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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 247, pp. 1--21.\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-2013/247\hfil
Well-posedness of discontinuous BVPs]
{Well-posedness of discontinuous boundary-value problems
for nonlinear elliptic complex equations in
multiply connected domains}

\author[G.-C. Wen \hfil EJDE-2013/247\hfilneg]
{Guo-Chun Wen} 

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

\thanks{Submitted  November 1, 2013. Published November 15, 2013.}
\subjclass[2000]{35J56, 35J25, 35J60, 35B45.}
\keywords{Well-posedness; discontinuous boundary value problem;
\hfill\break\indent nonlinear elliptic complex equation; A priori estimate;
existence of solutions}

\begin{abstract}
 In the first part of this article, we  study a discontinuous Riemann-Hilbert
 problem for nonlinear uniformly elliptic complex equations of first order
 in multiply connected domains. First we show  its well-posedness.
 Then we give the representation of solutions for a modified Riemann-Hilbert
 problem for the complex equations. Then we  obtain a priori
 estimates of the solutions and verify the solvability of the modified
 problem by using the Leray-Schauder theorem. Then the solvability
 of the original discontinuous Riemann-Hilbert boundary-value
 problem is obtained. In the second part, we study a discontinuous
 Poincar\'e boundary-value problem for nonlinear elliptic equations
 of second order in multiply connected domains.
 First we formulate the  boundary-value problem and show its new well-posedness.
 Next we obtain the representation of solutions and obtain a priori estimates
 for the solutions  of a modified Poincar\'e problem.
 Then with estimates and the method  of parameter extension, we obtain
 the solvability of the discontinuous Poincar\'e problem.
\end{abstract}

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

\section{Formulation of discontinuous Riemann-Hilbert problem}

Lavrent$'$ev and Shabat \cite{l1} introduced the Keldych-Sedov formula
for analytic functions in the upper half-plane, namely the representation of
solutions of the mixed boundary-value problem for analytic
functions, which is a special case of discontinuous boundary value problems
with the integer index. The authors also pointed out that this formula has
very important applications. However, for many problems in mechanics
and physics, for instance some free boundary problems and the Tricomi
problem for some mixed equations \cite{b1,w1,w2,w3,w4,w7,w8,w9,w10}, one needs to
apply more general discontinuous boundary-value problems
of analytic functions and some elliptic equations in
the simply and multiply connected domains.
In \cite{w1} the author solved the general discontinuous Riemann-Hilbert 
problems for analytic functions in simply connected domains,
but the general discontinuous boundary-value problems for elliptic equations 
in multiply connected domains have not been solved completely.
In this article, we study the general discontinuous Riemann-Hilbert problem and
discontinuous Poincar\'e problem and their new well-posedness for
nonlinear elliptic equations in multiply connected domains.

We study the nonlinear elliptic equations of first order
\begin{equation}
w_{\bar z}=F(z,w,w_{z}),\quad
F=Q_1w_z+Q_2\overline w_{\bar z}+A_1w+A_2\overline w+A_3,
\quad z\in D,\label{e1.1}
\end{equation}
where $z=x+iy$, $w_{\bar z}=[w_x+iw_y]/2$,
$Q_j=Q_j(z,w,w_z)$, $j=1,2$, $A_j=A_j(z,w)$, $j=1,2,3$ and assume that
equation \eqref{e1.1} satisfy the following conditions:
\begin{itemize}
\item[(C1)]
$Q_j(z,w,U)$, $A_{j}(z,w)$ $(j=1,2,3)$ are measurable in $z\in D$ for
all continuous functions $w(z)$ in $D^*=\overline D\backslash Z$ and all
measurable functions $U(z)\in L_{p_0}(D^*)$, and satisfy
\begin{equation}
L_{p}[A_j,\overline D]\le k_0,\quad j=1,2,\;L_{p}[A_3,\overline D]\le k_1,\label{e1.2}
\end{equation}
where $Z=\{t_1,\dots ,t_m\}$, $t_1,\dots ,t_m$ are different points on the
boundary $\partial D=\Gamma$ arranged according to the positive direction
successively, and $p,p_0,k_0,k_1$ are non-negative
constants, $2<p_0\le p$.

\item[(C2)] The above functions are continuous in $w\in{\mathbb{C}}$ for almost
every point $z\in D,\,U\in{\mathbb{C}}$. and
$Q_j=0$ $(j=1,2)$, $A_j=0$ $(j=1,2,3)$ for $z\in\mathbb{C}\backslash D$.

\item[(C3)]  The complex equation \eqref{e1.1} satisfies the uniform ellipticity
condition
\begin{equation}
|F(z,w,U_1)-F(z,w,U_2)|\le q_0|U_1-U_2|,\label{e1.3}
\end{equation}
for almost every point $z\in D$, in which $w,U_1,U_2\in{\mathbb{C}}$ and
$q_0$ is a non-negative constant, $q_0<1$.

\end{itemize}
Let $N\ge1$ and let $D$ be an $N+1$-connected bounded domain in $\mathbb{C}$
with the boundary $\partial D={\Gamma}=\cup^N_{j=0}{\Gamma}_j\in C^1_\mu$
$(0<\mu<1)$. Without loss of generality, we assume that $D$
is a circular domain in $|z|<1$, bounded by the $(N+1)$-circles
${\Gamma}_j:|z-z_j|=r_j,\,j=0,1,\dots ,N$ and
${\Gamma}_0={\Gamma}_{N+1}:|z|=1,\,z=0\in D$.
In this article, we use the same notation as in references
\cite{b1,h1,w1,w2,w3,w4,w5,w6,w7,w8,w9,w10}.
Now we formulate the
general discontinuous Riemann-Hilbert problem for equation \eqref{e1.1} as
follows.

\subsection*{Problem A}
The general discontinuous Riemann-Hilbert problem for \eqref{e1.1}
is to find a continuous solution $w(z)$ in $D^*$ satisfying the boundary
 condition:
\begin{equation}
\operatorname{Re}[\overline{\lambda(z)}w(z)]=c(z),\;z\in{\it\Gamma^*}=\it\Gamma\backslash Z,\label{e1.4}
\end{equation}
where $\lambda(z),c(z)$ satisfy the conditions
\begin{equation}
C_\alpha[\lambda(z),\hat{\it\Gamma_j}]\le k_0,\quad
C_\alpha[|z-t_{j-1}|^{\beta_{j-1}}
|z-t_j|^{\beta_j}c(z),\hat{\it\Gamma_j}]\le k_2,\quad
j=1, \dots ,m,\label{e1.5}
\end{equation}
in which $\lambda(z)=a(z)+ib(z)$, $|\lambda(z)|=1$ on
$\Gamma$, and $Z=\{t_1,\dots ,t_m\}$ are the first kind of
discontinuous points of $\lambda(z)$ on $\Gamma$, $\hat{\Gamma_j}$ is an
arc from the point $t_{j-1}$ to $t_{j}$ on $\Gamma$, and does not
include the end point $t_j$ $(j=1,2,\dots ,m)$, we can assume that
$t_j\in{\Gamma}_0$ $(j=1,\dots ,m_0)$,
$t_j\in{\Gamma}_1$ $(j=m_0+1,\dots ,m_1)$, \dots ,
$t_j\in {\Gamma}_N$ $(j=m_{N-1}+1\dots ,m)$ are all discontinuous points of
$\lambda(z)$ on $\Gamma$; If $\lambda(z)$ on ${\Gamma}_l\,(0\le l\le N)$
has no discontinuous point, then we can choose a point
$t_j\in{\Gamma}_l$ $(0\le l\le N)$ as a discontinuous point of
$\lambda(z)$ on $\Gamma_l$ $(0\le l\le N)$, in this case $t_j=t_{j+1}$;
$\alpha (1/2<\alpha<1)$, $k_0$, $k_2$, $\beta_j(0<\beta_j<1)$ are positive
constants and satisfy the conditions
$$
\beta_j+|\gamma_j|<1,\quad j=1,\dots ,m,
$$
where $\gamma_j$ $(j=1,\dots ,m)$ are as stated in \eqref{e1.6} below.

Denote by $\lambda(t_j-0)$ and $\lambda(t_j+0)$ the left limit and right
limit of $\lambda(t)$ as $t\to t_j$ $(j=1,2,\dots ,m)$ on $\Gamma$, and
\begin{equation}
\begin{gathered}
 e^{i\phi_j}=\frac{\lambda(t_j-0)}{\lambda(t_j+0)},\quad
\gamma_j =\frac1{\pi i}\ln\frac{\lambda(t_j-0)}{\lambda(t_j+0)}=
\frac{\phi_j}\pi-K_j, \\
K_j=\big[\frac{\phi_j}\pi\big]+J_j,\quad J_j=0
\text{ or }1,\quad j=1, \dots ,m,
\end{gathered} \label{e1.6}
\end{equation}
in which $0\le\gamma_j<1$ when $J_j=0$, and
$-1<\gamma_j<0$ when $J_j=1$, $j=1,\dots , m$. The index $K$ of Problem
A is defined as 
$$
K=\frac12(K_1+\dots+K_m)=\sum^m_{j=1}
[\frac{\phi_j}{2\pi}-\frac{\gamma_j} 2].
$$
If $\lambda(t)$ on $\Gamma$ is continuous, then
$K=\Delta_\Gamma\arg\lambda(t)/{2\pi}$ is a unique integer. Now the function
$\lambda(t)$ on $\Gamma$ is not continuous, we can choose $J_j=0$ or
$1\,(j=1,\dots ,m)$, hence the index $K$ is not unique. Later on there
is no harm in assuming that the partial indexes $K_l$ of $\lambda(z)$ on
$\Gamma_l$ $(l=1,\dots ,N_0\le N)$ are not integers, and the partial
indexes $K_l$ of $\lambda(z)$ on $\Gamma_l$ $(j=0,N_0+1,\dots ,N)$ are
integers; (if $K_0$ of $\lambda(z)$ on $\Gamma_0$ is not integer, then we
can similarly discuss). We can require that the solution $w(z)$
possesses the property
\begin{equation}
\begin{gathered}
R(z)w(z)\in C_\delta(\overline D),\quad
R(z)=\prod^{m}_{j=1}|z-t_j|^{\eta_j/\tau^2},
\\
\eta_j=\begin{cases}
 \beta_j+\tau, &\text{if }\gamma_j\ge0,\;\gamma_j<0,\;
\beta_j>|\gamma_j|, \\
|\gamma_j|+\tau,&\text{if } \gamma_j<0,\;\beta_j\le |\gamma_j|,
\end{cases}
\end{gathered} \label{e1.7}
\end{equation}
in which $\gamma_j$ $(j=1,\dots ,m)$ are
real constants as stated in \eqref{e1.6},
$\tau \le\min(\alpha,1-2/p_0)$ and
$\delta <\min(\beta_1,\dots ,\beta_m$, $\tau)$ are small positive
constants.

When the index $K<0$, Problem A may not be solvable,
when $K\ge 0$, the solution of Problem  A is not necessarily
unique. Hence we put forward a new concept of well-posedness of Problem A
with modified boundary conditions as follows.

\subsection*{Problem B}
Find a continuous solution $w(z)$ of
the complex equation \eqref{e1.1} in $D^*$ satisfying the boundary
condition
\begin{equation}
\operatorname{Re} [\overline{\lambda(z)}w(z)]=r(z)+h(z)\overline{\lambda(z)}X(z),\quad
z\in \Gamma^*,\label{e1.8}
\end{equation}
where $X(z)$ is as stated in \eqref{e1.9} below,
and
$$
h(z)=\begin{cases}
0, & z\in \Gamma_0, \; K\ge0 \\
h_j,& z\in\Gamma_{j},\;j=1,\dots ,N,\; K\ge0 \\
h_{j},& z\in\Gamma_j,\;j=1,\dots ,N,\; K<0\\
[1+(-1)^{2K}]h_{0} \\
+\operatorname{Re} \sum^{[|K|+1/2]-1}_{m=1}(h^{+}_{m}
+ i h^-_{m})z^{m}, & z\in \Gamma_{0},\; K<0
\end{cases}
$$
in which
$h_j$ $(j=[1-(-1)^{2K}]/2,\dots ,N)$,
$h^+_m,h^-_m$, $(m=1,\dots ,[|K|+1/2]-1)$
are unknown real constants to be determined appropriately, and
$h_{N+1}(=h_0)=0$, if $2|K|$ is an odd integer; and
\begin{align*}
Y(z)
&=\prod_{j=1}^{m_0}(z-t_j)^{\gamma_j}\prod_{l=l}^{N}(z-z_l)
^{-[\tilde{K}_l]}
\prod_{j=m_0+1}^{m_1}\Big(\frac{z-t_j}{z-z_1}
\Big)^{\gamma_j}\Big(\frac{z-t'_1}{z-z_1}\Big) \\
&\quad \times\prod_{j=m_{N_0-1}+1}^{m_{N_0}}
\Big(\frac{z-t_j}{z-z_{N_0}}\Big)^{\gamma_j}\Big(\frac{z-t'_{N_0}}
{z-z_{N_0}}\Big)
\prod_{j=m_{N_0}+1}^{m_{N_0+1}}\Big(\frac{z-t_j}{z-z_{N_0+1}}
\Big)^{\gamma_j}\dots\\
&\quad\times \prod_{j=m_{N-1}+1}^m
\Big(\frac{z-t_j}{z-z_N}\Big)^{\gamma_j},
\end{align*}
where $\tilde{K}_l=\sum^{m_l}_{j=m_{l-1}+1}K_j$  denote the partial
index on $\Gamma_l$ $(l=1,\dots ,N)$,
$t'_l$ $(\in \Gamma_l,\,l=,\dots ,N_0)$ are
fixed points, which are not the discontinuous points at $Z$; we must
give the attention that the boundary circles $\Gamma_j$ $(j=0,1,\dots ,N)$
of the domain $D$ are moved round the positive direct. Similarly to
 \cite[(1.7)--(1.12) Chapter V]{w1}, we see that
$$
\frac{\lambda(t_j-0)}{\lambda(t_j+0)}\overline{\Big[
\frac{Y(t_j-0)}{Y(t_j+0)}\Big]}
=\frac{\lambda(t_j-0)}{\lambda(t_j+0)}e^{-i\pi\gamma_j}=\pm1,\quad
j=1,\dots ,m,
$$
it only needs to charge the symbol on some arcs on $\Gamma$, then
$\lambda(z)\overline{Y(z)}/|Y(z)|$ on $\Gamma$ is continuous. In this case, its
index $$
\kappa=\frac1{2\pi}\Delta_\Gamma[\lambda(z)\overline{Y(z)}]=K-\frac{N_0}2
$$
is an integer; and
\begin{equation}
\begin{gathered}
 X(z)=\begin{cases}
iz^{[\kappa]}e^{iS(z)}Y(z), & z\in\Gamma_0, \\
ie^{i\theta_j}e^{iS(z)}Y(z), & z\in\Gamma_j,j=1,\dots ,N,
\end{cases}\\
\operatorname{Im}[\overline{\lambda(z)}X(z)] =0,\quad z\in\Gamma, \\
\operatorname{Re} S(z)=S_1(z)-\theta(t),\\
S_1(z)=\begin{cases}
\arg\lambda(z)- [\kappa]\arg z-\arg Y(z), & z\in\Gamma_0, \\
\arg\lambda(z)-\arg Y(z), & z\in\Gamma_j,\; j=1,\dots ,N,
\end{cases} \\
\theta(z)=\begin{cases}
0, & z\in\Gamma_0, \\
\theta_j, & z\in\Gamma_j,\; j=1,\dots ,N,
\end{cases} \\
\operatorname{Im}[S(1)]=0,
\end{gathered}\label{e1.9}
\end{equation}
in which $S(z)$ is a solution of the modified Dirichlet problem with
the above boundary condition for analytic functions,
$\theta_j$ $(j=1,\dots ,N)$ are real constants, and
$\kappa=K-N_0/2$.

In addition, we may assume that the solution $w(z)$ satisfies the
following point conditions
\begin{equation}
\operatorname{Im}[\overline{\lambda(a_j)}w(a_j)]=b_j,\quad
j\in J=\{1,\dots ,2K+1\},\quad \text{if } K\ge 0,\label{e1.10}
\end{equation}
where $a_{j}\in\Gamma_{0}\,(j\in J)$ are distinct points; and
$b_{j}(j\in J)$ are all real constants satisfying the conditions
$$
|b_{j}|\le k_{3},\quad j\in J
$$
with the positive constant $k_{3}$. Problem  B with
$A_3(z,w)=0$ in $D$, $c(z)=0$ on $\Gamma$ and $b_j=0$ $(j\in J)$ is
called Problem $\rm B_0$.

We mention that the undetermined real constants $h_j, h^\pm_m$ in
\eqref{e1.8} are for ensuring the existence of continuous solutions, and
the point conditions in \eqref{e1.10} are for ensuring the uniqueness of
continuous solutions in $D$. The condition $0<K<N$ is called the
singular case, which only occurs in the case of multiply connected
domains, and is not easy handled.

Now we introduce the previous well-posedness of the discontinuous
Riemann-Hilbert problem of elliptic complex equations, which are
we always use here.

\subsection*{Problem  C}
Find a continuous solution $w(z)$ in $D$ of \eqref{e1.1}
with the modified boundary condition \eqref{e1.8}, where
\begin{equation}
h(z)=\begin{cases} 0, & z\in\Gamma,\; K>N-1, \\
h_j, &z\in\Gamma_j,\;j=1,\dots ,N-K',\; 0\leq K\le N-1 \\
0, &z\in\Gamma_j,\;j=N-K'+1,\dots ,N-K'+[K]+1,\\
 &\quad  0\leq K\le N-1, \\
h_{j}, &z\in\Gamma_j,\,j=1,\dots ,N,\; K<0, \\
[1+(-1)^{2K}] h_0\\
+\operatorname{Re} \sum^{[|K|+1/2]-1}_{m=1}
 (h^{+}_{m}\\
+ i h^{-}_{m})z^{m},
&z\in\Gamma_0,\; K<0\,.
\end{cases}
\label{e1.11}
\end{equation}
in which $K'=[K+1/2]$, $[K]$  denotes
the integer part of $K$,
$h_0,h^+_m,h^-_m\,(m=1,\dots ,[|K|+1/2]-1)$
are unknown real constants to be determined appropriately, and
$h_{N+1}(=h_0)=0$, if $2|K|$ is an odd integer; and the solution
$w(z)$ satisfies the point conditions
\begin{equation}
\operatorname{Im}[\overline{\lambda(a_j)}w(a_j)]=b_j,\quad
j\in J=\begin{cases}
1,\dots ,2K-N+1,&\text{if } K>N-1, \\
1,\dots ,[K]+1, &\text{if } 0\le K\le N-1,
\end{cases}
\label{e1.12}
\end{equation}
in which $a_{j}\in\Gamma_{j+N_0}$ $(j=1,\dots ,N-N_0)$,
$a_j\in \Gamma_{0}$ $(j=N-N_0+1,\dots ,2K-N+1$, if $K\ge N)$ are
distinct points; and when
$[K]+1\le N-N_0$, $a_j$ ($\in\Gamma_{j+N-[K]-1}$, $j=1,\dots ,[K]+1$), otherwise
$a_j$ ($\in\Gamma_{j+N-N_0}$, $j=1,\dots ,N_0$), and
$a_j$ $(\in\Gamma_0,\,j=N_0+1,\dots ,[K]+1)$ are distinct points, and
$$
|b_{j}|\le k_{3},\quad j\in J
$$
with a non-negative constant $k_{3}$.

We can prove the equivalence of Problem  B and Problem  C for
for equation \eqref{e1.1}. From this, we see that the
advantages of the new well-posedness are as follows:
\begin{itemize}
\item[(1)] The statement of the new well-posedness is simpler than others (see
\cite{w1,w2,w9}).

\item[(2)] The point conditions in $\Gamma_0=\{|z|=1\}$ are similar to those
for the simple connected domain $D=\{|z|<1\}$.

\item[(3)] The new well-posedness statement does not distinguished the singular
case $0<K<N$ and non-singular case $K\ge N$.
\end{itemize}
We mention the equivalence of these well-posedness statements; 
i.e. if there exists the unique solvability under one well-posedness statement, 
then we can derive the unique solvability under under the other well-posedness.
Hence it is best to choose the simplest well-posedness statement.

To prove the solvability of Problem  B for the complex
equation \eqref{e1.1}, we need to give a representation theorem.

\begin{theorem} \label{thm1.1}
 Suppose that the complex equation
\eqref{e1.1} satisfies conditions {\rm (C1)--(C3)}, and $w(z)$ is a solution of
Problem {\rm B} for \eqref{e1.1}. Then $w(z)$ is representable as
\begin{equation}
w(z)=[\Phi(\zeta(z))+\psi(z)]e^{\phi(z)},\label{e1.13}
\end{equation}
where $\zeta(z)$ is a homeomorphism in $\overline D$, which maps
quasi-conformally $D$ onto the $N+1$-connected circular domain $G$
with boundary $L=\zeta(\Gamma)$ in $\{|\zeta|<1\}$, such that:
three points on $\Gamma$ are mapped into three points on $L$ respectively;
$\Phi(\zeta) $ is an analytic function in $G$;
$\psi(z),\phi(z),\zeta(z)$ 
and its inverse function $z(\zeta)$ satisfy the estimates
\begin{gather}
C_{\beta}[\psi,\overline D]\le k_4,\quad C_{\beta}[\phi,\overline D]\le k_4,\quad
C_{\beta}[\zeta(z),\overline D]\le k_4, \label{e1.14}\\
L_{p_0}[|\psi_{\bar{z}}|+|\psi_{z}|,\quad
\overline D]\le k_4,\,L_{p_0} [|\phi_{\overline {z}}|+|{\phi_{z}}|,\overline D]\le k_4,
\label{e1.15}\\
C_{\beta}[z(\zeta),\overline G]\le k_4,\quad
L_{p_0}[|\chi_{\bar z}|+|\chi_z|, \overline D]\le k_5,
\label{e1.16}
\end{gather}
in which $\chi(z)$ is as stated in
\eqref{e1.20} below,
$\beta=\min(\alpha,1-2/p_0)$,
$p_0$ $(2<p_0\leq p)$,
$k_j=k_j(q_0,p_0,\beta,k_0$, $k_1,D)$ $(j=4,5)$ are non-negative
constants depending on $q_0,p_0,\beta,k_0,k_1,D$. Moreover, the
function $\Phi[\zeta(z)]$ satisfies the estimate
\begin{equation}
C_\delta[R(z)\Phi[\zeta(z)],\overline D]\le M_1=M_1(q_0,p_0,\beta,k,D)<\infty,
\label{e1.17}
\end{equation}
in which $R(z)$, $\gamma_j$ $(j=1,\dots ,m)$ are as stated in
\eqref{e1.7} and
$\tau\le\min(\alpha,1-2/p_0)$, $\delta<\min(\beta_1,\dots ,
\beta_m,\tau)$ are small positive constants,
$k=k(k_0,k_1,k_2,k_3)$, and $M_1$ is a non-negative constant
dependent on $q_0,p_0,\beta,k,D$.
\end{theorem}

\begin{proof}
We substitute the solution $w(z)$ of
Problem B into the coefficients of equation \eqref{e1.1} and consider
the  system
\begin{equation}
\begin{gathered}
\phi_{\bar z}=Q\phi_{z}+A,\quad
A=\begin{cases} A_{1}+A_{2}\overline w/w &\text{for }w(z)\neq 0, \\
0&\text{for } w(z)=0\text{ or }z\not\in D,
\end{cases} \\
\psi_{\bar z}=Q\psi_{z}+A_{3}e^{-\phi(z)},\quad
Q=\begin{cases}
Q_{1}+Q_{2} \overline{w_{z}}/{w_{z}}&\text{for } w_{z}\neq 0, \\
0&\text{for } w_{z}=0\text{ or } z\not\in D,
\end{cases} \\
W_{\bar z}=QW_{z},\quad W(z)=\Phi[\zeta(z)]\quad \text{in } D.
\end{gathered}
\label{e1.18}
\end{equation}
By using the continuity method and the principle of contracting
mapping, we can find the solution
\begin{equation}
\begin{gathered}
\psi(z)=T_0f=-{\frac1{\pi}}\int\int_{D}
\frac{f(\zeta)}{\zeta-z}d\sigma_{\zeta}, \\
\phi(z)=T_0g,\,\zeta(z)=\Psi[\chi(z)],\quad
\chi(z)=z+T_0h
\end{gathered} \label{e1.19}
\end{equation}
of \eqref{e1.18}, in which $f(z),g(z),h(z)\in L_{p_0}(\overline D)$,
$2<p_{0}\le p$, $\chi(z)$ is a homeomorphic solution of the third equation in
\eqref{e1.18}, $\Psi(\chi)$ is a univalent analytic function, which
conformally maps $E=\chi(D)$ onto the domain $G$ (see \cite{v1,w2}), and
$\Phi(\zeta)$ is an analytic function in $G$. We can verify that
$\psi(z),\phi(z),\zeta(z)$ satisfy the estimates
\eqref{e1.14} and \eqref{e1.15}.
It remains to prove that $z=z(\zeta)$ satisfies the estimate in
\eqref{e1.16}. In fact, we can find a homeomorphic solution of the last
equation in \eqref{e1.18} in the form $\chi(z)=z+T_0h$ such that
$[\chi(z)]_z,[\chi(z)]_{\bar z}\in L_{p_0}(\bar D)$ (see \cite{v1}). Next,
we find a univalent analytic function $\zeta=\Psi(\chi)$, which maps
$\chi(D)$ onto $G$, hence $\zeta=\zeta(z)=\Psi[\chi(z)]$. By the
result on conformal mappings, applying the method of
\cite[Theorem 1.1,Chapter III]{w2} or
\cite[Theorem 1.1.1, Chapter I]{w9}, we can prove
that \eqref{e1.16} is true. It is easy to see that the function
$\Phi[\zeta(z)]$ satisfies the boundary conditions
\begin{equation}
\operatorname{Re}[\overline{\lambda(z)}e^{\phi(z)}\Phi(\zeta(z))]=c(z)+h(z)\overline{\lambda(z)}X(z)
-\operatorname{Re}[\overline{\lambda(z)}e^{\phi(z)}\psi(z)],\quad
z\in\Gamma^*.\label{e1.20}
\end{equation}
On the basis of the estimates \eqref{e1.14} and \eqref{e1.16}, and
use the methods of \cite[Theorem 1.1.1, Chapter I]{w9},
we can prove that $\Phi[\zeta(z)]$
satisfies the estimate \eqref{e1.17}.
\end{proof}

\section{Estimates for discontinuous
Riemann-Hilbert problems}


Now, we derive a priori estimates of solutions for Problem B
for the complex equation \eqref{e1.1}.

\begin{theorem} \label{thm2.1}
 Under the same conditions as in
Theorem \ref{thm1.1}, any solution $w(z)$ of Problem {\rm B} for \eqref{e1.1}
satisfies the estimates
\begin{gather}
\hat C_{\delta}[w(z),\overline D]=C_{\delta}[R(z)w(z),\overline D]\leq
M_{1}=M_{1}(q_{0},p_{0},\delta,k,D), \label{e2.1}\\
\hat L^1_{p_0}[w(z),\overline D]=L_{p_0}[|RSw_{\bar
z}|+|RSw_z|,\overline D]\leq M_{2}=M_{2}(q_{0},p_{0},\delta,k,D), \label{e2.2}
\end{gather}
where
$S(z)=\prod^{m}_{j=1}|z-t_j|^{1/\tau^2}$,
$k=k(k_0,k_1,k_2,k_3)$, $\delta<\min(\beta_1,\dots ,\beta_m,\tau)$,
$p_0,p$, $(2<p_{0}\le p)$, $M_j$ $(j=1,2)$ are positive constant only
depending on $q_{0},p_{0},\delta,k,D$.
\end{theorem}

\begin{proof}
 On the basis of Theorem \ref{thm1.1}, the solution
$w(z)$ of Problem  B can be expressed the formula as in \eqref{e1.13},
hence the boundary value problem $\rm B$ can be transformed into the
boundary value problem (Problem $\rm\tilde B$) for analytic functions
\begin{gather}
\operatorname{Re}[\overline{\Lambda(\zeta)}\Phi(\zeta)]=\hat r(\zeta)
+H(\zeta)\overline{\lambda(z(\zeta))}X[z(\zeta)],
\quad \zeta\in L^*=\zeta(\Gamma^*), \label{e2.3}
\\
H(\zeta)=\begin{cases}
 0, & \zeta\in L_0, \; K\ge0,\\
h_j, &\zeta\in L_j,\; j=1,\dots ,N,\; K\ge0, \\
 h_{j}, & \zeta\in L_j,\;j=1,\dots ,N,\; K<0,\\
[1+(-1)^{2K}]h_0\\
+\operatorname{Re} \sum^{[|K|+1/2] -1}_{m=1}(h^{+}_{m}+ih^{-}_{m})\zeta^{m},
& \zeta\in L_0,\; K<0,
\end{cases} \label{e2.4}
\\
\operatorname{Im}[\overline{\Lambda(a'_{j})}\Phi(a'_{j})]=b'_{j},\quad j\in J,
\label{e2.5}
\end{gather}
where
\begin{gather*}
\overline{\Lambda(\zeta)}=\overline{\lambda[z(\zeta)]} e^{\phi[z(\zeta)]}, \quad
\hat r(\zeta)=r[z(\zeta)]-\operatorname{Re}
\lbrace\overline{\lambda[z(\zeta)]}\psi[z(\zeta)]\rbrace, \\
a'_{j}=\zeta(a_{j}),\quad
b'_{j}=b_{j}-\operatorname{Im}[\overline{\lambda(a_{j})}
e^{\phi(a_j)}\psi(a_{j})],\quad j\in J.
\end{gather*}
 By \eqref{e1.5} and \eqref{e1.14}--\eqref{e1.16}, it can be seen that
$\Lambda(\zeta)$, $\hat r(\zeta)$, $b'_{j}$ $(j\in J)$ satisfy the
conditions
\begin{equation}
C_{\alpha\beta}[R[z(\zeta)]\Lambda(\zeta),L]\leq M_{3},\quad
C_{\alpha\beta} [R[z(\zeta)]\hat r(\zeta),L]\le M_{3},\quad
| b'_{j}|\leq M_{3},\,j\in J,\label{e2.6}
\end{equation}
where
$M_3=M_{3}(q_{0},p_{0},\beta,k,D)$. If we can prove that the
solution $\Phi(\zeta)$ of Problem $\rm\tilde B$ satisfies the
estimates
\begin{equation}
C_{\delta\beta}[R(z(\zeta))\Phi(\zeta),\overline G]\leq M_{4},\quad
C[R(z(\zeta))S(z(\zeta))\Phi'(\zeta),\tilde G]\leq
M_{5},\label{e2.7}
\end{equation}
where $\beta$ is the  constant as defined
in \eqref{e1.14}, $\tilde G=\zeta(\tilde D)$,
$M_j=M_j(q_{0},p_{0},\delta,k,D)$, $j=4,5$, then from the
representation \eqref{e1.13} of the solution $w(z)$ and the estimates
\eqref{e1.14}-\eqref{e1.16} and \eqref{e2.7}, the estimates \eqref{e2.1}
and \eqref{e2.2} can be derived.

It remains to prove that \eqref{e2.7} holds. For this, we first verify the
boundedness of $\Phi(\zeta)$; i.e.,
\begin{equation}
C[R(z(\zeta))\Phi(\zeta),\overline G]\leq M_{6}
=M_{6}(q_{0},p_{0},\beta,k,D). \label{e2.8}
\end{equation}
Suppose that \eqref{e2.8} is not true. Then there exist sequences of
functions $\{\Lambda_{n}(\zeta)\}$, $\{\hat r_{n}(\zeta)\}$,
$\{b_{jn}'\}$ satisfying the same conditions as $\Lambda(\zeta)$, $\hat
r(\zeta)$, $b'_{j}$, which  converge uniformly to
$\Lambda_{0}(\zeta)$, $\hat r_{0}(\zeta)$, $b'_{j0}\,(j\in J)$ on $L$
respectively. For the solution $\Phi_{n}(\zeta)$ of the boundary
value problem (Problem $\rm B_{n}$) corresponding to
$\Lambda_{n}(\zeta)$, $\hat r_{n}(\zeta)$, $b'_{jn}$ $(j\in J)$, we have
$I_{n}=C[R(z(\zeta))\Phi_{n}(\zeta),\overline G] \to\infty$ as $n\to\infty$.
There is no harm in assuming that $I_{n}\geq 1$, $n=1,2,\dots $.
Obviously $\tilde\Phi_{n}(\zeta)=\Phi_{n}(\zeta)/{I_{n}}$ satisfies the
boundary conditions
\begin{gather}
\operatorname{Re}[\overline{\Lambda_{n}(\zeta)}\tilde\Phi_{n}(\zeta)]
=[\hat r_{n}(\zeta)+H(\zeta)\overline{\lambda(z(\zeta))}X[z(\zeta)]]/{I_{n}},
\quad \zeta\in L^*, \label{e2.9}
\\
\operatorname{Im}[\overline{\Lambda_{n}(a'_{n})}\tilde\Phi_{n}(a'_{n})] =b'_{jn}/{I_{n}},\quad j\in J.
\label{e2.10}
\end{gather}
 Applying the Schwarz formula, the Cauchy formula and the
method of symmetric extension (see \cite[Theorem 4.3, Chapter IV]{w1}),
the estimates
\begin{equation}
C_{\delta\beta}[R(z(\zeta))\tilde\Phi_n(\zeta),\overline G]\le M_{7},\quad
C[R(z(\zeta))S(z(\zeta))\tilde\Phi'_n(\zeta),\overline G]\le M_{8},
\label{e2.11}
\end{equation}
can be obtained, where $\tilde G=\zeta(\tilde D)$, and
$M_j=M_j(q_{0},p_{0},\delta,k,D)$, $j=7,8$. Thus we can select a
subsequence of $\lbrace\tilde\Phi_{n}(\zeta)\rbrace$, which
converge uniformly  to an analytic function $\tilde\Phi_{0}(\zeta)$
in $G$, and $\tilde\Phi_{0}(\zeta)$ satisfies the homogeneous
boundary conditions
\begin{gather}
\operatorname{Re}[\overline{\Lambda_{0}(\zeta)}\tilde\Phi_{0}(\zeta)]=H(\zeta)
\overline{\lambda(z(\zeta))}X[z(\zeta)], \quad \zeta\in L^*, \label{e2.12}\\
\operatorname{Im}[\overline{\Lambda_{0}(a'_{j})}\tilde\Phi_{0}(a'_{j})]=0,\quad j\in J. \label{e2.13}
\end{gather}
On the basis of the uniqueness theorem
(see \cite[Theorems 3.2--3.4, Chapter IV]{w1}), we conclude that
$\tilde \Phi_{0}(\zeta)=0$, $\zeta\in \bar G$. However, from
$C[R(z(\zeta))\tilde\Phi_{n}(\zeta),\bar G]=1$, it follows that
there exists a point $\zeta_{*}\in \overline G$, such that
$|R(z(\zeta_*))\tilde\Phi_{0}(\zeta_{*})|=1$. This contradiction
proves that \eqref{e2.8} holds. Afterwards using the method which leads
from \eqref{e2.8} to \eqref{e2.11}, the estimate \eqref{e2.7} can be derived.
\end{proof}

\begin{theorem} \label{thm2.2}
Under the same conditions as in
Theorem \ref{thm2.1}, any solution $w(z)$ of Problem $\rm B$ for \eqref{e1.1}
satisfies
\begin{equation}
\begin{gathered}
\hat C_{\delta}[w(z),\overline D]=C_{\delta}[R(z)w(z),\overline D]\le M_{9}k_{*}, \\
\hat L^1_{p_0}[w,\overline D]=L_{p_{0}}[|RSw_{\bar z}|+|RSw_z|,\overline
D]\le M_{10}k_{*},
\end{gathered}\label{e2.14}
\end{equation}
where $\delta,p_{0}$ are as stated in Theorem \ref{thm2.1},
$k_{*}=k_{1} +k_{2}+k_{3}$,
$M_j=M_j(q_{0},p_{0},\delta,k_0,D)$ $(j=9,10)$.
\end{theorem}


\begin{proof}
 If $k_{*}=0$, i.e. $k_{1}=k_{2}=k_{3}=0$,
from Theorem \ref{thm2.3} below, it follows that $w(z)=0$, $z\in D$. If
$k_{*}>0$, it is easy to see that $W(z)=w(z)/{k_{*}}$ satisfies the
complex equation and boundary conditions
\begin{gather}
W_{\bar z}-Q_{1}W_{z}-Q_{2}\overline{W_{z}}-A_{1}W-A_{2}\overline W=A_{3}/{k_{*}},
\quad z\in D,\label{e2.15}\\
\operatorname{Re}[\overline{\lambda(z)}W(z)]=[r(z)+h(z)\overline{\lambda(z)}X(z)]/{k_{*}},\quad z\in\Gamma^*,
 \label{e2.16}\\
\operatorname{Im}[\overline{\lambda(a_{j})}W(a_{j})]=b_{j}/{k_{*}},\quad j\in J,\label{e2.17}
\end{gather}
Noting that
$L_{p}[A_{3}/{k_{*}},\overline D]\le 1$,
$C_{\alpha}[R(z)r(z)/{k_{*}},\Gamma] \leq 1$,
$| b_{j}/{k_{*}}| \le 1$, $j\in J$ and according to the
proof of Theorem \ref{thm2.1}, we have
\begin{equation}
\hat C_{\delta}[W(z),\overline D]\leq M_{9},\quad
\hat L^1_{p_{0}}[W(z),\overline D] \le M_{10}.\label{e2.18}
\end{equation}
From the above estimates, it
follows that \eqref{e2.14} holds.
\end{proof}

Next, we prove the uniqueness of solutions of Problem  B for
the complex equation \eqref{e1.1}. For this, we need to add the following
condition: For any continuous functions $w_1(z),w_2(z)$ in $D^*$ and
$U(z)\,(R(z)S(z)U(z)\in L_{p_0}(\overline{D})$, there is
\begin{equation}
F(z,w_1,U)-F(z,w_2,U)=Q(z,w_1,w_2,U)U_z+A(z,w_1,w_2,U)(w_1-w_2),\label{e2.19}
\end{equation}
in which $|Q(z,w_1,w_2,U)|\leq q_0\,(<1),\,A(z,w_1,w_2,U)\in
L_{p_0}(\overline{D})$. When \eqref{e1.1} is linear, \eqref{e2.19} obviously holds.

\begin{theorem} \label{thm2.3}
 If Condition {\rm C1--C3} and \eqref{e2.19} hold, then the solution
of Problem {\rm B} for \eqref{e1.1} is unique.
\end{theorem}

\begin{proof} Let $w_{1}(z),w_{2}(z)$ be
two solutions of Problem  B for \eqref{e1.1}. By Condition  (C1)--(C3) and
\eqref{e2.19}, we see that $w(z)=w_{1}(z)-w_{2}(z)$ is a solution of the
 boundary value problem
\begin{gather}
w_{\bar z}-\tilde Qw_{z}=\tilde A w,\quad z\in D,\quad \label{e2.20}\\
\operatorname{Re}[\overline{\lambda(z)}w(z)]=h(z)\overline{\lambda(z)}X(z),\quad z\in\Gamma^*, \label{e2.21}\\
\operatorname{Im}[\overline{\lambda(a_{j})}w(a_{j})]=0,\quad j\in J, \label{e2.22}
\end{gather}
where
\begin{gather*}
 \tilde Q=\begin{cases}
[F(z,w_{1},w_{1z})-F(z,w_{1},w_{2z})]
/(w_{1}-w_{2})_{z} &\text{for } w_{1z}\neq w_{2z}, \\
0 &\text{for } w_{1z }=w_{2z},\;z\in D,
\end{cases}
\\
\tilde A=\begin{cases}
[F(z,w_{1},w_{2z})-F(z,w_{2},w_{2z})]/{(w_{1}-w_{2})}
&\text{for } w_{1}(z)\neq w_{2}(z), \\
0&\text{for } w_{1}(z)=w_{2}(z),\;z\in D,
\end{cases}
\end{gather*}
and $|\tilde Q|\leq q_{0}<1$, $z\in D$, $L_{p_{0}}(\tilde A,\overline
D)<\infty$. According to the representation \eqref{e1.13}, we have
\begin{equation}
w(z)=\Phi[\zeta(z)]e^{\phi(z)},\quad\label{e2.23}
\end{equation}
where $\phi(z),\zeta(z),\Phi(\zeta)$ are as stated in Theorem \ref{thm2.1}.
It can be seen that the analytic function $\Phi(z)$ satisfies the
boundary conditions of Problem $\rm B_{0}$:
\begin{gather}
\operatorname{Re}[\overline{\Lambda(\zeta)}\Phi
(\zeta)]=H(\zeta)\overline{\lambda[z(\zeta)]}X[z(\zeta)],\quad
\zeta\in L^*= \zeta(\Gamma^*),\label{e2.24}
\\
\operatorname{Im}[\overline{\Lambda(a'_{j})}
\Phi(a'_{j})]=0,\quad j\in J,
\label{e2.25}
\end{gather}
where $\Lambda(\zeta)$, $H(\zeta)$ $(\zeta\in L)$, $a'_{j}$ $(j\in J)$ are as
stated in \eqref{e2.3}--\eqref{e2.5}. According to the method in the proof of
\cite[Theorem 1.2.4]{w9}, we can derive that $\Phi(\zeta)=0$,
$\zeta\in G=\zeta(D)$. Hence, $w(z)=\Phi[\zeta(z)]e^{\phi(z)}=0$; i.e.,
$w_{1}(z)=w_{2}(z)$, $z\in D$.
\end{proof}



\section{Solvability of discontinuous
Riemann-Hilbert problems}

Now we prove the existence of solutions of Problem  B for
equation \eqref{e1.1} by the Leray-Schauder theorem.

\begin{theorem} \label{thm3.1}
Suppose that \eqref{e1.1}
satisfies Conditions {\rm (C1)--(C3)} and \eqref{e2.19}.
Then the discontinuous boundary value problem, Problem {\rm B},
for \eqref{e1.1} has a solution.
\end{theorem}

\begin{proof} We discuss the complex equation \eqref{e1.1};
i.e.,
\begin{equation}
w_{\overline{\tilde z}}=F(z,w,w_z),\,F(z,w,w_z)
=Q_1w_z+Q_2\overline w_{\bar z}+A_1w+A_2\overline w+A_3\quad\text{in } D.\label{e3.1}
\end{equation}
To find a solution $w(z)$ of Problem B for equation \eqref{e3.1}
by the Leray-Schauder theorem, we consider
the equation \eqref{e3.1} with the parameter $t\in[0,1]$
\begin{equation}
w_{\overline{\tilde z}}=tF(z,w,w_z),\,F(z,w,w_z)=Q_1w_z+Q_2\overline w_{\bar z}
+A_1w+A_2\overline w+ A_3\quad \text{in }D,\label{e3.2}
\end{equation}
and introduce a bounded open set $B_M$ of the Banach space
$B=\hat C(\overline{D})\cap\hat L^1_{p_0}(\overline D)$, whose elements are
functions $w(z)$ satisfying the condition
\begin{equation}
\begin{aligned}
w(z)&\in\hat C(\overline D)\cap\hat L^1_{p_0}(\overline
D):\hat C[w,\overline{D}]+\hat L^1_{p_0}[w,\overline D] \\
&=C[R(z)w(z),\overline D]+L_{p_{0}}[|RSw_{\bar z}|+|RSw_z|,\overline
D]<M_{11},
\end{aligned} \label{e3.3}
\end{equation}
where $M_{11}=1+M_1+M_2,\,M_1,M_2,\delta$ are constants as stated
in \eqref{e2.1} and \eqref{e2.2}. We choose an arbitrary function
$W(z)\in\overline B_M$ and substitute it in the position of $w$ in $F(z,w,w_z)$.
 By using the method in the proof of \cite[Theorem 6.6, Chapter V]{w1}
 and \cite[Theorem 1.2.5]{w9},  a solution
$w(z)={\it\Phi}(z)+{\it\Psi}(z)=W(z)+T_0(tF)$ of Problem  B for
the complex equation
\begin{equation}
w_{\overline{\tilde z}}=tF(z,W,W_z)\label{e3.4}
\end{equation}
can be found. Noting that
$tR(z)S(z)F[z,W(z),W_z]\in L_\infty(\overline{D})$, the above solution
of Problem B for \eqref{e3.4} is unique.
Denote by $w(z)=T[W,t]\,(0\le t\le1)$ the mapping from
$W(z)$ to $w(z)$. From Theorem \ref{thm2.2}, we know that if $w(z)$ is a
solution of Problem B for the equation
\begin{equation}
w_{\overline{\tilde z}}=tF(z,w,w_z)\quad \text{in }D,\label{e3.5}
\end{equation}
then the function $w(z)$ satisfies the estimate
\begin{equation}
\hat C[w,\overline{D})]<M_{11}.\label{e3.6}
\end{equation}
Set $B_0=B_M\times[0,1]$. Now we verify the three conditions of the
Leray-Schauder theorem:

(1) For every $t\in[0,1]$, $T[W,t]$ continuously maps the Banach
space $B$ into itself, and is completely continuous in $\overline{B_M}$.
In fact, we arbitrarily select a sequence $W_{n}(z)$ in
$\overline{B_M}$, $n=0,1,2,\dots $, such that
$\hat C[W_{n}-W_{0},\overline{D}]\to 0$ as $n\to\infty$.
By Condition {\rm C}, we see that
$L_\infty[RS(F(z,W_n,W_{nz})$ $-F(z,W_{0},W_{0z})),\overline D]\to 0$ as
$n\to\infty$. Moreover, from $w_{n}=T[W_{n},t]$,
$w_{0}=T[W_{0},t]$,
it is easy to see that $w_{n}-w_{0}$ is a solution of Problem B for
the following complex equation
\begin{equation}
(w_{n}-w_0)_{\overline{\tilde z}}=t[F(z,W_n,W_{nz})-F(z,W_0,W_{0z})]\quad
\text{in }D,\label{e3.7}
\end{equation}
and then we can obtain the estimate
\begin{equation}
\hat C[w_{n}-w_m,\overline D)]\le2k_0\hat C[W_{n}(z)-W_{0}(z),\overline D].
\label{e3.8}
\end{equation}
Hence $\hat C[w_{n}-w_{0},\overline D]\to 0$ as $n\to\infty$. In
addition for $W_{n}(z) \in\overline{B_M},\,n=1,2,\dots $, we have
$w_n=T[W_n,t]$, $w_m=T[W_m,t]$, $w_n,w_m\in B_M$, and then
\begin{equation}
(w_{n}-w_m)_{\overline{\tilde z}}=t[F(z,W_n,W_{nz})
-F(z,W_m,W_{mz})]\quad \text{in }D,
\label{e3.9}
\end{equation}
where $L_\infty[RS(F(z,W_n,W_{nz})-F(z,W_m,W_{mz})),\overline D]\le
2k_0M_5$. Hence similarly to the proof of Theorem \ref{thm2.2}, we can obtain
the estimate
\begin{equation}
\hat C[w_{n}-w_m,\overline{D}]\le 2M_9k_0M_{11}.\label{e3.10}
\end{equation}
Thus there exists a function $w_0(z)\in B_M$, from $\{w_n(z)\}$ we
can choose a subsequence $\{w_{n_k}(z)\}$ such that
$\hat C[w_{n_k}-w_0,\overline{D}]\to0$ as $k\to\infty$. This shows that
$w=T[W,t]$ is completely continuous in $\overline{B_M}$. Similarly we can
prove that for $W(z)\in\overline{B_M}$, $T[W,t)$ is uniformly continuous
with respect to $t\in [0,1]$.

(2) For $t=0$, it is evident that $w=T[W,0]={\it\Phi}(z)\in B_M$.

(3) From the estimate \eqref{e2.14}, we see that $w=T[W,t]$
$(0\le t\le1)$ does not have a solution $w(z)$ on the boundary
 $\partial B_M=\overline{B_M}\backslash B_M$.

Hence by the Leray-Schauder theorem, we know that there exists a
function $w(z)\in B_M$, such that $w(z)=T[w(z),t]$, and the function
$w(z)\in\hat C_\delta(\overline {D})$ is just a solution of Problem B for
the complex equation \eqref{e1.1}.
\end{proof}

Moreover, we can derive the solvability result of Problem  A
for \eqref{e1.1} as follows.

\begin{theorem} \label{thm3.2}
Under the same conditions as in Theorem \ref{thm3.1}, the following statements hold.

$(1)$ If the index $K\ge N$, then Problem {\rm A} for \eqref{e1.1} is
solvable, if $N$ solvability conditions hold, under these
conditions, its general solution includes $2K+1$ arbitrary real
constants.

$(2)$ If $K<0$, then Problem {\rm A} for \eqref{e1.1} is solvable under
$-2K-1$ solvability conditions.
\end{theorem}

\begin{proof}
Let the solution $w(z)$ of Problem  B for \eqref{e1.1} be substituted
into the boundary condition \eqref{e1.8}. If
the function $h(z)=0$, $z\in\Gamma$; i.e.,
\begin{gather*}
 h_j=0,\quad j=1,\dots ,N,\quad \text{if }K\ge0, \\
h_j=0,\quad j=[1-(-1)^{2K}]/2,,\dots ,N, \quad\text{if } K<0, \\
h_m^\pm=0,\quad m=1,\dots, [|K|+1/2]-1,\quad\text{if } K<0,
\end{gather*}
then the function $w(z)$ is just a solution of Problem A
for \eqref{e1.1}. Hence the
total number of above equalities is just the number of solvability
conditions as stated in this theorem. Also note that the real
constants $b_j(j\in J)$ in \eqref{e1.10} are arbitrarily chosen. This shows
that the general solution of Problem A for \eqref{e1.1} includes
the number of arbitrary real constants as stated in the
theorem.
\end{proof}

The above theorem shows that the general solution of Problem A
for \eqref{e1.1} includes the number of arbitrary real constants as
stated in the above theorem. In fact, for the linear case of the
complex equation \eqref{e1.1} satisfying Conditions  (C1)--(C3), namely
\begin{equation}
w_{\bar z}=Q_1(z)w_z+Q_2(z)\bar w_{\bar z}+A_1(z)w
+A_2(z)\bar w+A_3(z)\quad\text{in } D,\label{e3.11}
\end{equation}
the general solution of Problem  A with
the index $K\ge0$ can be written as
\begin{equation}
w(z)=w_0(z)+\sum^{2K+1}_{n=1}d_nw_n(z),\label{e3.12}
\end{equation}
where $w_0(z)$ is a solution of nonhomogeneous boundary value
problem (Problem  A), and $d_n$ $(n=1,\dots ,2K+1)$ are the
arbitrary real constants, $w_n(z)$ $(n=1,\dots ,2K+1)$ are linearly
independent solutions of homogeneous boundary value problem (Problem
$\rm A_0$), which can be satisfied the point conditions
$$
\operatorname{Im}[\overline{\lambda(a_j)}w_n(a_j)]=\delta_{jn},\quad j,n=1,\dots ,2K+1,\;K\ge 0,
$$
where $\delta_{jn}=1$, if $j=n=1,\dots ,2K+1$ and $\delta_{jn}=0$,
if $j\ne n$, $1\le j,n\le 2K+1$.


\section{Formulation of the general discontinuous Poincar\'e problem}

Now we discuss the general discontinuous Poincar\'e problem for some
nonlinear elliptic equations of second order in multiply connected
domains and its new well-posedness.

Let $D$ be a bounded $(N+1)$-connected domain point with the
boundary $\Gamma=\cup^{N}_{j=0} \Gamma_{j}$ in $\mathbb{C}$ as stated in Section
1. We consider the nonlinear elliptic equation of second order in
the complex form
\begin{equation}
\begin{gathered}
u_{z\bar z}=F(z,u,u_{z},u_{zz}),\quad
F=\operatorname{Re}[Qu_{zz}+ A_{1}u_{z}]+A_{2}u+A_{3}, \\
Q=Q(z,u,u_{z},u_{zz}),\quad
A_{j}=A_{j}(z,u,u_{z}),\quad j=1,2,3,
\end{gathered}\label{e4.1}
\end{equation}
satisfying the following conditions.
\begin{itemize}
\item[(C4)]
$Q(z,u,w,U),A_{j}(z,u,w)$ $(j=1,2,3)$ are continuous in
$u\in{\mathbb{R}}$, $w\in{\mathbb{C}}$ for almost every point
$z\in D$, $U\in{\mathbb{C}}$, and $Q=0$, $A_{j}=0$ $(j=1,2,3)$
for $z\in\mathbb{C}\backslash D$.

\item[(C5)]  The above functions are measurable in $z\in D$ for all
continuous functions $u(z),w(z)$ in $D$, and satisfy
\begin{equation}
L_{p}[A_1(z,u,w),\overline D]\le k_0,\quad
L_{p}[A_1(z,u,w),\overline D]\le\varepsilon k_0,\quad
L_{p}[A_3(z,u,w),\overline D]\le k_1,
\label{e4.2}
\end{equation}
in which $p,p_0,k_{0},k_1$ are non-negative constants with
$2<p_{0}\le p$, $\varepsilon$ is a sufficiently small positive constant.

\item[(C6)] Equation \eqref{e4.1} satisfies the uniform ellipticity condition,
namely for any number $u\in {\mathbb{R}}$ and $w,U_{1},U_{2}\in{\mathbb{C}}$,
the inequality
\begin{equation}
|F(z,u,w,U_{1})-F(z,u,w,U_{2})|\le q_{0}|U_{1}-U_{2}|,\label{e4.3}
\end{equation}
holds for almost every point $z\in D$ holds, where $q_0<1$ is a
non-negative constant.
\end{itemize}

Now, we formulate the general discontinuous boundary value problem
as follows.

\subsection*{Problem P}
Find a solution $u(z)$ of  \eqref{e4.1}, which is continuously
differentiable in $D^*=\overline D\backslash Z$, and satisfies the boundary condition
\begin{equation}
\frac12\frac {\partial u}{\partial {\nu}}+c_{1}(z)u=c_{2}(z),\;{\rm i.e.}\;
\operatorname{Re}[\overline{\lambda(z)}u_z]+c_1(z)u=c_2(z),\quad
z\in\Gamma^*=\Gamma\backslash Z,\label{e4.4}
\end{equation}
in which $\lambda(z)=a(z)+ ib(z),\,|\lambda(z)|=1$ on
$\Gamma$, and $Z=\{t_1,t_2,\dots ,t_m \}$ are the first kind of
discontinuous points of $\lambda(z)$ on $\Gamma$, and $\lambda(z),c(z)$
satisfies the conditions
\begin{equation}
\begin{gathered}
C_\alpha[\lambda(z),\hat\Gamma_j]\le k_{0},\quad
C_{\alpha}[|z-t_{j-1}|^{\beta_{j-1}}
|z-t_j|^{\beta_j}c_1(z),\hat\Gamma_j]\le \varepsilon k_0, \\
C_{\alpha}[|z-t_{j-1}|^{\beta_{j-1}}
|z-t_j|^{\beta_j}c_2(z),\hat\Gamma_j]\le
k_2,\quad j=1,\dots ,m,
\end{gathered}\label{e4.5}
\end{equation}
in which ${\hat\Gamma}_j$ is an arc
from the point $t_{j-1}$ to $t_{j}$ on ${\hat\Gamma}$,
${\hat\Gamma}_j$, $(j=1,2,\dots ,m)$ does not include the end points, and
$\alpha,\varepsilon,\beta_j$ are positive constants
with $1/2<\alpha<1$ and $\beta_j<1$, $j=1,\dots ,m$.
Denote by $\lambda(t_j-0)$ and $\lambda(t_j+0)$ the left
limit and right limit of $\lambda(z)$ as $z\to t_j$ $(j=1,2,\dots ,m)$ on
$\Gamma$, and
\begin{equation}
\begin{gathered}
 {\rm e}^{ i\phi_j}= \frac{\lambda(t_j-0)}{\lambda(t_j+0)},\quad
\gamma_j =\frac1{\pi  i}\ln[\frac{\lambda(t_j-0)}{\lambda(t_j+0)}]=\frac{\phi_j}\pi-K_j,\\
K_j=[\frac{\phi_j}\pi]+J_j,\quad J_j=0\text{ or }1,\;j=1,
\dots ,m,
\end{gathered}\label{e4.6}
\end{equation}
in which $0\le\gamma_j<1$ when $J_j=0$, and
$-1<\gamma_j<0$ when $J_j=1$, $j=1,\dots , m$. The number
\begin{equation}
K=\frac1{2\pi}\Delta_\Gamma\arg\lambda(z)=\sum_{j=1}^m\frac{K_j}2\label{e4.7}
\end{equation}
is called the index of Problem  P.
Let $\beta_j+|\gamma_j|<1$ for $j=1,\dots ,m$, we require
that the solution $u(z)$ possess the property
\begin{equation}
\begin{gathered}
 R(z)u_z\in C_\delta(\overline D),\quad R(z)=\prod^{m}_{j=1}|z-t_j|^{\eta_j/\tau^2},
 \\
\eta_j =\begin{cases}
\beta_j+\tau, &\text{for }\gamma_j\ge0,\; \gamma_j<0,\beta_j\ge|\gamma_j|, \\
|\gamma_j|+\tau, &\text{for }\gamma_j<0,\;\beta_j<|\gamma_j|,\,j=1,\dots ,m,
\end{cases}
\end{gathered} \label{e4.8}
\end{equation}
in the neighborhood ($\subset D$) of $z_j$, where
$\tau\le\min(\alpha,1-2/p_0)$, $\delta<\min(\beta_1,\dots ,\beta_m,\tau$
are two small positive constants.

We mention that the first boundary value problem, second boundary
value problem and third boundary value problem; i.e., regular oblique
derivative problem are the special cases of Problem  P, because
their boundary conditions are the continuous boundary conditions,
and their indexes are equal to $K=N-1$. Now $2K$ can be
equal to any positive or negative integer, hence Problem P is a very
general boundary value problem. Because Problem P is not certainly
solvable, In the following, we introduce a new well-posedness of
discontinuous Poincar\'e boundary value problem for the nonlinear
elliptic equations of second order, namely

\subsection*{Problem Q}
 Find a continuous solution $[w(z),u(z)]$ of the complex equation
\begin{equation}
\begin{gathered}
w_{\bar z}=F(z,u,w,w_{z}),\quad z\in D, \\
F=\operatorname{Re}\,[Qw_{z}+A_{1}w]+A_{2}u+A_{3},
\end{gathered} \label{e4.9}
\end{equation}
satisfying the boundary condition
\begin{equation}
\operatorname{Re} [\overline{\lambda(z)}w(z)]+c_1(z)u
=c_2(z)+h(z)\overline{\lambda(z)}X(z),\quad z\in{\Gamma^*},\label{e4.10}
\end{equation}
and the relation
\begin{equation}
u(z)=\operatorname{Re}\int_{a_0}^z[w(z)+\sum_{j=1}^N\frac{{i}d_j}{z-z_j}{\rm d}z]
+b_0,\label{e4.11}
\end{equation}
where $a_0=1$, $b_0$ is a real constant,
$d_j$ $(j=1,\dots ,N)$ are appropriate real constants such that the
function determined by the integral in \eqref{e4.11} is single-valued in
$D$, and the undetermined function $h(z)$ is
\begin{equation}
h(z)=\begin{cases}
0, & z\in \Gamma_0,\; K\ge 0,  \\
h_j, &z\in \Gamma_{j},j=1,\dots ,N,\; K\ge 0, \\
h_{j}, &z\in\Gamma_j,\;j=1,\dots ,N,\; K<0, \\
[1+(-1)^{2K}]h_{0}\\
+\operatorname{Re} \sum^{[|K|+1/2]-1}_{m=1}(h^{+}_{m}
+  ih^-_{m})z^{m}, & z\in \Gamma_{0},\; K<0,
\end{cases} \label{e4.12}
\end{equation}
in which $h_j$ $(j=[1-(-1)^{2K}]/2,\dots ,N+1)$ are unknown real constants to
be determined appropriately, and $h_{N+1}(=h_0)=0$, if $2|K|$ is an
odd integer. And
\begin{equation}
\begin{aligned}
\Pi(z)&=\prod_{j=1}^{m_0}(z-t_j)^{\gamma_j}\prod_{l=l}^{N}(z-z_l)
^{-[\tilde{K}_l]}
\prod_{j=m_0+1}^{m_1}\Big(\frac{z-t_j}{z-z_1}\Big)^{\gamma_j}\dots\\
&\quad \times \prod_{j=m_{N_0-1}+1}^{m_{N_0}}
\Big(\frac{z-t_j}{z-z_{N_0}}\Big)^{\gamma_j}
\prod_{j=m_{N_0}+1}^{m_{N_0+1}}\Big(\frac{z-t_j}{z-z_{N_0+1}}\Big)^{
\gamma_j}\Big(\frac{z-t'_{N_0+1}}{z-z_{N_0+1}}\Big)\dots\\
&\quad \times \prod_{j=m_{N-1}+1}^m
\Big(\frac{z-t_j}{z-z_N}\Big)^{\gamma_j}\Big(\frac{z-t'_N}{z-z_N}
\Big),
\end{aligned} \label{e4.13}
\end{equation}
where $\tilde{K}_l=\sum^{m_l}_{j=m_{l-1}+1}K_{j}$
are denoted the partial indexes on $\Gamma_l$ $(l=1,\dots ,N)$;
$t_j\in\Gamma_0$ $(j=1,\dots ,m_0)$, $t_j\in\Gamma_1$
$(j=m_0+1,\dots ,m_1)$,\dots , $t_j\in\Gamma_N$, $(j=m_{N-1}+1\dots ,m)$
are all discontinuous points of $\lambda(z)$ on $\Gamma$.
If $\lambda(z)$ on $\Gamma_l$ $(0\le l\le N)$ has no discontinuous point,
then we can choose a point $t_j\in \Gamma_l\,(0\le l\le N)$ as a discontinuous
point of $\lambda(z)$ on $\Gamma_l$ $(0\le l\le N)$, in this case
$t_j=t_{j+1}$. There is in no harm assuming that the partial indexes
$K_l$ of $\lambda(z)$ on $\Gamma_l$ $(l=0,1,\dots ,N_0\,(\le N))$ are integers,
and the partial indexes $K_l$ of $\lambda(z)$ on
$\Gamma_l$ $(j=N_0+1,\dots ,N)$ are no integers, and we choose the points
$t'_l$ $(\in \Gamma_l,\,l=N_0+1,\dots ,N)$ are not discontinuous points on
$\Gamma_l$ $(l=N_0+1,\dots ,N)$ respectively. Similarly to
\eqref{e1.7}-\eqref{e1.12},
\cite[Chapter V]{w1}, we see that
$$
\frac{\lambda(t_j-0)}{\lambda(t_j+0)}\overline{\Big[
\frac{Y(t_j-0)}{Y(t_j+0)}\Big]}
=\frac{\lambda(t_j-0)}{\lambda(t_j+0)}e^{-i\pi\gamma_j}=\pm 1,
$$
it only needs to charge the symbol on some arcs on $\Gamma$, then
$\lambda(z)\overline{Y(z)}/|Y(z)|$ on $\Gamma$ is continuous. In this case, the
new index
$$
\kappa=\frac1{2\pi}\Delta_\Gamma[\lambda(z)\overline{Y(z)}]=K-\frac{N-N_0}2
$$
is an integer; and
\begin{equation}
\begin{gathered}
X(z)=\begin{cases}
iz^{[\kappa]}e^{iS(z)}Y(z), & z\in\Gamma_0, \\
ie^{i\theta_j}e^{iS(z)}Y(z), & z\in\Gamma_j,\; j=1,\dots ,N,
\end{cases}\\
\operatorname{Im}[\overline{\lambda(z)}X(z)] =0,\quad z\in\Gamma, \\
\operatorname{Re} S(z)=\begin{cases}
\arg\lambda(z)- [\kappa]\arg z-\arg Y(z),& z\in\Gamma_0, \\
\arg\lambda(z)-\arg Y(z)-\theta_j, & z\in\Gamma_j,\; j=1,\dots ,N,
\end{cases}\\
\operatorname{Im}[S(1)]=0,
\end{gathered} \label{e4.14}
\end{equation}
where $S(z)$ is a solution of the
modified Dirichlet problem with the above boundary condition for
analytic functions, $\theta_j\,(j=1,\dots ,N)$ are real constants.

 If $K\ge0$, we require that the solution $w(z)=u_z$ satisfy the
point conditions
\begin{equation}
\operatorname{Im}[\overline{\lambda(a_j)}w(a_j)]=b_j,\quad
j\in J=\{1,\dots ,2K+1\},\quad\text{if }K\ge 0, \label{e4.15}
\end{equation}
in which
$a_j\in\Gamma_0\,(j\in J)$ are distinct points; and $b_j$
$(j\in J)$, $b_0$ are real constants satisfying the conditions
\begin{equation}
|b_j|\le k_3,\quad j\in J\cup\{0\}\label{e4.16}
\end{equation}
with the a positive constant $k_3$.
This is the well-posedness of
Problem  P for equation \eqref{e4.1}.

 Problem Q with the conditions $A_3(z)=0$ in \eqref{e4.1},
$c_2(z)=0$ in \eqref{e4.10} and $b_j=0$ $(j\in J\cup\{0\})$ in
\eqref{e4.11}, \eqref{e4.15} will be called Problem $\rm Q_0$.

The undetermined real constants
$d_j,h_j$ $(j=[1-(-1)^{2K}]/2,\dots ,N)$, $h^\pm_m$ $(m=1\dots ,-K-1)$
in \eqref{e4.11}, \eqref{e4.12} are for ensuring the existence of continuous
solutions, and $b_j$ $(j=0,1,\dots ,2K+1)$ in \eqref{e4.11}, \eqref{e4.15} are
for ensuring the uniqueness of continuous solutions in $\overline D$.

 Now we introduce the previous well-posedness of the discontinuous
Poincar\'e problem of elliptic complex equations.

\subsection*{Problem  R}
Find a continuous solution $w(z)$ in $D$ of  \eqref{e4.9}
with the modified boundary condition \eqref{e4.10} and the
relation \eqref{e4.11},
where
\begin{equation}
h(z)=\begin{cases}
0, & z\in\Gamma,\;K>N-1, \\
 0, &z\in\Gamma_j,\;j=1,\dots ,[K]+1,\; 0\leq K\le N-1, \\
h_j, &z\in\Gamma_j,\;j=[K]+2,\dots ,[K]+1+N-K',\\
 &\quad 0\leq K\le N-1,\\
 h_{j}, & z\in\Gamma_j,\,j=1,\dots ,N,\;K<0, \\
[1+(-1)^{2K}]h_0\\
+\operatorname{Re}
\sum^{[|K|+1/2]-1}_{m=1}(h^{+}_{m}\\
+ ih^{-}_{m})z^{m},
& z\in\Gamma_0,\;K<0,
\end{cases} \label{e4.17}
\end{equation}
in which $K'=[K+1/2]$, $[K]$  denotes
the integer part of $K$, $h_0,h^+_m,h^-_m$, $(m=1,\dots ,[|K|+1/2]-1)$
are unknown real constants to be determined appropriately, and
$h_{N+1}(=h_0)=0$, if $2|K|$ is an odd integer; and the solution
$w(z)$ satisfies the point conditions
\begin{equation}
\operatorname{Im}[\overline{\lambda(a_j)}w(a_j)]=b_j,\quad
j\in J=\begin{cases} 1,\dots ,2K-N+1,&\text{if } K>N-1, \\
1,\dots ,[K]+1,&\text{if } 0\leq K\leq N-1,
\end{cases} \label{e4.18}
\end{equation}
in which where $a_j\in\Gamma_j$ $(j=1,\dots ,N_0)$,
$a_j\in\Gamma_0$ $(j=N_0+1,\dots ,2K-N+1,\text{if }K\ge N)$
are distinct points; and when
 $[K]+1>N_0$, $a_j\in \Gamma_j$ $(j=1,\dots ,N_0)$,
$a_j\in\Gamma_0$ $(j=N_0+1,\dots ,[K]+1, \text{ if } 0\le K<N)$, otherwise
$a_{j}\in\Gamma_j$ $(j=1,\dots ,[K]+1,\text{ if }0\le K<N)$ are
distinct points; and
$$
|b_{j}|\le k_{3},\quad j\in J
$$
with a non-negative constant $k_{3}$.

The equivalence of Problem  Q for  equation \eqref{e4.9} and
Problem  R for \eqref{e4.9} can be verified. We can see that the
advantages of the new well-posedness. We mention the equivalence of
these well-posedness, i.e. if there exists the unique solvability of
one well-posedness, then we can derive that another well-posedness
possesses the unique solution. Hence it is best to choose the most
simple well-posedness.


\section{Estimates for solutions of discontinuous
Poincar\'e problems}

First of all, we prove the following result.

\begin{theorem} \label{thm5.1}
Suppose that \eqref{e4.1} satisfies Conditions {\rm (C4)--(C6)}
and $\varepsilon$ in \eqref{e4.2}, \eqref{e4.5} is
small enough. Then Problem $\rm Q_0$ for equation \eqref{e4.1} in $D$ has
only the trivial solution.
\end{theorem}

\begin{proof}
 Let $[u(z),w(z)]$ be any
solution of Problem $\rm Q_0$ for  equation \eqref{e4.9}; i.e.,
$[w(z),u(z)]$ satisfies the  complex equation with boundary
conditions
\begin{gather}
w_{\bar z}+\operatorname{Re}[Qw_z+A_1w]=-A_2u\quad\text{in }D,\label{e5.1}
\\
\begin{gathered}
\operatorname{Re}[\overline{\lambda(z)}w(z)]+c_1(z)u=h(z)\overline{\lambda(z)}X(z),\quad
z\in\Gamma^*, \\
\operatorname{Im}[\overline{\lambda(a_j)}w(a_j)]=0,\quad j\in J,\;u(a_0)=0.
\end{gathered} \label{e5.2}
\end{gather}
and the relation
\begin{equation}
u(z)=\operatorname{Re}\int_{a_0}^z[\,w(z)+\sum_{j=1}^N\frac{{i}d_j}{z-z_j}{\rm
d}z],\label{e5.3}
\end{equation}
where $a_0=1$. From the three formulae in \eqref{e5.3}, we
see that
\begin{equation}
\begin{gathered}
d_j=\frac1{2\pi}\int_{\Gamma_j}w(z)d\theta,\quad j=1,\dots ,N,\\
C_\delta[R'(z)u(z),\overline D]\leq M_{12}C_\delta[R(z)w(z),\overline D],
\end{gathered}\label{e5.4}
\end{equation}
where $\delta$ ia a positive constant as stated in \eqref{e4.8},
$M_{12}=M_{12}(R,D)$ is a non-negative constant. From the conditions
\eqref{e4.2} and \eqref{e4.5}, we can obtain
\begin{equation}
\begin{gathered}
\begin{aligned}
L_{p_0}[R'A_2u,\overline D]
&\le L_{p_0}[A_2,\overline D]\,C[R'u,\overline D]
 \le \varepsilon k_0C[R'(z)u(z),\overline D]\\
&\le\varepsilon k_0C_\delta[R'(z)u(z),\overline D],
\end{aligned} \\
\begin{aligned}
C_\alpha[R(z)c_1(z)R'(z)u(z),\Gamma]
&\le C_\alpha[R(z)c_1(z),\Gamma ]C_\delta[R'(z)u(z),\overline D]\\
&\le\varepsilon k_0C_\delta[R'(z)u(z),\overline D],
\end{aligned}
\end{gathered} \label{e5.5}
\end{equation}
where $R(z)$ is as stated in \eqref{e4.8} and $|R(z)|\le1$
in $\overline D$. Thus by using the result of the Riemann-Hilbert boundary
value problem for the complex equation of first order (see
\cite[Theorems 3.2-3.4, Chapter V]{w1}
 and \cite[Theorem 6.1, Chapter VI]{w2}), the
following estimate of the solution $w(z)$ can be obtained, namely
\begin{equation}
C_\delta[R(z)w(z),\overline D]\le2\varepsilon k_0M_{13}C_\delta[R'(z)u(z),\overline D],\label{e5.6}
\end{equation}
where $M_{13}=M_{13}(q_0,p_0,\delta,k_0,D)$ is a non-negative constant.
 From the estimate \eqref{e5.4}, it follows the estimate about $u(z)$:
\begin{equation}
C_\delta[R'(z)u(z),\overline D]\le 2\varepsilon k_0M_{12}M_{13}C_\delta[R'(z)u(z),\overline D].
\label{e5.7}
\end{equation}
Provided that the positive number $\varepsilon$ in \eqref{e4.2} and \eqref{e4.5}
 is small enough, such that
\begin{equation}
2\varepsilon k_0M_{12}M_{13}<1,\label{e5.8}
\end{equation}
it can be derived that $u(z)\equiv0$ and then $w(z)\equiv0$ in $D$. Hence
Problem $\rm Q_0$ for  equation \eqref{e5.1} has only the trivial
solution. This completes the proof of Theorem \ref{thm5.1}.
\end{proof}

\begin{theorem} \label{thm5.2}
Let \eqref{e4.1} satisfy Conditions {\rm (C4)--(C6)}  and
\eqref{e4.2}, \eqref{e4.5} with the sufficiently
small positive number $\varepsilon$. Then any solution $[u(z),w(z)]$ of
Problem $\rm Q$ for \eqref{e4.9} satisfies the estimates
\begin{equation}
\begin{gathered}
\hat C^1_\delta[u,\overline D]=C_\delta[R'(z)u,\overline
D]+C_\delta[R(z)w(z), \overline D]\le M_{14}, \\
\hat L^1_{p_0}[w,\overline D]=L_{p_0}[|RSw_{\bar
z}|+|RSw_{z}|,\tilde D]\leq M_{15},
\end{gathered}\label{e5.9}
\end{equation}
where $R(z)$ and $,S(z)$ are
\begin{equation}
\begin{gathered}
R(z)=\prod^{m}_{j=1}|z-t_j|^{\eta_j/\tau^2},\quad
S(z)= \prod^{m}_{j=1}|z-t_j|^{1/\tau^2}, \\
\eta_j=\begin{cases} |\gamma_j|+\tau,&\text{if } \gamma_j<0,\;
\beta_j\leq |\gamma_j|, \\
\beta_j+\tau,&\text{if } \gamma_j\ge0,\; \gamma_j<0,\;
\beta_j>|\gamma_j|,
\end{cases}
\end{gathered}\label{e5.10}
\end{equation}
where $\gamma_j$ $(j=1,\dots ,m)$ are real constants as stated in \eqref{e4.6},
$\tau=\min(\alpha,1-2/p_0)$,
$\delta<\min[\beta_1,\dots , \beta_m, \tau]$ is a small positive constant,
 $k=k(k_0,k_1,k_2,k_3)$,
$M_j=M_j(q_0,p_0,\delta,k,D)$ $(j=14,15)$ are non-negative
constants only dependent on $q_0,p_0,\delta,k,D,\,j=3,4$.
\end{theorem}

\begin{proof}
 By using the reduction to
absurdity, we shall prove that any solution $u(z)$ of Problem Q
satisfies the estimate of bounded-ness
\begin{equation}
\hat C^1[u,\overline D]=
C[R'(z)u(z),\overline{D}]+C[R(z)w(z),\overline D]\le M_{16},\label{e5.11}
\end{equation}
in which $M_{16}=M_{16}(q_0,p_0,\delta,k,D)$ is a
non-negative constant. Suppose that \eqref{e5.11} is not true, then there
exist sequences of coefficients $\{A^{(m)}_j\}$ $(j=1,2,3)$,
$\{Q^{(m)}\}$, $\{\lambda^{(m)}(z)\}$, $\{c_j^{(m)}\}\,(j=1,2)$,
$b_j^{(m)}$ $(j\in  J\cup\{0\})$, which satisfy Conditions
(C4)--(C6) and \eqref{e4.5}, \eqref{e4.16}, such that
 $\{A^{(m)}_j\}$ $(j=1,2,3)$, $\{Q^{(m)}\}$,
$\{\lambda^{(m)}(z)\}$, $\{|z-t_{j-1}|^{\beta_{j-1}}
|z-t_j|^{\beta_j}c_j^{(m)}\}$ $(j=1,2)$ and $\{b_j^{(m)}\}$
$(j\in J\cup\{0\})$ in $\overline D,{\it\Gamma^*}$ converge weakly  or
 converge uniformly to $A^{(0)}_j$ $(j=1,2,3)$, $Q^{(0)}$,
$\lambda^{(0)}(z)$, $|z-t_{j-1}|^{\beta_{j-1}}
|z-t_j|^{\beta_j}c_j^{(0)}$ $(j=1,2)$, $b_j^{(0)}(j\in  J\cup\{0\})$
respectively, and the corresponding boundary value problem
\begin{equation}
w_{\bar z}-\operatorname{Re}[Q^{(m)}w_z+A^{(m)}_{1}w]-
A^{(m)}_2u= A^{(m)}_{3},\label{e5.12}
\end{equation}
 and
\begin{equation}
\begin{gathered}
\operatorname{Re}[\overline{\lambda(z)}w(z)]+c^{(m)}_1(z)u=c^{(m)}_2(z)+c(z)\overline{\lambda(z)}X(z)\quad
\text{on } \Gamma^*, \\
\operatorname{Im}[\overline{\lambda(a_j)}w(a_j)]=b^{(m)}_j,\quad j\in J,\quad
u(a_0)=b^{(m)}_0
\end{gathered}\label{e5.13}
\end{equation}
have the solutions
$\{u^{(m)}(z),w^{(m)}(z)\}$, but $\hat C^1[u^{(m)}(z),\overline D]$
$(m=1,2,\dots )$ are unbounded. Thus we can choose a subsequence of
$\{u^{(m)}(z),w^{(m)}(z)\}$ denoted by $\{u^{(m)}(z),w^{(m)}(z)\}$
again, such that $h_m=\hat C[u^{(m)}(z),\overline{D}]\to\infty$ as
$m\to\infty$, and assume that $H_m\ge\max[k_1,k_2,k_3,1]$. It is
easy to see that $\{\tilde u^{(m)}(z),\tilde
w^{(m)}(z)\}=\{u^{(m)}(z)/H_m,\tilde w^{(m)}(z)/H_m\}$ $(m=1,2,\dots )$
are solutions of the boundary value problems
\begin{gather}
\tilde w_{\bar z}-\operatorname{Re}[Q^{(m)}\tilde w_{z}+A^{(m)}_{1}\tilde w_{z}]
-A^{(m)}_{2} \tilde u=A^{(m)}_{3}/H_m, \label{e5.14}
\\
\begin{gathered}
\operatorname{Re}[\overline{\lambda(z)}\tilde{w}(z)]+c^{(m)}_1(z)
\tilde u=[c^{(m)}_2(z)+h(z)\overline{\lambda(z)}X(z)]/{H_m}\quad\text{on } \Gamma^*, \\
\operatorname{Im}[\overline{\lambda(a_j)}\tilde{w}(a_j)]=b^{(m)}_j/H_m,\quad j\in J,\quad
\tilde u(a_0)=b^{(m)}_0/H_m.
\end{gathered}\label{e5.15}
\end{gather}

We can see that the functions in the above equation and the boundary
conditions satisfy the condition (C4)--(C6), \eqref{e4.5},\eqref{e4.16} and
\begin{equation}
\begin{gathered}
|R'(z)u^{(m)}|/H_m\leq 1,\quad L_\infty[A^{(m)}_3/H_m,\overline{D}]\leq 1, \\
|R(z)c_2^{(m)}/H_m|\le1,\quad |b_j^{(m)}/H_m|\le1,\quad j\in J\cup\{0\},
\end{gathered}\label{e5.16}
\end{equation}
hence by using a similar method as in the proof of
\cite[Theorem 6.1, Chapter IV]{w2}, we can obtain the estimates
\begin{equation}
\hat C_{\delta}[\tilde u^{(m)}(z),\overline{D}]\le M_{17},\quad
\hat L^1_{p_0}[\tilde w^{(m)}(z),\overline D]\le M_{18},
\label{e5.17}
\end{equation}
where $M_j=M_j(q_0,p_0,\delta,k_0,D)$ $(j=17,18)$
are non-negative constants. Moreover from the sequence
$\{\tilde u^{(m)}(z)$, $\tilde w^{(m)}(z)\}$, we can choose a subsequence
denoted by $\{\tilde u^{(m)}(z),\tilde w^{(m)}(z)\}$ again, which in
$\overline D$ uniformly converge to $\tilde u_0(z),\tilde w_0(z)$
respectively, and $R(z)S(z)(\tilde w^{(m)})_{\bar z}$,
$R(z)S(z)(\tilde w^{(m)})_{z}$ in $D$ are weakly convergent. This
shows that $[\tilde u_0(z),\tilde w_0(z)]$ is a solution of the
boundary-value problem
\begin{gather}
\tilde w_{0\bar z}-\operatorname{Re}[Q^{(0)}\tilde w_{0z}+A^{(0)}_{1}\tilde
u_{0}]-A^{(0)}_{2}\tilde u_0=0,\label{e5.18}\\
\begin{gathered}
\operatorname{Re}[\overline{\lambda(z)}\tilde w_0(z)]+2
c^{(0)}_1(z)\tilde u_0=h(z)\overline{\lambda(z)}X(z)\quad \text{on }\Gamma^*, \\
\operatorname{Im}[\overline{\lambda(a_j)}\tilde{w}_{0}(a_j)]=0,\quad j\in J,\;\tilde u_0(a_0)=0.
\end{gathered}\label{e5.19}
\end{gather}
We see that \eqref{e5.18} is a
homogeneous equation, and \eqref{e5.19} is a homogeneous boundary
condition. On the basis of Theorem \ref{thm5.1}, the solution
$\tilde u_0(z)=0$, $\tilde{w}_0(z)=0$ however, from
$\hat C^1[\tilde u^{(m)}(z),\overline D]=1$, we can derive that there
exists a point $z^*\in\overline D$, such that
$|R'(z^*)\tilde u_0(z^*)|+|R(z^*)\tilde{w}_0(z^*)|\ne 0$. This is impossible.
This shows that the first estimate in \eqref{e5.9} are true.
Moreover it is not difficult to verify
the second estimate in \eqref{e5.9}.
\end{proof}

Now we prove the uniqueness  of solutions of Problem Q for
equation \eqref{e4.1} as follows.


\begin{theorem} \label{thm5.3}
 Suppose that \eqref{e4.1} satisfies conditions {\rm (C4)--(C6)} and the
following condition: for any real functions
 $R'(z)u_j(z)\in C(D^*),R(z)w_j(z)\in C(D^*)$,
$R(z)S(z)U(z)\in L_{p_0}(\overline D)$ $(j=1,2)$, the equality
\begin{equation}
F(z,u_1,w_1,U)-F(z,u_2,w_2,U)
=\operatorname{Re}[\tilde QU+\tilde A_1(w_1-w_2)]+\tilde A_2(u_1-u_2)
\label{e5.20}
\end{equation}
holds, where $|\tilde Q|\leq_0 <1$  in $D$,
$L_p[\tilde A_1,\overline D]\leq K_0$, $L_p[\tilde A_2,\overline D]\le\varepsilon k_0$ and \eqref{e4.5}
with the sufficiently small positive constant $\varepsilon$. Then Problem
$\rm Q$ for equation \eqref{e4.1} has at most one solution.
\end{theorem}

\begin{proof}
 Denote by $[u_j(z),w_j(z)](j=1,2)$ two solutions of Problem  Q for
\eqref{e4.9}, and substitute them into \eqref{e4.9}-\eqref{e4.11}
and \eqref{e4.15}, we see that
$[u(z),w(z)]=[u_1(z)-u_2(z),w_1(z)-w_2(z)]$ is a solution of
the  homogeneous boundary-value problem
\begin{gather*}
w_{\bar z}=\operatorname{Re}[\tilde Qw_z+\tilde A_1w]+\tilde A_2u,\quad z\in D,  \\
\operatorname{Re}[\overline{\lambda(z)}w(z)]+c_1(z)u(z)=h(z)\overline{\lambda(z)}X(z),\quad z\in\Gamma^*, \\
\operatorname{Im}[\overline{\lambda(a_j)}w(a_j)]=0,\quad j\in J, \\
u(z)=\int^z_{a_0}[w(z)dz+\sum^N_{j=1}\frac{id_j}
{z-z_j}dz]\quad \text{in }D,
\end{gather*}
the coefficients of which satisfy same
conditions of \eqref{e4.2},\eqref{e4.3},\eqref{e4.5} and \eqref{e4.16},
but $k_1=k_2=k_3=0$.
On the basis of Theorem \ref{thm5.1}, provided that $\varepsilon$ is sufficiently small,
we can derive that $u(z)=w(z)=0$ in $\overline{D}$; i.e.,
$u_1(z)=u_2(z)$, $w_1(z)=w_2(z)$ in $\overline{D}$.
\end{proof}

\section{Solvability results of discontinuous
Poincar\'e problem}

In this section, we shall prove the solvability of general
discontinuous Poincar\'e boundary value problem by the the method of
parameter extension.

\begin{theorem} \label{thm6.1}
 Suppose that the nonlinear elliptic equation \eqref{e4.1} satisfies
condition {\rm (C4)--(C6)}, \eqref{e5.20}, and $\varepsilon$ in
\eqref{e4.2}, \eqref{e4.5} is small enough.
Then there exists a solution $[u(z),w(z)]$ of Problem $\rm Q$ for
\eqref{e4.9} and
$[u(z),w(z)]\in B=\hat C^1_\delta(\overline{D})\cap\hat L^1_{p_0}(\overline D)$,
where $B=\hat C^1_\delta(\overline{D})\cap\hat L^1_{p_0}(\overline D)$ is a Banach
space; i.e., $\hat C^1_\delta[u,\overline{D}]<\infty,\,\hat L^1_{p_0}[w,\overline
D]<\infty$, and $p_0\,(>2)$ is stated as in \eqref{e5.9}.
\end{theorem}

\begin{proof}
 We introduce the nonlinear
elliptic equation with the parameter $t\in[0,1]$,
\begin{equation}
w_{\bar z}=tF(z,u,w,w_{z})+A(z),\label{e6.1}
\end{equation}
where $A(z)$ is any measurable function in $D$ and
$R(z)S(z)A(z)\in L_{p_0}(\overline{D})$, $2<p_0\le p$. Let $E$ be a subset of
$0\le t\le 1$ such that Problem Q is solvable for \eqref{e6.1} with
any $t\in E$ and any $R(z)S(z)A(z)\in L_{p_0}(\overline{D})$.
In accordance with the method in
the proof of \cite[Theorem 6.1, Chapter VI]{w2}, we can prove that when
$t=0$, Problem  Q has a unique solution $[u(z),w(z)]$
satisfying the complex equation and boundary conditions; i.e.,
\begin{gather}
w_{\bar z}=A(z),\;z\in D,\label{e6.2}\\
\begin{gathered}
\operatorname{Re} [\overline{\lambda(z)}w(z)]+c_1(z)u=
c_2(z)+h(z)\overline{\lambda(z)}X(z),\quad z\in{\Gamma^*}, \\
\operatorname{Im}[\overline{\lambda(a_j)}w(a_j)]=b_j,\,j\in J,
\end{gathered} \label{e6.3}
\end{gather}
and the relation
\begin{equation}
u(z)=\operatorname{Re}\int_{a_0}^z[w(z)+\sum_{j=1}^N\frac{{i}d_j}{z-z_j}{\rm d}z]
+b_0.\label{e6.4}
\end{equation}
This shows that the point set $E$ is not empty.

 From Theorem \ref{thm5.3}, We see that $[u(z),w(z)]\in B=\hat C^1_\delta(\overline
D)\cap\hat L^1_{p_0}(\overline D)$. Suppose that when
 $t=t_0\,(0\le t_0<1)$, Problem $\rm Q$ for the complex equation \eqref{e6.1}
has a unique solution, we shall prove that there exists a neighborhood of
$t_0$: $E=\{|t-t_0|\le\delta_0,0\le t\le 1,\delta_0>0\}$, so that for
every $t\in E$ and any function $R(z)S(z)A(z)\in L_{p_0}(\overline{D})$, Problem
Q for \eqref{e6.1} is solvable. In fact, the complex equation \eqref{e6.1}
can be written in the form
\begin{equation}
w_{\bar z}-t_0F(z,u,w,w_{z})=(t-t_0)F(z,u,w,w_{z})+A(z).
\label{e6.5}
\end{equation}
We  select an arbitrary function
$[u_0(z),w_0(z)]\in B=\hat C^1_\delta(\overline D)\cap\hat
L^1_{p_0}(\overline D)$, in particular $[u_0(z),w_0(z)]=0$ in $\overline{D}$.
Let $[u_0(z),w_0(z)]$ be replaced into the position of $u(z),w(z)$
in the right hand side of \eqref{e6.5}. By condition (C4)--(C6),
it is obvious that
$$
B_0(z)=(t-t_0)RSF(z,u_0,w_{0z},w_{0zz})+R(z)S(z)A(z)\in L_{p_0}(\overline{D}).
$$
Noting the \eqref{e6.5} has a solution $[u_1(z),w_1(z)]\in B$. Applying the
successive iteration, we can find out a sequence of functions:
$[u_n(z),w_n(z)]\in B$, $n=1,2,\dots $, which satisfy the complex
equations
\begin{equation}
w_{n+1z\bar z}-t_0F(z,u_{n+1},w_{n+1},w_{n+1z})
= (t-t_0)F(z,u_n,w_{n},w_{nz})+A(z),
\label{e6.6}
\end{equation}
for $n=,2,\dots$.
The difference of the above equations for $n+1$ and $n$ is as
follows:
\begin{equation}
\begin{aligned}
&(w_{n+1}-w_n)_{z\bar z}-t_0[F(z,u_{n+1},w_{n+1},w_{n+1z})-
F(z,u_{n},w_{n},w_{nz}] \\
&=(t-t_0)[F(z,u_{n},w_{n},w_{nz})-F(z,u_{n-1},w_{n-1},w_{n-1z})],\quad
n= 1,2,\dots .
\end{aligned}\label{e6.7}
\end{equation}
From conditions  (C4)--(C6), it can be seen that
\begin{equation}
\begin{gathered}
\begin{aligned}
&F(z,u_{n+1},w_{n+1},w_{n+1z})-F(z,u_{n},w_{n},w_{nz})=
F(z,u_{n+1},w_{n+1},w_{n+1z}) \\
&-F(z,u_{n+1},w_{n+1},w_{nz})+[F(z,u_{n+1},w_{n+1},w_{nz})
-F(z,u_{n},w_{n},w_{nz})] \\
&=\operatorname{Re}[\tilde Q_{n+1}(w_{n+1}-w_n)_{z}+\tilde A_{1n+1}(w_{n+1}-w_n)]+\tilde
A_{2n+1}(u_{n+1}-u_{n}),
\end{aligned} \\
|\tilde Q_{n+1}|\le q_0<1,\quad
L_{p_0}[\tilde A_{1n+1},\overline D]\le k_0,\quad
L_{p_0}[\tilde{A}_{2n+1},\overline D]\le\varepsilon k_0,
\end{gathered}
\label{e6.8}
\end{equation}
for $n=1,2,\dots$,
and
\begin{align*}
&L_{p_0}[RS(F(z,u_{n},w_{n},w_{nz})-F(z,u_{n-1},w_{n-1},
w_{n-1z})),\overline{D}] \\
&\le q_0L_{p_0}[RS(w_n-w_{n-1})_{z},\overline{D}]+k_0C_\delta[R(w_n-w_{n-1}),
\overline{D}] \\
&\le(q_0+k_0)[\hat C^1_\delta[u_n-u_{n-1},\overline{D}]+\hat
L^1_{p_0}[w_n-w_{n-1},\overline D]=(q_0+k_0)L_n.
\end{align*}
Moreover, $u_{n+1}(z)-u_n(z)$ satisfies the homogeneous boundary
conditions
\begin{equation}
\begin{gathered}
\operatorname{Re}[\overline{\lambda(z)}(w_{n+1}-w_{n})]+c_1(z)[u_{n+1}(z)
-u_n(z)]=h(z)\overline{\lambda(z)}X(z),\quad z\in\Gamma^*, \\
\operatorname{Im}[\overline{\lambda(a_j)}(w_{n+1}(a_j)-w_n(a_j))]=0,\quad
j\in J,\quad u_{n+1}(a_0)-u_n(a_0)=0.
\end{gathered} \label{e6.9}
\end{equation}
On the basis of Theorem \ref{thm5.2}, we have
\begin{equation}
 L_{n+1}=\hat C^1_\delta[u_{n+1}-u_n,\overline{D}]+\hat
L^1_{p_0}[w_{n+1}-w_{n},\overline{D}]\leq M_{19}|t-t_0|(q_0+k_0)
L_n,\label{e6.10}
\end{equation}
where
$M_{19}=(M_{17}+M_{18})k_*$, $M_{17}$ and $M_{18}$ are as stated in
\eqref{e5.17}. Provided $\delta_0>0$ is small enough, so that
$\sigma=\delta_0M_{19}(q_0+2k_0)<1$, it can be obtained that
\begin{equation}
L_{n+1}\le\sigma L_n\le \sigma^n L_1=\sigma^n[\hat C^1_\delta(u_1,\overline{D})
+\hat L^1_{p_0}(w_1,\overline{D})]\label{e6.11}
\end{equation}
for every $t\in E$. Thus
\begin{equation}
\begin{aligned}
&\hat C^1_\delta[u_n-u_m,\overline{D}]+\hat L^1_{p_0} [w_n-w_m,\overline{D}] \\
&\le L_n+L_{n-1}+\dots+L_{m+1}\le(\sigma^{n-1}+\sigma^{n-2}+\dots+\sigma^m) L_1 \\
&=\sigma^m(1+\sigma+\dots+\sigma^{n-m-1})L_1\\
&\le\sigma^{N+1} \frac{1-\sigma^{n-m}}{1-\sigma}L_1
 \le\frac{\sigma^{N+1}}{1-\sigma}L_1
\end{aligned}\label{e6.12}
\end{equation}
for $n\ge m>N$, where $N$ is a positive
integer. This shows that $S(u_n-u_m)\to0$ as $n,m\to\infty$.
Following the completeness of the Banach space
$B=\hat C^1_\delta(\overline{D})\cap\hat L^1_{p_0}(\overline D)$, there is a function
$w_*(z)\in B$, such that
$$
\hat C^1_\delta[u_n-u_*,\overline{D}]+\hat L^1_{p_0}[w_n-w_*,\overline{D}]\to0,\quad
\text{as } n\to\infty.
$$
By conditions  (C4)--(C6), from \eqref{e5.17} it
follows that $u_*(z)$ is a solution of Problem  Q for \eqref{e6.5};
i.e., \eqref{e6.1} for $t\in E$. It is easy to see that the positive
constant $\delta_0$ is independent of $t_0\,(0\le t_0<1)$. Hence from
Problem  Q for the complex equation \eqref{e6.5} with $t=t_0=0$ is
solvable, we can derive that when
$t=\delta_0,2\delta_0,\dots,[1/\delta_0]\delta_0,1$, Problem  Q for \eqref{e6.5}
are solvable, especially Problem  Q for \eqref{e6.1} with $t=1$ and
$A(z) =0$, namely Problem  Q for \eqref{e4.9} has a unique solution.

 From the above theorem, the solvability results of Problem P for
equation \eqref{e4.1} can be derived.
\end{proof}

\begin{theorem} \label{thm6.2}
 Under the same
conditions as in Theorem \ref{thm6.1}, the following statements hold.

$(1)$ When the index $K\ge 0$, Problem $\rm P$ for \eqref{e4.1} has $2N$
solvability conditions, and the solution of Problem $\rm P$ depends
on $2K+2$ arbitrary real constants.

$(2)$ When $K<0$, Problem ${\rm P}$ for \eqref{e4.1} is solvable under
$2N-2K-1$ conditions, and the solution of Problem $\rm P$ depends on
one arbitrary real constant.
\end{theorem}

\begin{proof}  Let the solution $[w(z),u(z)]$
of Problem Q for \eqref{e4.9} be substituted into the boundary
condition \eqref{e4.10}, \eqref{e4.12} and the relation \eqref{e4.11}.
If the function $h(z)=0$, $z\in\Gamma$; i.e.,
\begin{gather*}
h_j=0,\quad j=1,\dots ,N,\quad\text{if } K\ge 0, \\
h_j=0,\quad j=[1-(-1)^{2K}]/2,\dots ,N, \quad\text{if }K<0,\\
h_m^\pm=0,\quad m= 1,\dots ,[|K|+1/2] -1,\quad\text{if }K<0,
\end{gather*}
and $d_j=0$, $j=1,\dots ,N$, then we have $w(z)=u_z$ in $D$ and the function
$w(z)$ is just a solution of Problem  P for \eqref{e4.1}. Hence the
total number of above equalities is just the number of solvability
conditions as stated in this theorem. Also note that the real
constants $b_0$ in \eqref{e4.11} and $b_j$ $(j\in J)$ in \eqref{e4.15}
and \eqref{e4.16} are arbitrarily chosen. This shows that the general
solution of Problem P for \eqref{e4.1} includes the number of arbitrary
real constants as stated in the theorem.
\end{proof}

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\end{document}