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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 72, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/72\hfil 
Coefficients of degenerate elliptic equations]
{An approach for constructing coefficients
of degenerate elliptic complex equations}

\author[G.-C. Wen \hfil EJDE-2013/72\hfilneg]
{Guo Chun Wen}  % in alphabetical order

\address{Guo Chun Wen \newline
LMAM, School of Mathematical Sciences, Peking University, Beijing
100871, China}
\email{Wengc@math.pku.edu.cn}

\thanks{Submitted November 30, 2012. Published March 17, 2013.}
\subjclass[2000]{35J55, 35R30, 47G10}
\keywords{Degenerate elliptic complex equations;
  coefficients of equations; \hfill\break\indent 
  method of integral equations;  H\"older continuity of a singular integral}

\begin{abstract}
 This article deals with the inverse problem for degenerate
 elliptic systems of first order equations with Riemann-Hilbert
 type map in simply connected domains.
 Firstly the formulation and the complex form of the
 problem for the first-order elliptic systems with the degenerate
 rank 0 are given, and then the coefficients of the  systems are
 constructed by a new complex analytic method. Here we verify and
 apply the H\"older continuity of a singular integral operator.
\end{abstract}

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

\section{Formulation of the inverse problem for
degenerate elliptic complex equations of first order}

In \cite{b1,i1,k1,s1,t1,t2,w7,w8},  the authors  discussed the inverse
problem of second-order elliptic equations without degeneracy.
In this article, by using the methods of integral equations and
complex analysis, the existence of solutions of the inverse problem
for degenerate elliptic complex equations of first order with
Riemann-Hilbert type map is discussed.

Let $D(\supset\{0\})$ be a simply connected bounded domain in
the complex plane $\mathbb{C}$ with the boundary $\partial D=\Gamma\in
C^1_\mu(0<\mu<1)$. There is no harm in assuming that the domain $D$
is $\{|z|<1\}$ with  boundary $\Gamma=\{|z|=1\}$. Consider the
linear elliptic systems of first-order equations with degenerate
rank 0,
\begin{equation}
\begin{gathered}
H_1(y)u_{x}-H_2(y)v_{y}=au+bv  \quad \text{in }D\\
H_1(y)v_{x}+H_2(y)u_{y}=cu+dv \quad \text{in }D,
\end{gathered}\label{e1.1}
\end{equation}
in which $H_j(y)=|y|^{m_j/2}h_j(y),h_j(y)$ $(j = 1,2)$ are positive continuous
functions in $\overline D$, $m_j\,(j = 1,2,m_2 < \min(1,m_1))$ are
positive constants, and $a,b,c,d\,(j=1,2)$ are functions of
$x+iy\,(\in D)$ satisfying the conditions $a,b,c,d\in L_\infty(D)$,
which is called  Condition $C$. In this article, the notation
is  the same as in references \cite{v1,w1,w2,w3,w4,w5,w6,w7,w8}.
 The following degenerate
elliptic system is a special case of system \eqref{e1.1} with
$H_j(y)=|y|^{m_j/2}$ $(j=1,2)$:
\begin{equation}
\begin{gathered}
|y|^{m_1/2}u_{x}-|y|^{m_2/2}v_{y}=au+bv \quad \text{in }D,\\
|y|^{m_1/2}v_{x}+|y|^{m_2/2}u_{y}=cu+dv \quad \text{in }D,
\end{gathered} \label{e1.2}
\end{equation}
For convenience, we mainly discuss equation \eqref{e1.2}, and
equation \eqref{e1.1} can be similarly discussed.
From the elliptic condition in \eqref{e1.2}
(see \cite[(1.3), Chpater II]{w5}), namely
$$
J=4K_1K_4-(K_2+K_3)^2=4H^2(y)=4[H_1(y)/H_2(y)]^2>0\quad\text{in }\overline D\backslash\gamma
$$
and $J=0$ on $\gamma=\{-1<x<1,y=0\}$, hence system \eqref{e1.1} or \eqref{e1.2}
is elliptic system of first-order equations in $D$ with the
parabolic degenerate line $\gamma=(-1,1)$ on the $x$-axis in
$x + i y$-plane. Setting $Y=G(y)=\int^{y}_0H(t)dt$,
$Z=x+iY$ in $\overline{D}$,
if $H(y)=|y|^{m/2}h_1(y)/h_2(y),\,m=m_1 - m_2,\,Y = \int^{y}_0H(t)dt \le
|s_0y|^{(m+2)/2}$, where $s_0$ is a positive constant, thus we have
$s_0|y|\ge|Y|^{2/(m+2)}$. Denote
\begin{equation}
\begin{gathered}
 W(z)=u+iv,\\
\begin{aligned}
 W_{\overline{\tilde z}}&=\frac12[H_1(y)W_x+iH_2(y)W_y]
=\frac{H_1(y)}2[W_x + iW_Y] \\
&= H_1(y)W_{x-iY}
 = H_1(y)W_{\overline Z},
\end{aligned}
\end{gathered}\label{e1.3}
\end{equation}
where $dY = H(y)dy = H_1(y)dy/H_2(y),\,H_2u_y = H_1u_Y$, then the
system \eqref{e1.1} can be written in the complex form
\begin{equation}
\begin{gathered}
W_{\overline{\tilde z}}=H_1(y)W_{\overline Z}=A(z)W+B(z)\overline{W}
\quad\text{in } D,\\
A = \frac14[a+ic-ib + d],\quad
B = \frac14[a+ic+ib - d],
\end{gathered}\label{e1.4}
\end{equation}
in which $D_Z$ is the image domain of $D$
with respect to the mapping $Z=Z(z) =x+iY=x+iG(y)$ in $D$, and
denoted by $D$ again for simply, and $z=z(Z)$ is the inverse
function of $Z=Z(z)$. For convenience we only discuss the complex
equation \eqref{e1.4} about the number $Z$ replaced by $z$ in Sections 1
and 2 later on.

Introduce the Riemann-Hilbert boundary conditions for the equation
\eqref{e1.4} as follows:
\begin{equation}
\begin{gathered}
\operatorname{Re}[\overline{\lambda(z)}W(z)]=r(z)+f(z)=f_1(z),\quad z\in \Gamma, \\
\operatorname{Im}[\overline{\lambda(a_{j})}W(a_j)]=b_{j},\quad j=1,\dots ,2K+1,\; K\ge0,
\end{gathered}\label{e1.5}
\end{equation}
where
$$
f(z)=\begin{cases}
0, & K\ge0, \\
g_0+\operatorname{Re}\sum^{-K-1}_{m=1}(g^{+}_{m}+ig^{-}_{m})z^{m},
&K<0,
\end{cases}
$$
in which $\lambda(z)$ $(\ne0)$, $r(z)\in C_\alpha(L)$, $\alpha(0<\alpha<1)$ is a positive
constant, $g_0$, $g^{\pm}_{m}$ $(m=1,\dots ,-K - 1,K<0)$ are unknown real
constants to be determined appropriately,
$a_{j}(\in \Gamma=\{|z|=1\},\,j=1,\dots ,2K+1,\,K\ge0)$ are distinct points, and
$b_{j}(j=1,\dots ,2K+1)$ are all real constants, in which
$K=\frac1{2\pi}\Delta_\Gamma\arg\lambda(z)$ is called the index of $\lambda(z)$ on
$\Gamma$. The above Riemann-Hilbert boundary value problem is called
Problem $RH$ for equation \eqref{e1.4}. Under Condition $C$, the
solution $W(z)$ of Problem $RH$ in $D$ can be found.
From \cite[(5.114) and (5.115), Chapter VI]{v1},
 we see that Problem $RH$ of equation
\eqref{e1.4} possesses the important application to the shell and
elasticity.

It is clear that the above solution $W(z)$ satisfies the following
Riemann-Hilbert type boundary condition for the equation \eqref{e1.4}:
\begin{equation}
\operatorname{Im}[\overline{\lambda(z)}W(z)]=f_2(z)\quad\text{on }\Gamma,\label{e1.6}
\end{equation}
and then the boundary condition of Riemann-Hilbert to
Riemann-Hilbert type map can be written as follows
\begin{equation}
\begin{gathered}
\overline{\lambda(z)}W(z) = f_1(z)+if_2(z)\quad\text{on $\Gamma$, i.e.} \\
W(z)=h(z)=[f_1(z)+if_2(z)]/\overline{\lambda(z)}\quad\text{on }\Gamma,
\end{gathered}\label{e1.7}
\end{equation}
which will be called Problem $RR$ for the complex equation \eqref{e1.4}
(or \eqref{e1.1}), where $h(z)\in C_\alpha(\Gamma)$ is a complex function. Thus we
can define the Riemann-Hilbert to Riemann-Hilbert type map
$\Lambda :C_{\alpha}(\Gamma){\to}C_\alpha(\Gamma)$, i.e.
$f_1(z){\to}f_2(z)$ by $\Lambda{f_1}=f_2$

Our inverse problem is to determine the coefficient $a,b,c,d$
of equation \eqref{e1.1} (or $A(z), B(z)$ in \eqref{e1.4}) from the map
$\Lambda$.
Obviously the function $f_1(z)+if_2(z)$  corresponds to the
function $h(z)$ one by one. Denote by $R_h$ the set of $\{h(z)\}$.
It is clear that for any function $f_1(z)$ of the set $C_\alpha(\Gamma)$
in the Riemann-Hilbert boundary condition \eqref{e1.5}, there is a set
$\{f_2(z)\}$ of the functions of Riemann-Hilbert type boundary
condition \eqref{e1.6}, where $R_h=\{h(z)\}$ is corresponding to the
complex equation \eqref{e1.4}. Inversely from the set $R_h=\{h(z)\}$, one
complex equation in \eqref{e1.4} can be determined, which will be verified
later on.

In Section 3, we prove Theorems \ref{thm3.1} and \ref{thm3.2}, 
which are important results in the present paper. In fact we first assume that the
coefficients $A=B=0$, $H=H(y)$ of the complex equation \eqref{e1.4} in the
$\varepsilon$-neighborhood $D_\varepsilon=D\cap\{|\operatorname{Im} z|<\varepsilon\}$ of $D\cap\{\operatorname{Im}
z=0\}$, note that the above coefficients $A(z),\,B(z)$ weakly
converge to $A(z),B(z)$ in $D$ as $\varepsilon\to0$, and on the basis of
Theorem \ref{thm3.1} below, we see the H\"older continuity of solution
$W(Z)$ and $TW_{\overline Z}=T[AW+B\overline W]/H_1$ of the complex equation
\eqref{e1.4} with above coefficients and $TW_{\overline Z}=T[AW+B\overline W]/H_1$ 
(see \cite{v1,w3,w5},
 hence from $\{W(z)\}$ and $TW_{\overline Z}$, we can choose
the subsequences, which uniformly converges the H\"older
continuous functions in $\overline D$ respectively. From this, we can also
obtain the corresponding Pompeiu and Plemelj-Sokhotzki formulas
about $W(z)$ in $\overline D$.

\section{Existence of solutions of the inverse problem for
degenerate elliptic complex equations of first order}

We introduce a singular integral operator
$$
\tilde Tf(z)=T\Big(\frac
f{H_1}\big)=-\frac1\pi\int \int_{D}\frac{f(\zeta)/H_1(y)}{\zeta-Z}d\sigma_\zeta,
$$
where $|y|^\tau f(z)\in L_\infty(D)$ with
$\tau=\max(1-m_1/2,0),\,m_1$ is a positive constant,
$H_1(y)$ is as
stated in \eqref{e1.1}. Suppose that $f(z)=0$ in ${\mathbb{C}}\backslash\overline D$. Then
$|y|^\tau f(z)\in L_\infty({\mathbb{C}})$, from Theorem \ref{thm3.1} below, it
follows $(\tilde Tf)_{\bar z}=f(z)/H_1$ in $\mathbb{C}$.
We consider the first-order complex equation with singular coefficients
\begin{equation}
\begin{gathered}
H_1W_{\overline{z}}-A(z)W-B(z)\overline W=0,\quad \text{i.e.}, \\
H_1(y)[g(z)]_{\overline{z}}-A(z)g(z)-B(z)\overline{g(z)}=0\quad\text{in }\mathbb{C},
\end{gathered} \label{e2.1}
\end{equation}
where $G_1(y)=\int_0^{y}H_1(y)dy,\;g(z)=W(z)$. Applying the Pompeiu
formula (see \cite[Chapters I and III]{v1}), the corresponding integral
equation of the complex equation \eqref{e2.1} is as follows
\begin{equation}
g(z) - T[(Ag + B\overline{g})/H_1] = \frac1{2\pi i}\int_\Gamma \frac{g(\zeta)}{\zeta - z}
d\zeta\quad\text{in }D.\label{e2.2}
\end{equation}
For simplicity we can consider only the
 integral equation
$$
g(z)-T[(Ag+B\overline{g})/H_1]=1
$$
 or $i$ in $D$ later on.
 On the basis of Theorem \ref{thm3.1} below, we know that the
integral in \eqref{e2.2} is a completely continuous operator, hence by
using the similar method as in \cite[Sec. 5, Chapter III]{v1} and the
proof of \cite[Lemma 2.2]{w7}, we can verify that the above integral
equation has a unique solution.

We first prove the following lemma (see \cite{t2}).

\begin{lemma} \label{lem2.1}
The function
$g(z)=h_j(z)$ $(h_j(z),j=1,2)$ are a solutions of the integral
equations
\begin{equation}
\begin{gathered}
g(z)-T(A/H_1)g-T(B/H_1)\overline g
=\begin{cases}
1\\
i \end{cases}\quad \text{in } \overline D,\\
g(z)=\begin{cases}  h_1(z)\\
h_2(z) \end{cases}\quad \text{on }\Gamma,
\end{gathered}\label{e2.3}
\end{equation}
if and only if it is a solution of
the integral equation
\begin{equation}
\begin{gathered}
\frac12g(z)+\frac1{2\pi
i}\int_\Gamma\frac{g(\zeta)}{\zeta-z}d\zeta
=\begin{cases} 1,\\
i,
\end{cases}
\quad  g(\zeta)=\begin{cases} h_1(\zeta),\\
h_2(\zeta), \end{cases}
\quad\text{i.e.,}\\
\frac{h_1(z)}2 + \frac1{2\pi i} \int_\Gamma  \frac{h_1(\zeta)}{\zeta-z}d\zeta
 = 1,\quad
\frac{h_2(z)}2 + \frac1{2\pi
i} \int_\Gamma  \frac{h_2(\zeta)}
{\zeta-z}d\zeta = i\quad\text{on }\Gamma
\end{gathered} \label{e2.4}
\end{equation}
respectively.
\end{lemma}

\begin{proof} It is clear that we need to
discuss only the case of $h_1$.
If $g(z)$ is a solution of the first
integral equation in \eqref{e2.3}, then $g_{\overline z}=Ag/H_1+B\overline{g}/H_1$. On
the basis of the Pompeiu formula
\begin{equation}
g(z) = \frac1{2\pi i}\int_\Gamma \frac{g(\zeta)}{\zeta - z}
d\zeta + T[g(\zeta)]_{\overline\zeta} = \frac1{2\pi
i}\int_\Gamma \frac{g(\zeta)}{\zeta - z}d\zeta + T[Ag/H_1+B\overline{g}/H_1]
\label{e2.5}
\end{equation}
in $D$ (see \cite[Chapters I and III]{v1}), we have
\begin{equation}
g(z,k)-TAg/H_1-TB\overline g/H_1=1=\frac1{2\pi i}\int_\Gamma\frac{g(\zeta)}{\zeta-z}
d\zeta\quad\text{in }D,\label{e2.6}
\end{equation}
where $g(\zeta)=h_1(\zeta)$ on $\Gamma$.
Moreover by using the Plemelj-Sokhotzki formula for Cauchy type
integral (see \cite{m1,w1})
\[
1=\frac1{2\pi i}\int_\Gamma\frac{g(\zeta)}{\zeta-z}d\zeta
+\frac12g(z),\quad g(\zeta)=h_1(\zeta) \quad\text{on }Ga,
\]
this is the first formula in \eqref{e2.4}.

Conversely if the first integral equation in \eqref{e2.4} is true, then by
the conditions in Section 1, there exists a solution of equation
$g_{\overline z}=Ag/H_1+B\overline{g}/H_1$ in $\overline D$ with the boundary values
$g(\zeta)=h_1(\zeta)$ on $\Gamma$, thus we have \eqref{e2.5}, where the
integral $\frac1{2\pi i}\int_\Gamma\frac{g(\zeta)}{\zeta-z}d\zeta$ in
$D$ is analytic, whose boundary value on $\Gamma$ is
$$
\lim_{z'(\in D)\to z(\in\Gamma)}\frac1{2\pi i}
\int_\Gamma \frac{g(\zeta)} {\zeta-z'}d\zeta 
= \frac12g(z) + \frac1{2\pi i}\int_\Gamma \frac{g(\zeta)}{\zeta-z}d\zeta = 1,
$$
hence
$$
\frac1{2\pi i}\int_\Gamma\frac{g(\zeta)}{\zeta-z}
d\zeta=1\quad\text{in }D,
$$ and the first formula in \eqref{e2.3} is true.
\end{proof}

\begin{theorem} \label{thm2.2}
Under the above conditions, the functions $h_1(z),\,h_2(z)$ as
stated in Section $1$ are the solutions of the system of integral equations
\begin{equation}
\begin{gathered}
\frac{h_1}2+Sh_1=1,\quad \frac{h_2}2+Sh_2=i,\\
 Sh_1 = \frac1{2\pi i}\int_\Gamma\frac{h_1(\zeta)}{\zeta-t}d\zeta,\quad
 Sh_2 = \frac1{2\pi i}\int_\Gamma\frac{h_2(\zeta)}{\zeta-t}d\zeta.
\end{gathered} \label{e2.7}
\end{equation}
\end{theorem}

\begin{proof}
 From the theory of integral equations (see \cite{m1,t1,w7}),
 we can derive the solutions $h_1$ and $h_2$ of \eqref{e2.7}.
In fact, on the basis of Lemma \ref{lem2.1}, we can find the
solutions of the  integral equations
\begin{gather*}
W_1(z)= 1+T[(AW_1+B{\overline W_1})/H_1] \quad\text{in }D,\\
W_2(z)= i+T[(AW_2+B\overline{ W_2})/H_1] \quad\text{in }D.
\end{gather*}
 By using the Pompeiu formula, the above equations can be rewritten as
\begin{gather*}
W_1(z)=  \frac1{2\pi i}\int_L\frac{W_1(t)}{t-z}dt
-\frac1\pi\int  \int_{D}\frac{AW_1(\zeta)+B\overline{W_1(\zeta)}}{(\zeta-z)H_1}
d\sigma_\zeta\quad\text{in }D, \\
W_2(z)= \frac1{2\pi i}\int_L\frac{W_2(t)}{t-z}dt
-\frac1\pi\int  \int_{D}\frac{AW_2(\zeta)+B\overline{
W_2(\zeta)}}{(\zeta-z)H_1}d\sigma_\zeta \quad\text{in }D
\end{gather*}
 and
$W_1(z)=h_1(z)$ and $W_2(z)=h_2(z)$ on $\Gamma$. Because the functions
$\frac1{2\pi i}\int_L\frac{h_j(\zeta)}{\zeta-z}dt$ $(j=1,2)$ are
analytic in $D'=\mathbb{C}\backslash\overline D$ (see \cite{t1}), we can analytically extend
$h_j(z)\,(j=1,2)$ to the domain $D'$; i.e., define
\begin{equation}
\begin{gathered}
w_1(z)= 1-\frac1{2\pi i} \int_\Gamma\frac{h_1(\zeta)}{\zeta-z}d\zeta
\quad z\in\mathbb{C}\backslash\overline D  ,\\
w_2(z)=i-\frac1{2\pi i}\int_\Gamma\frac{h_2(\zeta)}{\zeta-z}d\zeta,
\quad z\in\mathbb{C}\backslash\overline D,
\end{gathered} \label{e2.8}
\end{equation}
which are analytic in $D'$ with the boundary values
$h_1(z),\,h_2(z)$ on $\Gamma$ respectively. According to the
Plemelj-Sokhotzki formula for Cauchy type integrals, we immediately
obtain the formulas
\begin{gather*}
 h_1(t) = 1 - \lim_{z(\in D')\to t(\in\Gamma)} \frac1{2\pi i}\int_\Gamma
\frac{h_1(\zeta)}{\zeta - z}d\zeta=1+\frac12
h_1(t)-Sh_1  \quad z\in\mathbb{C}\backslash\overline D,\\
 h_2(t) = i - \lim_{z(\in D')\to t(\in\Gamma)}\frac1{2\pi
i}\int_\Gamma\frac{h_2(\zeta)}{\zeta - z}d\zeta=i+\frac12
h_1(t)-Sh_2 \quad z\in\mathbb{C}\backslash\overline D.
\end{gather*}
This is just the formula \eqref{e2.7} with
$h_j(t),\,j=1,2$.
\end{proof}

\begin{theorem} \label{thm2.3}
For the inverse problem of Problem $RR$ for equation \eqref{e1.1}
with Condition $C$, we can reconstruct the coefficients
$a(z),b(z),c(z)$ and $d(z)$.
\end{theorem}

\begin{proof}
We shall find two solutions
$\phi_1(z)=W_1(z)$ and $i\phi_2(z)=W_2(z)$ of complex equation
\begin{equation}
[\phi]_{\bar z}-A/H_1\phi-B\overline{\phi}/H_1=0\quad\text{in }\mathbb{C}\label{e2.9}
\end{equation}
with the conditions
$\phi_1(z)\to 1$ and $i\phi_2(z)\to i$ as $z\to\infty$. In fact the
above solutions $F(z)=\phi_1(z),\,G(z)=i\phi_2(z)$ are also the
solutions of integral equations
\begin{equation}
 \begin{gathered}
F(z)-T[(AF+B\overline{F})/H_1\}=1 \quad\text{in } \mathbb{C}, \\
G(z)-T[(AG+B\overline{G})/H_1\}=i \quad\text{in } \mathbb{C}.
\end{gathered}\label{e2.10}
\end{equation}
As stated in Lemma  \ref{lem2.1} and Theorem \ref{thm2.2}, we can require that the above
solutions satisfy the boundary conditions
$$
F(z)=h_1(z),\,G(z)=h_2(z)\quad\text{on }\Gamma,
$$
where $h_1(z),\,h_2(z)\in R_h$.

Noting that $F(z),\,G(z)$ satisfy the complex equations
\begin{equation}
 \begin{gathered}
F_{\bar z}-\{(AF+B\overline{F})/H_1\}=0 \quad\text{in } \mathbb{C}, \\
G_{\bar z}-\{(AG+B\overline{G})/H_1\}=0 \quad\text{in } \mathbb{C}.
\end{gathered} \label{e2.11}
\end{equation}
Moreover, on the basis of Lemma  \ref{lem2.4} below, we have
\begin{equation}
\operatorname{Im}[F(z)\overline{G(z)}]=[F(z)\overline{G(z)}-\overline{F(z)}G(z)]/2i\ne0
\quad\text{in }D.
\label{e2.12}
\end{equation}
Thus from \eqref{e2.11}, the coefficients $A/H_1$ and $B/H_1$ can be determined
as follows
\begin{gather*}
A/H_1=\frac{F_{\bar z}\overline G-G_{\bar z}\overline F}{F\overline G-\overline F G},\quad
B/H_1=-\frac{F_{\bar z}G-G_{\bar z}F}{F\overline G-\overline F G}\quad\text{in $D$; i.e.,}\\
A=H_1\frac{F_{\bar z}\overline G-G_{\bar z}\overline F}{F\overline G-\overline F G},\quad
B=-H_1\frac{F_{\bar z}G-G_{\bar z}F}{F\overline G-\overline F G}\quad\text{in } D.
\end{gather*}
From the above formulas, the coefficients $a(z)$ and $b(z)$
of the system \eqref{e1.1} are obtained; i.e.,
$$
a(z)+ic(z)=2[A(z)+B(z)],\,d(z)-ib(z)=2[A(z)-B(z)]\quad\text{in }D.
$$
\end{proof}

\begin{lemma} \label{lem2.4}
For the solution $[F(z),G(z)]$ of the system  \eqref{e2.11},
we can get the inequality \eqref{e2.12}.
\end{lemma}

\begin{proof}
Suppose that \eqref{e2.12} is not true,
then there exists a point $z_0\in D$ such that
$\operatorname{Im}[\overline{F(z_0)}G(z_0)]=0$; i.e.,
$$
\begin{vmatrix} \operatorname{Re} F(z_0)& \operatorname{Im} F(z_0)\\
 \operatorname{Re} G(z_0)& \operatorname{Im} G(z_0)
\end{vmatrix}
=0.
$$
Thus we have two real constants $c_1,c_2$, which are not both equal
to 0, such that $c_1F(z_0)+c_2G(z_0)=0$.
Next, we prove that the equality of
$c_1F(z_0)+c_2G(z_0)=0$ can not be true. If
$W(z_0)=c_1F(z_0)+c_2G(z_0)=0$, then
$W(z)=\Phi(z)e^{\phi(z)}=(z-z_0)\Phi_0(z)e^{\phi(z)}$, where
$\Phi(z),\Phi_0(z)$ are analytic functions in $D$, and
\begin{align*}
&(z - z_0)\Phi_0(z)e^{\phi(z)} + \frac1\pi\int \int_{D}\frac{(\zeta - z_0)
\Phi_0(\zeta)e^{\phi(\zeta)}[A/H_1
 + B\overline{W(\zeta)}/H_1W(\zeta)]}{\zeta-z}d\sigma_\zeta \\
&=c_1+c_2i.
\end{align*}
 Letting $z\to z_0$, we have
$$
\frac1\pi\int \int_{D}\Phi_0(\zeta)e^{\phi(\zeta)}
[A/H_1+B\overline{W(\zeta)}/H_1W(\zeta)]d\sigma_\zeta
=c_1 +c_2i,
$$
and then
\begin{align*}
&c_1 + c_2i \\
&= (z - z_0)\Phi_0(z)e^{\phi(z)} \\
&\quad +\frac1\pi\int  \int_{D}\frac{(\zeta - z +
z - z_0)\Phi_0(\zeta)e^{\phi(\zeta)}[A/H_1
 + B\overline{W(\zeta)}/H_1W(\zeta)]}{\zeta-z}d\sigma_\zeta \\
&=(z-z_0)\{\Phi_0(z)e^{\phi(z)}
+\frac1\pi\int \int_{D}\frac{\Phi_0(\zeta)e^{\phi(\zeta)}[A/H_1+
B\overline{W(\zeta)}/H_1W(\zeta)]}
{\zeta-z}d\sigma_\zeta\} \\
&\quad +\frac1\pi\int \int_{D}\Phi_0(\zeta)e^{\phi(\zeta)}
 [A/H_1+B\overline{W(\zeta)}/H_1W(\zeta)]d\sigma_\zeta.
\end{align*}
The above equality implies
$$
\Phi_0(z)e^{\phi(z)}+\frac1\pi\int \int_{D}\frac{\Phi_0(\zeta)
 e^{\phi(\zeta)}[A/H_1+B\overline{W(\zeta)}
/H_1W(\zeta)]}{\zeta-z}d\sigma_\zeta=0\quad\text{in }D,
$$
and the above homogeneous integral equation only have the trivial solution,
namely $\Phi_0(z)=0\quad\text{in }D$, thus
$W(z)=\Phi(z)e^{\phi(z)}=(z-z_0)\Phi_0(z)e^{\phi(z)}\equiv0$ in $D$.
This is impossible.

In addition, by using another way, we can prove that the equality
$c_1F(z_0)+c_2G(z_0)=0$ can not be true. According to the method  in
\cite[Section 5, Chapter III]{v1}, we know that the integral equations
$$
W(z)-T[AW/H_1+B\overline{W}/H_1]=\begin{cases}
c_1+c_2i &\text{in }\overline D,\\
c_1+c_2i &\text{in }{\mathbb{C}},
\end{cases}
$$
 have the unique solutions
$W(z)=c_1F(z)+c_2G(z)$ in $\overline D$ and $\mathbb{C}$ respectively, where
$A,B\in L_p(\overline D)$ and $A=B=0$ in ${\mathbb{C}}\backslash\overline D$, this
shows that the function $W(z)$ in $\overline D$ can be continuously
extended in $\mathbb{C}$. Moreover according to the method in 
\cite{v1,w5}, the
solution $W(z)$ can be expressed as
$W(z)=\Phi(z)e^{T[A/H_1+B\overline W/H_1W]}$ in $\mathbb{C}$.
Note that $T[A/H_1+B\overline W/H_1W]\to0$ as
$z\to\infty$, and the entire function $\Phi(z)$ in $\mathbb{C}$ satisfies
the condition $\Phi(z)\to c_1+c_2i$ as $z\to\infty$, hence
$\Phi(z)=c_1+c_2i$ in $\mathbb{C}$, thus
$W(z)=(c_1+c_2i)e^{T[A/H_1+B\overline W/H_1W]}$ in $\overline D$
and $W(z_0)=c_1F(z_0)+c_2G(z_0)\ne0$. This
contradiction verifies that \eqref{e2.12} is true.

For the above discussion, we see that four real coefficients
$a(z),b(z),c(z),d(z)$ of  system \eqref{e1.1} or two complex
coefficients $A(z),\,B(z)$ of the complex equation \eqref{e1.4} can be
determined by two boundary functions $h_1(z),h_2(z)$ in the set
$R_h$.
\end{proof}


\section{H\"older continuity of a singular
integral operator}

It is clear that the complex equation
\begin{equation}
W_{\overline Z}=0\quad\text{in }\overline{D_Z}\label{e3.1}
\end{equation}
is a special case of equation \eqref{e1.4}, where $D_Z$ is a bounded simply
connected domain with boundary $\partial D\in C^1_\mu\,(0<\mu<1)$. On
the basis of \cite[Theorem 1.3, Chapter I]{w3}, we can find a unique
solution of Problem $RH$ for equation \eqref{e3.1} in $\overline{D_Z}$.

Now we consider the function $g(Z)\in L_\infty(D_Z)$, and first
extend the function $g(Z)$ to the exterior of $\overline{D_Z}$ in $\mathbb{C}$,
i.e. set $g(Z)  = 0$ in ${\mathbb{C}}\backslash\overline{D_Z}$, hence we can only
discuss the domain $D_0 = \{|x| < R_0\}$ $\cap\{\operatorname{Im}
Y\ne0\}\supset\overline{D_Z}$, here $Z=x+iY$ and  $R_0$ is an appropriately
large positive number. In the following we shall verify that the
integral
\begin{equation}
\begin{gathered}
\Psi(Z) = T\Big(\frac g{H_1}\Big)
 = -\frac1\pi \int \int_{D_0} \frac{g(t)/H_1(\operatorname{Im}\,t)}
{t - Z}d\sigma_t\quad\text{in }D_0, \\
L_\infty[g(Z),D_0]\le k_3,
\end{gathered}\label{e3.2}
\end{equation}
satisfies the estimate \eqref{e3.3} below, where
$H_j(y)=y^{m_j/2}h_j(y)$ $(m_j > 0,j=1,2,m_2 < \min(1,m_1))$ are
as stated in Section 1, and $H_1(y)=H_1[\operatorname{Im} z(Z)]$, $z(Z)$ is as
stated in \eqref{e1.3}. It is clear that the function
$g(Z)/H_1(y)=g(Z)/H_1[\operatorname{Im} z(Z)]$ belongs to the space $L_1(D_0)$ and
in general is not belonging to the space $L_p(D_0)\,(p>2)$, and the
integral $\Psi(Z_0)$ is definite when $\operatorname{Im} Z_0\ne0$. If $Z_0\in D_0$
and $\operatorname{Im} Z_0=0$, we can define the integral $\Psi(Z_0)$ as the limit
of the corresponding integral over $D_0\cap\{|\operatorname{Re} t -\operatorname{Re}
Z_0|\ge\varepsilon\}\cap\{|\operatorname{Im} t-\operatorname{Im} Z_0|\ge\varepsilon\}$ as $\varepsilon\to0$, where $\varepsilon$
is a sufficiently small positive number. The H$\rm\ddot{o}$lder
continuity of the singular integral will be proved by the following
method.

\begin{theorem} \label{thm3.1}
If the function $g(Z)$ in $D_Z$ satisfies the condition in \eqref{e3.2}, and
$H_1(y)=y^{m_1/2}h_1(y)$, where $m_1$ is a positive number, $h_1(y)$
is a continuous positive function, then the integral in \eqref{e3.2}
satisfies the estimate
\begin{equation}
C_{\beta}[\Psi(Z),\overline{D_Z}]\le M_1,\label{e3.3}
\end{equation}
in which $\beta = (2 - m_2)/(m + 2) - \delta$,
$m = m_1 - m_2$,  $\delta$ is a sufficiently small positive constant, and
$M_1=M_1(\beta,k_3,H_1,D_Z)$ is a positive constant.
\end{theorem}

\begin{proof}
We first give the estimates for
 $\Psi(Z)$ of \eqref{e3.2} in $D\cap\{\operatorname{Im} Y\ge0\}$, and verify the
boundedness of the function in \eqref{e3.2}. As stated Section 1, if 
$H_1(y) = y^{m_1/2}h_1(y)$, then $H_1(y) \ge sY^{m_1/(m+2)}$, where
$s$ is a positive constant. For any two points
$Z_0=x_0\in\gamma = (-1,1)$ on $x$-axis and 
$Z_1=x_1+iY_1(Y_1>0)\in D_0$ satisfying the condition 
$2\operatorname{Im} Z_1/\sqrt3\le|Z_1-Z_0|\le2\operatorname{Im} Z_1$, this means that the inner
angle at $Z_0$ of the triangle
$Z_0Z_1Z_2$ $(Z_2=x_0+iY_1\in D_0 )$ is not less than $\pi/6$ and not
greater than $\pi/3$, choose a sufficiently large positive number
$q$, from the H\"older inequality, we have
$L_1[\Psi(Z),D_0] \le L_q[g(Z),D_0]L_p[1/H_1(\operatorname{Im} t)(t-Z),D_0]$,
where $p =  q/(q - 1)$ $(>1)$ is close to 1. In fact we can
derive it as follows
\begin{equation}
\begin{aligned}
|\Psi(Z_0)|
&\le \big|\frac1\pi\int  \int_{D_0}  \frac{g(t)/H_1(\operatorname{Im}
t)}{t-Z_0}d\sigma_t\big| \\
&\le \frac1{s\pi}L_q[g(Z),D_0]
\Big[\int \int_{D_0} \Big|\frac1{t^{m_1/(m+2)}(t - Z_0)}\Big|^pd\sigma_t
\Big]^{1/p}\\
&= \frac1{s\pi}L_q[g(Z),D_0]\,J_1^{1/p},
\end{aligned}\label{e3.4}
\end{equation}
in which
\begin{align*}
 J_1& =  \int  \int_{D_0}  \Big|\frac1{t^{m_1/(m+2)}
(t - Z_0)}\Big|^pd\sigma_t\\
& \le  \int  \int_{D_0}\frac1{|t|^{pm_1/(m+2)}|\operatorname{Im}(t-Z_0)|^{p\beta_0}|
\operatorname{Re}(t-Z_0)|^{p(1-\beta_0)}}d\sigma_t\\
& \le  \Big|\int_0^{d_0} \frac1{Y^{pm_1/(m+2)}|Y - Y_0|^{p\beta_0}}dY
\int_{d_1}^{d_2}
\frac1{|x - x_0|^{p(1-\beta_0)}}dx\Big|\le k_4,
\end{align*}
where
$d_0 = \max_{Z\in\overline{D_0}}\operatorname{Im} Z$,
$d_1 = \min_{Z\in\overline{D_0}}\operatorname{Re} Z$,
$d_2 = \max_{Z\in\overline{D_0}}\operatorname{Re} Z$,
$\beta_0 = (2 - m_2)/(m+2) - \varepsilon$, $\varepsilon\,(<1/p-m_1/(m+2))$
is a sufficiently small positive constant, we can choose
$\varepsilon=2(p-1)/p\,(\le(2-m_2)/(m+2))$, such that $p(1-\beta_0)<1$ and
$p[m_1/(m+2)+\beta_0]<1$, and $k_4=k_4(\beta,k_3,H_1,D_0)$ is a
non-negative constant.

Next we estimate the H\"older continuity of the integral
$\Psi(Z)$ in $\overline{D_0}$; i.e.,
\begin{equation}
\begin{aligned}
&|\Psi(Z_1) - \Psi(Z_0)|\\
&\le \frac{|Z_1 - Z_0|}\pi
 \Big|\int  \int_{D_0}  \frac{g(t)/H_1(\operatorname{Im} t)}{(t - Z_0)(t - Z_1)}d\sigma_t\Big|\\
&\le \frac{|Z_1 - Z_0|}{s\pi}L_q[g(Z),D_0]\Big[\int  \int_{D_0}
\Big|\frac1{t^{m_1/(m+2)}(t - Z_0)(t - Z_1)}\Big|^pd\sigma_t\Big]^{1/p},
\end{aligned} \label{e3.5}
\end{equation}
and
\begin{align*}
 J_2& = \int \int_{D_0} \big|\frac1{t^{m_1/(m+2)}(t - Z_0)(t
- Z_1)}\big|^pd\sigma_t\\
& \le \int \int_{D_0} \frac{|\operatorname{Re}(t - Z_0)|^{p(\beta_0/2-1)}|\operatorname{Re}(t - Z_1)
|^{p(\beta_0/2-1)}}{|t|^{pm_1/(m+2)}|\operatorname{Im}(t - Z_0)|^{p\beta_0/2}|\operatorname{Im}(t - Z_1)|^{p\beta_0
/2}}d\sigma_t\\
& \le   \int_0^{d_0} \frac1{Y^{pm_1/(m+2)}|\operatorname{Im}(Y - Z_0)|^{p\beta_0/2}
|\operatorname{Im}(Y - Z_1)|^{p\beta_0/2}}dY\\
& \quad\times\int_{d_1}^{d_2}\frac1{|\operatorname{Re}(t - Z_0)|^{p(1-\beta_0/2)}
|\operatorname{Re}(t - Z_1)|^{p(1-\beta_0/2)}}d\operatorname{Re} t\\
& \le  k_5 \int_{d_1}^{d_2}\frac1{|x-x_0)|^{p(1-\beta_0/2)}|x-x_1|^{p(1
-\beta_0/2)}}dx,
\end{align*}
where $\beta_0=(2-m_2)/(m+2) - \varepsilon$ is
chosen as before and
$$
k_5 =  \max_{Z_0,Z_1\in D_0} \int_0^{d_0}[Y^{pm_1/(m+2)}|\operatorname{Im}(Y - Z_0)
|^{p\beta_0/2}|\operatorname{Im}(Y - Z_1)|^{p\beta_0/2}]^{-1}dY.
$$
Denote
$\rho_0=|\operatorname{Re}(Z_1-Z_0)|=|x_1-x_0|$,
$L_1=D_0\cap\{|x-x_0|\le2\rho_0,Y=Y_0 \}$ and
$L_2=D_0\cap\{2\rho_0<|x-x_0|\le2\rho_1<\infty,Y=Y_0\}\supset[d_1,d_2]
\backslash L_1$, where $\rho_1$ is a sufficiently large positive number, we
can derive
\begin{align*}
 J_2
& \le k_5\Big[\int_{L_1}\frac1{|x-x_0|^{p(1-\beta_0/2)}|x-x_1|^{p(1-
\beta_0/2)}}dx\\
& \quad +\int_{L_2}\frac1{|x-x_0|^{p(1-\beta_0/2)}|x-x_1|^{p(1-\beta_0/2)}}
dx\Big]\\
& \le
k_5\Big[|x_1 - x_0|^{1-2p+p\beta_0}\int_{|\xi|\le2}\frac1{|\xi|^{p(1-
\beta_0/2)}|\xi\pm1|^{p(1-\beta_0/2)}}d\xi\\
& \quad +k_6|\int_{2\rho_0}^{2\rho_1}\rho^{p\beta_0-2p}d\rho|\Big] \\
&\le k_7|x_1 - x_0|^{1-p(2-\beta_0)}\\
&= k_7|x_1-x_0|^{p((2-m_2)/(m+2)-\varepsilon+1/p-2)},
\end{align*}
in which we use
$|x-x_0|=\xi|x_1-x_0|$,
$|x-x_1|=|x-x_0-(x_1-x_0)|=|\xi\pm1| |x_1-x_0|$ if $x\in L_1$,
$|x-x_0|=\rho\le2|x-x_1|$ if $x\in L_2$,
choose that $p$ $(>1)$ is close to 1 such that $1-p(2-\beta_0)<0$,
and $k_j=k_j(\beta,k_3,H,D_0)\,(j=6,7)$ are non-negative constants.
Thus we obtain
\begin{equation}
|\Psi(Z_1) - \Psi(Z_0)| \le k_7|Z_1 - Z_0||x_1 - x_0
|^{(2-m_2)/(m+2)-\varepsilon+1/p-2}
\le k_8|Z_1 - Z_0|^\beta, \label{e3.6}
\end{equation}
in which we use that the inner angle at $Z_0$ of the triangle
$Z_0Z_1Z_2$  $(Z_2=x_0+iY_1\in D_0 )$ is not less than $\pi/6$ and not
greater than $\pi/3$, and choose
$\varepsilon = 2(p - 1)/p$, $\beta = (2 - m_2)/(m + 2)  - \delta$,
$\delta = 3(p- 1)/p$, $k_8 = k_8 (\beta,k_3,H_1,D_0)$ is a
non-negative constant. The above points $Z_0=x_0$, $Z_1 =x_1+iY_1$
can be replaced by $Z_0=x_0+iY_0$,
$Z_1=x_1+iY_1\in\overline{D_0}$, $0<Y_0<Y_1$ and
$2(Y_1-Y_0)/\sqrt3\le|Z_1-Z_0|\le2(Y_1-Y_0)$.

Finally we consider any two points $Z_1=x_1+iY_1$,
$Z_2=x_2+iY_1$ and
$x_1 <x_2$, from the above estimates, the following estimate can be
derived
\begin{equation}
\begin{aligned}
&|\Psi(Z_1)-\Psi(Z_2)|\le|\Psi(Z_1)-\Psi(Z_3)|+|\Psi(Z_3)-\Psi(Z_2)|
 \\
&\le k_8|Z_1 - Z_3|^{\beta}+k_8|Z_3 - Z_2|^{\beta}\le
k_9|Z_1 - Z_2 |^{\beta},
\end{aligned} \label{e3.7}
\end{equation}
where $Z_3=(x_1+x_2)/2+i[Y_1+(x_2-x_1)/(2\sqrt3)]$. If
$Z_1=x_1+iY_1$, $Z_2= x_1+iY_2$, $Y_1<Y_2$, and we choose
$Z_3=x_1+(Y_2-Y_1)/2\sqrt3+i(Y_2+Y_1)/2$, and can also get \eqref{e3.7}. If
$Z_1=x_1+iY_1$, $Z_2=x_2+iY_2$, $x_1<x_2,Y_1<Y_2$, and we choose
$Z_3=x_2 +iY_1$, obviously
$$
|\Psi(Z_1)-\Psi(Z_2)|\le|\Psi(Z_1)-\Psi(Z_3)|+|\Psi(Z_3)-\Psi(Z_2)|,
$$
 and $|\Psi(Z_1)-\Psi(Z_3)|,\,|\Psi(Z_3)-\Psi(Z_2)|$ can be
estimated by the above way, hence we can obtain the estimate of
$|\Psi(Z_1)-\Psi(Z_2)|$. For the function $\Psi(Z)$ of \eqref{e3.2} in
$D\cap\{\operatorname{Im} Y\le0\}$, the similar estimates can be also derived.
Hence we have the estimate \eqref{e3.3}.
\end{proof}

\begin{theorem} \label{thm3.2}
 If the condition $H_1(y)= y^{m_1/2}h_1(y)$ in Theorem \ref{thm3.1} 
is replaced by $H_1(y)=y^{\eta}h_1(y)$, herein 
$\eta$ is a positive constant
satisfying the inequality $\eta<(m+2)/(2-m_2)$, then by the same
method we can prove that the integral $\Psi(Z)=T(g/H_1)$ satisfies
the estimate
\begin{equation}
C_{\beta}[\Psi(Z),D_Z]\le M_1,\label{e3.8}
\end{equation}
in which $\beta=1 - \eta(2-m_2)/(m+2)-\delta$, $\delta$ is a sufficiently
small positive constant, and $M_1=M_1(\beta,k_3,H_1,D_Z)$ is a
positive constant. In particular if $H_1(y)=y$; i.e., $\eta=1$, then
we can choose $\beta=m_1/(m+2)-\delta$, $\delta$ is a sufficiently small
positive constant.
\end{theorem}


\begin{thebibliography}{00}

\bibitem{b1} R.-M. Brown, G. Uhlmann;
\emph{ Uniqueness in the inverse conductivity problem for smooth 
conductivities in two dimensions,}
 Comm in Partial Differential Equations, \textbf{22}
(1997): 1009-1027.

\bibitem{i1} V. Isakov; 
\emph{Inverse problems for partial
differential equations,} Springer-Verlag, New York, (2004).

\bibitem{k1} A. Kirsch; 
\emph{An introduction to the mathematical theory of inverse problems,} 
Springer-Verlag, New York, (1996).

\bibitem{m1} N.-I. Mushelishvili; 
\emph{Singular integral equations,} Noordhoff, Groningen, (1953).

\bibitem{s1} L.-Y. Sung; 
\emph{ An inverse scattering problem for
the Davey-Stewartson II equations, I, II, III,} J Math Anal Appl,
\textbf{183} (1994): 121-154, 389-325, 477-494.

\bibitem{t1}{A. Tamasan;
 \emph{On the scattering method for the
$\overline\partial$-equation and reconstruction of convection coefficients,}
Inverse Problem, \textbf{20} (2004): 1807-1817.

\bibitem{t2} Z.-L. Tong, J. Cheng, M. Yamamoto; 
\emph{A method for constructing the convection coefficients of 
elliptic equations from Dirichlet to Neumann map,} (in Chinese) 
Science in China, Ser A, \textbf{34} (2004): 752-766.

\bibitem{v1} I.-N. Vekua;  
\emph{Generalized analytic functions,} Pergamon, Oxford, (1962).

\bibitem{w1} G.-C, Wen; 
\emph{Conformal mappings and boundary value problems,} 
Translations of Mathematics Monographs 106, Amer.
Math. Soc., Providence, RI, (1992).

\bibitem{w2} G.-C, Wen; 
\emph{Linear and quasilinear complex equations of hyperbolic and mixed Type,} 
Taylor \& Francis, London, (2002).

\bibitem{w3} G.-C, Wen;
 \emph{Elliptic, hyperbolic and mixed complex
equations with parabolic degeneracy,} World Scientific, Singapore,(2008).

\bibitem{w4} G.-C, Wen;
 \emph{Recent progress in theory and
applications of modern complex analysis,} Science Press, Beijing,
(2010).}

\bibitem{w5} G.-C, Wen, H. Begehr; 
\emph{Boundary value problems for elliptic equations and systems,}
 Longman Scientific and Technical Company, Harlow, (1990).

\bibitem{w6} G.-C, Wen, D.-C, Chen, Z.-L. Xu; 
\emph{Nonlinear complex analysis and its applications,} 
Mathematics Monograph Series 12, Science Press, Beijing, (2008).

\bibitem{w7} G.-C. Wen, Z,-L. Xu; 
\emph{The inverse problem for linear elliptic systems of first order
 with Riemann-Hilbert type map in multiply connected domains,} 
Complex Variables and Elliptic Equations, \textbf{54} (2009): 811-823.

\bibitem{w8} G.-C. Wen, Z,-L. Xu,  F.-M. Yang;
 \emph{The inverse problem for elliptic equations from Dirichlet to
 Neumann map in multiply connected domains,} Boundary Value Problems, 
\textbf{305291} (2009): 1-16.

\end{thebibliography}

\end{document}