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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 202, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/202\hfil Existence and uniqueness]
{Existence and uniqueness for boundary-value problem with additional single
point conditions of the Stokes-Bitsadze system}

\author[M. Tahir \hfil EJDE-2012/202\hfilneg]
{Muhammad Tahir} 

\address{Muhammad Tahir \newline
Mathematics Department\\
HITEC University, Taxila Cantonment, Pakistan}
\email{mtahir@hitecuni.edu.pk}

\thanks{Submitted July 24, 2012. Published November 15, 2012.}
\subjclass[2000]{35J57}
\keywords{Bitsadze system; boundary value problem;
 single point conditions}

\begin{abstract}
 This article shows the uniqueness of a solution
 to a Bitsadze system of equations, with a boundary-value problem that
 has four additional single point conditions. It also shows how to
 construct the solution.
\end{abstract}

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

\section{Introduction}

The planar Stokes flow based on stream function $\psi(x,y)$ and
stress function $\phi(x,y)$, is expressed as
\begin{equation} \label{e1}
\begin{gathered}
\phi_{xx}-\phi_{yy}=-4\eta\psi_{xy},\\
-\phi_{xy}=\eta(\psi_{yy}-\psi_{xx}),
\end{gathered}
\end{equation}
where $\eta$ is a material constant, see for the details \cite{c1,d1,o2}.
The re-scaling (2$\eta\psi\to \psi$) reduces the system \eqref{e1} to
\begin{equation} \label{e2}
  \begin{gathered}
	\phi_{xx}-\phi_{yy}+2\psi_{xy}=0,\\
	\psi_{xx}-\psi_{yy}-2\phi_{xy}=0,
  \end{gathered}
\end{equation}
which is the famous second order elliptic system called the Bitsadze system
of equations and is identified as Stokes-Bitsadze system \cite{t1}.
In the literature Bitsadze appears to have been the first to question
the uniqueness and existence or even the well-posedness of \eqref{e2}
subject to certain boundary conditions, see for reference \cite{b2,b3,k1}.
Oshorov \cite{o1} finds well-posed problems for the Cauchy-Riemann system
and extends those to the Bitsadze system \eqref{e2}. Vaitekhovich \cite{v1}
 discusses Dirichlet and Schwarz problems for the inhomogeneous Bitsadze
equation for a circular ring domain. In the interior of unit
disc a boundary value problem for the Bitsadze equation is considered
by Babayan \cite{b1} and is proved to be Noetherian.
In his paper Babayan also proposes solvability conditions for the
inhomogeneous Bitsadze equation. The unique solvability in a unit
 disc for the inhomogeneous Bitsadze system is discussed in \cite{h1}.

 The Stokes-Bitsadze system \eqref{e2} can be expressed in the matrix form as
\begin{equation} \label{e3}
A\mathbf{U}_{xx}+2B\mathbf{U}_{xy}+C\mathbf{U}_{yy}=\mathbf{0},
\end{equation}
where
\[
A = \begin{pmatrix}
1 & 0 \\
0 & 1 \end{pmatrix},\quad
B = \begin{pmatrix}
0 & 1 \\
-1 & 0 \end{pmatrix},\quad
 C=-A, \quad
\mathbf{U}(x,y)= \begin{pmatrix}
\phi \\
\psi\end{pmatrix}.
\]
In a domain $\Omega\subset\mathbb{R}^2$ with boundary $\Gamma$ a linear
boundary value problem of Poincar\'{e} for the system \eqref{e3}
can be formulated as
\begin{equation} \label{e4}
p_1\mathbf{U}_{x}+p_2\mathbf{U}_{y}+q\mathbf{U}=\boldsymbol{\alpha}(x,y),\quad
(x,y)\in \Gamma
\end{equation}
where $p_1,p_2, q$ are real 2$\times$2 matrices and
$\boldsymbol{\alpha}(x,y)$ a real vector given on the boundary $\Gamma$.
The boundary-value problems of Poincar\'e for the Stokes-Bitsadze system
 will be discussed elsewhere. In this paper we are interested in a boundary
value problem with four additional single point conditions.

\section{A boundary value problem with additional single point conditions}

We consider the Stokes-Bitsadze system \eqref{e2} in domain
 $\Omega\subset\mathbb{R}^2$  with boundary $\Gamma$ subject to
the following boundary conditions.
\begin{equation} \label{e5}
	\psi=f, \quad 	\psi_{n}=g \quad \text{on } \Gamma,
\end{equation}
and
\begin{equation} \label{e6}
\phi=\phi^P,\quad
\nabla\phi=(\nabla\phi)^P,\quad
\Delta\phi=(\Delta\phi)^P, \quad
\text{at a single point $P \in \bar{\Omega}$}.
\end{equation}

\begin{theorem} \label{thm2.1}
For $f,g\in C(\Gamma)$, the boundary value problem \eqref{e5}--\eqref{e6}
for the Stokes-Bitsadze system \eqref{e2} has a unique solution
 $(\phi,\psi)\in C^4(\Omega)\times C^4(\Omega)$.
\end{theorem}

\begin{proof}
Suppose $\phi,\psi\in C^4(\Omega)$. If $(\phi,\psi)$ satisfies  \eqref{e2},
then $\phi$ and $\psi$ are biharmonic in $\Omega$, and for $f,g\in C(\Gamma)$
the problem
\begin{equation} \label{e7}
\begin{gathered}
	\Delta^2 \psi=0 \quad \text{in } \Omega\\
	\psi=f \quad \text{on } \Gamma \\
	\psi_n=g \quad \text{on } \Gamma
\end{gathered}
\end{equation}
has a unique solution $\psi\in C^4(\Omega)$,  \cite{t2},
that satisfies \eqref{e2} and \eqref{e5}.
Let the unique solution be denoted by $\widetilde{\psi}$.
Now we show that for the unique $\widetilde{\psi}$ if there exists $\phi$
satisfying \eqref{e2} and \eqref{e5}--\eqref{e6} then that $\phi$ is unique.
Assume that the pairs $(\phi_1,\widetilde{\psi})$ and $(\phi_2,\widetilde{\psi})$
with $\phi_1\neq \phi_2$ satisfy \eqref{e2} and \eqref{e5}--\eqref{e6}
and that $\delta = \phi_1-\phi_2$. Then from \eqref{e2} it immediately follows that
\begin{equation} \label{e8}
\delta_{xx}-\delta_{yy}=0, \quad
\delta_{xy}=0\quad
\text{on }\Omega.
\end{equation}
But \eqref{e6} then yields
\begin{equation} \label{e9}
\delta=0,\quad \nabla\delta=0,\quad \Delta\delta=0 \quad\text{at }P,
\end{equation}
and the general solution of the system \eqref{e8} becomes,
\begin{equation} \label{e10}
\delta=ax+by+c(x^{2}+y^{2})+d,
\end{equation}
which on imposing the conditions \eqref{e9} gives $\delta\equiv 0$
in $\bar{\Omega}$ and uniqueness of $\phi$ thus follows.
 Hence there exists at most one pair
$(\phi,\psi)\in C^4(\Omega)\times C^4(\Omega)$
that can satisfy \eqref{e2} and \eqref{e5}--\eqref{e6}.
We are now in a position to assume (without proof) that
$(\widetilde{\phi},\widetilde{\psi})$ is a solution of \eqref{e2}
and \eqref{e5}--\eqref{e6}.

 Next, we suppose that $P(x_{P},y_{P})$ and $Q(x,y_{P})$ are the points
in $\bar{\Omega}$, refer to the Figure \ref{fig1}.

\begin{figure}[hbtp]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1}
%\includegraphics[scale=0.5]{img}
\end{center}
\caption{Boundary conditions and additional single point conditions}
\label{fig1}
\end{figure}

At point $P$ the expressions \eqref{e2}(a) and \eqref{e6}(c)
respectively take the form
\begin{equation} \label{e11}
\begin{gathered}
\phi_{xx}^P-\phi_{yy}^P=-2\psi_{xy}^P,\\
\phi_{xx}^P+\phi_{yy}^P=\Delta\phi^P,
\end{gathered}
\end{equation}
from which it is obvious that $\phi_{xx}^P$ and $\phi_{yy}^P$ are
known at $P$. Since $(\widetilde{\phi},\widetilde{\psi})$ satisfies \eqref{e2}(b),
therefore
\begin{equation} \label{e12}
\widetilde{\phi}_{xyy}=\frac{1}{2}[\widetilde{\psi}_{xxy}-\widetilde{\psi}_{yyy}],\\
\end{equation}
and on integration along $PQ$ we have
\begin{gather} \label{e13}
\widetilde{\phi}_{yy}(x,y_{P})=\phi _{yy}^P+\frac{1}{2}\int_{x_P}^x
[\widetilde{\psi}_{xxy}(\lambda,y_P)-\widetilde{\psi}_{yyy}(\lambda,y_P)]d\lambda,\\
 \label{e14}
\widetilde{\phi}_y(x,y_P)=\phi_{y}^P+\frac{1}{2}\int_{x_P}^x
[\widetilde{\psi}_{xx}(\lambda,y_P)-\widetilde{\psi}_{yy}(\lambda,y_P)]d\lambda.
\end{gather}
Since all the terms on right hand sides of \eqref{e13} and \eqref{e14}
are known therefore $\widetilde{\phi}_{yy}$ and $\widetilde{\phi}_{y}$ are
known along $PQ$. Since $(\widetilde{\phi},\widetilde{\psi})$
satisfies \eqref{e2}(a), we have
\begin{equation} \label{e15}
\widetilde{\phi}_{xx}=\widetilde{\phi}_{yy}-2\widetilde{\psi}_{xy},
\end{equation}
and using \eqref{e13}, can further be expressed as
\begin{equation} \label{e16}
\widetilde{\phi}_{xx}(x,y_P)
=\phi_{yy}^P +\frac{1}{2}\int_{x_P}^x[\widetilde{\psi}_{xxy}(\lambda,y_P)
-\widetilde{\psi}_{yyy}(\lambda,y_P)]d\lambda-2 \widetilde{\psi}_{xy}(\lambda,y_P).
\end{equation}
Further on integration along $PQ$, we have
\begin{equation} \label{e17}
\begin{split}
\widetilde{\phi}_{x}(x,y_P)
&=\phi_{x}^P+ \int_{x_P}^x\bigg[\phi_{yy}^P
+\frac{1}{2}\int_{x_P}^\mu [ \widetilde{\psi}_{xxy}(\lambda,y_P)
-\widetilde{\psi}_{yyy}(\lambda,y_P)]\bigg ]\,d\lambda \,d\mu \\
&\quad  - 2\int _{x_P}^x\widetilde{\psi}_{xy}(\lambda,y_P)\,d\lambda,
\end{split}
\end{equation}
whence
\begin{equation} \label{e18}
\begin{split}
&\widetilde{\phi}(x,y_P)\\
&=\phi^P+(x-x_P)\phi_x^ P +\frac{1}{2}(x-x_P)^2\phi_{yy}^P
 - 2 \int_{x_P}^x \int_{x_P}^\mu \widetilde{\psi}_{xy}(\lambda,y_P)d\lambda\,\,d\mu\\
&\quad +\frac{1}{2}\int_{x_P}^x \int_{x_P}^\nu\int_{x_P}^\mu
\big [\widetilde{\psi}_{xxy}(\lambda,y_P)-\widetilde{\psi}_{yyy}(\lambda,y_P)
\big ] \,d\lambda\,\,d\mu\,\,d\nu.
\end{split}
\end{equation}
Since all the terms on right hand sides of \eqref{e15}, \eqref{e16},
\eqref{e17} are known therefore $\widetilde{\phi}_{xx}, \widetilde{\phi}_x $
and $\widetilde{\phi}$ are known along $PQ$ and hence we know
 $\widetilde{\phi}, \nabla\widetilde{\phi}$ and
$\Delta\widetilde{\phi}$ at $Q(x,y_{P})$.

 Now from the point $Q$ we draw the line $QR$ where $R(x,y)\in\bar{\Omega}$
is an arbitrary point. Again, since $(\widetilde{\phi},\widetilde{\psi})$
satisfies \eqref{e2}(b); therefore
\begin{equation} \label{e19}
\widetilde{\phi}_{xxy}=\frac{1}{2}[\widetilde{\psi}_{xxx}-\widetilde{\psi}_{xyy}],
\end{equation}
which on integration, along $QR$, gives
\begin{gather} \label{e20}
\widetilde{\phi}_{xx}(x,y)
=\widetilde{\phi}_{xx}(x,y_P)+\frac{1}{2}\int_{y_P}^y
[\widetilde{\psi}_{xxx}(x,\lambda)-\widetilde{\psi}_{xyy}(x,\lambda)]d\lambda,\\
 \label{e21}
\widetilde{\phi}_{x}(x,y)=\widetilde{\phi}_{x}(x,y_P)
+\frac{1}{2}\int_{y_P}^y[\widetilde{\psi}_{xx}(x,\lambda)
-\widetilde{\psi}_{yy}(x,\lambda)]d\lambda.
\end{gather}
But the following expression from \eqref{e2}(a)
\begin{equation} \label{e22}
\widetilde{\phi}_{yy}=\widetilde{\phi}_{xx}+2\widetilde{\psi}_{xy},
\end{equation}
on integration along $QR$ gives
\begin{equation} \label{e23}
\widetilde{\phi}_y(x,y)=\widetilde{\phi}_{y}(x,y_P)
 +\int_{y_P}^y[\widetilde{\phi}_{xx}(x,\lambda)
 +2\widetilde{\psi}_{xy}(x,\lambda)]\,d\lambda.
\end{equation}
Using \eqref{e14} and \eqref{e20} the expression \eqref{e23}
takes the form
\begin{equation} \label{e24}
\begin{split}
\widetilde{\phi}_y(x,y)
&=\phi_y^P+\frac{1}{2}\int_{x_P}^x[\widetilde{\psi}_{xx}(\lambda,y_P)
 -\widetilde{\psi}_{yy}(\lambda,y_P)]d\lambda+(y-y_P)\widetilde{\phi}_{xx}(x,y_P)\\
&\quad +\frac{1}{2}\int_{y_P}^y\int_{y_P}^\mu[\widetilde{\psi}_{xxx}(x,\lambda)
 -\widetilde{\psi}_{xyy}(x,\lambda)]d\lambda\,\,d\mu
 +2 \int_{y_P}^y \widetilde{\psi}_{xy}(x,\lambda)d\lambda.
\end{split}
\end{equation}
Integrating along $QR$ we obtain from \eqref{e24} as follows.
\begin{equation} \label{e25}
\begin{split}
\widetilde{\phi}(x,y)
&= \widetilde{\phi}(x,y_P)+(y-y_P)\phi _y^P
 + \frac{1}{2}(y-y_P)^2 \widetilde{\phi}_{xx}(x,y_P)\\
&\quad +\frac{1}{2}(y-y_P)\int_{x_P}^x[\widetilde{\psi}_{xx}(\lambda,y_P)
 -\widetilde{\psi}_{yy}(\lambda,y_P)]d\lambda\\
&\quad  +\frac{1}{2}\int_{y_P}^y \int_{y_P}^\nu \int _{y_P}^\mu
 [\widetilde{\psi}_{xxx}(x,\lambda)-\widetilde{\psi}_{xyy}(x,\lambda)]
 d\lambda \,d\mu d\nu\\
&\quad + 2\int_{y_P}^y \int_{y_P}^\mu\widetilde{\psi}_{xy}(x,\lambda)
 d\lambda \,d\mu.
\end{split}
\end{equation}
Using \eqref{e16} and \eqref{e18} we finally obtain the following expression
for $\widetilde{\phi}(x,y)$ at an arbitrary point $(x,y)\in \bar{\Omega}$.
\begin{equation} \label{e26}
\begin{split}
&\widetilde{\phi}(x,y)\\
&=\phi^P+(x-x_P)\phi_x^P+(y-y_P)\phi_y^P
 +\frac{1}{2}[(x-x_P)^2+(y-y_P)^2]\phi_{yy}^P\\
&\quad -(y-y_P)^2\widetilde{\psi}_{xy}(x,y_P)
 +\frac{1}{2}(y-y_P)\int_{x_P}^x[\widetilde{\psi}_{xx}(\lambda,y_P)
 -\widetilde{\psi}_{yy}(\lambda,y_P)]\,d\lambda\\
&\quad + \frac{1}{4}(y-y_P)^2 \int_{x_P}^x[\widetilde{\psi}_{xxy}(\lambda,y_P)
 -\widetilde{\psi}_{yyy}(\lambda,y_P)]d\lambda \\
&\quad -2\int_{x_P}^x\int_{x_P}^\mu \widetilde{\psi}_{xy}(\lambda,y_P)\,d\lambda \,d\mu
 +2\int_{y_P} ^y \int_{y_P}^\mu \widetilde{\psi}_{xy}(x,\lambda)d\lambda \,d\mu \\
&\quad +\frac{1}{2}\int_{x_P}^x\int_{x_P}^\nu \int_{x_P}^\mu
 [\widetilde{\psi}_{xxy}(\lambda,y_P)-\widetilde{\psi}_{yyy}(\lambda,y_P)]
 d\lambda\,\,d\mu\,\,d\nu\\
&\quad +\frac{1}{2}\int_{y_P}^y\int_{y_P}^\nu\int_{y_P}^\mu
 [\widetilde{\psi}_{xxx}(x,\lambda)-\widetilde{\psi}_{xyy}(x,\lambda)]
 d\lambda\,\,d\mu\,d\nu.
\end{split}
\end{equation}
Obviously we have obtained an explicit representation for $\widetilde{\phi}$
in terms of the point conditions and $\widetilde{\psi}$, on the assumption
 that $(\widetilde{\phi},\widetilde{\psi})$ satisfies \eqref{e2} and
\eqref{e5}--\eqref{e6}.
Next we show that $(\widetilde{\phi},\widetilde{\psi})$ actually satisfies
the Bitsadze system \eqref{e2} and the conditions \eqref{e6}.

 From expression \eqref{e26} it is easy to verify that
$\widetilde{\phi}(x_P,y_P )=\phi^P$. We use \eqref{e17} in
\eqref{e21} to obtain
\begin{align*}
\widetilde{\phi}_x(x,y)
&=\phi_x ^P+\int_{x_P}^x[\phi_{yy}^P+\frac{1}{2}
 \int_{x_P}^\mu [ \widetilde{\psi}_{xxy}(\lambda,y_P)
 -\widetilde{\psi}_{yyy}(\lambda,y_P)] d\lambda] \,d\mu \\
&\quad  - 2\int_{x_P}^x\widetilde{\psi}_{xy}(\lambda,y_P)\,d\lambda
 +\frac{1}{2}\int _{y_P}^y[\widetilde{\psi}_{xx}(x,\lambda)
 -\widetilde{\psi}_{yy}(x,\lambda)]d\lambda,
\end{align*}
and it can be easily verified that $\widetilde{\phi}_x(x_P,y_P)=\phi_x^P$.
Similarly from \eqref{e14} and \eqref{e24} we have
\[
\widetilde{\phi}_y(x,y)=\phi_y ^P+\frac{1}{2}\int_{x_P}^x
 [\widetilde{\psi}_{xx}(\lambda,y_P)-\widetilde{\psi}_{yy}(\lambda,y_P)]
\,d\lambda+\int_{y_P}^y[\widetilde{\phi}_{xx}(x,\lambda)
 +2\widetilde{\psi}_{xy}(x,\lambda)]\,d\lambda,
\]
and it follows that $\widetilde{\phi}_y(x_P,y_P)=\phi_y ^P$. Again,
 from \eqref{e16}and \eqref{e20} we obtain
\begin{align*}
\widetilde{\phi}_{xx}(x,y)
&=\phi_{yy}^P+\frac{1}{2}\int_{x_P}^x[\widetilde{\psi}_{xxy}(\lambda,y_P)
 -\widetilde{\psi}_{yyy}(\lambda,y_P)]\,d\lambda
 -2\widetilde{\psi}_{xy}(x,y_P)\\
&\quad +\frac{1}{2}\int_{y_P}^y[\widetilde{\psi}_{xxx}(x,\lambda)
 -\widetilde{\psi}_{xyy}(x,\lambda)]\,d\lambda,
\end{align*}
which at $P$ yields
\begin{equation} \label{e27}
\widetilde{\phi}_{xx}(x_P,y_P)=\phi_{yy}^P-2\widetilde{\psi}_{xy}(x_P,y_P),
\end{equation}
and from \eqref{e11}(a) we obtain $\widetilde{\phi}_{xx}(x_P,y_P)=\phi_{xx}^P$.
Also from \eqref{e22} it is obvious that
\begin{equation} \label{e28}
\widetilde{\phi}_{yy}(x_P,y_P)
=\widetilde{\phi}_{xx}(x_P,y_P)+2\widetilde{\psi}_{xy}(x_P,y_P),
\end{equation}
and \eqref{e27}--\eqref{e28} yield $\widetilde{\phi}_{yy}(x_P,y_P)={\phi}_{yy}^P$.

 Now we verify that $\widetilde{\phi}(x,y)$ satisfies \eqref{e2}(a).
Using \eqref{e14} in \eqref{e24} and then differentiating with respect
to $x$ we obtain
\begin{align*}
&\widetilde{\phi}_{xy}(x,y)
=\frac{1}{2}[\widetilde{\psi}_{xx}(x,y_P)-\widetilde{\psi}_{yy}
(x,y_P)]+\frac{1}{2}(y-y_P)[\widetilde{\psi}_{xxy}(x,y_P)
 -\widetilde{\psi}_{yyy}(x,y_P)]\\
&\quad -2(y-y_P)\widetilde{\psi}_{xxy}(x,y_P)
 + \frac{1}{2}\int_{y_p} ^y \int_{y_P}^\mu[\widetilde{\psi}_{xxxx}(x,\lambda)
 - \widetilde{\psi}_{xxyy}(x,\lambda)] \,d\lambda \,d\mu \\
&\quad +2\widetilde{\psi}_{xx}(x,y)-2\widetilde{\psi}_{xx}(x,y_P),
\end{align*}
which, since $\Delta^2\widetilde{\psi}=0$, can be simplified as
\begin{equation} \label{e29}
\begin{split}
&\widetilde{\phi}_{xy}(x,y)\\
&=- \frac{1}{2}[3\widetilde{\psi}_{xx}(x,y_P)+\widetilde{\psi}_{yy}(x,y_P)]
 -\frac{1}{2}(y-y_P)[3\widetilde{\psi}_{xxy}(x,y_P)
 +\widetilde{\psi} _{yyy}(x,y_P)]\\
&\quad  - \frac{1}{2}[3\widetilde{\psi}_{xx}(x,y_P)+\widetilde{\psi}_{yy}(x,y)]
 + \frac{1}{2}[3\widetilde{\psi}_{xx}(x,y_P)+\widetilde{\psi}_{yy}(x,y_P)]\\
&\quad +\frac{1}{2}(y-y_P)[3\widetilde{\psi}_{xxy}(x,y_P)
 +\widetilde{\psi} _{yyy}(x,y_P)]+2\widetilde{\psi}_{xx}(x,y),
\end{split}
\end{equation}
and we obtain
\begin{equation} \label{e30}
\begin{split}
\widetilde{\phi}_{xy}(x,y)=\frac{1}{2}[\widetilde{\psi}_{xx}(x,y)-\widetilde{\psi}_{yy}(x,y)].
\end{split}
\end{equation}
Then, to verify that $\widetilde{\phi}(x,y)$ satisfies
\eqref{e2}(b), we use \eqref{e26} to obtain
\begin{align*}
&\widetilde{\phi}_{xx}(x,y)-\widetilde{\phi}_{yy}(x,y)\\
&=-(y-y_P)^2\widetilde{\psi}_{xxxy}(x,y_P)
 +\frac{1}{2}(y-y_P)[\widetilde{\psi}_{xxx}(x,y_P)-\widetilde{\psi}_{xyy}(x,y_P)]
\\
&\quad +\frac{1}{4}(y-y_P)^2[\widetilde{\psi}_{xxxy}(x,y_P)
 -\widetilde{\psi}_{xyyy}(x,y_P)]\\
&\quad +2\int_{y_P}^y \int_{y_P}^\mu\widetilde{\psi}_{xxxy}(x,\lambda)d\lambda \,d\mu
 +\frac{1}{2}\int_{x_P}^x[\widetilde{\psi}_{xxy}(\lambda,y_P)
 -\widetilde{\psi}_{yyy}(\lambda,y_P)]\,d\lambda\\
&\quad +\frac{1}{2}\int_{y_P}^y\int_{y_P}^\nu\int_{y_P}^\mu
 [\widetilde{\psi}_{xxxxx}(x,\lambda)-\widetilde{\psi}_{xxxyy}(x,\lambda)]
 \,d\lambda \,d\mu \,d\nu \\
&\quad -\frac{1}{2}\int_{x_P}^x[\widetilde{\psi}_{xxy}(\lambda,y_P)
 -\widetilde{\psi}_{yyy}(\lambda,y_P)]d\lambda-2\widetilde{\psi}_{xy}(x,y)\\
&\quad -\frac{1}{2}\int_{y_P}^y[\widetilde{\psi}_{xxx}(x,\lambda)
 -\widetilde{\psi}_{xyy}(x,\lambda)]\,d\lambda,
\end{align*}
which can further be simplified to obtain
\begin{align*}
&\widetilde{\phi}_{xx}(x,y)-\widetilde{\phi}_{yy}(x,y)\\
&=-\frac{1}{4}(y-y_P)^2[3\widetilde{\psi}_{xxxy}(x,y_P)
 +\widetilde{\psi}_{xyyy}(x,y_P)]\\
&\quad -\frac{1}{2}(y-y_P)[3\widetilde{\psi}_{xxx}(x,y_P)
 +\widetilde{\psi}_{xyy}(x,y_P)]\\
&\quad -\frac{1}{2}\int_{y_P}^y[3\widetilde{\psi}_{xxx}(x,\lambda)
 +\widetilde{\psi}_{xyy}(x,\lambda)]\,d\lambda
 +\frac{1}{2}(y-y_P)[3\widetilde{\psi}_{xxx}(x,y_P)
 +\widetilde{\psi}_{xyy}(x,y_P)]\\
&\quad +\frac{1}{4}(y-y_P)^2[3\widetilde{\psi}_{xxxy}(x,y_P)
 +\widetilde{\psi}_{xyyy}(x,y_P)]\\
&\quad -2\widetilde{\psi}_{xy}(x,y)
 +\frac{1}{2}\int_{y_P}^y[3\widetilde{\psi}_{xxx}(x,\lambda)
 +\widetilde{\psi}_{xyy}(x,\lambda)]d\lambda,
\end{align*}
and finally we have
\[
\widetilde{\phi}_{xx}(x,y)-\widetilde{\phi}_{yy}(x,y)
=-2 \widetilde{\psi}_{xy}(x,y),
\]
which completes the proof.
\end{proof}

\subsection*{Conclusion}
It has been proved by construction that there exists a unique solution
 $(\widetilde{\phi},\widetilde{\psi})$ in $C^4(\Omega)\times C^4(\Omega)$
to the Stokes-Bitsadze system \eqref{e2} subject to the boundary
conditions \eqref{e5} along with additional single point conditions \eqref{e6}.


\subsection*{Acknowledgements}
The author is grateful to Professor A. Russell Davies,
 Head School of Mathematics, Cardiff University, United Kingdom
for his useful suggestions.

\begin{thebibliography}{00}

\bibitem{b1} A. H. Babayan;
\emph{A boundary value problem for Bitsadze equation in the unit disc},
J. Contemp. Math. Anal. \textbf{42}(4) (2007) 177-183.

\bibitem{b2} A. V. Bitsadze;
\emph{Some classes of partial differential equations},
 Gordon and Breach Science Publishers, New York, 1988.

\bibitem{b3} A. V. Bitsadze;
\emph{On the uniqueness of the solution of the Dirichlet problem
for the elliptic partial differential operators}, Uspekhi Mat. Nauk. \textbf{3}(6)
 (1948) 211-212.

\bibitem{c1} C. J. Coleman;
\emph{A contour integral formulation of plane creeping Newtonian flow},
Q. J. Mech. appl. Math. \textbf{XXXIV} (1981) 453-464.

\bibitem{d1} A. R. Davies, J. Devlin;
\emph{On corner flows of Oldroyd-B fluids},
J. Non-Newtonian Fluid Mech. \textbf{50} (1993) 173-191.

\bibitem{h1} S. Hizliyel, M. Cagliyan;
\emph{A boundary value problem for Bitsadze equation in matrix form}.
Turkish J. Math. \textbf{35}(1) (2011) 29-46.

\bibitem{k1} E. N. Kuzmin;
\emph{On the Dirichlet problem for elliptic systems in space.
 Differential Equations}, \textbf{3}(1) (1967) 78-79.

\bibitem{o1} B. B. Oshorov;
\emph{On boundary value problems for the Cauchy-Riemann and Bitsadze
 systems of equations}, Doklady Mathematics \textbf{73}(2) (2006) 241-244.

\bibitem{o2} R. G. Owens, T. N. Phillips;
\emph{Mass and momentum conserving spectral methods for Stokes flow},
J. Comput. Appl. Math. \textbf{53} (1994) 185-206.

\bibitem{t1} M. Tahir;
\emph{The Stokes-Bitsadze system}, Punjab Univ. J. Math.
 \textbf{XXXII} (1999) 173-180.

\bibitem{t2} A. N. Tikhonov, A. A. Samarskii;
\emph{Equations of Mathematical Physics}. Pergamon Press Ltd. Oxford, 1963.

\bibitem{v1} T. Vaitekhovich;
\emph{Boundary value problems to second order complex partial differential
equations in a ring domain}, Siauliai Math. Semin. \textbf{2} (10) (2007) 117-146.

\end{thebibliography}
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