we obtain
$$||Dv - u||_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})} \leq \liminf_{n \rightarrow \infty}||Du_{n} - u||_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})} = 0.$$
Thus $u = Dv$. Now we complete the proof of Theorem 3.2. \qed


\noindent\textbf{Theorem 3.3.}  \begin{itshape} We have $\big(L^{p(x)}(\Omega,\mathrm{C}\ell _{n})
\cap \mathrm{im}Q\big)^{\ast} = L^{p'(x)}(\Omega,\mathrm{C}\ell _{n}) \cap
\mathrm{im}Q$, that is, the linear operator
$$\Phi : DW^{1,p'(x)}_{0}(\Omega,\mathrm{C}\ell _{n}) \rightarrow \big(DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})\big)^{\ast}$$
given by
$$\Phi(Du)(D\varphi) = (D\varphi , Du)_{Sc}:= \int_{\Omega}[\overline{D\varphi}Du]_{0}dx$$
is a Banach space isomorphism.\end{itshape}


{\it Proof.}  First, in view of Theorem 3.2,
$DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ and
$DW^{1,p'(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ are reflexive Banach spaces since
they are closed in $L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$ and
$L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})$ respectively. The linearity of $\Phi$ is
clear. For injectivity, suppose
$$\Phi(Du)(D\varphi) = (D\varphi , Du)_{Sc} = 0\eqno(3.3)$$
for all $\varphi \in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ and some $u \in
W^{1,p'(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$. For any $\omega \in
L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$, according to Theorem 3.1, we may write
$\omega = \alpha + \beta$ with $\alpha \in \mathrm{ker}\widetilde{D}
\cap L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$ and $\beta \in
DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$. From Lemma 2.15 we obtain
$$(\omega , Du)_{Sc} = (\alpha + \beta , Du)_{Sc} = (\alpha , Du)_{Sc} + (\beta, Du)_{Sc} = (\beta, Du)_{Sc}.$$
This together with (3.3) gives $(\omega , Du)_{Sc} = 0$, hence
 $Du = 0$. It follows that $\Phi$ is injective. To get
surjectivity, let $f \in
\big(DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})\big)^{\ast}$. By the Hahn-Banach
theorem, there is $F \in \big(L^{p(x)}(\Omega,\mathrm{C}\ell _{n})\big)^{\ast}$
with $||F|| = ||f||$ and $F|_{DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})} = f$.
In terms of Lemma 2.4, there exists $\varphi \in
L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})$ such that $F(u) = (u, \varphi)_{Sc}$ for
any $u \in L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$. According to Theorem 3.1, we
can write $\varphi =\eta+ D\alpha$, where $\eta \in
\mathrm{ker}\widetilde{D} \cap L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})$, $D\alpha
\in DW^{1,p'(x)}_{0}(\Omega,\mathrm{C}\ell _{n}).$ For any $Du \in
DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$, from Lemma 2.15 we have
\begin{align*}
f(Du)&= (Du, \varphi)_{Sc} = (Du, \eta + D\alpha)_{Sc} = (Du, \eta)_{Sc} + (Du, D\alpha)_{Sc}\\
&= (Du, D\alpha)_{Sc} = \Phi (D\alpha)(Du).
\end{align*}
Consequently, $\Phi (D\alpha) = f$. It follows that $\Phi$ is
surjective. By means of Lemma 2.3 we have
$$|\Phi (Du)(D\varphi)| = |(D\varphi, Du)_{Sc}| \leq C||D\varphi||_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}||Du||_{L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})}.$$
This means that $\Phi$ is continuous. Furthermore, it is immediate that $\Phi^{-1}$ is continuous from the inverse function theorem.
This concludes the proof of Theorem 3.3. \qed


\section{Applications}


\subsection{A-Dirac equations with variable growth}
The Dirac equation arises in the study of nonlinear spinor fields in the unified theory of
elementary particles, see Heisenberg \cite{heise} and Weyl \cite{weyl}. The stationary states of the nonlinear Dirac field have been proposed as a model for elementary extended fermions and nucleons, see Thaller \cite{thaller}.

In this subsection, we are
interested in the existence of solutions to the following $A$-Dirac
equations:
$$DA(Du) = 0,\eqno(4.1)$$
where $A: \mathrm{C}\ell _{n}(\Omega) \rightarrow \mathrm{C}\ell _{n}(\Omega)$ satisfies the following conditions:\\
(A1) $|A(\xi) - A(\eta)| \leq C_{1}(|\xi| + |\eta|)^{p(x) - 2}|\xi -
\eta|;$\\
(A2) $\big[\overline{\big(A(\xi) - A(\eta)\big)}(\xi -
\eta)\big]_{0} \geq C_{2}(|\xi| + |\eta|)^{p(x) - 2}|\xi -
\eta|^{2};$\\
(A3) $A(0) \in L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})$, \\
where $\xi$ and $\eta$ are arbitrary elements in $\mathrm{C}\ell _{n}$, and
$C_{1}$ and $C_{2}$ are positive constants.
Of course, $A(Du)(x)=A(Du(x))$ and $DA(Du) = 0$ are meant in
the distributional sense.

Notice that when $A(\xi) = |\xi-\alpha|^{p(x)-2}(\xi-\alpha)$, where $\alpha \in L^{p(x)}
(\Omega, \mathrm{C}\ell _{n})$ is fixed, then equation (4.1) generalizes
the important case of the
 equation $D\big(|Du-\alpha|^{p(x)-2}(Du-\alpha)\big) = 0$. As $p(x)\equiv p$ and $\alpha\equiv 0$, $p$-Dirac equations were introduced and their conformal invariance was investigated in \cite{29}.
  These
equations are nonlinear generalizations of the Dirac Laplace equation as well
as generalizations of elliptic equations of $A$-harmonic type ${\rm div}\, A(x,\nabla u)= 0$.
The study of these equations is partially motivated by the fact that some
arise as the Euler-Lagrange equations to variational integrals.

 If $u$ is a real-valued function, then the $p(x)$-Dirac
 equation becomes the so-called $p(x)$-Laplacian equation. In recent
 years these equations have been extensively studied, see \cite{5, 15, 24} and the references therein.


In order to get the existence of a solution to the $A$-Dirac equations, we need a theorem of Minty-Browder as follows.\\


\noindent\textbf{Proposition 4.1.} (\cite[Theorem 5.16]{3}) \begin{itshape}Let
$X$ be a reflexive Banach space and let $G: X\rightarrow X^{\ast}$
be a
continuous nonlinear mapping such that \\
$(i)\ (strict\
monotonicity)\ (Gv-Gw, v-w)>0\ \ \forall\  v, w\in X,\
\ v\neq w.$\\
$(ii)\ (coerciveness)\  \lim_{\|v\| \rightarrow \infty}
{{\|v\|}^{-1}}{(Gv,v)}=\infty.$\\
Then for every $f\in X^{\ast}$ there exists a unique solution $u\in X$ of the equation $Gu =f$.\end{itshape}\\

 Now we are ready to prove our result as follows.\\

\noindent\textbf{Theorem 4.2.} \begin{itshape}Under conditions $(A1)$, $(A2)$ and
$(A3)$, there exists a weak solution
 $u \in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ to the A-Dirac equations $(4.1)$, hence there exists a
 Clifford-valued function $u \in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ such that $$\int_{\Omega}\overline{A(Du)}Dvdx=0$$ for any $v \in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$. Furthermore, the solution is unique up to a monogenic function.\end{itshape}\\

{\it Proof.}  We first claim that $A(u) \in
L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})$ for every $u \in L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$.
Indeed, from (A1) and (A3) we obtain
\begin{align}
\int_{\Omega}|A(u)|^{p'(x)}dx &= \int_{\Omega}(|A(u) - A(0) + A(0)|^{p'(x)})dx \nonumber\\
&\leq 2^{p'_{+}}\int_{\Omega}|A(u) - A(0)|^{p'(x)}dx + 2^{p'_{+}}\int_{\Omega}|A(0)|^{p'(x)}dx \nonumber \\
&\leq 2^{p'_{+}}\int_{\Omega}\Big(|u|^{p(x)-2}|u|\Big)^{p'(x)}dx+
2^{p'_{+}}\int_{\Omega}|A(0)|^{p'(x)}dx. \tag{4.2}
\end{align}
This estimate together with Remark 2.1 and Lemma 2.2 yields the
previous assertion.

Next we show that $QA\big(DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})\big) =
\big(DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})\big)^{\ast} =
DW^{1,p'(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$. Evidently, it follows from (3.2)
that $QA\big(DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})\big) \subset
DW^{1,p'(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$. By Theorem 3.2, we get that
$DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ is a reflexive Banach space. In the
following, to get surjectivity of the operator $QA$, we need to
verify the conditions of Proposition 4.1 respectively.

(1) The operator $QA$ is continuous. Suppose that $Du_{k}, Dv \in
DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ and $||Du_{k} -
Dv||_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})} \rightarrow 0$ as $k \rightarrow
\infty$. From (A1), we obtain
$$\big|\big||A(Du_{k}) - A(Dv)|\big|\big|_{L^{p'(x)}(\Omega)} \leq C_{1}\Big|\Big|\big(|Du_{k}| + |Dv|\big)^{p(x)-2}|Du_{k} - Dv|\Big|\Big|_{L^{p'(x)}(\Omega)}.$$
Divide $\Omega$ into two parts: $\Omega_{1}=\{x\in\Omega: p(x)\leq 2
\}$ and $\Omega_{2}=\{x\in\Omega: p(x)>2 \}$. Then
$\Omega=\Omega_{1}\cup \Omega_{2}$. On $\Omega_{1}$ we have
$$\int_{\Omega_{1}}\Big(\big(|Du_{k}| + |Dv|\big)^{p(x)-2}|Du_{k} -
Dv|\Big)^{p'(x)}dx \leq \int_{\Omega}\Big(|Du_{k} -
Dv|^{p(x)-1}\Big)^{p'(x)}dx.$$ On $\Omega_{2}$, according to
H\"older inequality we have
$$\aligned
& \int_{\Omega_{2}}\Big(\big(|Du_{k}| + |Dv|\big)^{p(x)-2}|Du_{k} - Dv|\Big)^{p'(x)}dx\\
& \leq 2\Big|\Big|\big(|Du_{k}| +
|Dv|\big)^{p'(x)(p(x)-2)}\Big|\Big|_{L^{\frac{p(x)-1}{p(x)-2}}(\Omega)}
\Big|\Big||Du_{k} -
Dv|^{p'(x)}\Big|\Big|_{L^{\frac{p(x)}{p'(x)}}(\Omega)}.
\endaligned$$ From Remark 2.1 and Lemma 2.2 we deduce that
$$\int_{\Omega}\Big(\big(|Du_{k}| + |Dv|\big)^{p(x)-2}|Du_{k} -
Dv|\Big)^{p'(x)}dx \rightarrow 0$$ as $k \rightarrow \infty$. By means of Remark 2.1 and Lemma 2.2, we obtain
$$\big|\big|A(Du_{k}) - A(Dv)\big|\big|_{L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})} \rightarrow 0$$
as $k \rightarrow \infty$. Finally, the continuity follows immediately
from the boundedness of the operator $Q$.

(2) The operator $QA$ is strictly monotone. In view of Theorem 3.1,
we have
$$ QA(Du) = A(Du) - PA(Du)$$
for each $u \in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$.
Thus, for any $u,v \in
W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$, (3.2) and Lemma 2.15 give
$$\big(QA(Du), Dv\big)_{Sc} = \big(A(Du), Dv\big)_{Sc} - \big(PA(Du), Dv\big)_{Sc} = \big(A(Du), Dv\big)_{Sc}.\eqno(4.3)$$
Then the condition (A2) yields
$$\aligned
\big(QA(Du) - QA(Dv), Du - Dv\big)_{Sc} &= \big(A(Du) - A(Dv), Du - Dv\big)_{Sc}\\
&= \int_{\Omega} \Big[\overline{\big(A(Du) - A(Dv)\big)}(Du - Dv)\Big]_{0}dx\\
&\geq C_{2}\int_{\Omega} \big(|Du| + |Dv|\big)^{p(x)-2}|Du -
Dv|^{2}dx > 0,
\endaligned$$
as $Du \neq Dv$.

(3) The operator $QA$ is coercive. By means of (4.3) and (A2) we
have
$$\aligned
 \frac{\big(QA(Du),
Du\big)_{Sc}}{\big|\big||Du|\big|\big|_{L^{p(x)}(\Omega)}} -
\frac{\big(QA(0),
Du\big)_{Sc}}{\big|\big||Du|\big|\big|_{L^{p(x)}(\Omega)}}
&= \frac{\big(A(Du) - A(0), Du\big)_{Sc}}{\big|\big||Du|\big|\big|_{L^{p(x)}(\Omega)}}\\
&= \frac{\displaystyle\int_{\Omega} \big[\overline{\big(A(Du) - A(0)\big)}(Du - 0)\big]_{0}dx}{\big|\big||Du|\big|\big|_{L^{p(x)}(\Omega)}}\\
&\geq C_{2}\frac{ \displaystyle\int_{\Omega}
|Du|^{p(x)}dx}{\big|\big||Du|\big|\big|_{L^{p(x)}(\Omega)}}.
\endaligned$$
Since
$$\frac{ \displaystyle\int_{\Omega} |Du|^{p(x)}dx}{\big|\big||Du|\big|\big|_{L^{p(x)}(\Omega)}} = \int_{\Omega}\bigg(\frac{|Du|}{2^{-1}\big|\big||Du|\big|\big|_{L^{p(x)}(\Omega)}}\bigg)^{p(x)} \cdot \frac{ \Big(2^{-1}\big|\big||Du|\big|\big|_{L^{p(x)}(\Omega)}\Big)^{p(x)}}{\big|\big||Du|\big|\big|_{L^{p(x)}(\Omega)}}dx.$$
When $\big|\big||Du|\big|\big|_{L^{p(x)}(\Omega)} \geq 1$, we have
$$\frac{ \displaystyle\int_{\Omega} |Du|^{p(x)}dx}{\big|\big||Du|\big|\big|_{L^{p(x)}(\Omega)}} \geq 2^{-p_{+}}\big|\big||Du|\big|\big|_{L^{p(x)}(\Omega)}^{p_{-}-1}.$$
Hence we get from Remark 2.1
$$\frac{\big(QA(Du), Du\big)_{Sc}}{||Du||_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}} - \frac{\big(QA(0), Du\big)_{Sc}}{||Du||_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}} \rightarrow \infty$$
as $||Du||_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})} \rightarrow \infty$. By Lemma
2.3, we obtain
$$\frac{\big(QA(0), Du\big)_{Sc}}{||Du||_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}} = \frac{\big(A(0), Du\big)_{Sc}}{||Du||_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}} = \frac{\displaystyle\int_{\Omega} \big[\overline{A(0)}Du\big]_{0}dx}{||Du||_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}} \leq C||A(0)||_{L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})}.$$
Therefore,
$$\frac{\big(QA(Du), Du\big)_{Sc}}{||Du||_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}} \rightarrow \infty$$
as $||Du||_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})} \rightarrow \infty$.

According to Proposition 4.1, let $X=DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$
and $G=QA$. Then the operator $QA$ is surjective. Consequently,
there exists $u \in W^{1,p(x)}_{0}(\Omega,Cl_{n})$ such that $QA(Du)
= 0$.
 Furthermore, Theorem 3.1 gives
 $$\int_{\Omega}\overline{A(Du)}D\varphi dx=\int_{\Omega}\overline{\big(QA(Du) + PA(Du)\big)}D\varphi dx=\int_{\Omega}\overline{\widetilde{D}PA(Du)}\varphi dx=0,$$
for any $\varphi \in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$. Therefore, $u$
is a weak solution of the $A$-Dirac equations (4.1).

Finally, if $u_{1},u_{2}$ are solutions to (4.1), then
$\big(A(Du_{i}), D\varphi\big)_{Sc} = 0 \ (i=1,2)$  for all
$\varphi \in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$.
 Set $\varphi = u_{1} - u_{2}$, then the condition (A2) yields
$$\aligned
0 &= \big(A(Du_{1}) - A(Du_{2}), Du_{1} - Du_{2}\big)_{Sc}\\
&= \int_{\Omega} \Big[\overline{\big(A(Du_{1}) - A(Du_{2})\big)}(Du_{1} - Du_{2})\Big]_{0}dx\\
&\geq C_{2}\int_{\Omega} \big(|Du_{1}| +
|Du_{2}|\big)^{p(x)-2}|Du_{1} - Du_{2}|^{2}dx \geq 0.
\endaligned$$
Thus, $Du_{1} = Du_{2}$, hence $u_{1} - u_{2} \in \mathrm{ker}\,D$. The proof is now complete. \qed


\subsection{Stokes equations in variable exponent spaces}
%{\section*{{\bf 5. Proof of main results}}}

In the study of the stationary Navier-Stokes equations, the
corresponding Stokes equations plays a crucial role. It can be said
that any open question about Navier-Stokes equations, such as global
existence of strong solutions, uniqueness and regularity of weak
solutions, and asymptotic behaviour, is closely related with the
qualitative and quantitative properties of the solutions of Stokes
equations, see \cite{4}.

The Stokes system consists in finding a pair of function $(u,\omega)$
solution of the following equations:
\begin{align}
&-\Delta u + \frac{1}{\eta}\nabla\omega = \frac{\rho}{\eta}f,\ \ \ \mathrm{in}\ \Omega, \tag{4.4}\\
&\mathrm{div}\  u = f_{0},\ \ \ \mathrm{in}\ \Omega, \tag{4.5}\\
&u = v_{0},\ \ \ \ \ \ \mathrm{on}\ \partial \Omega. \tag{4.6}
\end{align}

With $\displaystyle\int_{\Omega}f_{0}dx=\displaystyle\int_{\partial\Omega}n\cdot v_{0}dx$ the necessary compatibility condition for the solvability is given. Here, $u$ is the velocity, $\omega$ the hydrostatic pressure, $\rho$ the density, $\eta$ the viscosity, $f$ the vector of the external forces and the scalar function $f_{0}$ a measure of the compressibility of fluid.
 The boundary condition (4.6) describes the adhesion on the boundary for $v_{0} = 0$. This system describes the stationary flow of a homogeneous viscous incompressible fluid for small Reynold numbers. For more details we refer to \cite{18, 22, 26}.


In \cite{6}, L. Diening, D. Lengeler and M. Ru\v{z}i\v{c}ka proved the
existence and uniqueness of a weak solution in
$\big(W^{1,p(x)}(\Omega)\big)^{n} \times L^{p(x)}(\Omega)$ of the
Stokes problem (4.3--4.5), provided that the right-hand side
$f\in\big(W^{-1,p(x)}(\Omega)\big)^{n}$, $f_{0}\in L^{p(x)}(\Omega)$
and $v_{0}\in \mathrm{tr}((W^{1,p(x)}(\Omega))^{n})$. The result is
based on generalizations of the classical theories of
Calder\'{o}n-Zygmund and Agmon-Douglis-Nirenberg to variable
exponent spaces. In this subsection, with the help of our Hodge-type
decomposition, we obtain an analogous result in the context of
Clifford analysis. Furthermore, we also give a representation of the
solution.

We suppose that $f=\sum_{i=1}^{n}f_{i}\mathrm{e}_{i}$ and
$u=\sum_{i=1}^{n}u_{i}\mathrm{e}_{i}$. Then the Stokes system may be
written in the following hypercomplex formulation (See \cite{18, 19}):
\begin{align}
&-\Delta u + \frac{1}{\eta}D\omega = \frac{\rho}{\eta}f,\ \ \mathrm{in}\  \Omega, \tag{4.7}\\
&[Du]_{0} = f_{0},\ \ \ \mathrm{in}\  \Omega, \tag{4.8}\\
&u = v_{0},\ \ \ \mathrm{on}\ \partial \Omega. \tag{4.9}
\end{align}
For the sake of simplicity, we consider the following Stokes system:
\begin{align}
&\widetilde{D}D u + \widetilde{D}\omega =f,\ \ \ \mathrm{in}\ \Omega, \tag{4.10}\\
&[Du]_{0} =0,\ \ \ \mathrm{in}\  \Omega, \tag{4.11}\\
&u = 0,\ \ \ \ \ \ \mathrm{on}\ \  \partial \Omega. \tag{4.12}
\end{align}

In the following, we require two basic results. The first one is an
abstract algebraic result which is known as the ``Peetre-Tartar
Lemma".


\noindent \textbf{Proposition 4.3.} (\cite{peetre}, \cite[Lemma 11.1]{33}) \begin{itshape}Let $ E_{1}$ be a Banach space, $ E_{2}, E_{3}$ be
 normed spaces, $A$ an operator in
 $\mathscr{L}(E_{1},E_{2})$ and $B$ a compact operator in
 $\mathscr{L}(E_{1},E_{3})$. If $\|u\|_{E_{1}}$ is equivalent to
 $\|Au\|_{E_{2}}+\|Bu\|_{E_{3}}$ for each $u\in E_{1}$, then the
 range space $R(A)$ of the operator $A$ is a closed subspace of
 $E_{2}$.\end{itshape}\\

The second result we will use is a generalization of $\mathrm{Ne\check{c}as}$' theorem which implies an important equivalence of norms.
 We first give the following definition.\\

\noindent\textbf{Definition 4.4.} The operator
 $\widetilde{\nabla}:L^{p(x)}(\Omega)\rightarrow
 (W^{-1,p(x)}(\Omega))^{n}$ is defined by $$\langle\widetilde{\nabla} f,\varphi\rangle=-\langle f,\mathrm{div}\, \varphi\rangle:=-\int_{\Omega}{f\
\mathrm{div}\, \varphi}\,dx$$ for all $f \in L^{p(x)}(\Omega)$ and
$\varphi\in (C^{\infty}_{0}(\Omega))^{n}$.\\


 \noindent\textbf{Proposition 4.5.} (\cite[Theorem 14.3.18]{5}) \begin{itshape}Let $\Omega$ be a bounded Lipschitz
 domain of $\mathbb{R}^{n}$. Then there exists a positive constant
 $C=C(n,p,\Omega)$ such that
 $$||f||_{L^{p(x)}(\Omega)}\leq C(||\widetilde{\nabla} f||_{W^{-1,p(x)}(\Omega)}+||f||_{W^{-1,p(x)}(\Omega)}) \eqno(4.13)$$
 holds for every $f\in L^{p(x)}(\Omega)$. \end{itshape}\\

\noindent\textbf{Remark 4.6.} From Theorem 14.3.18 in \cite{5} we know that Proposition 4.5 holds for the domain $\Omega$ satisfying
the emanating chain condition. From \cite[p. 239]{5} it follows that a bounded domain satisfies the emanating chain condition if and only if
it is a John domain. From \cite[p. 237]{5} it follows that any bounded Lipschitz domain is a John domain.\\

 As a first application of these two results, the following
 corollary derives an important property of the above-mentioned gradient operator.\\

\noindent\textbf{Corollary 4.7.} \begin{itshape}Let $\Omega$ be a bounded
Lipschitz
 domain of $\mathbb{R}^{n}$. Then the range space of the operator $\widetilde{\nabla}
 \in \mathscr{L}({L^{p(x)}(\Omega)},(W^{-1,p(x)}(\Omega))^{n})$ is a closed
 subspace of $(W^{-1,p(x)}(\Omega))^{n}$.\end{itshape}

{\it Proof.}  To apply Proposition 4.3, we suppose
$E_{1}=L^{p(x)}(\Omega),E_{2}=(W^{-1,p(x)}(\Omega))^{n},E_{3}=W^{-1,p(x)}(\Omega)$
$A=\widetilde{\nabla}$, $B=I$ (the identity operator). Since the
domain is bounded, the canonical imbedding $B$ of $E_{1}$ into
$E_{3}$ is compact (see \cite{6}). On the other hand, by virtue of
H\"older's inequality, we have for every $f\in
L^{p(x)}(\Omega)$
$$\aligned
||\widetilde{\nabla}
f||_{W^{-1,p(x)}(\Omega)}&=\sup_{||g||_{W^{1,p'(x)}_{0}(\Omega)}\leq1}|\langle\widetilde{\nabla}
f,g\rangle |\\
&=\sup_{||g||_{W^{1,p'(x)}_{0}(\Omega)}\leq1}|\int_{\Omega}{f \
\mathrm{div} g}dx|\leq C||f||_{L^{p(x)}(\Omega)}.
\endaligned$$
Therefore
$$||\widetilde{\nabla} f||_{W^{-1,p(x)}(\Omega)}+||f||_{W^{-1,p(x)}(\Omega)} \leq
C||f||_{L^{p(x)}(\Omega)} \ \ \mbox{for\ all}\ f\in
L^{p(x)}(\Omega).\eqno(4.14)$$
Then (4.13) and (4.14) yield the
equivalence of the norms of both sides above. Thus, the
desired conclusion follows from Proposition 4.3.\qed



Now we begin our investigations with the following lemma
which is a generalization of De Rham's theorem to the variable exponent spaces.\\

\noindent\textbf{Lemma 4.8.}\begin{itshape} Let $\Omega$ be a bounded Lipschitz
 domain of $\mathbb{R}^{n}$. Let $f\in (W^{-1,p(x)}(\Omega))^{n}$
 satisfy:
 $$\langle f,\varphi\rangle :=\int_{\Omega}{f\cdot \varphi}dx=0,$$
 for any
 $\varphi\in \mathscr{W}(\Omega):=\big\{v\in
 (W^{1,p'(x)}_{0}(\Omega))^{n}:\mathrm{div}\ v=0 \big\}$. Then there exists
 $q\in L^{p(x)}(\Omega)$ such that $f=\widetilde{\nabla} q$.\end{itshape}


{\it Proof.}  Since $\langle -\widetilde{\nabla} u,g\rangle =\langle u,\mathrm{div}\, g\rangle$ for every $u\in
L^{p(x)}(\Omega)$ and $g\in(W^{1,p'(x)}_{0}(\Omega))^{n}$, it
follows that the operator $-\widetilde{\nabla} \in
\mathscr{L}({L^{p(x)}(\Omega)},(W^{-1,p(x)}(\Omega))^{n})$ is
 the adjoint of the operator $$\mathrm{div} \in
 \mathscr{L}((W^{1,p'(x)}_{0}(\Omega))^{n},{L^{p'(x)}(\Omega)}).$$
 But according to Corollary 4.7, the range space $R(\widetilde{\nabla} )$ is a
 closed subspace of $(W^{-1,p(x)}(\Omega))^{n}$. Then the closed
 range theorem of Banach (see \cite{34}) implies that
 $$R(\widetilde{\nabla})=(\mathrm{ker(div)})^{\perp}:=\big\{y\in (W^{-1,p(x)}(\Omega))^{n}:\
 \langle y,v\rangle =0,\forall\
 v\in \mathscr{W}(\Omega)\big\}.$$ This implies the desired conclusion.\qed


\noindent\textbf{Remark 4.9.}
Note that Lemma 4.8 remains valid if the set
$\mathscr{W}(\Omega)$ is replaced by
$$\tilde{\mathscr{W}}(\Omega)=\{v\in
(C^{\infty}_{0}(\Omega))^{n}:\mathrm{div}\ v=0\}.$$ The proof is
analogous to that of Theorem 2.8 in \cite{2} and thus we omit it.\\



\noindent\textbf{Definition 4.10.} We call $(u,\omega) \in
W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n}) \times L^{p(x)}(\Omega)$ a solution
of (4.10--4.12) provided that it satisfies the system (4.10--4.12) for
every $f\in W^{-1,p(x)}(\Omega, \mathrm{C}\ell _{n})$.\\



\noindent\textbf{Lemma 4.11.}\begin{itshape} Assume that $f \in W^{-1,p(x)}(\Omega,
\mathrm{C}\ell _{n})$. Then for every solution $(u,\omega) \in
W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n}) \times L^{p(x)}(\Omega)$ of system
$(4.10)$--$(4.12)$ we have the representation
$$TQ\widetilde{T}f= u + TQ\omega.$$\end{itshape}

{\it Proof.}  Let $\varphi_{n} \in W^{1,p(x)}_{0}(\Omega,
\mathrm{C}\ell _{n})$ with $\varphi_{n} \rightarrow \varphi$ in $L^{p(x)}(\Omega,
\mathrm{C}\ell _{n})$. By Lemma 2.12, we have
$$TQT(D\varphi_{n}) = TQ\varphi_{n}.$$
Since $W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n})$ is dense in $L^{p(x)}(\Omega,
\mathrm{C}\ell _{n})$, it follows that $TQ\widetilde{T}\widetilde{D}\varphi =
TQ\varphi$. Thus, for $u \in W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n})$ and
$\omega \in L^{p(x)}(\Omega)$ we obtain
$$TQ\widetilde{T}f = TQ\widetilde{T}(\widetilde{D}Du + \widetilde{D}\omega) = u + TQ\omega,$$
as desired.\qed

We are now in a position to prove our result as follows.


\noindent\textbf{Theorem 4.12.}\begin{itshape} Suppose $f \in
W^{-1,p(x)}(\Omega, \mathrm{C}\ell _{n})$. Then the Stokes system $(4.10)$--$(4.12)$
has a unique solution $(u,\omega) \in W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n})
\times L^{p(x)}(\Omega)$ in the form
 $$u + TQ\omega = TQ\widetilde{T}f,$$
with respect to the estimate
$$||u||_{W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})} + ||Q\omega||_{L^{p(x)}(\Omega)} \leq C||f||_{W^{-1,p(x)}(\Omega, \mathrm{C}\ell _{n})}.$$
Here, the hydrostatic pressure $\omega$ is unique up to a
constant.\end{itshape}


{\it Proof.}  Lemma 4.11 implies that our system (4.10)--(4.12) is
equivalent to the system
\begin{align}
&u + TQ\omega = TQ\widetilde{T}f, \tag{4.15}\\
&[Q\omega]_{0} = [Q\widetilde{T}f]_{0}. \tag{4.16}
\end{align}
Observe that the equality (4.15) is equivalent to the following
equality
$$Du + Q\omega = Q\widetilde{T}f. \eqno(4.17)$$
Indeed, due to Lemma 2.12 we know that when $D$ is applied to
(4.15), this leads to (4.17). When $T$ is applied to (4.17), this
reduces to (4.15).

Therefore we need to show that for each $f\in W^{-1,p(x)}(\Omega,
\mathrm{C}\ell _{n}^{1})$, the function $QTf$ can be decomposed into two
functions $Du$ and $Q\omega$. Notice that we are not decomposing the
whole space $L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$ but only the subspace
$L^{p(x)}(\Omega, \mathrm{C}\ell _{n}^{1}) \cap \mathrm{im}Q$. Therefore, suppose
$Du + Q\omega=0$ for $u \in W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n}^{1}) \cap
\mathrm{ker \ div}$ and $\omega \in L^{p(x)}(\Omega)$. Then (4.11)
gives $[Q\omega]_{0} = 0$. Notice that the operator $Q$ maps the
space $L^{p(x)}(\Omega)$ into itself. Thus, $Q\omega = 0$. Hence,
$Du = Q\omega = 0$.
 This means that $Du + Q\omega$ is a direct sum, which is a subset of $\mathrm{im}Q$.

Next we have to ask about the existence of a functional $\mathcal
{F}\in (L^{p(x)}(\Omega, \mathrm{C}\ell _{n}^{1}) \cap \mathrm{im}Q)^{\ast}$ with
$\mathcal {F}(Du)=0$ and $\mathcal {F}(Q\omega)=0$ but $\mathcal
{F}(Q\widetilde{T}f)\neq 0$. This amounts to asking if there exists
$g \in W^{-1,p'(x)}(\Omega, \mathrm{C}\ell _{n}^{1})$, such that for all $u \in
W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n}^{1}) \cap \mathrm{ker \ div}$ and
$\omega \in L^{p(x)}(\Omega)$
$$(Du, Q\widetilde{T}g)_{Sc} = 0, \eqno(4.18)$$
$$(Q\omega, Q\widetilde{T}g)_{Sc} = 0, \eqno(4.19)$$
but $(Q\widetilde{T}f, Q\widetilde{T}g)_{Sc} \neq 0$. Here, we have applied Theorem
3.3 and Lemma 2.13.

Thus, let us consider the system (4.18) and (4.19) with $g \in
W^{-1,p'(x)}(\Omega, \mathrm{C}\ell _{n}^{1})$. Observe that, with the help of
Lemma 2.14 and Lemma 2.15, (4.19) yields
$$(Du, Q\widetilde{T}g)_{Sc} = (u, \widetilde{D}Q\widetilde{T}g)_{Sc} = (u, \widetilde{D}\widetilde{T}g - \widetilde{D}P\widetilde{T}g)_{Sc} = (u, g)_{Sc} = 0,$$
which implies $g = \widetilde{\nabla} h = \widetilde{D}h$ with $h
\in L^{p'(x)}(\Omega)$ because of Lemma 4.7. Notice that, from Lemma
2.14, (4.19) gives
$$(Q\omega, Q\widetilde{T}g)_{Sc} = (Q\omega, Q\widetilde{T}\widetilde{D}h)_{Sc} = (Q\omega, Qh)_{Sc} = 0$$
holds for each $\omega \in L^{p(x)}(\Omega)$. Hence,
$Q\omega=|Qh|^{p'(x)-2}Qh$ gives $Qh = 0$. Then we obtain
$$g =\widetilde{D}h= \widetilde{D}Qh + \widetilde{D}Ph= 0.$$ Furthermore, we
get
$$(Q\widetilde{T}f, Q\widetilde{T}g)_{Sc} = 0,\ \ \forall f \in W^{-1,p(x)}(\Omega, \mathrm{C}\ell _{n}^{1}).$$
Finally, since (4.17) yields
$$||Du||_{L^{p(x)}(\Omega, \mathrm{C}\ell _{n})} + ||Q\omega||_{L^{p(x)}(\Omega)} \geq ||Q\widetilde{T}f||_{L^{p(x)}(\Omega, \mathrm{C}\ell _{n})}.$$
By the norm equivalence theorem, we obtain
$$||Du||_{L^{p(x)}(\Omega, \mathrm{C}\ell _{n})} + ||Q\omega||_{L^{p(x)}(\Omega)} \leq C||Q\widetilde{T}f||_{L^{p(x)}(\Omega, \mathrm{C}\ell _{n})}.$$
By Remark 2.1, Lemma 2.13 and the boundedness of the operator $Q$,
we get
$$||u||_{W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n})} + ||Q\omega||_{L^{p(x)}(\Omega)} \leq C||f||_{W^{-1,p(x)}(\Omega, \mathrm{C}\ell _{n})}.\eqno(4.20)$$
From (4.20) the uniqueness of the solution follows. Notice that
$Q\omega = 0$ implies $\omega \in \mathrm{ker}\widetilde{D}$.
Therefore, $\omega$ is unique up to a constant. Now we complete the proof of Theorem 4.11.\qed



The system (4.10)--(4.12) is just an example for the general
way of treating such kinds of problems with the help of our
Hodge-type decomposition. We expect that this study can be extended to
investigate Navier-Stokes problems.

\medskip
\noindent\textbf{Acknowledgments.} Binlin Zhang would like
to express his sincere thanks to Professor U. K\"{a}hler for his
willingness to give many helps.  V.~R\u adulescu acknowledges the support through Grant CNCS PCE-47/2011.


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