\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 310, pp. 1--5.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/310\hfil  Riemann-Gilbert problem]
{Riemann-Hilbert problem for the Moisil-Teodorescu system in \\
 multiple connected domains}

\author[V. A. Polunin, A. P. Soldatov \hfil EJDE-2016/310\hfilneg]
{Viktor A. Polunin, Alexandre P. Soldatov}

\address{Viktor A. Polunin \newline
Belgorod State University, Belgorod, Russia}
\email{polunin@bsu.edu.ru}

\address{Alexandre P. Soldatov \newline
Belgorod State University, Belgorod, Russia}
\email{soldatov48@gmail.com}

\thanks{Submitted August 14, 2016. Published December 5, 2016.}
\subjclass[2010]{35J46, 45B05, 45E05}
\keywords{Riemann-Gilbert problem; elliptic system; integral of
Cauchy type; \hfill\break\indent Fredholm integral operator;
Moisil-Teodorescu system}

\begin{abstract}
 In this article we obtain a new integral representation of the general
 solution of the Moisil-Teodorescu system in a multiply connected domain.
 Also we give applications of this representation to Riemann-Hilbert problem.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\allowdisplaybreaks

\section{Introduction}

Consider the Moisil-Teodorescu system \cite{Bi72}
\begin{equation}
M\big(\frac{\partial}{\partial x}\big)u(x)=0,\quad
 M(\zeta)=\begin{pmatrix}
0&\zeta_1&\zeta_2&\zeta_3 \\
\zeta_1&0&-\zeta_3&\zeta_2\\
\zeta_2&\zeta_3&0&-\zeta_1\\
\zeta_3&-\zeta_2&\zeta_1&0
\end{pmatrix},\label{e1}
\end{equation}
for a vector $u(x)=(u_1,u_2,u_3,u_4)$.
The identity
 $M^\top(\zeta)M(\zeta)=|\zeta|^2$ shows that the components of this
vector are harmonic functions. Note also that using the notation
\begin{equation}
u=(u_1,v)\label{e2}
\end{equation}
system \eqref{e1} can be written in the form
\begin{equation}
\operatorname{div}v=0,\quad \operatorname{rot}v
+\operatorname{grad}u_1=0.\label{e3}
\end{equation}

It is well known \cite{Bi72} that the matrix-valued function $M^\top(x)/|x|^3$,
where $\top$ stands for the transposed, is the fundamental solution of the
 differential operator $M(D)$. Thus the Cauchy type integral
\begin{equation}
(I\psi)(x)=\frac{1}{2\pi}\int_\Gamma
\frac{M^\top(y-x)}{|y-x|^3}M[n(y)]\psi(y)d_2y,\quad x\notin
\Gamma,\label{e4}
\end{equation}
where $\Gamma$ is a closed smooth surface and $n(y)$ is a unit normal, defines
a solution of \eqref{e1}.

 Let $\Gamma$ be a boundary of a finite domain $D$ for which $n$ is an
exterior normal, $D' = \mathbb{R}^3 \setminus D$ be an open set and for
consistency the notation $D^+ = D$, $D^- =D'$ are introduced.
 Let $\Gamma$ be a Lyapunov surface, $u=I\psi$ and suppose that the function
$\psi$ satisfies the Holder condition.
 Then there exist the limit values
 $$
u^\pm(y_0)=\lim_{x\to y_0,x\in D^\pm}u(x),\quad y_0\in \Gamma,
$$
and the analogue of Plemelj-Sokhotskyii formula
\begin{equation}
u^\pm=\pm\psi+u^*.\label{e5}
\end{equation}
holds. Here $u^*=I^*\psi$ is defined by the singular integral
$$
(I^*\psi)(y_0)=\frac{1}{2\pi}\int_\Gamma
\frac{M^\top(y-y_0)}{|y-y_0|^3}M[n(y)]\psi(y)d_2y.
$$
 These formulas were first obtained by Bitsadze \cite{Bi53}.
 Based on the minimal requirements on the smoothness of the surface this
result made precise in \cite{PoS11a}:
 if $\Gamma$ belongs to the class $C^{1,\nu},\;0<\nu<1$, then the operator
$I$ is bounded $C^\mu(\Gamma)\to C^\mu(\overline{D})$, $0<\mu<\nu$.

 Let the matrix-valued function
$$
B=\begin{pmatrix}
B_{11}&B_{12}&B_{13}&B_{14} \\
B_{21}&B_{22}&B_{23}&B_{24}
\end{pmatrix}
$$
be continuous on $\Gamma$ and be of the rank $2$ at any point $y\in \Gamma$.
 We consider the following analogue of the Riemann-Hilbert boundary value
problem % (problem \eqref{e6})
\begin{equation}
Bu^+=f,\label{e6}
\end{equation}
for the system \eqref{e1}.
A natural approach for the study of this problem (in case of special matrices
$B$) using the Cauchy type integrals was proposed by Bitsadze\cite{Bi55}.
A complete study of problem \eqref{e6} for the domains homeomorphic to the
ball was done by Shevchenko \cite{Sh66, Sh70}. Another approach based on
the integral representation of special type was described in \cite{PoS10, PoS11b}.

In this article, we consider the case of arbitrary multiply connected domain.
Taking into account a general elliptic theory \cite{GiT89, AgD64},
 problem \eqref{e6} is Fredholm one under a so called complementarity condition.
This condition can be defined as follows \cite{Sh70,PoS11b}.
Consider the vector
 $s=(s_1,s_2,s_3)$ with components
$$
s_1=b^{12}+b^{34},\quad s_2=b^{13}-b^{24},\quad s_3=b^{14}+b^{23},
$$
where $b^{kj}=b_{1k}b_{2j}-b_{1j}b_{2k}$ are the corresponding minors of the matrix
 $B$. Then complementarity condition can be expressed in the form
\begin{equation}
s(y)n(y)\ne0,\quad y\in \Gamma.\label{e7}
\end{equation}

As shown in \cite{Sh70}, if $\Gamma$ is homeomorphic to a ball, then under the
above condition the operator $R$ has a Fredholm property and its index equals
to $-1$. In the case of a arbitrary multiply connected domain $D$ only
 the Fredholm property of this problem can be stated.

\begin{theorem} \label{thm1}
 Suppose the surface $\Gamma$ belongs to the class $C^{1,\nu}$ and the matrix-valued
function $B\in C^\nu(\Gamma)$ satisfies \eqref{e7}.
Then the operator $R:C^\mu(\Gamma)\to C^\mu(\overline{D})$ of the
problem \eqref{e1}, \eqref{e6} has a Fredholm property.
\end{theorem}

\begin{proof}
Every two-component vector $\varphi=(\varphi_1,\varphi_2)$ corresponds to a
four-component vector $\psi=\widetilde{\varphi}$ by the formula
$\widetilde{\varphi}=(\varphi_1,n\varphi_2)$ and we put
\begin{equation}
(I_0\varphi)(x)=(I\widetilde{\varphi})(x),\quad x\in D.\label{e8}
\end{equation}
So, the operator $I_0$ acts from the space $C^\mu(\overline{D})$ of
two-component vector- valued functions to the space
 $C^\mu(\overline{D})$ of solutions of the system \eqref{e1} in the domain $D$.
We first prove that this operator has a Fredholm property.

 For this purpose we consider the special case of problem \eqref{e6},
which is defined by the boundary value condition
\begin{equation}
Cu^+=f\label{e9}
\end{equation}
where
$$
C=\begin{pmatrix}
1&0&0&0 \\
0&n_1&n_2&n_3
\end{pmatrix}.
$$
We verify, that the kernel of this problem has a finite dimension.

Indeed if $Cu^+=0$ then using notation \eqref{e2} we have
\begin{equation}
u_1^+=0,\quad v^+n=0.\label{e10}
\end{equation}
 Since the function $u_1$ is harmonic in the domain $D$, then $u_1=0$ and
the second equality \eqref{e3} becomes $ \operatorname{rot} v=0$.
Hence, in each simply-connected subdomain $D_0\subseteq D$ the function $v$
can be defined as $\operatorname{grad}w_0$ of some function $w_0$,
which is harmonic by virtue of the first equality of \eqref{e3}.
If $D_1$ denotes another simply-connected subdomain
$D$ with corresponding representation $v=\operatorname{grad}w_1$,
then $w_0-w_1$ is a locally constant function on the open set $D_0\cap D_1$
because its gradient equals 0. In the whole multiple-connected domain $D$,
the harmonic function $w$ such that $v=\operatorname{grad}w$ is a
multi-valued function. It follows from the second equality of \eqref{e10} that
\begin{equation}
\frac{\partial w^+}{\partial n}=0.\label{e11}
\end{equation}

To avoid the multipli-connectedness of the domain $D$ let us consider its cuts.
 By definition the cut is a simply-connected smooth surface $R\subseteq \overline{D}$
with smooth boundary $\partial L$, such that
$R\cap \Gamma=\partial L$. There exist disjoint cuts $R_1,\ldots, R_m$,
such that the set
$$
D_R=D\setminus R,\quad R=R_1\cup\ldots\cup R_m,
$$
 is a simply connected domain. In this domain the function $w$ is a single-valued
and its boundary values satisfy the relation
\begin{equation}
 (w^+-w^-)\bigr|_{R_i}=c_i,\quad 1\le i\le m,\label{e12}
\end{equation}
 with some constants $c_i$. Nevertheless equalities $c_1=\ldots=c_m=0$
indicate that $w$ is univalent function. So it is harmonic in the whole
domain $D$, while in a view of \eqref{e11} this is possible only if $w$ is constant.
These arguments prove that the space of solutions of homogeneous problem
\eqref{e9} is finite dimensional space.

 We denote the operator of the problem \eqref{e9} by $S$ and consider the
composition $SI_0$, which is acting within the space of two-component
vector-functions in the space $C^\mu(\Gamma)$. Note that the product
$CC^\top$ is the unit $2\times2$-matrix. Also note that \eqref{e6} can be written
as $\widetilde{\varphi}=C^\top\varphi$. So by virtue of \eqref{e4}, \eqref{e5}
we have the equality $SI_0=1+K_0$ with integral operator $K_0$, according
to the formula
\begin{equation}
(K_0\varphi)(y_0)=\frac{1}{2\pi }
\int_\Gamma \frac{k_0(y_0,y)}{|y-y_0|^2}\varphi(y)d_2y,\quad
 y_0\in \Gamma,\label{e13}
\end{equation}
with the matrix-valued function
$$
k_0(y_0,y)=C(y_0)M^\top(\xi)M[n(y)]C^\top(y),\quad \xi=\frac{y-y_0}{|y-y_0|}.
$$
It is easy to see that
\begin{equation}
M^\top(\xi)M(n)C^\top
=\begin{pmatrix}
 n\xi & 0 \\
 {[n,\xi]}_1 & \xi_1 \\
 {[n,\xi]}_2 & \xi_2 \\
 {[n,\xi]}_3 & \xi_3 \\
\end{pmatrix},\label{e14}
\end{equation}
where in the sequel brackets denote the vector product, a product without
brackets is a scalar product, and $[n,\xi]_k$ are components of the vector
$[n,\xi]$. Therefore we get the following expression in  explicit form
$$
k_0(y_0,y)=
 \begin{pmatrix}
 n(y)\xi & 0 \\
 n(y_0)[n(y),\xi] & n(y_0)\xi \\
 \end{pmatrix}
= \begin{pmatrix}
 n(y)\xi & 0 \\
 \,[n(y_0),n(y)]\xi & n(y_0)\xi \\
 \end{pmatrix}.
$$
It was stated in \cite{PoS11a} under assumption $\Gamma\in C^{1,\nu}$
that the function $k_0(t_0,t)$ belongs to $C^\nu(\Gamma\times\Gamma)$ )
and equals zero at $t=t_0$. So a kernel of the operator $K_0$ has weak
singularity, and the proper operator is compact in the space $C^\mu(\Gamma)$.
According to Riesz theorem we conclude that the image $\mathrm{im}\,(SI_0)$
is closed subspace of finite co-dimension.
 Since $\mathrm{im}\,S\supseteq\mathrm{im}\,(SI_0)$, then an image of the
 operator $S$ has the same property. Therefore the operator $S$ has the
Fredholm property, and taking into account the Fredholm property of the
product $SI_0=1+K$ this implies that $I_0$ is Fredholm operator.
\end{proof}

Let us next turn to the original problem \eqref{e6}. 
As before it is obvious that $RI_0=G+K$ with the matrix -valued function 
$G=BC^\top$ and the integral operator $K$ defined similar \eqref{e13}
with respect to the function $k(y_0,y)=B(y_0)M^\top(\xi)M[n(y)]C^\top(y)$, 
contrary to the previous case, this operator is singular operator.

Since $I_0$ is Fredholm operator, the operator $R$ of our problem is Fredholm 
equivalent to the operator $N=G+K$.
 For a surface $\Gamma$, homeomorphic to a ball, the inequality \eqref{e7} 
provides the Fredholm property of the singular operator $N$. 
Since the Fredholm criterion for this operator has local property \cite{Mi62} 
the similar result is true for arbitrary surface also, and this completes the proof.

Note that the expressions for the matrices
 $G(y_0)$ and $k(y_0,y)$ can be simplified. To see this we write  the 
matrix $B=(B_{ij})$ in the form
$$
B=\begin{pmatrix}
 B_{11} & b_1 \\
 B_{21} & b_2 
\end{pmatrix}
$$
with the vectors $b_k=(B_{k2},B_{k3},B_{k4})$.
Then taking into account \eqref{e14} we obtain
\begin{gather*}
G=\begin{pmatrix}
 B_{11} & b_1n \\
 B_{21} & b_2n 
\end{pmatrix}, \\
\begin{aligned}
k(y_0,y)&=\begin{pmatrix}
 B_{11}(y_0)n(y)\xi+b_1(y_0)[n(y),\xi] & b_1(y)\xi \\
 B_{21}(y_0)n(y)\xi+b_2(y_0)[n(y),\xi] & b_2(y),\xi
\end{pmatrix} \\
&=
\begin{pmatrix}
 B_{11}(y_0) n(y)\xi +[b_1(y_0),n(y)]\xi & b_1(y)\xi \\
 B_{21}(y_0) n(y)\xi +[b_2(y_0),n(y)]\xi & b_2(y)\xi
\end{pmatrix}
\end{aligned} \\
\xi=\frac{y-y_0}{|y-y_0|}.
\end{gather*}


\begin{thebibliography}{00}

\bibitem{AgD64} Agmon, S.; Douglis, A.; Nirenberg, L.;
\emph{Estimates near the boundary for solutions of elliptic partial differential
equations satisfying general boundary conditions. II}, Comm. Pure Appl.
Math. ,1964, 17, 35-92.

\bibitem{Bi72}  Bitsadze, A. V.;
\emph{Foundation of the theory of analytic functions of a complex variable},
 Nauka, 1972  (in Russian).

\bibitem{Bi53}  Bitsadze, A. V.;
\emph{Spatial analogue of an integral of Cauchy type and some of its applications},
 Izv. Akad. Nauk SSSR Ser. Mat., 17:6 (1953), 525–538 (in Russian)

\bibitem{Bi55} Bitsadze, A. V.;
\emph{On two-dimensional integral of Cauchy type},
 Soob. Akad. nauk Gruz. SSR, 1955, 16, 3, 177-184 in (Russian)

\bibitem{GiT89}  Gilbarg, D.; Trudinger, N. S.;
\emph{Elliptic Partial Differential Equations of Second Order Springer},
 1989 - p. 464.

\bibitem{Mi62} Mikhlin, S. G.;
\emph{Multidimensional singular integrals and integral
equations.}, Fizmatgiz, 1962 (in Russian)

\bibitem{PoS11a} Polunin, V. A.; Soldatov, A. P.;
\emph{Three-dimensional analogue of the Cauchy type integral}, 
Differential Equations, 2011, vol. 47. No.3,
pp. 1-10 DOI: 10.1134/S0012266111030074

\bibitem{PoS10}  Polunin, V. A.; Soldatov, A. P.;
\emph{The Riemann- Hilbert problem for Moisil-Theodorescu
system in bounded domains}, Non-classical equations of mathematical physics.
 - Novosibirsk: 2010 pp. 192-201 (in Russian)

\bibitem{PoS11b}  Polunin, V. A.; Soldatov, A. P.;
\emph{On the integral representation of solutions for
 Moisil-Theodorescu}, Vedomosty. BSU, 2011., 5(118). pp. 84-92. 2011 (in Russian)

\bibitem{Sh66}  Shevchenko, V. I.;
\emph{On the Riemann - Hilbert problem for a holomorphic vector},
 Doklady Akad. Nauk SSSR, 1966. Vol. 169, 6.  pp. 1285-1288 (in Russian)

\bibitem{Sh70}  Shevchenko, V. I.;
\emph{Some boundary value problems for holomorphic vector}, Sborn
 Math. Phys. Kiev. - 1970. Vol. 8. - pp. 172-187 (in Russian)

\end{thebibliography}

\end{document}