# Oblique derivative boundary value problems on families of planar domains

Ziming Shi  Department of Mathematics, Rutgers University - New Brunswick,
Piscataway, NJ 08854<EMAIL_ADDRESS>
###### Abstract.

We consider second-order elliptic equations with oblique derivative boundary
conditions, defined on a family of bounded domains in $\mathbb{C}$ that depend
smoothly on a real parameter $\lambda\in[0,1]$. We derive the precise
regularity of the solutions in all variables, including the parameter
$\lambda$. More specifically we show that the solution and its derivatives are
continuous in all variables, and the Hölder norms of the space variables are
bounded uniformly in $\lambda$.

###### Key words and phrases:

Boundary value problem, oblique derivative, generalized analytic functions,
singular integral equation

###### 2020 Mathematics Subject Classification:

30E25, 30G20, 35J25, 45E05

Supported in part by NSF grant DMS-1500162

###### Contents

  1. 1 Introduction
  2. 2 Hölder spaces with parameter
  3. 3 A review of Vekua’s theory on fixed domains
  4. 4 Families of simply-connected domains
  5. 5 Kernels with parameter
  6. 6 Families of multiply-connected domains
    1. 6.1 The $\mathcal{C}^{\nu^{-},0}$ estimate
    2. 6.2 The $\mathcal{C}^{\nu,0}$ estimate
    3. 6.3 Differentiability in space variable: The $\mathcal{C}^{k+1+\mu,0}$ estimate
    4. 6.4 Differentiability in parameter: the $\mathcal{C}^{k+1+\mu,j}$ estimate
  7. 7 Appendix
    1. 7.1 Estimates for the Cauchy-Green/ $\overline{\partial}$ operator
    2. 7.2 Estimates for Cauchy type operators
    3. 7.3 Theory of singular integral operators on curves
    4. 7.4 Some other results

##  1\. Introduction

In this paper we study the domain dependence of solutions to the oblique
boundary value problem on the plane. We consider the following problem on a
family of “smoothly varying” bounded domains $\Omega^{\lambda}$ in
$\mathbb{C}$ that depend on some parameter $\lambda\in[0,1]$:

(1.1) $\mathcal{L}^{\lambda}u^{\lambda}=f^{\lambda}\quad\text{in
$\Omega^{\lambda}$
},\qquad\frac{du^{\lambda}}{dl^{\lambda}}=\gamma^{\lambda}\quad\text{on
$b\Omega^{\lambda}$. }$

Here $\mathcal{L}^{\lambda}$ is a second-order elliptic operator defined by

$\mathcal{L}^{\lambda}u^{\lambda}=a^{\lambda}(x,y)u^{\lambda}_{xx}+2b^{\lambda}(x,y)u^{\lambda}_{xy}+c^{\lambda}(x,y)u^{\lambda}_{yy}+d^{\lambda}(x,y)u^{\lambda}_{x}+e^{\lambda}(x,y)u^{\lambda}_{y},$

and $l^{\lambda}$ is an arbitrary non-vanishing directional vector field along
$b\Omega^{\lambda}$. We assume $a^{\lambda}>0$ and the uniform ellipticity
condition:

$a^{\lambda}c^{\lambda}-(b^{\lambda})^{2}\geq\delta_{0}>0,\quad\forall\lambda\in[0,1].$

We show that if all the coefficient functions
$a^{\lambda},b^{\lambda},c^{\lambda},d^{\lambda},e^{\lambda},F^{\lambda},\gamma^{\lambda}$
depend on the parameter $\lambda$ “smoothly”. Then with certain suitable
conditions to guarantee the existence and uniqueness of the problem (1.1), the
family of solutions $u^{\lambda}$ will also depend on the parameter
“smoothly”.

We now make the concept of domain dependence precise. Let $\Omega$ be a
bounded domain in $\mathbb{C}$, and set $k,j$ to be integers such that $k\geq
j\geq 0$. We say that a function $u^{\lambda}$ is in the space
$\mathcal{C}^{k+\alpha,j}(\overline{\Omega})$ if for every integer $i$ with
$0\leq i\leq j$, $[\lambda\mapsto\partial{}_{\lambda}^{i}u^{\lambda}]$ is a
continuous map from $[0,1]$ to $C^{k-i}(\overline{\Omega})$ and a bounded map
from $[0,1]$ to $C^{k-i+\alpha}(\overline{\Omega})$. Our main result is

######  Theorem 1.1.

Let integers $k$ and $j$ satisfy $k\geq j\geq 0$. Let $0<\mu<1$. Let $\Omega$
be a bounded domain in $\mathbb{C}$ with $\mathcal{C}^{k+2+\mu}$ boundary. Let
$\Gamma^{\lambda}$ be a family of maps that embed $\Omega$ to
$\Omega^{\lambda}$, with
$\Gamma^{\lambda}\in\mathcal{C}^{k+2+\mu,j}(\overline{\Omega})$. Suppose that
$a^{\lambda},b^{\lambda},c^{\lambda},d^{\lambda},e^{\lambda},f^{\lambda}$ are
functions in $\Omega^{\lambda}$, such that
$a^{\lambda}\circ\Gamma^{\lambda},b^{\lambda}\circ\Gamma^{\lambda},c^{\lambda}\circ\Gamma^{\lambda}\in\mathcal{C}^{k+1+\mu,j}(\overline{\Omega})$,
$d^{\lambda}\circ\Gamma^{\lambda},e^{\lambda}\circ\Gamma^{\lambda},f^{\lambda}\circ\Gamma^{\lambda}\in\mathcal{C}^{k+\mu,j}(\overline{\Omega})$,
and suppose that $l^{\lambda},\gamma^{\lambda}$ are functions on
$b\Omega^{\lambda}$ such that
$l^{\lambda}\circ\Gamma^{\lambda},\gamma^{\lambda}\circ\Gamma^{\lambda}\in\mathcal{C}^{k+1+\mu,j}(b\Omega)$.
Suppose certain well-posedness conditions (see Proposition 3.5) are satisfied
so that (1.1) has a unique solution for each $\lambda$. Then
$u^{\lambda}\circ\Gamma^{\lambda}\in\mathcal{C}^{k+2+\mu,j}(\overline{\Omega})$.

The existence and uniqueness theory for fixed $\lambda$ are classical results
and we shall review them in Section 3.

In Bertrand-Gong [B-G14] the analogous results were proved for the Dirichlet
and Neumann problems for the Laplacian equation. As a corollary, they proved a
parameter version of the Riemann mapping theorem, namely, for any family of
simply connected domains $\Omega^{\lambda}$ given by the embeddings
$\Gamma^{\lambda}:\overline{\Omega}\to\overline{\Omega^{\lambda}}$ with
$\Gamma^{\lambda}\in\mathcal{C}^{k+1+\mu,j}(\overline{\Omega})$, there exists
a family of Riemann mappings $R^{\lambda}:\Omega^{\lambda}\to\mathbb{D}$ such
that
$R^{\lambda}\circ\Gamma^{\lambda}\in\mathcal{C}^{k+1+\mu,j}(\overline{\Omega})$.
Here $\mathbb{D}$ is the unit disk. We shall use this result in our proof of
Theorem 1.1 when the domains are simply-connected.

For fixed $\lambda$, we use Vekua’s classical construction to solve equation
(1.1). The idea is to first find an isothermal coordinates in which the
problem takes the form

(1.2) $\Delta U+a(x,y)U_{x}+b(x,y)U_{y}=f(x,y),\quad\alpha U_{x}-\beta
U_{y}=\gamma.$

By setting $w=u+iv$ for $u=U_{x}$ and $v=-U_{y}$, (1.2) takes the form

(1.3) $\displaystyle\overline{\partial}_{z}w+Aw+B\overline{w}=F,\quad\text{in
$\Omega$,}$ (1.4) $\displaystyle Re[\overline{l}w]=\gamma,\quad\text{on
$b\Omega$,}$

where we assume $l=\alpha+i\beta$ is nowhere vanishing on $b\Omega$. Problem
(1.3)-(1.4) is called a _generalized Riemann-Hilbert problem_. (The classical
Riemann-Hilbert problem is the special case when $A,B,F\equiv 0$.) Vekua
tackled this problem by systematically developing the theory of generalized
analytic functions. His idea is to find an integral representation formula for
a solution $w$ satisfying equation (1.3). The boundary condition (1.4) can
then be reduced to a singular integral equation on curves for which there is
well-known theory.

For our problem on domains with parameter, we will first reduce the problem to
the $\overline{\partial}$ form (1.3)-(1.4). If the domains are simply-
connected, we can pull back the problem onto the fixed unit disk $\mathbb{D}$
by using the parameter version of the Riemann mapping theorem proved in
[B-G14]. The problem can be reduced to a Fredholm integral equation on the
boundary circle, and the smoothness in parameter follows from a standard
compactness argument.

The problem becomes more difficult for families of multiply-connected domains.
In this case one may also pull back the equations onto some fixed domain.
However now the pull back maps are no longer biholomorphic, so this procedure
will destroy the $\overline{\partial}$ form (1.2). Thus we are forced to deal
with variable domains in this case. We use Vekua’s idea to reduce the problem
to a singular integral equation on $b\Omega^{\lambda}$, and we need to reduce
it first to a Fredholm equation in order to use compactness argument.

We mention that one could certainly obtain an estimate of the form

$\|w^{\lambda}\|_{C^{k+1+\alpha}(\Omega^{\lambda})}\leq
C_{\lambda}\|f^{\lambda}\|_{C^{k+\alpha}(\Omega^{\lambda})},$

for each fixed $\lambda$, by solving the problem on each fixed domain. However
this estimate is not useful since in general we do not have any information as
to how the the constant $C_{\lambda}$ depends on $\lambda$.

Following Vekua, we will henceforth call (1.3)-(1.4) Problem A for
$\overline{\partial}$ or simply Problem A. There is a vast literature on
oblique derivative boundary value problem. See for example Miranda [MA70,
Chapter 3] and Hörmander [Hor63, Chapter 10].

The paper is organized as follows:

In Section 2, we introduce the notations used in our paper. We review the
definition of Hölder spaces with parameter as were used in [N-W63] and
[B-G14]. We then prove some basic properties of these spaces. In particular we
prove a result (Proposition 2.3) which will be used repeatedly throughout the
paper to simplify our arguments. In the end we give a parameter version of the
isothermal coordinates in Proposition 2.10.

Section 3 contains an exposition of Vekua’s theory on the solvability and
regularity of Problem A without parameter. The solvability properties are
determined by the index of the problem (see Definition 3.1) and the number of
connected components of the boundary $b\Omega$. All the results in this
section can be found in [VA62].

In Section 4, we prove Theorem 1.1 for families of simply-connected domains
$\Omega^{\lambda}$.

In Section 5 we examine how the kernel in Vekua’s integral solution formula
depends on $\lambda$. These results will be used in Section 6 to prove Theorem
1.1 for families of multiply-connected domains $\Omega^{\lambda}$. Most of the
work are used to show the solution is continuous in the $C^{0}$ norm and
bounded in some Hölder norm with respect the parameter. For our proof we are
able to avoid taking derivatives on the kernel as was done in [B-G14].

In the end we include an appendix, where we provide either complete proofs or
precise references to the various claims used in earlier proofs. For the
constants appearing in our estimate, we use the notation $C_{1,0}$ (resp.
$C_{k+\mu,j}$) to indicate that it depends on
$\|\Gamma^{\lambda}\|_{\mathcal{C}^{1,0}(\Omega)}$ (resp.
$\|\Gamma^{\lambda}\|_{\mathcal{C}^{k+\mu,j}(\Omega)}$). For a family of
functions $u^{\lambda}$ defined on $\Omega^{\lambda}$, we shall write either
$u^{\lambda}\in\mathcal{C}^{k+\mu,j}(\Omega^{\lambda})$ or
$u^{\lambda}\circ\Gamma^{\lambda}\in\mathcal{C}^{k+\mu,j}(\Omega)$. The
parameter space $[0,1]$ can be replaced

######  Acknowledgment.

I am deeply grateful to my advisor Professor Xianghong Gong for introducing
this problem to me, and for offering me innumerable inspiration, guidance and
help, without which this project could never be completed. I also want to
thank Jingrui Cheng and Liding Yao for many helpful comments and discussions.

##  2\. Hölder spaces with parameter

In this section we introduce Hölder spaces with parameter as in [B-G14], which
can be used to study boundary value problems on families to domains.

We denote by $C^{r}(\overline{\Omega})$, $r\geq 0$ the space of complex-valued
functions which are Hölder continuous with exponent $r$ on a bounded domain
$\Omega\subseteq\mathbb{C}$. When the domain $\Omega$ in question is clear, we
also write $|f|_{r}$ in place of $|f|_{C^{r}(\overline{\Omega})}$. We write
$\mathbb{D}$ for the unit disk in $\mathbb{C}$, and we write
$dA(z)=\frac{i}{2}dz\wedge d\overline{z}$ for the area element in
$\mathbb{C}$. The Cauchy-Green or $\overline{\partial}$ solution operator on a
domain $\Omega\subseteq\mathbb{C}$ is defined by

$T_{\Omega}f(z):=-\frac{1}{\pi}\iint_{\Omega}\frac{f(\zeta)}{\zeta-z}\,dA(\zeta),\quad
z\in\Omega.$

For $1\leq p\leq\infty$ and $\nu\in\mathbb{R}$, let $L_{p,\nu}(\mathbb{C})$
denote the space of functions $f$ satisfying

(2.1) $f\in L^{p}(\mathbb{D}),\quad\quad
f_{\nu}(z):=|z|^{-\nu}f\left(\frac{1}{z}\right)\in L^{p}(\mathbb{D}),\quad$

with the norm
$|f|_{L_{p,\nu}(\mathbb{C})}=|f|_{L^{p}(\mathbb{D})}+|f_{\nu}|_{L^{p}(\mathbb{D})}$.
The space $L_{p,\nu}(\mathbb{C})$ is a Banach space with this norm.
Intuitively the lower index $\nu$ describes the rate of growth of $f$ at
$\infty$, and we have $L_{p,\nu_{1}}(\mathbb{C})\subset
L_{p,\nu_{2}}(\mathbb{C})$, if $\nu_{1}>\nu_{2}$. It is easy to see that
$f_{\nu}\in L^{p}(\mathbb{D})$ if and only if

$\int_{\mathbb{C}\setminus\mathbb{D}}|u|^{\nu p-4}|f(u)|^{p}\,dA(u)<\infty.$

For $0<\alpha<1$, we let $L_{\alpha}^{p}(\Omega)$ denote the set of functions
$f$ that satisfy $f\in L^{p}(\Omega)$, $f\equiv 0$ outside $\Omega$, and

$B(f,\Omega,\alpha,p)=\sup_{h\in\mathbb{C}}\frac{\left(\iint_{\Omega}\left|f(z+h)-f(z)\right|^{p}\,dA(z)\right)^{\frac{1}{p}}}{|h|^{\alpha}}<\infty.$

Then $L_{\alpha}^{p}(\Omega)$ is a Banach space with the norm:

(2.2) $|f|_{L_{\alpha}^{p}(\Omega)}=|f|_{L^{p}(\Omega)}+B(f,\Omega,\alpha,p).$

We say that $w$ is a solution to the equation

(2.3) $\partial{}_{\overline{z}}w+Aw+B\overline{w}=F,\quad\text{in $\Omega$},$

if $w,\partial{}_{\overline{z}}w\in L^{1}(\Omega)$, and the above equation
holds in the sense of distributions in $\Omega$. We denote by
$\mathcal{L}_{p}(A,B,F,\Omega)$ (resp. $\mathcal{L}_{p,2}(A,B,F)$) the set of
$w\in L^{1}(\Omega)$ that satisfy (2.3) with $A,B,F\in L^{p}(\Omega)$ (resp.
$L_{p,2}(\mathbb{C})$) If $F\equiv 0$, we write $\mathcal{L}_{p}(A,B,\Omega)$
(resp. $\mathcal{L}_{p,2}(A,B)$) for the spaces defined as above.

We adopt the following notations for differences of functions:

$\delta_{z,\zeta}(f(\cdot,\lambda)):=f(z,\lambda)-f(\zeta,\lambda),\quad\quad\delta^{\lambda_{1},\lambda_{2}}f(z,\cdot):=f(z,\lambda_{1})-f(z,\lambda_{2}).$

We denote the difference quotient with respect to the parameter by

$D^{\lambda,\lambda_{0}}f(\cdot):=\frac{f(\cdot,\lambda)-f(\cdot,\lambda_{0})}{\lambda-\lambda_{0}}.$

To study boundary value problems with parameter, we use the Hölder space with
parameter $\mathcal{C}^{k+\mu,j}(\overline{\Omega_{\Gamma}})$ from [B-G14].
First we define spaces for a fixed domain:

######  Definition 2.1.

Let $k$ and $j$ be nonnegative integers. Let $0\leq\alpha<1$. Let $\Omega$ be
a bounded domain in $\mathbb{C}$ with $C^{k+\alpha}$ boundary, and let
$\\{u^{\lambda}\\}_{\lambda\in[0,1]}$ be a family of functions on $\Omega$. We
say that $\\{u^{\lambda}\\}_{\lambda\in[0,1]}$ is in the class
$\mathcal{C}^{k+\alpha,j}_{\ast}(\overline{\Omega})$, if for all $0\leq i\leq
j$, $|\partial{}_{\lambda}^{i}u^{\lambda}|_{k+\alpha}$ are uniformly bounded
in $\lambda$, and $\partial{}_{\lambda}^{i}u^{\lambda}$ is continuous in
$\lambda$ in the $C^{k}(\overline{\Omega})$ norm. The norm on
$\mathcal{C}^{k+\alpha,j}_{\ast}(\overline{\Omega})$ is defined to be

(2.4) $|u|_{k+\alpha,j}:=\sup_{\begin{subarray}{c}0\leq i\leq j,\\\
\lambda\in[0,1]\end{subarray}}\\{|\partial{}_{\lambda}^{i}u^{\lambda}|_{k+\alpha}\\}.$

Similarly we define $\mathcal{C}^{k+\alpha,j}_{\ast}(b\Omega)$ by replacing
$\Omega$ with $b\Omega$ in the above expressions.

For our problem with parameter, we consider functions defined on a family of
domains $\Omega^{\lambda}$, for $\lambda\in[0,1]$. We assume there is an
embedding $\Gamma^{\lambda}:\Omega\to\Omega^{\lambda}$, where $\Gamma^{0}$ is
the identity map from $\Omega$ to itself.

Now we make the following observation. Suppose for each $\lambda\in[0,1]$,
$u^{\lambda}$ is defined on an open set $V$ with
$\cup_{\lambda}\Omega^{\lambda}\subset\subset V$. Let
$\Gamma^{\lambda}\in\mathcal{C}^{k+\alpha,j}_{\ast}(\overline{\Omega})$, and
$u^{\lambda}\in\mathcal{C}^{k+\alpha,j}_{\ast}(V)$. But then
$\widetilde{u}^{\lambda}(z):=u^{\lambda}(\Gamma^{\lambda}(z))$ is not in the
space $\mathcal{C}^{k+\alpha,j}_{\ast}(\overline{\Omega})$, since when one
takes derivative in $\lambda$, by the chain rule one needs to take the space
derivative of $u^{\lambda}$. Hence instead of
$\mathcal{C}^{k+\alpha,j}_{\ast}(\overline{\Omega})$, we shall adopt the
following definition in [B-G14] for a family of functions defined on a family
of domains:

######  Definition 2.2.

Let $k$ and $j$ be non-negative integers satisfying $k\geq j$, and
$0<\alpha<1$. Let
$\\{\Gamma^{\lambda}:\overline{\Omega}\to\overline{\Omega^{\lambda}}\\}_{\lambda\in[0,1]}$
be in the class $\mathcal{C}^{k+\alpha,j}_{\ast}(\overline{\Omega})$. Let
$u^{\lambda}$ be a family of functions defined on $\Omega^{\lambda}$, and put
$\widetilde{u}(z,\lambda):=u^{\lambda}(\Gamma^{\lambda}(z))$. We say that

$u^{\lambda}\in\mathcal{C}_{\ast}^{k+\alpha,j}(\overline{\Omega_{\Gamma}})\quad\text{
if
$\widetilde{u}(z,\lambda)\in\mathcal{C}_{\ast}^{k+\alpha,j}(\overline{\Omega})$},$
$u^{\lambda}\in\mathcal{C}_{\ast}^{k+\alpha,j}(b\Omega_{\Gamma})\quad\text{if
$\widetilde{u}(z,\lambda)\in C_{\ast}^{k+\alpha,j}(b\Omega)$}.$

For $0\leq j\leq k$, define the spaces

$\mathcal{C}^{k+1+\alpha,j}(\overline{\Omega}_{\Gamma}):=\bigcap_{i=0}^{j}\mathcal{C}_{\ast}^{k-i+\alpha,i}(\overline{\Omega}_{\Gamma}),\quad\mathcal{C}^{k+1+\alpha,j}(b\Omega_{\gamma}):=\bigcap_{i=0}^{j}\mathcal{C}_{\ast}^{k-i+\alpha,i}(b\Omega_{\Gamma}),$

The norm on $\mathcal{C}^{k+1+\alpha,j}(\overline{\Omega}_{\Gamma})$ is
defined to be

$\|u\|_{k+\alpha,j}:=\max_{0\leq i\leq
j}\\{|\widetilde{u}(z,\lambda)|_{k-i+\alpha,i}\\},$

where $|u|_{k-i+\alpha,i}$ is defined in (2.4). In other words, we say that
$u\in\mathcal{C}^{k+1+\alpha,j}(\overline{\Omega}_{\Gamma})$ if
$|\partial{}_{\lambda}^{i}\widetilde{u}|_{k-i+\alpha}$ is uniformly bounded in
$\lambda$, and $\partial{}_{\lambda}^{i}\widetilde{u}$ is continuous in the
$C^{k-i}(\overline{\Omega})$ norm, for each $i$ with $0\leq i\leq j$. Since
$|\partial{}_{\lambda}^{i^{\prime}}u|_{k-i+\alpha}\leq|\partial{}_{\lambda}^{i^{\prime}}u|_{k-i^{\prime}+\alpha}$,
for any $i^{\prime}$, $0\leq i^{\prime}\leq i$, we have

$\|u\|_{k+\alpha,j}=\max_{0\leq i\leq
j}\\{|\partial{}_{\lambda}^{i}\widetilde{u}(z,\lambda)|_{k-i+\alpha}\\}.$

We denote the restriction of $\Gamma^{\lambda}$ to $b\Omega$ by
$\rho^{\lambda}(s)$, where $s$ is the arclength element on $b\Omega$. Then
$dt^{\lambda}=(\rho^{\lambda})^{\prime}(s)ds$.

######  Remark.

In the definition of $\mathcal{C}^{k+\alpha,j}_{\ast}(\overline{\Omega})$ with
$0<\alpha<1$, the requirement that $u^{\lambda}$ is continuous in $\lambda$ in
the $C^{k}(\overline{\Omega})$ norm is superficial. In fact, it is an easy
calculus fact that if $u^{\lambda}$ is continuous in $\lambda$ in the
$C^{0}$-norm and bounded uniformly in $\lambda$ in the $C^{k+\alpha}$-norm,
then $u^{\lambda}$ is continuous in $\lambda$ in the
$C^{k+\alpha^{\prime}}$-norm, for any $0<\alpha^{\prime}<\alpha$. We put this
observation in a proposition:

######  Proposition 2.3.

Let $0<\beta<1$, and $\Omega$ a bounded domain in $\mathbb{R}^{n}$.
(i) Let $\\{f^{\lambda}\\}$ be a family of functions in the class
$C^{\beta}(\overline{\Omega})$ such that $|f^{\lambda}|_{\beta}$ is uniformly
bounded in $\lambda$, and $f^{\lambda}$ is continuous in $\lambda$ in the
$|\cdot|_{0}$ -norm. Then $f^{\lambda}$ is continuous in $\lambda$ in the
$|\cdot|_{\alpha}$ -norm, for any $0<\alpha<\beta$. Furthermore, for every
$\varepsilon$ such that $0<\varepsilon<\frac{\beta-\alpha}{\beta}$, the
following holds:

(2.5) $|f^{\lambda_{1}}-f^{\lambda_{2}}|_{\alpha}\leq
C(\Omega,\alpha,\beta,\varepsilon)\|f\|_{\beta,0}|f^{\lambda_{1}}-f^{\lambda_{2}}|_{0}^{\varepsilon},$

where $\|f\|_{\beta,0}=\sup_{\lambda}|f^{\lambda}|_{\beta}$.
(ii) For $k\geq 1$, let $\\{f^{\lambda}\\}$ be a family of functions in the
class $C^{k+\beta}(\overline{\Omega})$ such that $|f^{\lambda}|_{k+\beta}$ is
uniformly bounded in $\lambda$, and $f^{\lambda}$ is continuous in $\lambda$
in the $|\cdot|_{0}$ -norm. Then $f^{\lambda}$ is continuous in $\lambda$ in
the $|\cdot|_{k+\alpha}$-norm, for any $0<\alpha<\beta$.

######  Remark.

The Arzela-Ascoli theorem says that the Hölder space
$C^{\beta}(\overline{\Omega})$ is relatively compact in
$C^{0}(\overline{\Omega})$. Inequality (2.5) implies that for any
$0<\alpha<\beta$, $C^{\beta}(\overline{\Omega})$ is relatively compact in
$C^{\alpha}(\overline{\Omega})$.

###### Proof.

(i) In order to prove (2.5) it is sufficient to show that if $\\{g_{n}\\}$ is
a sequence that converges to $0$ uniformly on $\overline{\Omega}$, and
$|g_{n}|_{\beta}$ is bounded uniformly. Then $|g_{n}|_{\alpha}$ converges to
$0$. Taking $\varepsilon<\frac{\beta-\alpha}{\beta}$ (so that
$\beta(1-\varepsilon)-\alpha>0$), we have

$\displaystyle\sup_{x,y\in\Omega}\frac{|g_{n}(x)-g_{n}(y)|}{|x-y|^{\alpha}}$
$\displaystyle\leq\sup_{x,y\in\Omega}\frac{|g_{n}(x)-g_{n}(y)|^{1-\varepsilon}}{|x-y|^{\alpha}}\sup_{x,y\in\Omega}|g_{n}(x)-g_{n}(y)|^{\varepsilon}$
$\displaystyle\leq
C\left(\|g_{n}\|_{\beta,0}\right)\sup_{x,y\in\Omega}|x-y|^{\beta(1-\varepsilon)-\alpha}|g_{n}|_{0}^{\varepsilon}$
$\displaystyle\leq
C\left(\|g_{n}\|_{\beta,0}\right)|g_{n}|_{0}^{\varepsilon}\quad\longrightarrow
0\quad\text{as $n\to\infty$,}$

where the constant $C$ depends on $\Omega$, $\alpha,\beta,\varepsilon$.

(ii) Seeking contradiction, suppose $|f^{\lambda}|_{k+\alpha}$ is not
continuous at some $\lambda_{0}$. Then there exists a sequence
$\lambda_{j}\to\lambda_{0}$ such that
$|f^{\lambda_{j}}-f^{\lambda_{0}}|_{k+\alpha}\geq\delta>0$. Since
$|f^{\lambda_{j}}|_{1+\beta}$ is bounded, passing to a subsequence if
necessary, $f^{\lambda_{j}}$ converges in the $|\cdot|_{k+\alpha}$ -norm to
some function $\widehat{f}$, and thus

(2.6) $|\widehat{f}-f^{\lambda_{0}}|_{k+\alpha}\geq\delta>0.$

On the other hand, we have

$|\widehat{f}-f^{\lambda_{0}}|_{0}\leq|\widehat{f}-f^{\lambda_{j}}|_{0}+|f^{\lambda_{j}}-f^{\lambda_{0}}|_{0}.$

The right-hand side converges to $0$ by assumption, so
$\widehat{f}=f^{\lambda_{0}}$, which contradicts (2.6). Hence $f^{\lambda}$ is
continuous in $\lambda$ in the $|\cdot|_{k+\alpha}$ norm. ∎

The following result shows that the definition
$\mathcal{C}^{k+\alpha,j}(\overline{\Omega}_{\Gamma})$ is independent of the
embedding $\Gamma^{\lambda}$. For this reason we shall from now on write
$\mathcal{C}^{k+\alpha,j}(\overline{\Omega^{\lambda}})$ instead of
$\mathcal{C}^{k+\alpha,j}(\overline{\Omega}_{\Gamma})$.

######  Proposition 2.4.

Let $k,j$ be non-negative integers satisfying $k\geq j$, and let $0<\mu<1$.
Let $\overline{\Omega}$ be a bounded domain with $b\Omega\in
C^{k+\alpha,j}\cap C^{1}$. Let $\Gamma^{\lambda}_{1}$ and
$\Gamma^{\lambda}_{2}$ be two embeddings from $\overline{\Omega}$ to
$\overline{\Omega^{\lambda}}$. If
$\Gamma_{i}^{\lambda}\in\mathcal{C}^{k+\alpha,j}(\overline{\Omega})\cap\mathcal{C}^{1,0}(\overline{\Omega})$.
Then
$\mathcal{C}^{k+\alpha,j}(\overline{\Omega}_{\Gamma_{1}})=\mathcal{C}^{k+\alpha,j}(\overline{\Omega}_{\Gamma_{2}})$.

The reader can find the proof in [B-G14].

######  Lemma 2.5.

Let $k$ and $j$ be non-negative integers satisfying $k\geq j$, and $0<\mu<1$.
Let $\Omega,\Omega^{\prime}\subset\mathbb{R}^{2}$, and
$F:\overline{\Omega}\times[0,1]\to\overline{\Omega^{\prime}}\times[0,1]$ be a
$\mathcal{C}^{k+1+\alpha,j}$-mapping, defined by

$F(x,y,\lambda)=(f_{1}(x,y,\lambda),f_{2}(x,y,\lambda),\lambda):=(\xi,\eta,\tau).$

Suppose that
$f(\cdot,\cdot,\lambda)=(f_{1}(\cdot,\cdot,\lambda),f_{2}(\cdot,\cdot,\lambda))$
are bijections from $\overline{\Omega}$ to $\overline{\Omega^{\prime}}$ for
each fixed $\lambda$, and the Jacobian matrix $DF$ has non-vanishing
determinant in $\overline{\Omega}\times[0,1]$. Then
$F^{-1}\in\mathcal{C}^{k+1+\alpha,j}(\overline{\Omega^{\prime}}\times[0,1])$.

The proof can be done by induction on the index $k$ and $j$. The details are
left to the reader.

######  Proposition 2.6.

Let $k$ and $j$ be non-negative integers or $\infty$ satisfying $0\leq j\leq
k$, and $0<\alpha<1$. Let $\Omega$ be a simply-connected domain in
$\mathbb{C}$ with $b\Omega\in C^{k+1+\alpha}$, and let
$\Gamma^{\lambda}:\overline{\Omega}\to\overline{\Omega^{\lambda}}$ be in the
class $\mathcal{C}^{k+1+\alpha,j}(\overline{\Omega})$. Then there exists a
family of Riemann mappings $R^{\lambda}:\Omega^{\lambda}\to\mathbb{D}$ such
that
$\widetilde{R}(z,\lambda):=R^{\lambda}\circ\Gamma^{\lambda}(z)\in\mathcal{C}^{k+1+\alpha,j}(\overline{\Omega}).$

For proof see [B-G14]*p. 211.

######  Proposition 2.7.

Given the assumptions in the last theorem. Let $R^{\lambda}$ be the family of
Riemann mappings in Proposition 2.6. For each $\lambda\in[0,1]$, let
$(R^{\lambda})^{-1}:\mathbb{D}\to\Omega^{\lambda}$ be the inverse of
$R^{\lambda}$. Then
$(R^{\lambda})^{-1}\in\mathcal{C}^{k+1+\alpha,j}(\overline{\mathbb{D}})$.

###### Proof.

Define
$F\in\mathcal{C}^{k+1+\alpha,j}(\overline{\Omega}\times[0,1]):\overline{\Omega}\times[0,1]\to\overline{\mathbb{D}}\times[0,1]$
by

$F(z,\lambda)=(\widetilde{R}(z,\lambda),\lambda):=(R(\Gamma(z,\lambda),\lambda),\lambda).$

Since $\Gamma(\cdot,\lambda)$ is an embedding from $\Omega$ to
$\Omega^{\lambda}$, and $R(\cdot,\lambda)$ is a Riemann mapping, the map
$\widetilde{R}$ is a bijection for each fixed $\lambda$, and its Jacobian is
invertible. It follows that the Jacobian of $F$ is invertible (when $j\geq
0$.) By lemma (2.5), $F$ is invertible and the inverse $F^{-1}$ is in
$\mathcal{C}^{k+1+\alpha,j}(\overline{\mathbb{D}}\times[0,1])$. $F^{-1}$ is
given by

$F^{-1}(w,\tau)=(\widetilde{R}^{-1}(w,\tau),\tau).$

Write

$R^{-1}(z,\lambda)=\Gamma(\widetilde{R}^{-1}(z,\lambda),\lambda).$

Since $\Gamma\in\mathcal{C}^{k+1+\alpha,j}(\overline{\Omega})$, and
$\widetilde{R}^{-1}\in\mathcal{C}^{k+1+\alpha,j}(\overline{\mathbb{D}})$, we
obtain $R^{-1}\in\mathcal{C}^{k+1+\alpha,j}(\overline{\mathbb{D}})$. ∎

In order to reduce our oblique derivative boundary value problem to Problem A
for $\overline{\partial}$, we need to change coordinates to reduce the second-
order terms in $\mathcal{L}u$ in the elliptic operator to Laplacian. This is
the classical problem of finding isothermal coordinate on the plane. We state
some of the results which are well-known.

######  Proposition 2.8.

Let $\Omega$ be a bounded domain in $\mathbb{C}$, and consider the following
elliptic equation defined on $\Omega$:

(2.7)
$a(x,y)\frac{\partial{}^{2}u}{\partial{}x^{2}}+2b(x,y)\frac{\partial{}^{2}u}{\partial{}x\partial{}y}+c(x,y)\frac{\partial{}^{2}u}{\partial{}y^{2}}+F\left(x,y,u,\frac{\partial
u}{\partial x},\frac{\partial u}{\partial y}\right)=0,$

where the derivatives are taken in the sense of distributions. We assume the
following uniform ellipticity condition:

$d=ac-b^{2}\geq d_{0}>0\quad\text{a.e. in $\Omega$ . }$

(i) Suppose $\Omega$ has Lipschitz boundary, and $a,b,c\in W^{1,p}(\Omega)$,
for $2<p<\infty$. Then there exists a 1-1 map $\psi:z=x+iy\mapsto w=\xi+i\eta$
on $\Omega$ such that $\psi\in W^{2,p}(\Omega)\subset
C^{1+\alpha_{p}}(\overline{\Omega})$, for $\alpha_{p}=\frac{p-2}{p}$.
Moreover,and if we set $u(x,y)=\widetilde{u}(\psi(x,y))$, then $u$ is a
solution to equation (2.7) if and only if $\widetilde{u}$ is a solution to the
following equation

(2.8)
$\displaystyle\frac{\partial{}^{2}\widetilde{u}}{\partial{}\xi^{2}}+\frac{\partial{}^{2}\widetilde{u}}{\partial{}\eta^{2}}+F_{1}\left(\xi,\eta,\widetilde{u},\frac{\partial\widetilde{u}}{\partial\xi},\frac{\partial\widetilde{u}}{\partial\eta}\right)=0$

on the domain $\psi(\Omega)$.
(ii) For $k\geq 0$ and $0<\mu<1$, let $\Omega$ be a bounded domain in
$\mathbb{C}$ with $C^{k+1+\mu}$ boundary. Suppose $a,b,c\in
C^{k+1+\mu}(\overline{\Omega})$. Then the coordinate map $\psi$ constructed in
$(i)$ belongs to the class $C^{k+2+\mu}(\overline{\Omega})$.
(iii) The Jacobian of the map $\psi$ is positive at each point of $\Omega$:

$J(z)=\left|\frac{\partial w}{\partial z}\right|^{2}-\left|\frac{\partial
w}{\partial\overline{z}}\right|^{2}>0,\quad z\in\Omega.$

The reader can refer to [VA62] Theorems 2.5, 2.9 and 2.12 for the proofs of
above statements.

######  Lemma 2.9.

Let $\Omega$ be a bounded domain in $\mathbb{C}$ with $C^{1}$ boundary, and
let $\psi$ be the map from Proposition 2.8. In particular $\psi\in
C^{1}(\overline{\Omega})$. Then $\psi$ defines a one-to-one onto homeomorphism
from $\Omega$ to $\Omega^{\prime}$. Let $I$ be a non-vanishing vector field
along $b\Omega$, and $\psi_{\ast}(I)$ be the push-forward of $I$ by $\psi$, so
that $\psi_{\ast}$ is a non-vanishing vector field along $b\Omega^{\prime}$.
Then

$[I]_{b\Omega}=[\psi_{\ast}I]_{b\Omega^{\prime}}.$

where $[I]_{b\Omega}$ denotes the change in the argument of $I(z)$ as $z$
moves along $b\Omega$ once in the positive direction.

The proof is left to the reader.

We now prove a result on the parameter version of isothermal coordinates:

######  Proposition 2.10.

Let $k,j$ be non-negative integers satisfying $k\geq j$, and $0<\mu<1$. Let
$\Omega$ be a bounded domain in $\mathbb{C}$ with $C^{k+2+\mu}$ boundary, and
$\Gamma^{\lambda}:\Omega\to\Omega^{\lambda}$ be a family of embeddings in the
class $\mathcal{C}^{k+2+\mu,j}(\overline{\Omega})$, with $\Gamma^{0}$ being
the identity map. Consider the second-order elliptic equation with oblique
derivative boundary condition:

(2.9) $\displaystyle
a^{\lambda}(x,y)u^{\lambda}_{xx}+2b^{\lambda}(x,y)u^{\lambda}_{xy}+c^{\lambda}(x,y)u^{\lambda}_{yy}+d^{\lambda}(x,y)u^{\lambda}_{x}+e^{\lambda}(x,y)u^{\lambda}_{y}=f^{\lambda}(x,y),\quad\text{in
$\Omega^{\lambda}$.}$ (2.10)
$\displaystyle\alpha^{\lambda}u_{x}^{\lambda}+\beta^{\lambda}u_{y}^{\lambda}=\gamma^{\lambda},\quad\text{on
$b\Omega^{\lambda}$.}$

Suppose
$a^{\lambda},b^{\lambda},c^{\lambda}\in\mathcal{C}^{k+1+\mu,j}(\overline{\Omega^{\lambda}})$,
$d^{\lambda},e^{\lambda},f^{\lambda}\in\mathcal{C}^{k+\mu,j}(\overline{\Omega^{\lambda}})$,
$\alpha^{\lambda},\beta^{\lambda},\gamma^{\lambda}\in\mathcal{C}^{k+1+\mu,j}(\overline{\Omega^{\lambda}})$.
Then for all $|\lambda|<\varepsilon_{0}$ for some small $\varepsilon_{0}$,
there exists a family of coordinate maps
$\varphi^{\lambda}:\Omega^{\lambda}\to D^{\lambda}$ with
$\varphi^{\lambda}\circ\Gamma^{\lambda}\in\mathcal{C}^{k+1+\mu,j}(\overline{\Omega})$,
and the functions
$\mathcal{U}^{\lambda}(\tau^{\lambda}):=u^{\lambda}((\varphi^{\lambda})^{-1}(\tau^{\lambda}))$
($\tau^{\lambda}=s^{\lambda}+it^{\lambda}$) satisfies:

$\displaystyle\Delta\,\mathcal{U}^{\lambda}+p^{\lambda}(s^{\lambda},t^{\lambda})\,\mathcal{U}^{\lambda}_{s^{\lambda}}+q^{\lambda}(s^{\lambda},t^{\lambda})\,\mathcal{U}^{\lambda}_{t^{\lambda}}=h^{\lambda}(s^{\lambda},t^{\lambda}),\quad\text{in
$D^{\lambda}$, }$
$\displaystyle\nu_{1}^{\lambda}\,\mathcal{U}^{\lambda}_{s}+\nu_{2}^{\lambda}\,\mathcal{U}^{\lambda}_{t}=g^{\lambda}(s^{\lambda},t^{\lambda}),\quad\text{on
$bD^{\lambda}$, }$

where
$p^{\lambda},q^{\lambda},h^{\lambda}\in\mathcal{C}^{k+\mu,j}(\overline{D^{\lambda}})$,
$\nu_{1}^{\lambda},\nu_{2}^{\lambda},g^{\lambda}\in\mathcal{C}^{k+1+\mu,j}(\overline{D^{\lambda}})$.
Moreover, the winding number of the vector field
$\Upsilon^{\lambda}=(\alpha^{\lambda},\beta^{\lambda})$ along
$b\Omega^{\lambda}$ is the same as the winding number of
$\Upsilon_{1}^{\lambda}=(\nu_{1}^{\lambda},\nu_{2}^{\lambda})$ along
$bD^{\lambda}$.

###### Proof.

First we pull back equations (2.9)-(2.10) by the maps $\Gamma^{\lambda}$ to
the fixed domain $\Omega$. Let
$v^{\lambda}(\xi,\eta):=u^{\lambda}\circ\Gamma^{\lambda}(\xi,\eta)$. Then
$v^{\lambda}$ satisfies

(2.11)
$\displaystyle\widetilde{a}^{\lambda}(\xi,\eta)v^{\lambda}_{\xi\xi}+2\widetilde{b}^{\lambda}(\xi,\eta)v^{\lambda}_{\xi\eta}+\widetilde{c}^{\lambda}(\xi,\eta)v^{\lambda}_{\eta\eta}+\widetilde{d}^{\lambda}(\xi,\eta)v^{\lambda}_{\xi}+\widetilde{e}^{\lambda}(\xi,\eta)v^{\lambda}_{\eta}=\widetilde{f}^{\lambda}(\xi,\eta),\;\text{in
$\Omega$.}$ (2.12)
$\displaystyle\widetilde{\alpha}^{\lambda}v_{\xi}^{\lambda}+\widetilde{\beta}^{\lambda}v_{\eta}^{\lambda}=\widetilde{\gamma}^{\lambda},\quad\text{on
$b\Omega$.}$

The second-order coefficients
$\widetilde{a}^{\lambda},\widetilde{b}^{\lambda},\widetilde{c}^{\lambda}$ are
given by

(2.13)
$\displaystyle\widetilde{a}^{\lambda}(\xi,\eta)=(a^{\lambda}\circ\Gamma^{\lambda})(\xi^{\lambda}_{x})^{2}+2(b^{\lambda}\circ\Gamma^{\lambda})\xi^{\lambda}_{x}\xi^{\lambda}_{y}+(c^{\lambda}\circ\Gamma^{\lambda})(\xi^{\lambda}_{y})^{2},$
(2.14)
$\displaystyle\widetilde{b}^{\lambda}(\xi,\eta)=(a^{\lambda}\circ\Gamma^{\lambda})\xi^{\lambda}_{x}\eta^{\lambda}_{x}+(b^{\lambda}\circ\Gamma^{\lambda})\left(\xi^{\lambda}_{x}\eta^{\lambda}_{y}+\xi^{\lambda}_{y}\eta^{\lambda}_{x}\right)+(c^{\lambda}\circ\Gamma^{\lambda})(\xi^{\lambda}_{y}\eta^{\lambda}_{y}),$
(2.15)
$\displaystyle\widetilde{c}^{\lambda}(\xi,\eta)=(a^{\lambda}\circ\Gamma^{\lambda})(\eta^{\lambda}_{x})^{2}+2(b^{\lambda}\circ\Gamma^{\lambda})\eta^{\lambda}_{x}\eta^{\lambda}_{y}+(c^{\lambda}\circ\Gamma^{\lambda})(\eta^{\lambda}_{y})^{2}.$

Hence
$\widetilde{a}^{\lambda},\widetilde{b}^{\lambda},\widetilde{c}^{\lambda}\in\mathcal{C}^{k+1+\mu,j}(\overline{\Omega})$.
The first-order coefficients $\widetilde{d}^{\lambda},\widetilde{e}^{\lambda}$
are linear combinations of the products of
$a^{\lambda},b^{\lambda},c^{\lambda},d^{\lambda},e^{\lambda}$ and the first
and second derivatives of $(\Gamma^{\lambda})^{-1}$, and hence
$\widetilde{d}^{\lambda},\widetilde{e}^{\lambda}\in\mathcal{C}^{k+\mu,j}(\overline{\Omega})$.
Since $\widetilde{f}^{\lambda}=f^{\lambda}\circ\Gamma^{\lambda}$, we have
$\widetilde{f}^{\lambda}\in\mathcal{C}^{k+\mu,j}(\overline{\Omega})$.

For the boundary coefficients, we have

$\begin{pmatrix}[1.5]\widetilde{\alpha}^{\lambda}(\xi,\eta)\\\
\widetilde{\beta}^{\lambda}(\xi,\eta)\end{pmatrix}=\begin{pmatrix}[1.5]\frac{\partial(\Gamma^{\lambda})^{-1}}{\partial
x}(\Gamma^{\lambda}(\xi,\eta))&\frac{\partial(\Gamma^{\lambda})^{-1}}{\partial
y}(\Gamma^{\lambda}(\xi,\eta))\\\
\frac{\partial(\Gamma^{\lambda})^{-1}}{\partial
x}(\Gamma^{\lambda}(\xi,\eta))&\frac{\partial(\Gamma^{\lambda})^{-1}}{\partial
y}(\Gamma^{\lambda}(\xi,\eta))\end{pmatrix}\begin{pmatrix}[1.5]\alpha^{\lambda}(\Gamma^{\lambda}(\xi,\eta))\\\
\beta^{\lambda}(\Gamma^{\lambda}(\xi,\eta))\end{pmatrix},$

or put it differently,
$\widetilde{\Upsilon}^{\lambda}=(\Gamma^{\lambda})^{-1}_{\ast}(\Upsilon^{\lambda})$.
It follows that
$\widetilde{\alpha}^{\lambda},\widetilde{\beta}^{\lambda}\in\mathcal{C}^{k+1+\mu,j}(\overline{\Omega})$,
and
$\widetilde{\gamma}^{\lambda}=\gamma^{\lambda}\circ\Gamma^{\lambda}\in\mathcal{C}^{k+1+\mu,j}(\overline{\Omega})$.
Since $\Gamma^{0}$ is the identity map, the above matrix is close to the
identity matrix (in matrix norm) when $\lambda$ is close to $0$. Consequently
we observe that for $|\lambda|$ small, the winding number of the vector field
$\widetilde{\Upsilon}^{\lambda}(\widetilde{\alpha}^{\lambda},\widetilde{\beta}^{\lambda})$
along $b\Omega$ is the same as that of $(\alpha^{\lambda},\beta^{\lambda})$
along $b\Omega^{\lambda}$:

$\frac{1}{2\pi}[\widetilde{\Upsilon}^{\lambda}]_{b\Omega}=\frac{1}{2\pi}[\Upsilon^{\lambda}]_{b\Omega^{\lambda}}=\chi_{0}.$

Since $\Gamma^{0}$ is the identity map, in view of expressions (2.13), (2.14)
and (2.15), we have $\widetilde{a}^{0}=a^{0}$, $\widetilde{b}^{0}=b^{0}$ and
$\widetilde{c}^{0}=c^{0}$. Therefore for $|\lambda|<\varepsilon$, we have
$\widetilde{d}^{\lambda}:=\widetilde{a}^{\lambda}\widetilde{c}^{\lambda}-(\widetilde{b}^{\lambda})^{2}\geq\varepsilon_{0}>0$,
in other words equation (2.11) is uniformly elliptic for $\lambda$ close to
$0$. Applying Proposition 2.8 for $\lambda=0$, we get a map
$\Psi_{0}:\Omega=\Omega^{0}\to D^{0}$, such that $\Psi_{0}\in
C^{k+2+\mu}(\overline{\Omega_{0}})$. Writing
$\Psi_{0}(\xi,\eta)=(\xi^{\prime},\eta^{\prime})$, one can transform equations
(2.11) and (2.12) to

(2.16)
$\displaystyle\Delta\mathcal{V}^{\lambda}+a_{0}^{\lambda}\mathcal{V}_{\xi^{\prime}\xi^{\prime}}^{\lambda}+2b_{0}^{\lambda}\mathcal{V}_{\xi^{\prime}\eta^{\prime}}^{\lambda}+c_{0}^{\lambda}\mathcal{V}_{\eta^{\prime}\eta^{\prime}}^{\lambda}+d_{0}^{\lambda}\mathcal{V}^{\lambda}_{\xi^{\prime}}+e_{0}^{\lambda}\mathcal{V}^{\lambda}_{\eta^{\prime}}=f_{0}^{\lambda},\quad\text{in
$D^{0}$, }$ (2.17)
$\displaystyle\alpha_{0}^{\lambda}\mathcal{V}^{\lambda}_{\xi^{\prime}}+\beta_{0}^{\lambda}\mathcal{V}^{\lambda}_{\xi^{\prime}}=\gamma_{0}^{\lambda},\quad\text{on
$bD^{0}$.}$

Here
$\mathcal{V}(\xi^{\prime},\eta^{\prime})=v\circ\Psi_{0}^{-1}(\xi^{\prime},\eta^{\prime})$,
$a_{0}^{0}=b_{0}^{0}=c_{0}^{0}=0$, and
$|a_{0}^{\lambda}|,|b_{0}^{\lambda}|,|c_{0}^{\lambda}|<\varepsilon$ if
$|\lambda|<\delta_{0}$ for some small positive $\delta_{0}$. By expressions
similar to (2.13), (2.14), and (2.15), we see that the second-order
coefficients $a_{0}^{\lambda},b_{0}^{\lambda},c_{0}^{\lambda}$ are linear
combinations of the products of
$\widetilde{a}^{\lambda}\circ\Psi^{-1}_{0},\widetilde{b}^{\lambda}\circ\Psi^{-1}_{0},\widetilde{c}^{\lambda}\circ\Psi^{-1}_{0}$
and the first space derivatives of $\Psi_{0}$, and thus they are in the class
$\mathcal{C}^{k+1+\mu,j}(\overline{D_{0}})$. The first-order coefficients
$d_{0}^{\lambda},e_{0}^{\lambda}$ are linear combinations of products of
$\widetilde{a}^{\lambda}\circ\Psi^{-1}_{0},\widetilde{b}^{\lambda}\circ\Psi^{-1}_{0},\widetilde{c}^{\lambda}\circ\Psi^{-1}_{0},\widetilde{d}^{\lambda}\circ\Psi^{-1}_{0},\widetilde{e}^{\lambda}\circ\Psi^{-1}_{0}$
and the first and second derivatives of $\Psi_{0}$, so
$d_{0}^{\lambda},e_{0}^{\lambda}\in\mathcal{C}^{k+\mu,j}(\overline{D_{0}})$.
In a similar way $f^{\lambda}_{0}\in\mathcal{C}^{k+\mu,j}(\overline{D^{0}})$
and
$\alpha_{0}^{\lambda},\beta_{0}^{\lambda},\gamma_{0}^{\lambda}\in\mathcal{C}^{k+1+\mu,j}(\overline{D^{0}})$.

Let $\Upsilon_{0}^{\lambda}=(\alpha^{\lambda}_{0},\beta^{\lambda}_{0})$. Then
$\Upsilon_{0}^{\lambda}=(\Psi_{0})_{\ast}\widetilde{\Upsilon}^{\lambda}$. By
Lemma 2.9, the winding number of
$\Upsilon_{0}^{\lambda}=(\alpha^{\lambda}_{0},\beta^{\lambda}_{0})$ is the
same as the winding number of $\widetilde{\Upsilon}^{\lambda}$:

$\frac{1}{2\pi}[\Upsilon_{0}^{\lambda}]_{bD^{0}}=\frac{1}{2\pi}[\widetilde{\Upsilon}^{\lambda}]_{b\Omega}=\chi_{0}.$

Rewrite equation (2.16) as

(2.18)
$a_{1}^{\lambda}\mathcal{V}_{\xi^{\prime}\xi^{\prime}}^{\lambda}+2b_{0}^{\lambda}\mathcal{V}_{\xi^{\prime}\eta^{\prime}}^{\lambda}+c_{1}^{\lambda}\mathcal{V}_{\eta^{\prime}\eta^{\prime}}^{\lambda}+d_{0}^{\lambda}\mathcal{V}^{\lambda}_{\xi^{\prime}}+e_{0}^{\lambda}\mathcal{V}^{\lambda}_{\eta^{\prime}}=f_{0}^{\lambda},\quad\text{in
$D^{0}$,}$

where $a_{1}^{\lambda}=1+a_{0}^{\lambda}$ and
$c_{1}^{\lambda}=1+c_{0}^{\lambda}$. For $|\lambda|<\delta_{1}$, the
determinant for the above equation satisfies

$d_{0}^{\lambda}=a_{1}^{\lambda}c_{1}^{\lambda}-2b_{0}^{\lambda}\geq\varepsilon_{1}>0.$

By above we have that
$a_{1}^{\lambda},b_{0}^{\lambda},c_{1}^{\lambda},d^{\lambda}_{0}\in\mathcal{C}^{k+1+\mu,j}(D^{0})$.
Let
$\psi^{\lambda}:\zeta^{\prime}=\xi^{\prime}+i\eta^{\prime}\mapsto\tau^{\lambda}=s^{\lambda}+it^{\lambda}$
be a family of coordinate systems that satisfies:

$\tau^{\lambda}_{\overline{\zeta^{\prime}}}+q^{\lambda}(\zeta^{\prime})\tau^{\lambda}_{\zeta^{\prime}}=0,\quad
q^{\lambda}=\frac{a_{1}^{\lambda}-\sqrt{d_{0}^{\lambda}}-ib_{0}^{\lambda}}{a_{1}^{\lambda}+\sqrt{d_{0}^{\lambda}}+ib_{0}^{\lambda}}.$

Note that for $\lambda=0$, the map $\psi^{0}$ is the identity map,
$q^{0}\equiv 0$, and the above equation is trivially satisfied.

By Lemma 2.9, $\psi^{\lambda}\in\mathcal{C}^{k+2+\mu,j}(\overline{D^{0}})$.
Let $\psi^{\lambda}:D^{0}\to D^{\lambda}$. Then equations (2.18) and (2.17)
are transformed into

$\displaystyle\Delta\,\mathcal{U}^{\lambda}+p^{\lambda}(s^{\lambda},t^{\lambda})\,\mathcal{U}^{\lambda}_{s^{\lambda}}+q^{\lambda}(s^{\lambda},t^{\lambda})\,\mathcal{U}^{\lambda}_{t^{\lambda}}=h^{\lambda}(s^{\lambda},t^{\lambda}),\quad\text{in
$D^{\lambda}$.}$
$\displaystyle\nu_{1}^{\lambda}\,\mathcal{U}^{\lambda}_{s}+\nu_{2}^{\lambda}\,\mathcal{U}^{\lambda}_{t}=g^{\lambda}(s^{\lambda},t^{\lambda}),\quad\text{on
$bD^{\lambda}$, }$

where

$\displaystyle
p^{\lambda}=\frac{J^{\lambda}}{4\sqrt{d_{0}^{\lambda}}}\left(1+a_{1}^{\lambda}s^{\lambda}_{\xi^{\prime}\xi^{\prime}}+2b_{0}^{\lambda}s^{\lambda}_{\xi^{\prime}\eta^{\prime}}+c_{1}^{\lambda}s^{\lambda}_{\eta^{\prime}\eta^{\prime}}+d_{0}^{\lambda}s^{\lambda}_{\xi^{\prime}}+e^{\lambda}_{0}s^{\lambda}_{\eta^{\prime}}\right),$
$\displaystyle
q^{\lambda}=\frac{J^{\lambda}}{4\sqrt{d_{0}^{\lambda}}}\left(1+a_{1}^{\lambda}t^{\lambda}_{\xi^{\prime}\xi^{\prime}}+2b_{0}^{\lambda}t^{\lambda}_{\xi^{\prime}\eta^{\prime}}+c_{1}^{\lambda}t^{\lambda}_{\eta^{\prime}\eta^{\prime}}+d_{0}^{\lambda}t^{\lambda}_{\xi^{\prime}}+e^{\lambda}_{0}t^{\lambda}_{\eta^{\prime}}\right),$
$\displaystyle
h^{\lambda}(s^{\lambda},t^{\lambda})=\frac{J^{\lambda}f^{\lambda}_{0}}{4\sqrt{d_{0}^{\lambda}}}.$

Hence
$p^{\lambda},q^{\lambda}\in\mathcal{C}^{k+\mu,j}(\overline{D^{\lambda}})$.
Since $f^{\lambda}_{0}\in\mathcal{C}^{k+\mu,j}(\overline{D^{0}})$, we have
$h^{\lambda}\in\mathcal{C}^{k+\mu,j}(\overline{D^{\lambda}})$. Similarly it is
easy to see
$\nu_{1}^{\lambda},\nu_{2}^{\lambda},g^{\lambda}\in\mathcal{C}^{k+1+\mu,j}(\overline{D^{\lambda}})$.
Let $\Upsilon_{1}^{\lambda}=(\nu_{1}^{\lambda},\nu_{2}^{\lambda})$. Then
$\Upsilon_{1}^{\lambda}=\psi^{\lambda}_{\ast}(\Upsilon_{0}^{\lambda})$. Since
$\psi^{0}_{\ast}$ is the identity matrix, we see that the winding number of
$\Upsilon_{1}^{\lambda}$ is the same as that of $\Upsilon_{0}^{\lambda}$ for
$|\lambda|$ small:

$\frac{1}{2\pi}[\Upsilon^{\lambda}]_{b\Omega^{\lambda}}=\frac{1}{2\pi}[\Upsilon_{1}^{\lambda}]_{bD^{\lambda}}=\chi_{0}.$

We now put together the series of transformation we did:

$\varphi^{\lambda}=\psi^{\lambda}\circ\Psi^{0}\circ(\Gamma^{\lambda})^{-1}:\Omega^{\lambda}\longrightarrow
D^{\lambda}.$

Since $\Gamma^{\lambda}\in\mathcal{C}^{k+2+\mu,j}(\overline{\Omega})$,
$\Psi^{0}\in\mathcal{C}^{k+2+\mu,j}(\overline{\Omega_{0}})$, and
$\psi^{\lambda}\in\mathcal{C}^{k+2+\mu,j}(\overline{D_{0}})$, we have
$\varphi^{\lambda}\circ\Gamma^{\lambda}:\Omega\to
D^{\lambda}\in\mathcal{C}^{k+2+\mu,j}(\overline{\Omega})$. ∎

##  3\. A review of Vekua’s theory on fixed domains

In this section we review Vekua’s theory for solving oblique derivative
boundary value problem.

First we introduce Problem A (as in [VA62]) on a fixed domain.
Problem A (Generalized Riemann-Hilbert problem). Find in a bounded domain
$\Omega\subset\mathbb{C}$ the solution $w(z)=u(z)+iv(z)$ of the equation

(3.1)
$\begin{cases}\mathcal{L}(w)\equiv\partial{}_{\overline{z}}w+A(z)w(z)+B(z)\overline{w(z)}=F(z)\quad\text{in
$\Omega$;}\\\ \alpha u+\beta v\equiv Re[\overline{l(z)}w]=\gamma\quad\text{on
$b\Omega$, }\end{cases}$

where $l=\alpha+i\beta$ and is assumed to have unit length.

######  Definition 3.1.

The _index of Problem A_ , denoted by $n$, is the winding number of $l$ along
$b\Omega$:

$n:=\frac{1}{2\pi}[l]_{b\Omega}.$

Here we denote by $[l]_{b\Omega}$ the change in the argument of $l(t)$ as $t$
loops around the boundary $b\Omega$ once counterclockwise.

For now let us assume that

(3.2) $\displaystyle b\Omega\in C^{1+\mu},\quad A,B,F\in L^{p}(\Omega),\quad
2<p<\infty;$ (3.3) $\displaystyle l,\gamma\in C^{\mu}(\overline{\Omega}),\quad
0<\mu<1,\quad\nu=\min(\mu,\frac{p-2}{p}).$

A priori, we seek a solution $w$ in the class $C^{0}(\overline{\Omega})$.

First let us consider the case when $\Omega$ is simply-connected. By the
Riemann-mapping theorem it suffices to assume $\Omega=\mathbb{D}$. In this
case $n=\frac{1}{2\pi}[l]_{S^{1}}$, where $S^{1}$ is the boundary circle. We
now reduce the problem to a simpler form. Set $q(t)=-\arg(l(t))+n\,\arg(t)$.
Then $q$ is a well-defined function on $S^{1}$ and $q\in C^{\mu}(S^{1})$. We
have

$\displaystyle\overline{l(z)}$ $\displaystyle=e^{-i\arg(l(z))}$
$\displaystyle=e^{p(z)+i(-\arg(l(z))+n\arg z)}e^{-p(z)}e^{-i(n\arg(z))}$
$\displaystyle=e^{\chi(z)}e^{-p(z)}z^{-n}.$

Here $\chi$ is the holomorphic function in $\mathbb{D}$ taking the value
$p+iq$ on $S^{1}$. It is defined by the Schwarz integral:

(3.4)
$\chi(z):=\frac{1}{2\pi}\int_{S^{1}}q(t)\frac{t+z}{t-z}\frac{dt}{t},\quad\quad
z\in\mathbb{D}.$

By Proposition 7.12, $\chi\in C^{\mu}(\overline{\Omega})$. Using the
substitutions

(3.5) $\overline{l(z)}=z^{-n}e^{\chi(z)}e^{-p(z)},\quad
w_{\ast}(z)=e^{\chi(z)}w(z),$

$w_{\ast}$ is then a solution to the problem

(3.6)
$\begin{cases}\partial{}_{\overline{z}}w_{\ast}+A_{\ast}w_{\ast}+B_{\ast}\overline{w_{\ast}}=F_{\ast},\quad\text{in
$\mathbb{D}$; }\\\ Re[z^{-n}w_{\ast}]=\gamma_{\ast},\quad\text{on
$S^{1}$,}\end{cases}$

where

(3.7) $A_{\ast}=A,\quad B_{\ast}=Be^{2iIm\,\chi(z)},\quad
F_{\ast}=Fe^{\chi(z)},\quad\gamma_{\ast}(z)=\gamma(z)e^{p(z)}\quad\text{in
$\mathbb{D}$.}$

Conversely, given any solution $w_{\ast}$ to the problem (3.6), a solution to
the original problem (3.1) is given by $w(z)=e^{-\chi(z)}w_{\ast}(z)$. From
now on we can assume the problem takes the form (3.6), and we write
$w,A,B,F,\gamma$ without the $\ast$.

By assumption $\partial{}_{\overline{z}}w\in L^{1}(\overline{\Omega})$ and
Proposition 7.5, $w$ satisfies the integral equation

(3.8) $w+T_{\mathbb{D}}(Aw+B\overline{w})=\Phi+T_{\mathbb{D}}F,\quad\text{in
$\mathbb{D}$,}$

where $\Phi$ is some function holomorphic in $\mathbb{D}$. Now in view of
(7.1), any continuous solution $w$ of the above equation satisfies

$\partial{}_{\overline{z}}w+A(z)w+B(z)\overline{w(z)}=F(z).$

We would like to choose $\Phi$ in such a way that the solution to equation
(3.8) also satisfies the boundary condition $Re[z^{-n}w]=\gamma$ on $S^{1}$,
where $n$ is the index of (3.1). Then such solution will solve the original
Problem A.

First suppose $n\geq 0$. We write $\Phi$ in the form

(3.9) $\displaystyle\Phi(z)$
$\displaystyle=\Phi_{0}(z)+\frac{z^{2n+1}}{\pi}\iint_{\mathbb{D}}\frac{\overline{A(\zeta)w(\zeta)}+\overline{B(\zeta)}w(\zeta)}{1-\overline{\zeta}z}\,dA(\zeta)$
$\displaystyle\quad-\frac{z^{2n+1}}{\pi}\iint_{\mathbb{D}}\frac{\overline{F(\zeta)}}{1-\overline{\zeta}z}\,dA(\zeta),$

where $\Phi_{0}$ is some function to be determined, and it is holomorphic in
$\mathbb{D}$. Substituting (3.9) into equation (3.8), we obtain the following
integral equation for $w$:

(3.10) $w(z)+P_{n}(Aw+B\overline{w})=\Phi_{0}(z)+P_{n}F,$

where $P_{n}$ is the operator defined by:

(3.11)
$P_{n}\varphi:=-\frac{1}{\pi}\iint_{\mathbb{D}}\left(\frac{\varphi(\zeta)}{\zeta-z}+\frac{z^{2n+1}\overline{\varphi(\zeta)}}{1-\overline{\zeta}z}\right)\,dA(\zeta).$

The integral equation (3.10) is of Fredholm type, which we now describe.
Denoting by $\mathbb{D}^{c}$ the exterior of the unit disk, we can write
$P_{n}\varphi$ in the form

(3.12)
$P_{n}\varphi(z)=T_{\mathbb{D}}\varphi(z)+z^{2n+1}T_{\mathbb{D}^{c}}\varphi_{1}(z),$

where

(3.13)
$T_{\mathbb{D}^{c}}\varphi_{1}(z)=-\frac{1}{\pi}\iint_{\mathbb{D}^{c}}\frac{\varphi_{1}(\zeta)}{\zeta-z}\,dA(\zeta),\quad\varphi_{1}(\zeta)=\frac{\overline{\varphi\left(\frac{1}{\overline{\zeta}}\right)}}{\zeta\overline{\zeta}^{2}}.$

If $\varphi\in L^{p}(\mathbb{D}),2<p\leq\infty$, $\varphi_{1}$ satisfies

$\int_{\mathbb{D}}\left|\varphi_{1}\left(\frac{1}{\zeta}\right)\right|^{p}|\zeta|^{-2p}dA(\zeta)=\int_{\mathbb{D}}\left|\varphi\left(\zeta\right)\right|^{p}|\zeta|^{3p-2p}dA(\zeta)<\infty.$

According to Proposition 7.6, $T_{\mathbb{D}^{c}}\varphi_{1}$ is
$\alpha_{p}$-Hölder continuous in the entire plane, for
$\alpha_{p}=\frac{p-2}{p}$. By Proposition 7.4 the function
$T_{\mathbb{D}}\varphi$ is $\alpha$-Hölder continuous in the plane. Hence from
(3.12) $P_{n}\varphi$ is $\alpha$-Hölder continuous. It is easy to check that

$Re[z^{-n}P_{n}\varphi]=0\quad\quad\text{on $S^{1}$.}$

By assumption, $F\in L^{p}(\mathbb{D})$ for $2<p<\infty$, so by letting $F$ be
the $\varphi$ above, we see that $P_{n}F$ is Hölder continuous in the entire
plane, and $Re[z^{-n}P_{n}F(z)]=0$ on $S^{1}$. As we will see below, the
solution $w$ is Hölder continuous in $\overline{\mathbb{D}}$, and thus
$Aw+B\overline{w}\in L^{p}(\mathbb{D})$, for $p>2$ and
$Re[z^{-n}P_{n}(Aw+B\overline{w})]=0$ on $S^{1}$.

Now in view of equation (3.10), if the holomorphic function $\Phi_{0}$
satisfies the boundary condition

(3.14) $Re[z^{-n}\Phi_{0}(z)]=\gamma\quad\quad\text{on $S^{1}$,}$

then the solution of the integral equation (3.10) is a solution of the
boundary value problem (3.6).

By Proposition 7.20, the general solution of the problem (3.14) is given by
the formula:

$\Phi_{0}(z)=\frac{z^{n}}{2\pi
i}\int_{S^{1}}\gamma(t)\frac{t+z}{t-z}\frac{dt}{t}+\sum_{k=0}^{2n}c_{k}z^{k},$

where the $c_{k}$’s are complex numbers satisfying the relations

(3.15) $c_{2n-k}=-\overline{c_{k}},\quad k=0,1,\dots,n.$

Thus for $n\geq 0$, the problem is reduced to the following equivalent
integral equation of Fredholm type:

(3.16) $w+Q_{n}w=P_{n}F+\frac{z^{n}}{2\pi
i}\int_{S^{1}}\gamma(t)\frac{t+z}{t-z}\frac{dt}{t}+\sum_{k=0}^{2n}c_{k}z^{k},$

(see [VA62]*p. 225, (1.10)), where $Q_{n}$ is the operator defined by

(3.17) $Q_{n}w:=P_{n}(Aw+B\overline{w}),$

and

(3.18)
$\sum_{k=0}^{2n}c_{k}z^{k}=\sum_{k=0}^{n-1}a_{k}(z^{k}-z^{2n-k})+ib_{k}(z^{k}+z^{2n-k})+ic_{0}z^{n},$

with $\alpha_{k},\beta_{k},c_{0}$ being arbitrary real constants. As shown
earlier, $P_{n}F\in C^{\alpha_{p}}(\overline{\mathbb{D}})$ for
$\alpha_{p}=\frac{p-2}{p}$. The second term on the right-hand side of equation
(3.16) can be written as $z^{n}\mathcal{S}\gamma$, where $\mathcal{S}$ is the
Schwarz integral operator. By Proposition 7.12 applied to $\gamma\in
C^{\mu}(S^{1})$, one has $\mathcal{S}\gamma\in
C^{\mu}(\overline{\mathbb{D}})$. Hence the right-hand side of the equation
lies in the class $C^{\nu}(\overline{\mathbb{D}})$, for
$\nu=\min(\alpha_{p},\mu)$.

It is shown in [VA62]*p. 296 that equation (3.16) has a unique solution $w$ in
the class $L^{q}(\mathbb{D})$, $\frac{p}{p-1}<q<\infty$, for any right-hand
side function in the class $L^{q}(\mathbb{D})$. By Theorem 1.25 in [VA62]*p.
50, one has $Q_{n}w\in C^{0}(\overline{\mathbb{D}})$, and so it follows from
equation (3.16) that $w\in C^{0}(\overline{\mathbb{D}})$. By Proposition 7.4
and Proposition 7.6 applied to expression (3.12) where we replace $\varphi$ by
$w$, we have $Q_{n}w\in C^{\alpha_{p}}(\overline{\mathbb{D}})$. By equation
(3.16) again we obtain that $w\in C^{\nu}(\overline{\mathbb{D}})$,
$\nu=\min(\alpha_{p},\mu)$.

Therefore when the index $n\geq 0$, the solution to the non-homogeneous
Problem A exists for any boundary data, and the homogeneous Problem A
($F\equiv 0$, $\gamma\equiv 0$) admits $2n+1$ $\mathbb{R}$-linearly
independent solutions.

If $n<0$. Referring to [VA62]*p. 302, the solution to Problem A (see (3.1))
exists if and only if the following conditions are satisfied:

(3.19) $\frac{1}{2i}\int_{b\Omega}l(t)w^{\prime}_{i}(t)\gamma(t)\,dt-
Re\iint_{\Omega}w^{\prime}_{i}(z)F(z)\,dx\,dy=0,$

for $i=1,\dots,2k-1$, $k=-n$, and $(w^{\prime}_{1}),\dots,(w^{\prime}_{2k-1})$
are linearly-independent solutions of the homogeneous adjoint problem A’:

(3.20) $\displaystyle\partial_{\overline{z}}w-Aw-\overline{Bw}=0,\quad\text{in
$\Omega$; }$ (3.21) $\displaystyle Re[l(z)z^{\prime}(s)w(z)]=0,\quad\text{ on
$b\Omega$. }$

Here we write $z^{\prime}(s)=\frac{dz(s)}{ds}$. In this case if the solution
for Problem A (3.1) exists, then it is unique, and the solution can be
expressed as the unique solution to the following integral equation (see
[VA62]*p. 300 for details):

(3.22) $w+(Q_{k}^{\ast})w=P_{k}^{\ast}F+\frac{1}{\pi
i}\int_{b\Omega}\frac{\gamma(t)\,dt}{t^{k}(t-z)},\quad k=-n,$

where $P_{k}^{\ast}$ and $Q_{k}^{\ast}$ are defined to be

(3.23)
$P_{k}^{\ast}f:=-\frac{1}{\pi}\iint_{\Omega}\left(\frac{f(\zeta)}{\zeta-z}+\frac{\overline{\zeta}^{2k-1}\overline{f(\zeta)}}{1-z\overline{\zeta}}\right)\,dA(\zeta),$
$(Q_{k}^{\ast})f:=P_{k}^{\ast}(B\overline{f}).$

The above method of reducing the problem to a Fredholm integral equation on
the domain fails for multiply-connected domains. The geometry of the circle
allows for the vanishing of the term $Re[z^{-n}P_{n}\varphi]$ on $S^{1}$.
Vekua introduced the theory of generalized analytic functions to deal with the
general case. We state here a few of his results without much elaborations.
The reader can refer to [VA62] for details.

We say a function $w$ belongs to the class $\mathcal{L}_{p,2}(A,B,\mathbb{C})$
if $w$ satisfies

$\partial{}_{\overline{z}}w+Aw+B\overline{w}=0$

in the sense of distributions, for $A,B\in\mathcal{L}_{p,2}(\mathbb{C})$. (The
latter is defined in (2.1).) In our application, $A,B$ are defined only on
$\Omega$ in which case we extend $A,B$ trivially to $\mathbb{C}$ by setting
them to be $0$ on $\mathbb{C}\setminus\overline{\Omega}$.

Following Vekua, we call such $w$ a _generalized analytic function_ of the
class $\mathcal{L}_{p,2}(A,B,\mathbb{C})$. For fixed $t\in\mathbb{C}$, and
$A,B\in L_{p,2}(\mathbb{C})$, consider the following integral equations
([VA62]*p 167):

(3.24)
$X_{1}(z,t)-\frac{1}{\pi}\iint_{\mathbb{C}}\frac{A(\zeta)X_{1}(\zeta,t)+B(\zeta)\overline{X_{1}(\zeta,t)}}{\zeta-z}\,dA(\zeta)=\frac{1}{2(t-z)},$
(3.25)
$X_{2}(z,t)-\frac{1}{\pi}\iint_{\mathbb{C}}\frac{A(\zeta)X_{2}(\zeta,t)+B(\zeta)\overline{X_{2}(\zeta,t)}}{\zeta-z}\,dA(\zeta)=\frac{1}{2i(t-z)}.$

It is proved in [VA62]*p. 156 that equation (3.24) (respectively (3.25))
admits a unique solution in the class $L_{q,0}(\mathbb{C})$ if $A,B\in
L_{p,2}(\mathbb{C})$ and the right-hand side lies in the class
$L_{q,0}(\mathbb{C})$, for $q\geq\frac{p}{p-1}=p^{\prime}$. Since
$2<p<\infty$, we have $1<p^{\prime}<2$, and $\frac{1}{t-z}\in
L_{q,0}(\mathbb{C})$, for any $p^{\prime}\leq q<2$. So equations (3.24) and
(3.25) admit solutions $X_{1}\in L_{q,0}(\mathbb{C})$ and $X_{2}\in
L_{q,0}(\mathbb{C})$ respectively, for any $q\in[p^{\prime},2)$. For $z\neq
t$, $X_{i}$ satisfies the equation

$\overline{\partial{}_{z}}X_{i}(z,t)+A(z)X_{i}(z,t)+B(z)\overline{X_{i}(z,t)}=0,\quad
i=1,2.$

Furthermore, we have the following representation for $X_{i}$:

(3.26) $X_{1}(z,t)=\frac{e^{\omega_{1}(z,t)}}{2(t-z)},\quad\quad
X_{2}(z,t)=\frac{e^{\omega_{2}(z,t)}}{2i(t-z)},$

where for $i=1,2$, we have

(3.27)
$\omega_{i}(z,t)=\frac{t-z}{\pi}\iint_{\mathbb{C}}\frac{1}{(\zeta-z)(t-\zeta)}\left[A(\zeta)+B(\zeta)\frac{\overline{X_{i}(\zeta,t)}}{X_{i}(\zeta,t)}\right]\,dA(\zeta).$

The $\omega_{i}$ satisfies the following estimates:

(3.28) $|\omega_{i}(z_{1},t)-\omega_{i}(z_{2},t)|\leq
C_{p}|z_{1}-z_{2}|^{\alpha_{p}},\quad\alpha_{p}=\frac{p-2}{p}.$ (3.29)
$|\omega_{i}(z,t)|\leq C_{p}|z-t|^{\alpha},\quad i=1,2.$

Hence in view of (3.26), $X_{j}(\cdot,t)$ is in fact Hölder continuous
everywhere in the plane except at $z=t$.

Define $G_{i}$, the _fundamental kernels of the class_
$\mathcal{L}_{p,2}(A,B,\mathbb{C})$ as follows:

(3.30) $\displaystyle G_{1}(z,\zeta)=X_{1}(z,\zeta)+iX_{2}(z,\zeta),$
$\displaystyle G_{2}(z,\zeta)=X_{1}(z,\zeta)-iX_{2}(z,\zeta).$

By (3.26) we have

(3.31) $\displaystyle
G_{1}(z,\zeta)=\frac{e^{\omega_{1}(z,\zeta)}+e^{\omega_{2}(z,\zeta)}}{2(\zeta-z)},\qquad
G_{2}(z,\zeta)=\frac{e^{\omega_{1}(z,\zeta)}-e^{\omega_{2}(z,\zeta)}}{2(\zeta-z)}.$

The $G_{i}^{\prime}s$ satisfy the relation:

(3.32)
$\begin{split}\overline{\partial{}_{z}}G_{1}(z,\zeta)+A(z)G_{1}(z,\zeta)+B(z)\overline{G_{2}(z,\zeta)}=0,\\\
\overline{\partial{}_{z}}G_{2}(z,\zeta)+A(z)G_{2}(z,\zeta)+B(z)\overline{G_{1}(z,\zeta)}=0,\end{split}$

for any $z,\zeta\in\mathbb{C}$, $z\neq\zeta$. In addition we have the
estimates

(3.33) $G_{1}(z,\zeta)=\frac{1}{\zeta-z}+O(|z-\zeta|^{-\frac{2}{p}}),\quad
G_{2}(z,\zeta)=O(|z-\zeta|^{-\frac{2}{p}}).$

Hence $G_{1}$ behaves like the Cauchy kernel, modulo a mild singularity.

Denote by $G_{1}^{\prime}(z,\zeta)$ and $G_{2}^{\prime}(z,\zeta)$ the adjoint
fundamental kernel of the adjoint equation
$\mathcal{L}^{\prime}(w^{\prime})=\overline{\partial{}_{z}}w^{\prime}-Aw^{\prime}-\overline{B}\overline{w}^{\prime}=F^{\prime}$.
The following relations hold: ([VA62]*p.174)

(3.34) $G_{1}(z,\zeta)=-G_{1}^{\prime}(\zeta,z),\quad\quad
G_{2}(z,\zeta)=-\overline{G_{2}^{\prime}(\zeta,z)}.$

We will denote by $G_{1}(z,t,\Omega)$ and $G_{2}(z,t,\Omega)$ the fundamental
kernels of the class $\mathcal{L}_{p,2}(A,B,\mathbb{C})$, if $A\equiv B\equiv
0$ in $\Omega^{c}=\mathbb{C}\setminus\overline{\Omega}$. Given $A,B\in
L^{p}(\Omega)$, we let $\widetilde{A},\widetilde{B}$ be the trivial extensions
which vanish outside $\Omega$.

For our proof of the parameter problem, we need the following representation
formula for generalized analytic functions.

######  Proposition 3.2.

Suppose $w$ is continuous in $\overline{\Omega}$ and satisfies the equation

$\overline{\partial{}_{z}}w+A(z)w(z)+B(z)\overline{w(z)}=F(z),\quad
z\in\Omega$

where $A,B,F\in L^{p}(\Omega),p>2$. Then $w$ has the representation formula:

(3.35) $\displaystyle w(z)$ $\displaystyle=\frac{1}{2\pi
i}\int_{b\Omega}G_{1}(z,\zeta)w(\zeta)\,d\zeta-
G_{2}(z,\zeta)\overline{w(\zeta)}\,\overline{d\zeta}$
$\displaystyle\quad-\frac{1}{\pi}\iint_{\Omega}G_{1}(z,\zeta)F(\zeta)+G_{2}(z,\zeta)\overline{F(\zeta)}\,dA(\zeta),\quad
z\in\Omega.$

###### Proof.

See [VA62]*p. 176 formula (10.9). ∎

######  Remark.

When $A\equiv B\equiv 0$, by formula (3.27) we have
$\omega_{1}(z,t)\equiv\omega_{2}(z,t)\equiv 0$. Hence from (3.31) we get
$G_{1}=\frac{1}{\zeta-z}$ and $G_{2}\equiv 0$, and (3.35) reduces to the
Cauchy-Green formula.

Next we state some existence and uniqueness results for Problem A. For
homogeneous boundary value problems, we call a finite set of linearly
independent solutions which spans the solution space a _complete system_.

######  Proposition 3.3.

Consider Problem A on a multiply-connected domain $\Omega$ whose boundary
$b\Omega$ has $m+1$ connected components. Let $h,h^{\prime}$ be the numbers of
linearly-independent solutions to the homogeneous Problem A and the
homogeneous adjoint Problem A’ (3.20)-(3.21), respectively. Let $n$ be the
index of Problem A.

Suppose that $n<0$. Then

$h=0,\quad h^{\prime}=m-2n-1.$

The non-homogeneous Problem A has a (unique) solution if and only if the
following relations are satisfied:

(3.36) $\frac{1}{2i}\int_{b\Omega}l(t)w^{\prime}_{i}(t)\gamma(t)\,dt-
Re\iint_{\Omega}w^{\prime}_{i}F(z)\,dxdy=0,$

for $i=1,...,m-2n-1$, and $w_{1}^{\prime},...,w^{\prime}_{m-2n-1}$ is a
complete system of solutions of the adjoint homogeneous Problem A’
(3.20)-(3.21).

Suppose that $n>m-1$. Then

$h=2n+1-m,\quad h^{\prime}=0.$

The non-homogeneous Problem A is always solvable and its general solution is
given by:

(3.37) $w(z)=w_{0}(z)+\sum_{j=1}^{2n+1-m}c_{j}w_{j},\quad\quad\text{$c_{j}$
are real constant, }$

where $\\{w_{1},...,w_{2n+1-m}\\}$ is a complete system of solutions of the
homogeneous problem A, and $w_{0}$ is a particular solution of the non-
homogeneous problem A.

For the proof see [VA62]*p. 253 Theorem 4.10 and Theorem 4.12.

######  Remark.

When the domain is simply-connected, we have $m=0$, and the above theorem
reduces to the results we obtained earlier. When $0\leq n\leq m-1$ ($m\geq
1$), the existence and uniqueness of solutions to problem A are more subtle,
and we refer the reader to [VA62]*chapter 4, §5.

From now on we only consider the case $n>m-1$. We can impose some additional
conditions on the solution of problem A to make it unique. This will be
necessary when later on we add parameter since regularity in parameter depends
on the uniqueness of these solutions.

######  Definition 3.4.

Let $\Omega$ be a bounded domain in $\mathbb{C}$ whose boundary $b\Omega$
contains $m+1$ connected components, for $m$ a non-negative integer. Let
$z_{1},...,z_{h_{0}}$ and $z_{1}^{\prime},...,z_{h_{1}}^{\prime}$ be fixed
points of $\Omega$ and its boundary $b\Omega$, respectively, satisfying the
following conditions:
(i) The number $h_{0}$ and $h_{1}$ satisfy the relation:

$2h_{0}+h_{1}=2n+1-m.$

(ii) There are $m$ curves, e.g. $\Gamma_{i_{1}},...,\Gamma_{i_{m}}$, among the
$m+1$ boundary curves $b\Omega_{0},...,b\Omega_{m}$, on each of which an odd
number of $z_{j}^{\prime}$ are situated.
Then we call $\\{\\{z_{r}\\}_{r=1}^{h_{0}}$,
$\\{z_{s}^{\prime}\\}_{s=1}^{h_{1}}\\}$ a _normally distributed set_ ,
henceforth denoted by $(h_{0},h_{1},\overline{\Omega})$.

######  Remark.

In our choice of $h_{0}$ and $h_{1}$, two extreme cases are possible:
(1) $h_{0}=0$ and $h_{1}=2n+1-m$.
(2) $h_{0}=n-m$ and $h_{1}=m+1$.
Consequently, we have $0\leq h_{0}\leq n-m$, $m+1\leq h_{1}\leq 2n+1-m$. If
$m=0$, i.e. the domain is simply-connected, condition (i) becomes
$2h_{0}+h_{1}=2n+1$, and thus in this case $h_{1}$ needs to be odd.

We can specify on $(h_{0},h_{1},\overline{\Omega})$ the values of the unknown
solution of Problem A:

(3.38) $\begin{split}w(z_{r})=a_{r}+ib_{r},\quad\quad r=1,...,h_{0},\\\
w(z_{s}^{\prime})=l(z_{s}^{\prime})(\gamma(z_{s}^{\prime})+ic_{s}),\quad\quad
s=1,...,h_{1}.\end{split}$

######  Proposition 3.5.

Let $n$ be the index of Problem A on a domain $\Omega$ in $\mathbb{C}$, and
let $m$ be the number of connected components of $b\Omega$. Assume that on a
normally distributed point set $(h_{0},h_{1},\overline{\Omega})$ conditions of
the form (3.38) are given. Then for $n>m-1$, Problem A has always a solution
satisfying these conditions, and this solution is uniquely determined.

The reader can refer to Vekua [VA62]*p. 285 - 287 for the proof.

The following regularity result for simply-connected domains follow from the
above construction of the solution by Vekua, though it is not explicitly
stated in his book. So for readers’ convenience we provide details of the
proof here. On the other hand the same estimates hold for multiply-connected
domains and one can prove it using a trick of Vekua to reduce the simply-
connected case. (See [VA62] p. 228 and p. 336.)

######  Proposition 3.6.

(i) Let $\Omega\subset\mathbb{C}$ be a domain with $C^{1+\mu}$ boundary, for
$0<\mu<1$. Let $2<p<\infty$. Suppose $A,B,F\in L^{p}(\Omega)$, $l,\gamma\in
C^{\mu}(b\Omega)$. Then the solution to Problem A (3.1), if it exists, belongs
to the class $C^{\nu}(\overline{\Omega})$, for
$\nu=\text{min}\\{\alpha_{p},\mu\\}$, $\alpha_{p}=\frac{p-2}{p}$, and there
exists a constant $C$ depending on the coefficients $A,B,l,p$, such that

(3.39) $|w|_{\nu}\leq
M(A,B,l,p)\left(|w|_{0}+|F|_{L^{p}(\Omega)}+|\gamma|_{\mu}\right).$

(ii) Let $k\geq 0$, and $0<\mu<1$. Let $\Omega$ be a domain in $\mathbb{C}$
with $C^{k+1+\mu}$ boundary. Suppose $A,B,F\in C^{k+\mu}(\overline{\Omega})$,
$l,\gamma\in C^{k+1+\mu}(b\Omega)$. Then the solution $w$ of Problem A, if it
exists, belongs to the class $C^{k+1+\mu}(\overline{\Omega})$, and there
exists a constant $C_{k}$ depending on the coefficients $A,B,l$ such that

$|w|_{k+1+\mu}\leq
C_{k}(A,B,l)\left(|w|_{0}+|F|_{k+\mu}+|\gamma|_{k+1+\mu}\right).$

We remark that the regularity results are in [VA62]*p. 228 and p. 336. The
main purpose of the following proof is to derive the above two estimates. The
constants $C_{k}(A,B,l)$ are unknown here since the proof uses the open
mapping theorem.

Proof. We treat only the non-negative index case. The proof is similar if the
index is negative.
(i) By the Riemann mapping theorem it suffices to take $\Omega=\mathbb{D}$.
Moreover by the earlier remarks it suffices to assume that the Problem A has
the form (3.6). Let $\nu=\min(\alpha,\mu)$, $\alpha_{p}=\frac{p-2}{p}$. In
view of equation (3.16), any solution $w$ has the form

(3.40) $w=w_{\ast}+\sum_{s=0}^{2n}d_{s}w_{s}$

where $d_{s}$ are complex constants and $w_{\ast}$ is the (unique) solution to
the integral equation

(3.41) $w_{\ast}+Q_{n}w_{\ast}=P_{n}F+\mathcal{S}(\gamma),$

and $w_{s}$ is the (unique) solution to the integral equation

(3.42) $w_{s}+Q_{n}w_{s}=g_{s},\quad 0\leq s\leq 2n,$

where $g_{s}$ are the set of $2n+1$ functions
$\\{z^{k}-z^{2n-k},z^{k}+z^{2n-k},z^{n}\\}_{k=0}^{n-1}$ from expression
(3.18).

First we estimate $w_{\ast}$. Note that since the operator $I+Q_{n}$ is a
bijective bounded linear operator from the space $C^{0}(\overline{\Omega})$ to
itself, by the open mapping theorem it has a bounded inverse $(I+Q_{n})^{-1}$,
whose operator norm depend only on $A$ and $B$. Hence in view of Proposition
7.7 and Proposition 7.12 we have

$\displaystyle|w_{\ast}|_{0}$ $\displaystyle\leq
M(A,B)|P_{n}F+\mathcal{S}(\gamma)|_{0}$ $\displaystyle\leq
M(A,B)\left(|P_{n}F|_{0}+|\mathcal{S}(\gamma)|_{0}\right)$ $\displaystyle\leq
M^{\prime}(A,B)\left(|F|_{L^{p}}+|\gamma|_{\mu}\right).$

Next we estimate the Hölder norm of $w_{\ast}$. From (3.41) we have

$\displaystyle|w_{\ast}|_{\nu}$
$\displaystyle\leq|Q_{n}w_{\ast}|_{\alpha_{p}}+|P_{n}F|_{\alpha_{p}}+|\mathcal{S}\gamma|_{\mu}.$

By Proposition 7.7 we have

$\displaystyle|Q_{n}w_{\ast}|_{\alpha}$
$\displaystyle=|P_{n}(Aw_{\ast}+B\overline{w_{\ast}})|_{0}$ $\displaystyle\leq
C(p)|Aw_{\ast}+B\overline{w_{\ast}}|_{L^{p}(\mathbb{D})}$ $\displaystyle\leq
C(A,B,p)|w_{\ast}|_{0},$

and

$|P_{n}F|_{\alpha_{p}}\leq C(p)|F|_{L^{p}(\mathbb{D})}.$

By Proposition 7.12 we have

$|S\gamma|_{\mu}\leq C|\gamma|_{\mu}.$

Hence

(3.43) $\displaystyle|w_{\ast}|_{\nu}$ $\displaystyle\leq
C_{0}(A,B,p)\left(|w_{\ast}|_{0}+|F|_{L^{p}(\mathbb{D})}+|\gamma|_{\mu}\right)$
$\displaystyle\leq
C_{1}(A,B,p)\left(|F|_{L^{p}(\mathbb{D})}+|\gamma|_{\mu}\right).$

We now estimate $w_{s}$. Applying open mapping theorem to (3.42) we have

$|w_{s}|_{0}\leq C_{s}^{\prime}(A,B),\quad 0\leq s\leq 2n.$

Then

$|w_{s}|_{\alpha_{p}}\leq|Q_{n}w_{s}|_{\alpha_{p}}+|g_{s}|_{\alpha_{p}}\leq
C(A,B,p)(|w_{s}|_{0}+1)\leq C^{\prime\prime}_{s}(A,B,p).$

It remains to estimate the coefficients $d_{s}$ in (3.40). By the remarks in
[VA62]*p. 286-288, equation (6.10) (see also Proposition 4.4 below), they can
be expressed as in the form $\frac{P}{Q}$, where $Q\neq 0$ and $P$ and $Q$ are
linear combinations of the products of $w_{s}(z_{i}),\>w_{s}(z_{j}^{\prime})$
with terms like
$a_{i},\>b_{i},\>c_{j},\>\gamma(z_{j}^{\prime}),\>w_{\ast}(z_{i}),\>w_{\ast}(z_{j}^{\prime})$.
Since $|(a_{r},b_{r},c_{s})|\leq|w|_{0}$, we have

(3.44) $\displaystyle|d_{s}|$ $\displaystyle\leq
M_{s}(A,B)\left(|w|_{0}+|\gamma|_{0}+|w_{\ast}|_{0}\right)$ $\displaystyle\leq
M_{s}^{\prime}(A,B)\left(|w|_{0}+|F|_{L^{p}(\mathbb{D})}+|\gamma|_{\mu}\right).$

Hence in view of (3.40) and combining the above estimates we get

$\displaystyle|w|_{\nu}$
$\displaystyle\leq|w_{\ast}|_{\nu}+\sum_{s=0}^{2n}|d_{s}||w_{s}|_{\alpha_{p}}$
$\displaystyle\leq
M(A,B,p)\left(|w|_{0}+|F|_{L^{p}(\mathbb{D})}+|\gamma|_{\mu}\right).$

Finally let $\widetilde{w}$ be a solution to the original Problem A (3.1) on
$\mathbb{D}$:

(3.45)
$\begin{cases}\mathcal{L}(\widetilde{w})\equiv\partial{}_{\overline{z}}\widetilde{w}+\widetilde{A}(z)\widetilde{w}(z)+\widetilde{B}(z)\overline{\widetilde{w}(z)}=\widetilde{F}(z)\quad\text{in
$\Omega$.}\\\ \alpha u+\beta v\equiv
Re[\overline{l(z)}\widetilde{w}]=\widetilde{\gamma}\quad\text{on $b\Omega$.
}\end{cases}$

Then we have the following relations:

(3.46) $\displaystyle A(z)=\widetilde{A}(z),\quad
B(z)=\widetilde{B}e^{2i\chi(z)},\quad
F(z)=\widetilde{F}(z)e^{\chi(z)},\>z\in\mathbb{D};$ (3.47)
$\displaystyle\widetilde{w}(z)=e^{-\chi(z)}w(z),\>z\in
D,\quad\gamma(\zeta)=\widetilde{\gamma}(\zeta)e^{p(\zeta)},\>\zeta\in bS^{1}.$

where $\chi$ and $p=Re(\chi)$ are related to $l$ by formula (3.4). Hence

$|\widetilde{w}|_{\nu}\leq
C(\widetilde{A},\widetilde{B},l,p)\left(|\widetilde{w}|_{0}+|\widetilde{F}|_{L^{p}(\mathbb{D})}+|\widetilde{\gamma}|_{\mu}\right).$

(ii) Again let us first consider Problem A in the reduced form (3.6). Write
(3.41) as

(3.48) $w_{\ast}=-Q_{n}w_{\ast}+P_{n}F+\mathcal{S}(\gamma)$

First let us prove the statement for $k=0$. We have

$\displaystyle|w_{\ast}|_{1+\mu}$ $\displaystyle\leq
C\left(|Q_{n}w_{\ast}|_{1+\mu}+|P_{n}F|_{1+\mu}+|\mathcal{S}\gamma|_{1+\mu}\right)$
$\displaystyle=\left(|P_{n}(Aw_{\ast}+B\overline{w_{\ast}})|_{1+\mu}+|P_{n}F|_{1+\mu}+|\mathcal{S}\gamma|_{1+\mu}\right)$
$\displaystyle\leq
C^{\prime}\left(|Aw_{\ast}+B\overline{w_{\ast}}|_{\mu}+|F|_{\mu}+|\gamma|_{1+\mu}\right)$
$\displaystyle\leq
C^{\prime}\left\\{\left(|A|_{\mu}+|B|_{\mu}\right)|w_{\ast}|_{\mu}+|F|_{\mu}+|\gamma|_{1+\mu}\right\\}.$

In view of estimate (3.43), we know that

$|w_{\ast}|_{\mu}\leq C(A,B)\left(|F|_{\mu}+|\gamma|_{\mu}\right).$

Hence

$|w_{\ast}|_{1+\mu}\leq C_{0}(A,B)\left(|F|_{\mu}+|\gamma|_{1+\mu}\right).$

For $k>0$ we apply induction. By Proposition 7.8 and Proposition 7.12 applied
to equation (3.48) one gets

$\displaystyle|w_{\ast}|_{k+1+\mu}$ $\displaystyle\leq
C\left(|Q_{n}w_{\ast}|_{k+1+\mu}+|P_{n}F|_{k+1+\mu}+|\mathcal{S}\gamma|_{k+1+\mu}\right)$
$\displaystyle=\left(|P_{n}(Aw_{\ast}+B\overline{w_{\ast}})|_{k+1+\mu}+|P_{n}F|_{k+1+\mu}+|\mathcal{S}\gamma|_{k+1+\mu}\right)$
$\displaystyle\leq
C^{\prime}\left(|Aw_{\ast}+B\overline{w_{\ast}}|_{k+\mu}+|F|_{k+\mu}+|\gamma|_{k+1+\mu}\right)$
$\displaystyle\leq
C^{\prime}\left\\{\left(|A|_{k+\mu}+|B|_{k+\mu}\right)|w_{\ast}|_{k+\mu}+|F|_{k+\mu}+|\gamma|_{k+1+\mu}\right\\}.$

By the induction hypothesis $w_{\ast}$ satisfies the estimate

$|w_{\ast}|_{k+\mu}\leq
C_{k-1}(A,B)\left(|F|_{k-1+\mu}+|\gamma|_{k+\mu}\right).$

Hence

$|w_{\ast}|_{k+1+\mu}\leq
C_{k}(A,B)\left(|F|_{k+\mu}+|\gamma|_{k+1+\mu}\right).$

Next we estimate $w_{s}$, $0\leq s\leq 2n$. From equation (3.42) we have

(3.49) $w_{s}=-Q_{n}w_{s}+g_{s}.$

First we prove for $k=0$, so that $A,B\in C^{\mu}(\overline{\mathbb{D}})$. By
part (i), we know that $w_{s}\in C^{\frac{p-2}{p}}(\overline{D})$ for any
$2<p<\infty$. Hence $w_{s}\in C^{1-}(\overline{\Omega})$. By Proposition 7.8
we have

$\displaystyle|w_{s}|_{1+\mu}$ $\displaystyle\leq C(|Q_{n}w_{s}|_{1+\mu}+1)$
$\displaystyle=C\left(|P_{n}(Aw_{s}+B\overline{w_{s}})|_{1+\mu}+1\right)$
$\displaystyle\leq C^{\prime}\\{(|A|_{\mu}+|B|_{\mu})|w_{s}|_{\mu}+1\\}$
$\displaystyle\leq C_{0}^{\prime}(A,B).$

By (3.49) again and iterate the process we get $w_{s}\in C^{k+1+\mu}$ and

$|w_{s}|_{k+1+\mu}\leq C_{k}^{\prime}(A,B).$

Now for the coefficients $d_{s}$ we have estimate (3.44)

$|d_{s}|\leq
M_{s}^{\prime}(A,B)\left(|w|_{0}+|F|_{\mu}+|\gamma|_{\mu}\right).$

Finally we put together the estimates and we get

$\displaystyle|w|_{k+1+\mu}$
$\displaystyle\leq|w_{\ast}|_{k+1+\mu}+\sum_{s=0}^{2n}|d_{s}||w_{s}|_{k+1+\mu}$
$\displaystyle\leq
C_{k}(A,B)\left(|w|_{0}+|F|_{k+\mu}+|\gamma|_{k+1+\mu}\right).$

Finally we let $\widetilde{w}$ be the solution to the original Problem A
(3.45) on $\mathbb{D}$. In view of the relations (3.46) we easily get

$|\widetilde{w}|_{k+1+\mu}\leq
C_{k}(\widetilde{A},\widetilde{B},l)\left(|\widetilde{w}|_{0}+|\widetilde{F}|_{k+\mu}+|\widetilde{\gamma}|_{k+1+\mu}\right).$

This proves the statement. ∎

######  Remark.

In our study of the boundary value problems with parameter, the above
estimates imply that for each fixed $\lambda$, the solution $w^{\lambda}$
satisfies

$|w^{\lambda}|_{k+1+\mu}\leq
C(A,B,l,\lambda)\left(|w^{\lambda}|_{0}+|F^{\lambda}|_{k+\mu}+|\gamma^{\lambda}|_{k+1+\mu}\right)$

on $b\Omega^{\lambda}$, and our claim on the boundedness of the Hölder norm
will be proved if one can show the constant $C(A,B,l,\lambda)$ is independent
of $\lambda$. However the above proof does not yield bounds which are
independent of $\lambda$.

We end this section by connecting the oblique derivative boundary value
problem for the Laplacian elliptic operator (hereafter called Problem B) to
Problem A.
Problem B Consider the following boundary value problem:

(3.50) $\displaystyle\Delta U+a(x,y)U_{x}+b(x,y)U_{y}=f(x,y);$ (3.51)
$\displaystyle\alpha U_{x}+\beta U_{y}=\gamma$

To solve this problem we follow Vekua and reduce it to Problem A. Let

$u=U_{x},\quad v=-U_{y},\quad w=u+iv.$

The equation (3.50) reduces to the following specialized Problem $A$:

(3.52)
$\displaystyle\partial_{\overline{z}}w+\frac{1}{4}(a+ib)w+\frac{1}{4}(a-ib)\overline{w}=\frac{1}{2}f,\quad\text{in
$\Omega$,}$ (3.53) $\displaystyle\qquad Re[\overline{l}w]=\gamma,\quad
l=\alpha-i\beta,\quad\text{on $b\Omega$. }$

If $w$ is the solution to (3.52)-(3.53). Then the solution to Problem $B$
((3.50) - (3.51)) is given by

(3.54) $\displaystyle U(x,y)$
$\displaystyle=c_{0}+Re\int_{z_{0}}^{z}w(\zeta)\,d\zeta$
$\displaystyle=c_{0}+\int_{z_{0}}^{z}U_{x}(x,y)\,dx+U_{y}(x,y)\,dy.$

Now, $U$ is globally well-defined only if the closed form
$U_{x}\,dx+U_{y}\,dy$ is exact. This occurs when the domain is simply-
connected. Thus in the case of simply-connected domain, problem $B$ is
completely equivalent to problem $A$.

In the case $b\Omega$ has $m+1$ connected components, for $m>0$, in order to
guarantee the right-hand side of (3.54) is single-valued, it is necessary and
sufficient to add $m$ solubility conditions:

(3.55)
$Re\int_{\Gamma_{j}}w\,d\zeta=\int_{\Gamma_{j}}u\,dx-v\,dy=0,\quad(j=1,2,\dots,m)$

where $\Gamma_{0},\Gamma_{1},\dots,\Gamma_{m}$ are simple closed contours
bounding the domain $\Omega$, $\Gamma_{1},\dots,\Gamma_{m}$ being situated
inside $\Gamma_{0}$. Hence a solution $w$ to the specialized Problem A (3.52)
- (3.53) is a solution to the original Problem B if and only if it satisfies
conditions (3.55).

##  4\. Families of simply-connected domains

In this section we prove Theorem 1.1 for families of simply-connected domains.
First we need to define the index for oblique derivative boundary value
problem:

Given the oblique derivative boundary value problem

(4.1) $\mathcal{L}^{\lambda}u^{\lambda}=f^{\lambda}\quad\text{in
$\Omega^{\lambda}$
},\qquad\frac{du^{\lambda}}{d\Upsilon^{\lambda}}=\gamma^{\lambda}\quad\text{on
$b\Omega^{\lambda}$, }$

where $\mathcal{L}^{\lambda}$ is a second-order, uniformly elliptic
differential operator and $\Upsilon^{\lambda}=\xi^{\lambda}+i\eta^{\lambda}$
is a non-vanishing vector field along $b\Omega^{\lambda}$. We define the
_index $\varkappa$_ of problem (4.1) as the winding number of
$\overline{\Upsilon^{\lambda}}$:

(4.2)
$\varkappa:=\frac{1}{2\pi}\left[\overline{\Upsilon^{\lambda}}\right]_{b\Omega^{\lambda}}=\frac{1}{2\pi}\left[\xi^{\lambda}-i\eta^{\lambda}\right],$

Note that since $\varkappa$ is an integer, it is fixed under small
perturbation of $\lambda$.

We formulate our results in two parts depending on the sign of the index
$\varkappa$. First we consider the case of non-negative index, $\varkappa\geq
0$.

######  Theorem 4.1 (Simply-connected, non-negative index case).

Let $k,j$ be non-negative integers satisfying $k\geq j$, and let $0<\mu<1$.
Let $\Omega$ be a bounded, simply-connected domain in $\mathbb{C}$ with
$\mathcal{C}^{k+2+\mu}$ boundary. Let $\Gamma^{\lambda}$ be a family of maps
that embed $\Omega$ to $\Omega^{\lambda}$, with
$\Gamma^{\lambda}\in\mathcal{C}^{k+2+\mu,j}(\overline{\Omega})$. Consider the
following oblique derivative boundary value problem

(4.3) $\mathcal{L}^{\lambda}u^{\lambda}=f^{\lambda}\quad\text{in
$\Omega^{\lambda}$
},\qquad\frac{du^{\lambda}}{d\Upsilon^{\lambda}}=\gamma^{\lambda}\quad\text{on
$b\Omega^{\lambda}$, }$

with non-negative index $\varkappa\geq 0$. Here $\mathcal{L}^{\lambda}$ is a
second-order elliptic operator defined by

$\mathcal{L}^{\lambda}u^{\lambda}=a^{\lambda}(x,y)u^{\lambda}_{xx}+2b^{\lambda}(x,y)u^{\lambda}_{xy}+c^{\lambda}(x,y)u^{\lambda}_{yy}+d^{\lambda}(x,y)u^{\lambda}_{x}+e^{\lambda}(x,y)u^{\lambda}_{y},$

and $\Upsilon^{\lambda}=\xi^{\lambda}+i\eta^{\lambda}$ is an arbirary non-
vanishing directional vector field along $b\Omega^{\lambda}$. Suppose
$a^{\lambda},b^{\lambda},c^{\lambda},d^{\lambda},e^{\lambda},f^{\lambda}$ are
functions in $\Omega^{\lambda}$ such that
$a^{\lambda},b^{\lambda},c^{\lambda}\in\mathcal{C}^{k+1+\mu,j}(\overline{\Omega^{\lambda}})$,
and
$d^{\lambda},e^{\lambda},f^{\lambda}\in\mathcal{C}^{k+\mu,j}(\overline{\Omega^{\lambda}})$.
Suppose that $\Upsilon^{\lambda},\gamma^{\lambda}$ are functions in
$b\Omega^{\lambda}$ such that
$\Upsilon^{\lambda},\gamma^{\lambda}\in\mathcal{C}^{k+1+\mu,j}(b\Omega^{\lambda})$.
For each $\lambda\in[0,1]$, let $\\{\\{z_{r}^{\lambda}\\}_{r=1}^{h_{0}}$,
$(z_{s}^{\lambda})^{\prime}\\}_{s=1}^{h_{1}}\\}$ be a normally distributed set
for $\Omega^{\lambda}$, and let $u^{\lambda}$ be the unique solution to (4.3)
on $\Omega^{\lambda}$ satisfying

$\displaystyle
u^{\lambda}_{x}(z_{r}^{\lambda})-iu^{\lambda}_{y}(z_{r}^{\lambda})=a_{r}(\lambda)+ib_{r}(\lambda)\quad\quad(r=1,...,h_{0}),$
$\displaystyle
u^{\lambda}_{x}((z_{s}^{\prime})^{\lambda})-iu^{\lambda}_{y}((z_{s}^{\prime})^{\lambda})=l^{\lambda}((z_{s}^{\prime})^{\lambda})(\gamma^{\lambda}((z_{s}^{\prime})^{\lambda})+ic_{s}(\lambda)),\quad\quad(s=1,...,h_{1}),$
$\displaystyle u^{\lambda}(z_{0}^{\lambda})=g(\lambda),\quad
z_{0}^{\lambda}\in\Omega^{\lambda}.$

for some functions $a_{r},b_{r},c_{s},g\in C^{j}([0,1])$. Then
$u^{\lambda}\in\mathcal{C}^{k+2+\mu,j}(\overline{\Omega^{\lambda}})$.

By Proposition 2.10, we can reduce the problem to the following“canonical”
form (Problem B) with the same index:

$\displaystyle\Delta
U^{\lambda}+a^{\lambda}(x,y)U^{\lambda}_{x}+b^{\lambda}(x,y)U^{\lambda}_{y}=f^{\lambda}(x,y),\quad\text{in
$\Omega^{\lambda}$, }$
$\displaystyle\alpha^{\lambda}U_{x}^{\lambda}+\beta^{\lambda}U^{\lambda}_{y}=\gamma^{\lambda},\quad\text{on
$b\Omega^{\lambda}$,}$

where
$a^{\lambda},b^{\lambda},f^{\lambda}\in\mathcal{C}^{k+\mu,j}(\overline{\Omega^{\lambda}})$,
$\alpha^{\lambda},\beta^{\lambda},\gamma^{\lambda}\in\mathcal{C}^{k+1+\mu,j}(\overline{\Omega^{\lambda}})$.
Furthermore, by the remark in the last section (see (Remark) ) if one sets
$w^{\lambda}=U_{x}^{\lambda}-iU_{y}^{\lambda}$ then

(4.4)
$U^{\lambda}(x,y)=c_{0}(\lambda)+Re\int_{z_{0}}^{z}w^{\lambda}(\zeta)\,d\zeta.$

Hence it suffices to consider the corresponding Problem A for $w^{\lambda}$:

$\displaystyle\partial{}_{\overline{z}}w^{\lambda}+A^{\lambda}w^{\lambda}+B^{\lambda}\overline{w^{\lambda}}=F^{\lambda},\quad\text{in
$\Omega^{\lambda}$, }$ $\displaystyle
Re[\overline{l^{\lambda}}w^{\lambda}]=\gamma^{\lambda},\quad
l^{\lambda}=\alpha^{\lambda}-i\beta^{\lambda},\quad\text{on
$b\Omega^{\lambda}$,}$

with the corresponding index
$n=\frac{1}{2\pi}[l^{\lambda}]_{b\Omega^{\lambda}}=\frac{1}{2\pi}[\overline{\Upsilon^{\lambda}}]_{b\Omega^{\lambda}}=\varkappa\geq
0$,
$A^{\lambda},B^{\lambda},F^{\lambda}\in\mathcal{C}^{k+\mu,j}(\overline{\Omega^{\lambda}})$,
and
$l^{\lambda},\gamma^{\lambda}\in\mathcal{C}^{k+1+\mu,j}(\overline{\Omega^{\lambda}})$.

Our strategy is to use a smooth family of Riemann mappings proved in [B-G14]
to reduce the problem to one on a fixed domain, where only the coefficients
and the right-hand side of the equation depend on the parameter.

Let us first consider the following homogeneous Problem A on a fixed domain:

(4.5)
$\begin{cases}\mathcal{L}(w^{\lambda})\equiv\overline{\partial{}_{z}}w^{\lambda}+A^{\lambda}(z)w^{\lambda}+B^{\lambda}(z)\overline{w^{\lambda}}=0\quad\text{in
$\Omega$; }\\\ Re[\overline{l^{\lambda}(z)}w^{\lambda}]=0\quad\text{on
$b\Omega$. }\end{cases}$

Let $w_{s}^{\lambda}$, $1\leq s\leq 2n+1$ be the solutions to the integral
equation:

$w^{\lambda}+Q_{n}^{\lambda}w^{\lambda}=g_{s},$

where $g_{s}$ is one of the functions:

(4.6) $z^{k}-z^{2n-k},\quad i(z^{k}+z^{2n-k}),\quad iz^{n},\quad k=0,\cdots
n-1.$

In view of (3.16) and (3.18), $\\{w_{s}^{\lambda}\\}_{s=1}^{2n+1}$ forms a
basis of $\mathbb{R}$ linearly-independent solutions to problem (4.5). We now
prove regularity of $w^{\lambda}_{s}$ with respect to the parameter.

######  Proposition 4.2.

Let $k$ be a non-negative integer and let $0<\mu<1$. Let
$\Omega\subset\mathbb{C}$ be a simply connected domain with $C^{k+1+\mu}$
boundary. Let $w_{1}^{\lambda},\cdots,w_{2n+1}^{\lambda}$ be linearly-
independent solutions to the homogeneous Problem A (4.5) with parameter as
given above, with index $n\geq 0$. Suppose
$A^{\lambda},B^{\lambda}\in\mathcal{C}^{k+\mu,0}(\overline{\Omega})$,
$l^{\lambda}\in\mathcal{C}^{k+1+\mu,0}(\overline{\Omega})$. Then
$w_{s}^{\lambda}\in\mathcal{C}^{k+1+\mu,0}(\overline{\Omega})$, for
$s=1,\dots,2n+1$.

###### Proof.

First we reduce the problem to one on the unit disk. Since $\Omega$ is a
simply-connected domain, there exists a biholomorphic map
$\Gamma:\Omega\to\mathbb{D}$ which is $C^{k+1+\mu}$ up to the boundary. Define
$\widetilde{w}$ on $\mathbb{D}$ by $w(z)=\widetilde{w}\circ\Gamma(z)$. Then
$\widetilde{w}(\zeta)$ is the solution to the problem:

(4.7)
$\begin{cases}\overline{\partial{}_{\zeta}}\widetilde{w}^{\lambda}+\widetilde{A}^{\lambda}(\zeta)\widetilde{w}^{\lambda}+\widetilde{B}^{\lambda}(\zeta)\overline{\widetilde{w}^{\lambda}}=0\quad\text{in
$\mathbb{D}$;}\\\
Re[\overline{\widetilde{l}^{\lambda}(\zeta)}\widetilde{w}^{\lambda}]=0\quad\text{on
$S^{1}$, }\end{cases}$

where

$\widetilde{A}^{\lambda}=A^{\lambda}D^{-1},\quad\widetilde{B}^{\lambda}=B^{\lambda}D^{-1},\quad
D(\zeta)=\overline{\partial{}_{z}\Gamma}\circ\Gamma^{-1}(\zeta)\neq 0.$

Since $\Gamma\in C^{k+1+\mu}(\overline{\Omega})$ and $\Gamma^{-1}\in
C^{k+1+\mu}(\overline{\mathbb{D}})$, we have
$A^{\lambda},B^{\lambda},\in\mathcal{C}^{k+\mu,0}(\overline{\mathbb{D}})$, and
$\widetilde{l}^{\lambda}\in\mathcal{C}^{k+1+\mu,0}(\overline{\mathbb{D}}).$
Thus the problem is reduced to $\Omega=\mathbb{D}$.

As before it suffices to consider the reduced boundary value problem

(4.8)
$\begin{cases}\partial{}_{\overline{z}}w^{\lambda}+A^{\lambda}w^{\lambda}+B^{\lambda}\overline{w^{\lambda}}=0,\quad\text{in
$\mathbb{D}$; }\\\ Re[z^{-n}w^{\lambda}]=0,\quad\text{on $S^{1}$,
}\end{cases}$

where

$w^{\lambda}(z)=e^{\chi^{\lambda}(z)}\widetilde{w}^{\lambda}(z),\quad
A^{\lambda}=\widetilde{A}^{\lambda},\quad
B^{\lambda}=\widetilde{B}^{\lambda}e^{2iIm\,\chi^{\lambda}(z)},$

and $\chi^{\lambda}(z)$ is defined in (3.4). In view of Proposition 7.12, we
have $\chi^{\lambda}\in\mathcal{C}^{k+1+\mu,0}(\overline{\Omega})$, and hence
$A^{\lambda},B^{\lambda}\in\mathcal{C}^{k+\mu,0}(\overline{\Omega})$. In view
of equation (3.16), for $s=1,2,\dots,2n+1$, $w_{s}^{\lambda}$ is the unique
solution satisfying the integral equation

(4.9) $w_{s}^{\lambda}+Q_{n}^{\lambda}w_{s}^{\lambda}=g_{s},$

where
$Q_{n}^{\lambda}w^{\lambda}_{s}=P_{n}(A^{\lambda}w^{\lambda}+B^{\lambda}\overline{w^{\lambda}})$,
and $g_{s}$ are given by (4.6).

Let us first prove the statement for $k=0$. We claim $|w_{s}^{\lambda}|_{0}$
is bounded uniformly in $\lambda$. Suppose not, that there exists some
sequence $\lambda_{j}\in[0,1]$ with
$N_{j}=|w_{s}^{\lambda_{j}}|_{0}\to\infty$. Since $[0,1]$ is compact, passing
to a subsequence if necessary we can assume that $\lambda_{j}$ converges to
some $\lambda_{0}$. Dividing each side of the integral equation (4.9) by
$N_{j}$, and writing
$\widehat{w}_{s}^{\lambda_{j}}=w_{s}^{\lambda_{j}}/N_{j}$, we obtain

(4.10)
$\widehat{w_{s}}^{\lambda_{j}}+Q_{n}^{\lambda_{j}}\widehat{w_{s}}^{\lambda_{j}}=\frac{1}{N_{j}}g_{s},\quad\quad|\widehat{w_{s}}^{\lambda_{j}}|_{0}=1.$

By Proposition 7.7, we have

(4.11) $|Q_{n}^{\lambda}\widehat{w_{i}}^{\lambda_{j}}|_{\tau}\leq
C\left(|A^{\lambda}|_{0}+|B^{\lambda}|_{0}\right)|\widehat{w_{i}}^{\lambda_{j}}|_{0}\leq
C^{\prime},$

for any $0<\tau<1$. By Arzela-Ascoli theorem, and passing to a subsequence if
necessary, we have $Q_{n}^{\lambda_{j}}\widehat{w_{s}}^{\lambda_{j}}$
converges to some function uniformly. It follows from equation (4.10) that
$\widehat{w_{s}}^{\lambda_{j}}$ converges to some function $\widehat{w}_{s}$
uniformly. We show that $Q_{n}^{\lambda_{j}}\widehat{w_{s}}^{\lambda_{j}}$
converges uniformly to $Q_{n}^{\lambda_{0}}\widehat{w}$. Recall that

$Q_{n}^{\lambda}w^{\lambda}=P_{n}(A^{\lambda}w^{\lambda}+B^{\lambda}\overline{w^{\lambda}}),$

where $P_{n}$ is given by formula (3.11). Hence

$\displaystyle
Q_{n}^{\lambda_{1}}w^{\lambda_{1}}-Q_{n}^{\lambda_{2}}w^{\lambda_{2}}$
$\displaystyle=\left(Q_{n}^{\lambda_{1}}-Q_{n}^{\lambda_{2}}\right)w^{\lambda_{1}}+Q_{n}^{\lambda_{2}}(w^{\lambda_{1}}-w^{\lambda_{2}})$
$\displaystyle=P_{n}\left((A^{\lambda_{1}}-A^{\lambda_{2}})w^{\lambda_{1}}\right)+P_{n}\left((B^{\lambda_{1}}-B^{\lambda_{2}})\overline{w^{\lambda_{1}}}\right)$
$\displaystyle\quad+P_{n}\left(A^{\lambda_{2}}(w^{\lambda_{1}}-w^{\lambda_{2}})\right)+P_{n}\left(B^{\lambda_{2}}(w^{\lambda_{1}}-w^{\lambda_{2}})\right).$

By Proposition 7.7, we have

$\displaystyle\left|Q_{n}^{\lambda_{j}}\widehat{w_{s}}^{\lambda_{j}}-Q_{n}^{\lambda_{0}}\widehat{w}\right|_{\tau}$
$\displaystyle\leq\left(\left|A^{\lambda_{j}}-A^{\lambda_{0}}\right|_{0}+\left|B^{\lambda_{j}}-B^{\lambda_{0}}\right|_{0}\right)|\widehat{w_{s}}^{\lambda_{j}}|_{0}$
$\displaystyle\quad+\left(|A^{\lambda_{0}}|_{0}+|B^{\lambda_{0}}|_{0}\right)\left|\widehat{w_{s}}^{\lambda_{j}}-\widehat{w_{s}}\right|_{0}$

for any $0<\tau<1$. Since
$A^{\lambda},B^{\lambda}\in\mathcal{C}^{\mu,0}(\overline{\mathbb{D}})$, we
have $A,B$ are continuous in $\lambda$ in the sup norm. Hence by taking
$\tau=0$ in the above expression we get
$\left|Q_{n}^{\lambda_{j}}\widehat{w_{s}}^{\lambda_{j}}-Q_{n}^{\lambda_{0}}\widehat{w}\right|_{0}\longrightarrow
0$. Letting $\lambda_{j}\to\lambda_{0}$ in equation (4.10) we get

$\widehat{w}_{s}+Q_{n}^{\lambda_{0}}\widehat{w}_{s}=0.$

By Vekua [VA62]*p. 296-298, there does not exist non-trivial continuous
solutions for the above equation, thus $\widehat{w}_{s}=0$. But we have
$|\widehat{w}_{s}|_{0}=\lim_{j\to\infty}|\widehat{w_{s}}^{\lambda_{j}}|_{0}=1$,
which is a contradiction. Thus $|w_{s}^{\lambda}|_{0}$ is bounded by some
constant $C_{0}$ uniform in $\lambda$. It follows then

$|Q_{n}^{\lambda}w_{s}^{\lambda}|_{\tau}\leq
C(|A^{\lambda}|_{0}+|B^{\lambda}|_{0})|w_{s}^{\lambda}|_{0}\leq
C_{0}^{\prime},$

for any $0<\tau<1$. By equation (4.9) we obtain $|w^{\lambda}|_{\tau}$ is
bounded uniformly in $\lambda$ by some constant $C_{0}^{\prime\prime}$.

For $k>0$ we apply induction. By equation (4.9), and Proposition 7.8 one has

$\displaystyle|w_{s}^{\lambda}|_{k+1+\mu}$
$\displaystyle\leq|Q_{n}^{\lambda}w_{s}^{\lambda}|_{k+1+\mu}+M_{k}$
$\displaystyle=|P_{n}(A^{\lambda}w_{s}^{\lambda}+B^{\lambda}\overline{w_{s}^{\lambda}})|_{k+1+\mu}+M_{k}$
$\displaystyle\leq
C_{k}\left(|A^{\lambda}|_{k+\mu}+|B^{\lambda}|_{k+\mu}\right)|w_{s}^{\lambda}|_{k+\mu}+M_{k}$
$\displaystyle\leq C_{k}^{\prime}.$

Hence $|w_{s}^{\lambda}|_{k+1+\mu}$ is bounded uniformly in $\lambda$.

Finally we show that $w_{s}^{\lambda}$ is continuous in the $C^{0}$-norm.
Suppose not, we can then find a sequence $\lambda_{j}\to\lambda_{0}$,
$|w_{s}^{\lambda_{j}}-w_{s}^{\lambda_{0}}|_{0}\geq\delta>0$. Since
$|w_{s}^{\lambda_{j}}|_{k+1+\alpha}$ is bounded, passing to a subsequence if
necessary, $w_{s}^{\lambda_{j}}$ converges to some function $\widehat{w_{s}}$
in the $C^{k+1}$-norm. We have shown above that
$Q_{n}^{\lambda_{j}}w_{s}^{\lambda_{j}}$ converges uniformly to
$Q_{n}^{\lambda_{0}}\widehat{w_{s}}$. Letting $\lambda_{j}\to\infty$ in
equation (4.9) we have

$\widehat{w_{s}}+Q_{n}^{\lambda_{0}}\widehat{w_{s}}=g_{s}.$

This equation has a unique solution $w^{\lambda_{0}}_{s}$, thus
$\widehat{w_{s}}=w^{\lambda_{0}}_{s}$. But this contradicts with the fact that
$|\widehat{w_{s}}-w^{\lambda_{0}}_{s}|_{0}\geq\delta>0$. Thus we have proved
that $w_{s}^{\lambda}$ is continuous in $\lambda$ in the
$C^{0}(\overline{\mathbb{D}})$-norm. The continuity of $w_{s}^{\lambda}$ in
the $C^{k+1}(\overline{\mathbb{D}})$-norm then follows from Proposition 2.3. ∎

Next we show the smooth dependence in parameter for a particular solution of
the non-homogenous Problem A.

######  Proposition 4.3.

Let $k$ be a non-negative integer and let $0<\mu<1$. Consider Problem A with
parameter on a simply-connected domain $\Omega$:

(4.12)
$\begin{cases}\mathcal{L}(w^{\lambda})\equiv\overline{\partial{}_{z}}w^{\lambda}+A^{\lambda}(z)w^{\lambda}+B^{\lambda}(z)\overline{w^{\lambda}}=F^{\lambda}\quad\text{in
$\Omega$;}\\\
Re[\overline{l^{\lambda}(z)}w^{\lambda}]=\gamma^{\lambda}\quad\text{on
$b\Omega$, }\end{cases}$

where the index $n\geq 0$. Suppose
$A^{\lambda},B^{\lambda},F^{\lambda}\in\mathcal{C}^{k+\mu,0}(\overline{\Omega})$,
$l^{\lambda},\gamma^{\lambda}\in\mathcal{C}^{k+1+\mu,0}(b\Omega)$. Then there
exists a solution $w_{0}$ in the class
$\mathcal{C}^{k+1+\mu,0}(\overline{\Omega})$. Moreover there exists some
constant $C$ independent of $\lambda$ such that

(4.13)
$\displaystyle\|w_{0}^{\lambda}\|_{0,0}:=\sup_{\lambda\in[0,1]}|w_{0}^{\lambda}|_{0}\leq
C\left(\|F^{\lambda}\|_{0,0}+\|\gamma^{\lambda}\|_{\mu,0}\right),$ (4.14)
$\displaystyle\|w_{0}^{\lambda}\|_{k+1+\mu,0}\leq
C\left(\|F^{\lambda}\|_{k+\mu,0}+\|\gamma^{\lambda}\|_{k+1+\mu,0}\right).$

###### Proof.

To prove estimate (4.13) (resp.(4.14)), it suffices to show that
$\|w_{0}\|_{0,0}\leq C$ (resp. $\|w_{0}\|_{k+1+\mu,0}\leq C$ ) if each norm on
the right-hand side is bounded by $1$, since the equations are linear. As
before we can assume the problem takes the form (3.6):

(4.15)
$\begin{cases}\partial{}_{\overline{z}}w^{\lambda}+A^{\lambda}w^{\lambda}+B^{\lambda}\overline{w^{\lambda}}=F^{\lambda},\quad\text{in
$\mathbb{D}$; }\\\ Re[z^{-n}w^{\lambda}]=\gamma^{\lambda},\quad\text{on
$S^{1}$. }\end{cases}$

In view of equation (3.16), we can take $w_{0}^{\lambda}$ to be the unique
solution in $C^{0}(\overline{\mathbb{D}})$ to the integral equation in
$\mathbb{D}$:

(4.16)
$w^{\lambda}+Q_{n}^{\lambda}w^{\lambda}=P_{n}F^{\lambda}+\frac{z^{n}}{2\pi
i}\int_{S^{1}}\gamma^{\lambda}(t)\frac{t+z}{t-z}\frac{dt}{t},$

where

$P_{n}F^{\lambda}=T_{\mathbb{D}}F^{\lambda}+z^{2n+1}T_{\mathbb{D}^{c}}F_{1}^{\lambda},\quad
Q_{n}^{\lambda}w^{\lambda}=P_{n}(A^{\lambda}w^{\lambda}+B^{\lambda}\overline{w^{\lambda}}),$

where $F_{1}^{\lambda}$ and $F^{\lambda}$ are related by (3.13).

The proof is identical to that of the previous theorem, once we show that the
right-hand side of equation (4.16) belongs to the class
$\mathcal{C}^{k+1+\mu,0}(\overline{\mathbb{D}})$. By Proposition 7.8,

$|P_{n}F^{\lambda}|_{k+1+\mu}\leq C|F^{\lambda}|_{k+\mu}.$

The second term is $z^{n}$ times $\mathcal{S}\gamma^{\lambda}$, the Schwarz
transform of $\gamma^{\lambda}$ on $\mathbb{D}$. By Proposition 7.12,

$|\mathcal{S}\gamma^{\lambda}|_{k+1+\mu}\leq C|\gamma^{\lambda}|_{k+1+\mu}.$

Thus $|P_{n}F^{\lambda}+z^{n}\mathcal{S}\gamma^{\lambda}|_{k+1+\mu}$ is
bounded. Fix $\tau\in(0,1)$ we have

$|P_{n}F^{\lambda_{1}}-P_{n}F^{\lambda_{2}}|_{0}\leq|P_{n}F^{\lambda_{1}}-P_{n}F^{\lambda_{2}}|_{\tau}\leq
C|F^{\lambda_{1}}-F^{\lambda_{2}}|_{0},$

and

$|\mathcal{S}\gamma^{\lambda_{1}}-\mathcal{S}\gamma^{\lambda_{2}}|_{0}\leq|\mathcal{S}\gamma^{\lambda_{1}}-\mathcal{S}\gamma^{\lambda_{2}}|_{\tau}\leq
C|\gamma^{\lambda_{1}}-\gamma^{\lambda_{2}}|_{\tau}.$

By the above two inequalities and the assumptions, we see that
$P_{n}F^{\lambda}+z^{n}\mathcal{S}\gamma^{\lambda}$ is continuous in the
$C^{0}$-norm. By Proposition 2.3,
$P_{n}F^{\lambda}+S\gamma^{\lambda}\in\mathcal{C}^{k+1+\mu,0}(\overline{\mathbb{D}})$.
∎

We can now prove the following result for Problem A on fixed domain:

######  Proposition 4.4.

Let $k$ be a non-negative integer and let $0<\mu<1$. Consider the non-
homogeneous Problem A (4.12) on a simply-connected domain $\Omega$, with index
$n\geq 0$. Suppose $A,B,F\in\mathcal{C}^{k+\mu,0}(\overline{\Omega})$,
$l,\gamma\in\mathcal{C}^{k+1+\mu,0}(b\Omega)$, $a_{i},b_{i},c_{j}\in
C^{0}([0,1])$. Then for each $\lambda$ there exists a unique solution
$w^{\lambda}$ satisfying the following conditions on a normally distributed
set (3.4).

(4.17)
$\begin{split}w^{\lambda}(z_{i})=a_{i}(\lambda)+ib_{i}(\lambda),\quad\quad
i=1,...,N),\\\
w^{\lambda}(z_{j}^{\prime})=l^{\lambda}(z_{j}^{\prime})(\gamma^{\lambda}(z_{j}^{\prime})+ic_{j}(\lambda)),\quad\quad(j=1,...,N^{\prime})\end{split}$

where $2N+N^{\prime}=2n+1$. The family $w^{\lambda}$ belongs to the class
$\mathcal{C}^{k+1+\mu,0}(\overline{\Omega})$. Moreover there exists some
constant $C$ independent of $\lambda$ so that

(4.18) $\displaystyle\|w^{\lambda}\|_{0,0}\leq
C\left(\|F^{\lambda}\|_{0,0}+\|\gamma^{\lambda}\|_{\mu,0}+\sum_{i}|a_{i}|_{0}+|b_{i}|_{0}+\sum_{j}|c_{j}|_{0}\right).$
(4.19) $\displaystyle\|w^{\lambda}\|_{k+1+\mu,0}\leq
C\left(\|F^{\lambda}\|_{k+\mu,0}+\|\gamma^{\lambda}\|_{k+1+\mu,0}+\sum_{i}|a_{i}|_{0}+|b_{i}|_{0}+\sum_{j}|c_{j}|_{0}\right).$

where we denote $|a_{i}|_{0}=\sup_{\lambda\in[0,1]}|a_{i}(\lambda)|$ and
similarly for $|b_{i}|_{0}$ and $|c_{j}|_{0}$.

Proof. Let $w_{0}^{\lambda}$ be the particular solution to the non-homogeneous
Problem A given in Proposition 4.3, and let $w^{\lambda}_{s}$-s be the $2n+1$
$\mathbb{R}$-linearly independent solution to the homogeneous Problem A given
in Proposition 4.2. Then $w^{\lambda}$ is given by the formula:

(4.20) $\displaystyle
w^{\lambda}(z)=w_{0}^{\lambda}(z)+\sum_{s=1}^{2n+1}d_{s}(\lambda)w_{s}^{\lambda}(z),$

where $d_{s}$ are real-valued functions on $[0,1]$.

From (4.17), we have the following linear system of equations for the
determination of the functions $d_{s}(\lambda)$:

(4.21)
$\displaystyle\sum_{s=1}^{2n+1}d_{s}(\lambda)w_{s}^{\lambda}(z_{i})=a_{i}(\lambda)+ib_{i}(\lambda)-w^{\lambda}_{0}(z_{i})\quad\quad(i=1,...,h_{0})$
(4.22)
$\displaystyle\sum_{s=1}^{2n+1}d_{s}(\lambda)w_{s}^{\lambda}(z_{j}^{\prime})=l^{\lambda}(z_{j}^{\prime})(\gamma^{\lambda}(z_{j}^{\prime})+ic_{j}(\lambda))-w^{\lambda}_{0}(z_{j}^{\prime})\qquad(j=1,...,h_{1}).$

We can write the above as $2h_{0}+2h_{1}^{\prime}$ real equations. However,
there are in fact only $2h_{0}+h_{1}=2n+1$ equations, since the imaginary
parts of equations (4.22) are obtained from the real parts of equation (4.22),
by multiplying
$-\frac{\alpha^{\lambda}(z_{j}^{\prime})}{\beta^{\lambda}(z_{j}^{\prime})}$
(assume that $\beta^{\lambda}(z_{j}^{\prime})\neq 0$.) Indeed, we have for
$z_{j}^{\prime}\in b\Omega$,

$\displaystyle
0=Re[\overline{l^{\lambda}}w_{\ell}^{\lambda}(z_{j}^{\prime})]=\alpha^{\lambda}(z_{j}^{\prime})Re[w_{\ell}^{\lambda}(z_{j}^{\prime})]+\beta^{\lambda}(z_{j}^{\prime})Im[w_{\ell}^{\lambda}(z_{j}^{\prime})],$
$\displaystyle\gamma^{\lambda}(z_{j}^{\prime})=Re[\overline{l^{\lambda}}w_{0}^{\lambda}(z_{j}^{\prime})]=\alpha^{\lambda}(z_{j}^{\prime})Re[w_{0}^{\lambda}(z_{j}^{\prime})]+\beta^{\lambda}(z_{j}^{\prime})Im[w_{0}^{\lambda}(z_{j}^{\prime})],$

where we set
$l^{\lambda}(z_{j}^{\prime})=\alpha^{\lambda}(z_{j}^{\prime})+i\beta^{\lambda}(z_{j}^{\prime})$.
Thus we can view the linear system (4.21)-(4.22) as:

$\displaystyle\begin{pmatrix}Re[w^{\lambda}_{1}(z_{1})]&\ldots&Re[w^{\lambda}_{2n+1}(z_{1})]\\\
Im[w^{\lambda}_{1}(z_{1})]&\ldots&Im[w^{\lambda}_{2n+1}(z_{1})]\\\
\vdots&\vdots&\vdots\\\
Re[w^{\lambda}_{1}(z_{k})]&\ldots&Re[w^{\lambda}_{2n+1}(z_{k})]\\\
Im[w^{\lambda}_{1}(z_{k})]&\ldots&Im[w^{\lambda}_{2n+1}(z_{k})]\\\
Re[w^{\lambda}_{1}(z_{1}^{\prime})]&\ldots&Re[w^{\lambda}_{2n+1}(z_{1}^{\prime})]\\\
\vdots&\vdots&\vdots\\\
Re[w^{\lambda}_{1}(z_{k^{\prime}}^{\prime})]&\ldots&Re[w^{\lambda}_{2n+1}(z_{k^{\prime}}^{\prime})]\end{pmatrix}\times\begin{pmatrix}d_{1}(\lambda)\\\
\vdots\\\
d_{2n+1}(\lambda)\end{pmatrix}=\begin{pmatrix}Re(h^{\lambda}(z_{1}))\\\
Im(h^{\lambda}(z_{1}))\\\ \vdots\\\ Re(h^{\lambda}(z_{k}))\\\
Im(h^{\lambda}(z_{k}))\\\ \vdots\\\ Re(e^{\lambda}(z_{1}^{\prime}))\\\
\vdots\\\ Re(e^{\lambda}(z_{k^{\prime}}))\end{pmatrix}$

where we use $h^{\lambda}(z_{j})$ and $e^{\lambda}(z_{j}^{\prime})$ to denote
the right-hand side of (4.21) and (4.22), respectively. Write the above matrix
equation as $Mx=\textbf{v}$. By the result of Vekua [VA62]*p. 286-287, for
fixed $\lambda$ and a normally distributed set
$\\{h_{0},h_{1},\overline{\Omega}\\}$, $M$ is invertible, thus we can write
$d_{s}(\lambda)$ as $\frac{P}{Q}$, $Q\neq 0$, and $P$ and $Q$ are linear
combination of products of

(4.23) $Re[w_{s}^{\lambda}(z_{i})],\quad Im[w_{s}^{\lambda}(z_{i})],\quad
Re[w_{s}^{\lambda}(z_{j}^{\prime})],\quad
Im[w_{s}^{\lambda}(z_{j}^{\prime})],$

and

(4.24) $Re,Im[a_{i}(\lambda)+ib_{i}(\lambda)-w^{\lambda}_{0}(z_{i})],\quad
Re[l^{\lambda}(z_{j}^{\prime})(\gamma^{\lambda}(z_{j}^{\prime})+ic_{j}(\lambda))-w^{\lambda}_{0}(z_{j}^{\prime})].$

In view of the assumption, Proposition 4.2 and 4.3, for fixed $z_{i}$ and
$z_{j}^{\prime}$ the above functions are continuous in $\lambda$, thus
$d_{s}$-s are continuous functions of $\lambda$, and moreover in view of
estimate (4.13) we have

(4.25) $|d_{s}|_{0}\leq
C\left(\sum_{i}|a_{i}|_{0}+|b_{i}|_{0}+\sum_{j}|c_{j}|_{0}+\|F\|_{0,0}+\|\gamma\|_{\mu,0}\right).$

Since
$w_{0}^{\lambda},w_{s}^{\lambda}\in\mathcal{C}^{k+1+\mu,0}(\overline{\Omega^{\lambda}})$,
it follow from (4.20) that
$w^{\lambda}\in\mathcal{C}^{k+1+\mu,0}(\overline{\Omega})$. Putting together
estimates (4.13), (4.14) and (4.25) we get (4.18) and (4.19). ∎

######  Remark.

Since our proof uses a contradiction argument, the constant $C$ in (4.18) and
(4.19) is not explicit. In particular we do not know how it depends on the
coefficient functions $A^{\lambda}$, $B^{\lambda}$ and $l^{\lambda}$.

We now prove the analog of Proposition 4.4 when the index is negative.

######  Proposition 4.5.

Let $k$ be a non-negative integer, and let $0<\mu<1$. Consider Problem A
(4.12) with negative index. Suppose for each $\lambda$, conditions (3.19) is
fulfilled. Suppose
$A^{\lambda},B^{\lambda}\in\mathcal{C}^{k+\mu,0}(\overline{\Omega})$,
$l^{\lambda},\gamma^{\lambda}\in\mathcal{C}^{k+1+\mu,0}(b\Omega)$. Then for
each $\lambda$ there exists a unique solution $w^{\lambda}$ on $\Omega$. The
family $w^{\lambda}$ lies in the class
$\mathcal{C}^{k+1+\mu,0}(\overline{\Omega})$, and the following estimate hold

(4.26) $\displaystyle\|w\|_{k+1+\mu,0}\leq
C(A,B,l)\left(\|F\|_{k+\mu,0}+\|\gamma\|_{k+1+\mu,0}\right).$

###### Proof.

Reducing the problem to the canonical form (4.15). By Vekua’s result [VA62]*p.
300, (7.32), (7.33), $w^{\lambda}$ satisfies the integral equation (3.22)

(4.27)
$w^{\lambda}+(Q_{n^{\prime}}^{\ast})^{\lambda}w^{\lambda}=P_{k}^{\ast}F^{\lambda}+\frac{1}{\pi
i}\int_{S^{1}}\frac{\gamma^{\lambda}(t)\,dt}{t^{k}(t-z)},\quad n^{\prime}=-n,$

where
$Q^{\ast}_{n^{\prime}}w=P^{\ast}_{n^{\prime}}(A^{\lambda}w^{\lambda}+B^{\lambda}\overline{w^{\lambda}})$
and $P^{\ast}_{n^{\prime}}$ is given by (3.23). Vekua proved that for fixed
$\lambda$, the above equation is uniquely soluble for $w^{\lambda}$ given any
right-hand side in $L^{q}(\overline{\mathbb{D}})$, for $q\geq\frac{p}{p-1}$,
and $Q^{\ast}_{n^{\prime}}$ is a completely continuous operator on
$L^{q}(\mathbb{D})$ satisfying the estimate

$|(Q_{n^{\prime}}^{\ast})^{\lambda}f|_{k+1+\mu}\leq
C|B^{\lambda}|_{k+\mu}|f|_{k+1+\mu}.$

Following the same argument as for the positive index case, it suffices to
show the right-hand side of (4.27) lies in the space
$\mathcal{C}^{k+1+\mu,0}(\overline{\mathbb{D}}\times[0,1])$. For the first
term we have the estimate

$|P_{k}^{\ast}F^{\lambda}|_{k+1+\mu}\leq C|F^{\lambda}|_{k+\mu},$
$|P_{k}^{\ast}F^{\lambda_{1}}-P_{k}^{\ast}F^{\lambda_{2}}|\leq
C|F^{\lambda_{1}}-F^{\lambda_{2}}|_{0}.$

The second term is the Cauchy transform of the function
$g^{\lambda}=\gamma^{\lambda}/t^{k}$, and we have the estimate by Proposition
7.12:

$|\mathfrak{C}g^{\lambda}|_{k+1+\mu}\leq C|\gamma^{\lambda}|_{k+1+\mu},$
$|\mathfrak{C}g^{\lambda_{1}}-\mathfrak{C}g^{\lambda_{2}}|_{0}\leq
C|\gamma^{\lambda_{1}}-\gamma^{\lambda_{2}}|_{1}.$

Since $F^{\lambda}\in\mathcal{C}^{k+\mu,0}(\mathbb{D})$, and
$\gamma^{\lambda}\in\mathcal{C}^{k+1+\mu,0}(S^{1})$, we have
$P^{\ast}_{n^{\prime}}F^{\lambda}$ and $\mathcal{C}(g^{\lambda})$ in the class
$\mathcal{C}^{k+1+\mu,0}(\overline{\mathbb{D}})$. ∎

We now combine Proposition 4.4 and Proposition 2.6 to prove the following
result for Problem A on families of simply-connected domains. We state and
prove the theorem only for $n\geq 0$. For $n<0$, we have the same conclusions
if we assume conditions (3.19) hold. The proof can be done similarly as in
Theorem 4.6 by using Proposition 4.5 and we leave the details to readers.

######  Theorem 4.6.

Let $k$ and $j$ be non-negative integers such that $k\geq j$, and $0<\mu<1$.
Let $\Omega^{\lambda}$ be a $\mathcal{C}^{k+1+\alpha,j}$ family of bounded
simply-connected domains. That is, there exists a family of maps
$\Gamma^{\lambda}:\overline{\Omega}\to\overline{\Omega^{\lambda}}$, with
$\Gamma^{\lambda}\in\mathcal{C}^{k+1+\mu,j}(\overline{\Omega})$. For each
$\lambda$, consider the following problem A on $\Omega^{\lambda}$:

(4.28)
$\displaystyle\overline{\partial}_{z^{\lambda}}w^{\lambda}(z^{\lambda})+A^{\lambda}(z^{\lambda})w^{\lambda}(z^{\lambda})+B^{\lambda}(z^{\lambda})\overline{w^{\lambda}(z^{\lambda})}=F^{\lambda},\quad\text{in
$\Omega^{\lambda}$}$ (4.29) $\displaystyle
Re[\overline{l^{\lambda}(z^{\lambda})}w^{\lambda}(z^{\lambda})]=\gamma^{\lambda}(z^{\lambda}),\quad\text{on
$b\Omega^{\lambda}$}.$

(i) Suppose the index $n\geq 0$. Let $\\{z_{r}\in\Omega,z_{s}^{\prime}\in
b\Omega\\}$ be a normally distributed set in $\overline{\Omega}$ (Definition
3.4). Set $z^{\lambda}_{r}=\Gamma^{\lambda}(z_{r})$,
$(z^{\prime}_{s})^{\lambda}=\Gamma^{\lambda}(z^{\prime}_{s})$. Then
$\\{z^{\lambda}_{r},(z^{\prime}_{s})^{\lambda}\\}$ is a normally distributed
set in $\overline{\Omega^{\lambda}}$. Let $w^{\lambda}$ be the unique solution
to problem A (4.28)-(4.29) satisfying the following:

(4.30) $\displaystyle
w^{\lambda}(z_{r}^{\lambda})=a_{r}(\lambda)+ib_{r}(\lambda),\quad\quad(r=1,...,h_{0}),$
$\displaystyle
w^{\lambda}((z_{s}^{\prime})^{\lambda})=l^{\lambda}((z_{s}^{\prime})^{\lambda})[\gamma^{\lambda}((z_{s}^{\prime})^{\lambda})+ic_{s}(\lambda)],\quad\quad(s=1,...,h_{1})$

where $2h_{0}+h_{1}=2n+1$. Suppose
$A^{\lambda},B^{\lambda},F^{\lambda}\in\mathcal{C}^{k+\alpha,0}(\overline{\Omega^{\lambda}})$,
and
$l^{\lambda},\gamma^{\lambda}\in\mathcal{C}^{k+1+\alpha,0}(b\Omega^{\lambda})$,
and $a_{r},b_{r},c_{s}\in C^{0}([0,1])$. Then
$w^{\lambda}\in\mathcal{C}^{k+1+\alpha,0}(\overline{\Omega^{\lambda}})$, and
there exists some constant $C$ independent of $\lambda$ such that

(4.31) $\displaystyle\|w^{\lambda}\|_{0,0}\leq
C\left(\|F^{\lambda}\|_{0,0}+\|\gamma^{\lambda}\|_{\mu,0}+\sum_{r=1}^{h_{0}}|a_{r}|_{0}+|b_{r}|_{0}+\sum_{s=1}^{h_{1}}|c_{s}|_{0}\right),$
(4.32) $\displaystyle\|w^{\lambda}\|_{k+1+\mu,0}\leq
C\left(\|F^{\lambda}\|_{k+\mu,0}+\|\gamma^{\lambda}\|_{k+1+\mu,0}+\sum_{r=1}^{h_{0}}|a_{r}|_{0}+|b_{r}|_{0}+\sum_{s=1}^{h_{1}}|c_{s}|_{0}\right).$

(ii) Let $k$ and $j$ be non-negative integers satisfying $k\geq j$. Suppose
$A^{\lambda},B^{\lambda},F^{\lambda}\in\mathcal{C}^{k+\alpha,j}(\overline{\Omega^{\lambda}})$,
and
$l^{\lambda},\gamma^{\lambda}\in\mathcal{C}^{k+1+\alpha,j}(\overline{b\Omega^{\lambda}})$,
Let $\\{\zeta_{r},\zeta_{s}^{\prime}\\}$ be a normally distributed set in
$\mathbb{D}$. Set $\zeta_{r}^{\lambda}=(R^{\lambda})^{-1}(\zeta_{r})$,
$\zeta_{s}^{\lambda}=(R^{\lambda})^{-1}(\zeta_{s}^{\prime})$, where
$R^{\lambda}$ is the family of Riemann mappings given in Proposition 2.6. Let
$w^{\lambda}$ be the solution to problem (4.28)- (4.29) satisfying the
following condition

$\displaystyle
w^{\lambda}(\zeta_{r}^{\lambda})=a_{r}(\lambda)+ib_{r}(\lambda),\quad\quad(r=1,...,h_{0}),$
$\displaystyle
w^{\lambda}((\zeta_{s}^{\prime})^{\lambda})=l^{\lambda}((\zeta_{s}^{\prime})^{\lambda})[\gamma^{\lambda}((\zeta_{s}^{\prime})^{\lambda})+ic_{s}(\lambda)],\quad\quad(s=1,...,h_{1})$

where $a_{r},b_{r},c_{s}\in C^{j}([0,1])$. Then
$w\in\mathcal{C}^{k+1+\alpha,j}(\overline{\Omega^{\lambda}})$.

###### Proof.

(i) By Proposition 2.6, there exists a family of Riemann mappings
$R^{\lambda}:\Omega^{\lambda}\to\mathbb{D}$ with
$\widetilde{R}^{\lambda}(z):=R^{\lambda}(\Gamma^{\lambda}(z))\in\mathcal{C}^{k+1+\alpha,j}(\overline{\Omega})$.
By Proposition 2.7,
$(R^{\lambda})^{-1}\in\mathcal{C}^{k+1+\alpha,j}(\overline{\mathbb{D}})$. i.e.
$(R^{\lambda})^{-1}:\mathbb{D}\to\overline{\Omega^{\lambda}}$ is also a
$\mathcal{C}^{k+1+\alpha,j}$ family of embeddings. Define
$v^{\lambda}(\zeta)=w^{\lambda}\circ(R^{\lambda})^{-1}(\zeta)$,
$\zeta\in\mathbb{D}$, and write $z^{\lambda}=(R^{\lambda})^{-1}(\zeta)$. Since
$(R^{\lambda})^{-1}$ is holomorphic, we have the boundary value problem for
$v^{\lambda}$ on $\mathbb{D}$:

(4.33)
$\begin{cases}\overline{\partial}_{\zeta}v^{\lambda}(\zeta)+A^{\lambda}_{1}(\zeta)v^{\lambda}(\zeta)+B^{\lambda}_{1}(\zeta)\overline{v^{\lambda}(\zeta)}=F^{\lambda}_{1}(\zeta)&\text{in
$\mathbb{D}$,}\\\
Re[\overline{l_{1}^{\lambda}}(\zeta)v^{\lambda}(\zeta)]=\gamma_{1}^{\lambda}(\zeta)&\text{on
$S^{1}$, }\end{cases}$

where $l^{\lambda}_{1}:=l^{\lambda}\circ(R^{\lambda})^{-1}$,
$\gamma^{\lambda}_{1}:=\gamma^{\lambda}\circ(R^{\lambda})^{-1}$, and

(4.34) $\displaystyle
A_{1}^{\lambda}=A^{\lambda}\circ(R^{\lambda})^{-1}(\zeta)D^{\lambda}(\zeta),\quad
B_{1}^{\lambda}=B^{\lambda}\circ(R^{\lambda})^{-1}(\zeta)D^{\lambda}(\zeta),\quad$
$\displaystyle
F_{1}^{\lambda}=F^{\lambda}\circ(R^{\lambda})^{-1}(\zeta)D^{\lambda}(\zeta),\quad
D^{\lambda}=\overline{\partial}_{\zeta}\overline{(R^{\lambda})^{-1}}.$

By Proposition 2.4,
$A^{\lambda}_{1},B^{\lambda}_{1},F^{\lambda}_{1}\in\mathcal{C}^{k+\alpha,j}(\overline{\mathbb{D}})$,
and $\overline{l_{1}^{\lambda}}$,
$\gamma_{1}^{\lambda}\in\mathcal{C}^{k+1+\alpha,j}(\overline{\mathbb{D}})$.
Also $D^{\lambda}\in\mathcal{C}^{k+\alpha,j}(\overline{\mathbb{D}})$. Hence
$A_{1}^{\lambda}$,
$B_{1}^{\lambda},F_{1}^{\lambda}\in\mathcal{C}^{k+\alpha,j}(\overline{\mathbb{D}})$.
The index $n=\frac{1}{2\pi}[l^{\lambda}]_{b\Omega}$ is an integer independent
of $\lambda$. Let

$\zeta_{r}^{\lambda}:=R^{\lambda}\circ\Gamma^{\lambda}(z_{r}),\quad(\zeta_{s}^{\prime})^{\lambda}:=R^{\lambda}\circ\Gamma^{\lambda}(z_{s}).$

Then for each $\lambda$,
$\\{\zeta^{\lambda}_{i},(\zeta^{\prime}_{j})^{\lambda}\\}$ is a normally
distributed set in $\Omega^{\lambda}$. $v^{\lambda}$ satisfies the condition

(4.35) $\displaystyle
v^{\lambda}(\zeta_{r}^{\lambda})=w^{\lambda}(z_{r}^{\lambda})=a_{r}(\lambda)+ib_{r}(\lambda)\quad(r=1,...,h_{0}),$
(4.36) $\displaystyle
v^{\lambda}((\zeta_{s}^{\prime})^{\lambda})=w^{\lambda}((z_{j}^{\prime})^{\lambda})=l^{\lambda}((z_{s}^{\prime})^{\lambda})[\gamma^{\lambda}((z_{s}^{\prime})^{\lambda})+ic_{s}(\lambda)]\quad(s=1,...,h_{1}).$

Since
$v^{\lambda}=v_{0}^{\lambda}+\sum_{\ell}d_{\ell}(\lambda)v_{\ell}^{\lambda}$,
we have

(4.37)
$\displaystyle\sum_{\ell=1}^{2n+1}d_{\ell}(\lambda)v_{\ell}^{\lambda}(\zeta_{r}^{\lambda})=a_{r}(\lambda)+ib_{r}(\lambda)-v_{0}^{\lambda}(\zeta_{r}^{\lambda}),$
(4.38)
$\displaystyle\sum_{\ell=1}^{2n+1}d_{\ell}(\lambda)v_{\ell}^{\lambda}((\zeta_{s}^{\prime})^{\lambda})=l^{\lambda}((z_{s}^{\prime})^{\lambda})[\gamma^{\lambda}((z_{s}^{\prime})^{\lambda})+ic_{s}(\lambda)]-v_{0}^{\lambda}((\zeta_{s}^{\prime})^{\lambda}),$

where $v_{0}^{\lambda}$ is a particular solution to the non-homogeneous
problem (4.33) as given in Proposition 4.3, and
$v_{1}^{\lambda},...,v_{2n+1}^{\lambda}$ are linearly independent solutions to
the corresponding homogeneous problem as given in Proposition 4.2. We have
$v^{\lambda}_{0},v^{\lambda}_{1}$,
$\dots,v^{\lambda}_{2n+1}\in\mathcal{C}^{k+1+\alpha,0}(\overline{\mathbb{D}})$.
Since $\zeta^{\lambda}_{r}$ and $(\zeta^{\prime}_{s})^{\lambda}$ depend on
$\lambda$ continuously, $v_{0}^{\lambda}(\zeta_{r}^{\lambda})$,
$v_{\ell}^{\lambda}(\zeta_{r}^{\lambda})$,
$v_{0}^{\lambda}((\zeta_{s}^{\prime})^{\lambda})$,
$v_{\ell}^{\lambda}((\zeta_{s}^{\prime})^{\lambda})$ depend on $\lambda$
continuously. Same reasoning as in the proof of Proposition 4.4 then shows
that $d_{l}\in C^{0}([0,1])$, and consequently
$v^{\lambda}\in\mathcal{C}^{k+1+\mu,0}(\overline{\mathbb{D}})$, and we have

$\|v^{\lambda}\|_{k+1+\mu,0}\leq
C\left(\|F_{1}^{\lambda}\|_{k+\mu,0}+\|\gamma_{1}^{\lambda}\|_{k+1+\mu,0}+\sum_{r=1}^{h_{0}}|a_{r}|_{0}+|b_{r}|_{0}+\sum_{s=1}^{h_{1}}|c_{s}|_{0}\right).$

Now $w^{\lambda}:=v^{\lambda}\circ R^{\lambda}$ is the unique solution to
Problem A (4.28)-(4.29),
$w^{\lambda}\circ(R^{\lambda})^{-1}\in\mathcal{C}^{k+1+\mu,0}(\overline{\mathbb{D}})$,
and $w^{\lambda}$ satisfies the estimate (4.32). By Lemma 2.4,
$w^{\lambda}\circ\Gamma^{\lambda}\in\mathcal{C}^{k+1+\mu,0}(\overline{\Omega})$,
or $w^{\lambda}\in\mathcal{C}^{k+1+\mu,0}(\overline{\Omega^{\lambda}})$. ∎

(ii) The $j=0$ case is done in (i). Assume now that $j\geq 1$, and we use
induction. Write

$D^{\lambda,\lambda_{0}}v(\zeta)=\frac{v^{\lambda}(\zeta)-v^{\lambda_{0}}(\zeta)}{\lambda-\lambda_{0}}.$

In view of (4.33) $D^{\lambda,\lambda_{0}}v(\zeta)$ is a solution to

(4.39)
$\begin{cases}\overline{\partial}_{\zeta}u+A^{\lambda_{0}}_{1}u+B_{1}^{\lambda_{0}}\overline{u}=F_{\ast}^{\lambda,\lambda_{0}}\quad&\text{in
$\mathbb{D}$, }\\\
Re[\overline{\widetilde{l}^{\lambda_{0}}}u]=\gamma_{\ast}^{\lambda,\lambda_{0}}&\text{on
$S^{1}$,}\end{cases}$

where

$\displaystyle
F_{\ast}^{\lambda,\lambda_{0}}:=D^{\lambda,\lambda_{0}}F_{1}-(D^{\lambda,\lambda_{0}}A_{1})v^{\lambda}-(D^{\lambda,\lambda_{0}}B_{1})\overline{v^{\lambda}},$
$\displaystyle\gamma_{\ast}^{\lambda,\lambda_{0}}:=D^{\lambda,\lambda_{0}}\widetilde{\gamma}-Re[(D^{\lambda,\lambda_{0}}\overline{\widetilde{l}})v^{\lambda}].$

We also have

(4.40) $\displaystyle
D^{\lambda,\lambda_{0}}v(\zeta_{r})=h_{r}^{\lambda,\lambda_{0}}:=D^{\lambda,\lambda_{0}}a_{r}+iD^{\lambda,\lambda_{0}}b_{r},$
(4.41) $\displaystyle
D^{\lambda,\lambda_{0}}v(\zeta_{s}^{\prime})=g_{s}^{\lambda,\lambda_{0}}:=D^{\lambda,\lambda_{0}}\left\\{l^{\bullet}((\zeta_{s}^{\prime})^{\bullet})[\gamma^{\bullet}((\zeta_{s}^{\prime})^{\bullet})+ic_{s}(\bullet)]\right\\}.$

For every $\lambda_{0}$, denote by $\partial{}_{\lambda}v(\cdot,\lambda_{0})$
the solution to the problem:

(4.42)
$\begin{cases}\overline{\partial}_{\zeta}u+A^{\lambda_{0}}_{1}u+B_{1}^{\lambda_{0}}\overline{u}=F_{\ast}^{\lambda_{0}}\quad&\text{in
$\mathbb{D}$. }\\\
Re[\overline{\widetilde{l}^{\lambda_{0}}}u]=\gamma_{\ast}^{\lambda_{0}}&\text{on
$S^{1}$,}\end{cases}$

satisfying the condition

(4.43) $u^{\lambda_{0}}(\zeta_{r})=h_{r}(\lambda_{0}),\quad
u(\zeta^{\prime}_{s},\lambda_{0})=g_{s}(\lambda_{0}),$

where

(4.44) $\displaystyle
F_{\ast}^{\lambda_{0}}:=\partial{}_{\lambda}F_{1}(\cdot,\lambda_{0})-\partial{}_{\lambda}A_{1}(\cdot,\lambda_{0})v^{\lambda_{0}}-\partial{}_{\lambda}B_{1}(\cdot,\lambda_{0})\overline{v^{\lambda_{0}}},$
(4.45)
$\displaystyle\gamma_{\ast}^{\lambda_{0}}:=\partial{}_{\lambda}\widetilde{\gamma}(\cdot,\lambda_{0})-Re[(\partial{}_{\lambda}\overline{\widetilde{l}}(\cdot,\lambda_{0})v^{\lambda_{0}}],$
(4.46) $\displaystyle
h_{r}(\lambda_{0})=a_{r}^{\prime}(\lambda_{0})+ib_{r}^{\prime}(\lambda_{0}),$
(4.47) $\displaystyle
g_{s}(\lambda_{0})=\partial{}_{\lambda_{0}}\\{\widetilde{l^{\lambda}}(\zeta_{s}^{\prime})[\widetilde{\gamma^{\lambda}}(\zeta_{s}^{\prime})+ic_{s}(\lambda)]\\}.$

Note that the solution to (4.42) exists since the index is
$\frac{1}{2\pi}[\widetilde{l}^{\lambda_{0}}]$ which is non-negative.

By the induction hypothesis, we have
$w^{\lambda}\in\mathcal{C}^{k+\mu,j-1}(\overline{\Omega^{\lambda}})$, and
therefore
$v^{\lambda}=w^{\lambda}\circ(R^{\lambda})^{-1}(\zeta)\in\mathcal{C}^{k+\mu,j-1}(\overline{\mathbb{D}})$.
As shown in (i),
$A_{1},B_{1},F_{1}\in\mathcal{C}^{k+\mu,j}(\overline{\mathbb{D}})$, thus
$\partial{}_{\lambda}A_{1}$, $\partial{}_{\lambda}B_{1}$,
$\partial{}_{\lambda}F_{1}\in\mathcal{C}^{k-1+\mu,j-1}(\overline{\mathbb{D}})$.
From (4.44) we have
$F_{\ast}^{\lambda_{0}}\in\mathcal{C}^{k-1+\mu,j-1}(\overline{\mathbb{D}})$.
On the other hand we have
$\partial{}_{\lambda}\widetilde{\gamma},\partial{}_{\lambda}\overline{\widetilde{l}}\in\mathcal{C}^{k+\mu,j-1}(\overline{\mathbb{D}})$,
thus by (4.45),
$\gamma_{\ast}^{\lambda_{0}}\in\mathcal{C}^{k+\mu,j-1}(\overline{\mathbb{D}})$.

Since $a_{r}^{\prime},b_{r}^{\prime},c_{s}^{\prime}\in C^{j-1}([0,1])$, by
(4.46) and (4.47), $h_{r},g_{s}\in\mathcal{C}^{j-1}([0,1])$. Applying (i) to
problem (4.42)-(4.43), we obtain
$\partial{}_{\lambda}v(\cdot,\lambda_{0})\in\mathcal{C}^{k+\mu,j-1}(\overline{\mathbb{D}})$.

We now show that $\partial{}_{\lambda}v$ is indeed the $\lambda$-derivative of
$v^{\lambda}$, namely $D^{\lambda,\lambda_{0}}v$ converges to
$\partial{}_{\lambda}v(\cdot,\lambda_{0})$ uniformly on $\mathbb{D}$, as
$\lambda\to\lambda_{0}$.

Now
$u^{\lambda}:=D^{\lambda,\lambda_{0}}v-\partial{}_{\lambda}v(\cdot,\lambda_{0})$
is the unique solution to the boundary value problem

(4.48)
$\begin{cases}\overline{\partial}_{\zeta}u^{\lambda}+A_{1}^{\lambda_{0}}u^{\lambda}+B_{1}^{\lambda_{0}}\overline{u^{\lambda}}=F^{\lambda,\lambda_{0}}_{\ast}-F^{\lambda_{0}}_{\ast}&\text{in
$\mathbb{D}$,}\\\
Re[\overline{\widetilde{l^{\lambda_{0}}}}u^{\lambda}]=\gamma_{\ast}^{\lambda,\lambda_{0}}-\gamma^{\lambda_{0}}_{\ast}\quad\quad\text{on
$S^{1}$,}\end{cases}$

which satisfies the condition:

(4.49)
$\begin{split}u^{\lambda}(\zeta_{r})=h_{r}^{\lambda,\lambda_{0}}-h_{r}(\lambda_{0})\quad(r=1,...,h_{0}),\\\
u^{\lambda}(\zeta_{s}^{\prime})=g_{s}^{\lambda,\lambda_{0}}-g_{s}(\lambda_{0})\quad(s=1,...,h_{1}),\end{split}$

where the right-hand sides are defined by (4.40) and (4.41). We now show that
$|u^{\lambda}|_{0}$ converges to $0$ as $\lambda\to\lambda_{0}$. Reducing the
problem as before to the form

(4.50)
$\begin{cases}\overline{\partial}_{\zeta}\widehat{u}^{\lambda}+A^{\lambda_{0}}_{0}\widehat{u}^{\lambda}+B^{\lambda_{0}}_{0}\overline{\widehat{u}^{\lambda}}=F^{\lambda,\lambda_{0}}_{0}&\text{in
$\mathbb{D}$.}\\\
Re[\zeta^{-n}\widehat{u}^{\lambda}]=\gamma_{0}^{\lambda,\lambda_{0}}&\text{on
$S^{1}$}\end{cases}$

satisfying the conditions:

(4.51)
$\begin{split}\widehat{u}^{\lambda}(\zeta_{r})=\widehat{h}_{r}(\lambda),\quad(r=1,...,k)\\\
\widehat{u}^{\lambda}(\zeta_{s}^{\prime})=\widehat{g}_{s}(\lambda),\quad(s=1,...,k^{\prime})\end{split}$

where

$\displaystyle\widehat{u}^{\lambda}=e^{\chi^{\lambda_{0}}}u^{\lambda},\quad
A_{0}^{\lambda_{0}}=A_{1}^{\lambda_{0}},\qquad
B_{0}^{\lambda_{0}}=B_{1}^{\lambda_{0}}e^{2iIm(\chi^{\lambda_{0}})},$ (4.52)
$\displaystyle
F_{0}^{\lambda,\lambda_{0}}=\left(F^{\lambda,\lambda_{0}}_{\ast}-F^{\lambda_{0}}_{\ast}\right)e^{\chi^{\lambda_{0}}},\qquad\gamma_{0}^{\lambda,\lambda_{0}}=\left(\gamma_{\ast}^{\lambda,\lambda_{0}}-\gamma^{\lambda_{0}}_{\ast}\right)e^{p^{\lambda_{0}}},$
(4.53)
$\displaystyle\widehat{h}_{r}(\lambda)=h_{r}^{\lambda,\lambda_{0}}-h_{r}(\lambda_{0}),\qquad\widehat{g}_{s}(\lambda)=g_{s}^{\lambda,\lambda_{0}}-g_{s}(\lambda_{0}),$

where as before $\chi^{\lambda_{0}}$ is the holomorphic function in
$\mathbb{D}$ defined by (3.4). The solution to (4.50)-(4.51) is given by

(4.54)
$\widehat{u}^{\lambda}=u_{0}^{\lambda}+\sum_{\ell=1}^{2n+1}d_{\ell}(\lambda)u_{\ell}^{\lambda},$

where $u_{0}^{\lambda}$ is a particular solution which uniquely solves the
integral equation:

(4.55)
$u_{0}^{\lambda}+Q_{n}^{\lambda_{0}}u_{0}^{\lambda}=P_{n}(F_{0}^{\lambda})+\frac{z^{n}}{2\pi
i}\int_{b\mathbb{D}}\gamma_{0}^{\lambda}(t)\frac{t+z}{t-z}\frac{dt}{t}.$

$Q_{n}^{\lambda_{0}}$ and $P_{n}^{\lambda_{0}}$ are the compact operators in
the space $C^{0}(\overline{\mathbb{D}})$ defined by (3.17) and (3.12),
respectively. Since $I+Q^{\lambda_{0}}_{n}$ is a injective (Vekua [VA62]*p.
268), by Fredholm alternative, it is also surjective. By the open mapping
theorem, the inverse $(I-Q^{\lambda_{0}}_{n})^{-1}$ is continuous. Denoting
its operator norm by $M_{0}$, we get

$|u_{0}^{\lambda}|_{0}\leq
M_{0}\left(|F_{0}^{\lambda,\lambda_{0}}|_{0}+|\gamma_{0}^{\lambda,\lambda_{0}}|_{0}\right).$

Since $v\in C^{0,0}(\overline{\mathbb{D}})$ from part (i), we obtain from
expressions (4.52) that the right-hand side of the above equation converges to
$0$ as $\lambda\to\lambda_{0}$. Hence $|u_{0}^{\lambda}|_{0}\to 0$.

Finally we show $d_{\ell}(\lambda)$ converge to $0$. We have

$\displaystyle\widehat{h}_{r}(\lambda)=\widehat{u}^{\lambda}(\zeta_{r})=u_{0}^{\lambda}(\zeta_{r})+\sum_{\ell=1}^{2n+1}d_{\ell}(\lambda)u_{\ell}^{\lambda}(\zeta_{r}),\quad
r=1,...,h_{0}.$
$\displaystyle\widehat{g}_{s}(\lambda)=\widehat{u}^{\lambda}(\zeta_{s}^{\prime})=u_{0}^{\lambda}(\zeta_{s}^{\prime})+\sum_{\ell=1}^{2n+1}d_{\ell}(\lambda)u_{\ell}^{\lambda}(\zeta_{s}^{\prime}),\quad
s=1,...,h_{1}.$

In view of (4.53), $\widehat{h}_{r}(\lambda)$, $\widehat{g}_{s}(\lambda)$
converge to $0$ as $\lambda\to\lambda_{0}$. Also $u_{0}^{\lambda}(\zeta_{r})$
and $u_{0}^{\lambda}(\zeta_{s}^{\prime})$ all converge to $0$ as
$\lambda\to\lambda_{0}$, and therefore same reasoning as before shows that
$d_{\ell}(\lambda)$ converges to $0$. By (4.54), $|\widehat{u}^{\lambda}|_{0}$
converges to $0$, which implies $D^{\lambda,\lambda_{0}}v$ converges to
$\partial{}_{\lambda}v(\cdot,\lambda_{0})\in\mathcal{C}^{k+\alpha,j-1}(\overline{\mathbb{D}})$
in the sup-norm. This completes the induction and we obtain
$v\in\mathcal{C}^{k+1+\mu,j}(\overline{\mathbb{D}})$. Since
$w^{\lambda}=v^{\lambda}\circ R^{\lambda}$,
$w\in\mathcal{C}^{k+1+\mu,j}(\overline{\Omega^{\lambda}})$. ∎

Proof of Theorem 4.1. Proposition 4.6 gives a solution
$w^{\lambda}\in\mathcal{C}^{k+1+\alpha,j}(\overline{\Omega^{\lambda}})$. To
finish the proof we need to integrate $w^{\lambda}$ (as in (4.4)) to get a
solution $U^{\lambda}$ for the original elliptic boundary value problem.
Denote the resulting integration constant by $c_{0}(\lambda)$. Then by
combining
$w^{\lambda}\in\mathcal{C}^{k+1+\alpha,j}(\overline{\Omega^{\lambda}})$ and
the condition $u^{\lambda}(z_{0}^{\lambda})\in C^{j}([0,1])$ in the statement
of the theorem, we get $c_{0}\in C^{j}[0,1]$, and the proof of Theorem 4.1 is
complete. ∎

##  5\. Kernels with parameter

Starting in this section we study the parameter problem on families of
multiply-connected domains. As explained in the introduction, we can no longer
fix the domain while preserving $\overline{\partial}$ in the highest order
term, since there is no Riemann mapping theorem in this case. More
significantly, unlike in the simply-connected case where the solutions can be
shown to satisfy some Fredholm equation, the solutions on multiply-connected
domains in general can not be written as solutions of integral equations
directly. Instead we will follow Vekua to represent solutions by integral
formulas (3.35). In this section we show how the kernels $G_{1}$ and $G_{2}$
depend on $\lambda$.

For a function $f^{\lambda}$ defined on the domain $\Omega^{\lambda}$ we will
adopt the following notation:

$f(z,\lambda):=f^{\lambda}\circ\Gamma^{\lambda}(z),\quad
z\in\Omega,\quad\lambda\in[0,1].$

######  Lemma 5.1.

Let $\Omega\subset\mathbb{C}$ be a bounded domain with $C^{1}$ boundary, and
$\Gamma^{\lambda}:\overline{\Omega}\to\overline{\Omega^{\lambda}}$ be a
$C^{1,0}$-embedding. Let $1\leq r\leq 2$. For each $\lambda$, suppose
$f^{\lambda}\in L^{r}(\Omega^{\lambda})$ and
$|f(\cdot,\lambda)|_{L^{r}(\Omega)}$ is uniformly bounded in $\lambda$. Define

$T_{\Omega^{\lambda}}f(\cdot,\lambda):=\iint_{\Omega}\frac{f(\zeta,\lambda)}{\zeta^{\lambda}-z^{\lambda}}\,dA(\zeta^{\lambda}).$

Then the following inequality hold:

(5.1)
$|T_{\Omega^{\lambda}}f(\cdot,\lambda)|_{L^{\gamma}_{\alpha}(\Omega)}\leq
C_{1,0}C(r,\gamma,\Omega)|f(\cdot,\lambda)|_{L^{r}(\Omega)},$

where $\gamma$ is an arbitrary number satisfying the inequality

$1<\gamma<\frac{2r}{2-r},$

and

$\alpha=\frac{1}{\gamma}-\frac{2-r}{2r}>0.$

The constant $C(r,\gamma,\Omega)$ depends only on $r,\gamma$ and $\Omega$ and
does not depend on $\lambda$.

###### Proof.

Recall definition (2.2) for the norm $|\cdot|_{L^{\gamma}_{\alpha}}$, so we
have

$\displaystyle|T_{\Omega^{\lambda}}f^{\lambda}(\cdot)|_{L^{\gamma}_{\alpha}(\Omega^{\lambda})}$
$\displaystyle=\left(\int_{\Omega}\left|\frac{f(\zeta,\lambda)}{\zeta^{\lambda}-z^{\lambda}}\right|^{\gamma}a^{\lambda}(\zeta)\,dA(\zeta)\right)^{\frac{1}{\gamma}}$
$\displaystyle\quad+\sup_{h\in\mathbb{C}}\frac{1}{|h|^{\alpha}}\left(\int_{\Omega}|f(\zeta+h,\lambda)-f(\zeta,\lambda)|^{\gamma}a^{\lambda}(\zeta)\,dA(\zeta)\right)^{\frac{1}{\gamma}}$
$\displaystyle\leq
C_{1,0}\left\\{\left(\int_{\Omega}\left|\frac{f(\zeta,\lambda)}{\zeta-z}\right|^{\gamma}\,d|\zeta|\right)^{\frac{1}{\gamma}}+\sup_{h\in\mathbb{C}}\frac{1}{|h|^{\alpha}}\left(\int_{\Omega}|f(\zeta+h,\lambda)-f(\zeta,\lambda)|^{\gamma}\,dA(\zeta)\right)^{\frac{1}{\gamma}}\right\\}$
$\displaystyle=C_{1,0}|T_{\Omega}f(\cdot,\lambda)|_{L^{\gamma}_{\alpha}(\Omega)}$

where $a^{\lambda}(\zeta)=|\det(\Gamma^{\lambda})|$. By the result in
[VA62]*p. 47, for $\gamma,\alpha,r$ satisfying the above conditions, we have

$|T_{\Omega}f^{\lambda}(\cdot,\lambda)|_{L^{\gamma}_{\alpha}(\Omega)}\leq
C(r,\gamma,\Omega)|f(\cdot,\lambda)|_{L^{r}(\Omega)},$

from which estimate (5.1) then follows. ∎

######  Lemma 5.2.

Let $2<p<\infty$. Suppose $\lambda\mapsto A(\cdot,\lambda),B(\cdot,\lambda)$
are continuous maps from $[0,1]$ to $L^{p}(\mathbb{C})$, where for each
$\lambda\in[0,1]$, $A(\cdot,\lambda),B(\cdot,\lambda)\equiv 0$ outside
$\Omega$. For $j=1,2$, let

(5.2)
$\omega_{j}(z,t,\lambda)=\frac{t^{\lambda}-z^{\lambda}}{\pi}\iint_{\Omega}\frac{A(\zeta,\lambda)X_{j}(\zeta,t,\lambda)+B(\zeta,\lambda)\overline{X_{j}(\zeta,t,\lambda)}}{(\zeta^{\lambda}-z^{\lambda})(t^{\lambda}-\zeta^{\lambda})X_{j}(\zeta,t,\lambda)}\,dA(\zeta^{\lambda}).$

Then we have

(5.3) $|\omega_{j}(\cdot,t,\lambda)|_{\alpha}\leq
C_{1,0}C(p,\Omega)\left(|A(\cdot,\lambda)|_{L^{p}(\Omega)}+|B(\cdot,\lambda)|_{L^{p}(\Omega)}\right),$

where the constants do not depend on $\lambda$ and $t\in K$, for any compact
subset $K$ of $\mathbb{C}$. In particular, if
$A^{\lambda},B^{\lambda}\in\mathcal{C}^{0,0}(\overline{\Omega^{\lambda}})$,
then

(5.4) $|\omega_{j}(\cdot,t,\lambda)|_{\tau}\leq
C_{1,0}C(\tau,\Omega)\left(\|A^{\lambda}\|_{0,0}+\|B^{\lambda}\|_{0,0}\right),\quad\quad\text{for
any $0<\tau<1$}.$

Proof. Let $a^{\lambda}(\zeta)=|\det\Gamma^{\lambda}(\zeta)|$. We can rewrite
$\omega_{j}$ as

$\displaystyle\omega_{j}(z,t,\lambda)$
$\displaystyle=\frac{1}{\pi}\iint_{\Omega}\left[\frac{1}{\zeta^{\lambda}-z^{\lambda}}-\frac{1}{\zeta^{\lambda}-t^{\lambda}}\right]$
$\displaystyle\quad\times\left[A(\zeta,\lambda)+B(\zeta,\lambda)\frac{\overline{X_{j}(\zeta,t,\lambda)}}{X_{j}(\zeta,t,\lambda)}\right]a^{\lambda}(\zeta)\,dA(\zeta).$

By Hölder’s inequality and Proposition 7.4, we have for any
$z,t\in\mathbb{C}$,

(5.5) $|\omega_{j}(z,t,\lambda)|\leq
C_{1,0}C(p,\Omega)\left(|A(\cdot,\lambda)|_{L^{p}(\Omega)}+|B(\cdot,\lambda)|_{L^{p}(\Omega)}\right),$
(5.6) $|\omega_{j}(z,t,\lambda)|\leq
C_{1,0}C_{p}\left(|A(\cdot,\lambda)|_{L^{p}(\Omega)}+|B(\cdot,\lambda)|_{L^{p}(\Omega)}\right)|z-t|^{\alpha},$
(5.7) $\displaystyle|\omega_{j}(z_{1},t,\lambda)-\omega_{j}(z_{2},t,\lambda)|$
$\displaystyle\quad\leq
C_{1,0}C_{p}\left(|A(\cdot,\lambda)|_{L^{p}(\Omega)}+|B(\cdot,\lambda)|_{L^{p}(\Omega)}\right)|z_{1}-z_{2}|^{\alpha}$

for any $z_{1},z_{2},t\in\mathbb{C}$. ∎

######  Lemma 5.3.

Let $\Omega$ be a bounded domain in $\mathbb{C}$ with $C^{1}$ boundary, and
$\Gamma:\Omega\to\Omega^{\lambda}$ be a $\mathcal{C}^{1,0}$ embedding. Let
$2<p<\infty$, and $p^{\prime}=\frac{p}{p-1}$ be the conjugate of $p$, with
$1<p^{\prime}<2$. Suppose $\lambda\mapsto A(\cdot,\lambda),B(\cdot,\lambda)$
are continuous maps from $[0,1]$ to $L^{p}(\mathbb{C})$, where for each
$\lambda\in\mathcal{P}$, $A(\cdot,\lambda)$, $B(\cdot,\lambda)$ $\equiv 0$
outside $\Omega$. Fix $t$ in some compact subset $K$ of $\mathbb{C}$. For any
$q$ with $1<p^{\prime}\leq q<2$, and for each $\lambda\in[0,1]$, let
$X_{1}(z,t,\lambda)$ and $X_{2}(z,t,\lambda)$ be the unique solution in
$L_{q,0}(\mathbb{C})$ to the integral equations (see Proposition 7.21):

(5.8)
$X_{1}(z,t,\lambda)-\frac{1}{\pi}\iint_{\Omega}\frac{A(\zeta,\lambda)X_{1}(\zeta,t,\lambda)+B(\zeta,\lambda)\overline{X_{1}(\zeta,t,\lambda)}}{\zeta^{\lambda}-z^{\lambda}}\,dA(\zeta^{\lambda})=\frac{1}{2(t^{\lambda}-z^{\lambda})},$
(5.9)
$X_{2}(z,t,\lambda)-\frac{1}{\pi}\iint_{\Omega}\frac{A(\zeta,\lambda)X_{2}(\zeta,t,\lambda)+B(\zeta,\lambda)\overline{X_{2}(\zeta,t,\lambda)}}{\zeta^{\lambda}-z^{\lambda}}\,dA(\zeta^{\lambda})=\frac{1}{2i(t^{\lambda}-z^{\lambda})}.$

Then we have the following:
(i) For each $t\in\mathbb{C}$, $\lambda\mapsto X_{j}(\cdot,t,\lambda)$ is a
continuous map from $[0,1]$ to $L^{q}(\Omega)$. Moreover, for all $t\in K$
where $K$ is some compact subset of $\mathbb{C}$, we have

(5.10) $\left|X_{j}(\cdot,t,\lambda)\right|_{L^{q}(\Omega)}\leq
C_{1,0}C(p,\Omega)\left(\left|A(\cdot,\lambda)\right|_{L^{p}(\Omega)}+\left|B(\cdot,\lambda)\right|_{L^{p}(\Omega)}\right).$

where the constants on the right-hand side does not depend on $\lambda$ and
$t\in K$.
(ii) The following holds

$\sup_{t\in
K}|X_{j}(\cdot,t,\lambda_{1})-X_{j}(\cdot,t,\lambda_{2})|_{L^{q}(\Omega)}\to
0\quad\text{as $|\lambda_{1}-\lambda_{2}|\to 0$.}$

(iii) For each $\varepsilon>0$, let
$E_{\varepsilon}(t)=\\{z\in\Omega:|z-t|>\varepsilon\\}$. Then we have

$\sup_{t\in
K}|X_{j}(\cdot,t,\lambda_{1})-X_{j}(\cdot,t,\lambda_{2})|_{C^{0}(E_{\varepsilon}(t))}\to
0\quad\text{as $|\lambda_{1}-\lambda_{2}|\to 0$.}$

###### Proof.

(i) We shall only prove the lemma for $X_{1}$, as the proof for $X_{2}$ is
identical. First, we show the $L^{q}$ norm of $X_{j}$ is uniformly bounded in
$\lambda$ and $t\in K$. Seeking contradiction, suppose there exists a sequence
$\lambda_{j}\in[0,1]$ such that
$N_{j}:=|X_{1}(\cdot,t,\lambda_{j})|_{L^{q}(\Omega)}\to\infty$ as
$j\to\infty$. Since $[0,1]$ is compact we can assume that
$\lambda_{j}\to\lambda_{0}\in[0,1]$ for some $\lambda_{0}$. Dividing both
sides of equation (5.8) by $N_{j}$, and writing
$\widehat{X_{1}}(\cdot,t,\lambda_{j})=\frac{1}{N_{j}}X_{1}(\cdot,t,\lambda_{j})$,
we obtain

(5.11)
$\widehat{X_{1}}(\cdot,t,\lambda_{j})-P^{\lambda_{j}}\widehat{X_{1}}(\cdot,t,\lambda_{j})=\frac{1}{2(t^{\lambda_{j}}-z^{\lambda_{j}})N_{j}},$

where $|\widehat{X_{1}}(\cdot,t,\lambda_{j})|_{L^{q}(\Omega)}=1$ and we set

$\displaystyle
P^{\lambda}f(z,t,\lambda)=T_{\mathbb{C}}[A(\cdot,\lambda)f(\cdot,t,\lambda)+B(\cdot,\lambda)\overline{f(\cdot,t,\lambda)}](z,t,\lambda).$

Setting $r=1$ in Lemma 5.1 and applying Hölder’s inequality, we have

$\displaystyle\left|P^{\lambda_{j}}\widehat{X_{1}}(\cdot,t,\lambda_{j})\right|_{L^{\gamma}_{\alpha}(\Omega)}\leq
C(p,\gamma,\Omega)\left|(A+B)(\cdot,\lambda_{j})\widehat{X}_{1}(\cdot,t,\lambda_{j})\right|_{L^{1}(\Omega)}$
$\displaystyle\qquad\leq
C(p,\gamma,\Omega)\left(|A(\cdot,\lambda_{j})|_{L^{p}(\Omega)}+|B(\cdot,\lambda_{j})|_{L^{p}(\Omega)}\right)|\widehat{X}_{1}(\cdot,t,\lambda_{j})|_{L^{p^{\prime}}(\Omega)}$

for $1<\gamma<\frac{2\cdot 1}{2-1}=2$, and
$\alpha=\frac{1}{\gamma}-\frac{1}{2}>0$. In particular we can choose
$\gamma=q$. (Since $1<p^{\prime}\leq q<2$.) By Proposition 7.22 applied to the
sequence $P^{\lambda_{j}}\widehat{X_{1}}(\cdot,t,\lambda_{j})$, and passing to
a subsequence if necessary, we can assume that
$P^{\lambda_{j}}\widehat{X_{1}}(\cdot,t,\lambda_{j})$ converges to some limit
in $L^{q}(\Omega)$. Meanwhile the right-hand side of equation (5.11) converges
to $0$ in $L^{q}(\Omega)$. Therefore by (5.11)
$\widehat{X_{1}}(\cdot,t,\lambda_{j})$ converges to some
$\widehat{X_{1}}(\cdot,t)$ in $L^{q}(\Omega)$.

We show that
$|P^{\lambda_{j}}\widehat{X_{1}}(\cdot,t,\lambda_{j})-P^{\lambda_{0}}\widehat{X_{1}}(\cdot,t)|_{L^{q}(\Omega)}\to
0$ as $\lambda_{j}\to\lambda_{0}$. In what follows we write
$a^{\lambda}(\zeta)$ for $|\det D\Gamma^{\lambda}(\zeta)|$. Writing
$D^{\lambda_{j}}\widehat{X_{1}}(\zeta,t)=\widehat{X_{1}}(\zeta,t,\lambda_{j})-\widehat{X_{1}}(\zeta,t)$,
we have

(5.12)
$|P^{\lambda_{j}}\widehat{X_{1}}(\cdot,t,\lambda_{j})-P^{\lambda_{0}}\widehat{X_{1}}(\cdot,t)|_{L^{q}(\Omega)}\leq
D_{1}+D_{2}+D_{3}+D_{4}.$

where

$\displaystyle
D_{1}=\left|\iint_{\Omega}\frac{(\delta^{\lambda_{j},\lambda_{0}}A)\widehat{X_{1}}(\zeta,t,\lambda_{j})+A(\zeta,\lambda_{0})(D^{\lambda_{j}}\widehat{X_{1}}(\zeta,t))}{\zeta^{\lambda_{j}}-z^{\lambda_{j}}}\,dA(\zeta^{\lambda_{j}})|\right|_{L^{q}(\Omega)},$
$\displaystyle
D_{2}=\left|\iint_{\Omega}\frac{(\delta^{\lambda_{j},\lambda_{0}}B)\widehat{X_{1}}(\zeta,t,\lambda_{j})+B(\zeta,\lambda_{0})\overline{D^{\lambda_{j}}\widehat{X_{1}}(\zeta,t)}}{\zeta^{\lambda_{j}}-z^{\lambda_{j}}}\,dA(\zeta^{\lambda_{j}})|\right|_{L^{q}(\Omega)},$
$\displaystyle
D_{3}=\left|\iint_{\Omega}\frac{A(\zeta,\lambda_{0})\widehat{X_{1}}(\zeta,t,\lambda_{0})+B(\zeta,\lambda_{0})\overline{\widehat{X_{1}}(\zeta,t,\lambda_{0})}}{\zeta^{\lambda_{j}}-z^{\lambda_{j}}}\left\\{a^{\lambda_{j}}(\zeta)-a^{\lambda_{0}}(\zeta)\right\\}\,dA(\zeta)\right|_{L^{q}(\Omega)},$

and

$\displaystyle D_{4}$
$\displaystyle=\left|\iint_{\Omega}\left(A(\zeta,\lambda_{0})\widehat{X_{1}}(\zeta,t,\lambda_{0})+B(\zeta,\lambda_{0})\overline{\widehat{X_{1}}(\zeta,t,\lambda_{0})}\right)\right.$