# Optimal estimates for transmission problems including relative
conductivities with different signs

Hongjie Dong  and  Zhuolun Yang  Division of Applied Mathematics, Brown
University, 182 George Street, Providence, RI 02912, USA
<EMAIL_ADDRESS>Institute for Computational and Experimental Research
in Mathematics, Brown University, 121 South Main Street, Providence, RI 02903,
USA<EMAIL_ADDRESS>
###### Abstract.

We study the gradient and higher order derivative estimates for the
transmission problem in the presence of closely located inclusions. We show
that in two dimensions, when relative conductivities of circular inclusions
have different signs, the gradient and higher order derivatives are bounded
independent of $\varepsilon$, the distance between the inclusions. We also
show that for general smooth strictly convex inclusions, when one inclusion is
an insulator and the other one is a perfect conductor, the derivatives of any
order is bounded independent of $\varepsilon$ in any dimensions $n\geq 2$.

H. Dong is partially supported by Simons Fellows Award 007638, NSF Grant
DMS-2055244, and the Charles Simonyi Endowment at the Institute for Advanced
Study.

Z. Yang is partially supported by Simons Foundation Institute Grant Award ID
507536.

##  1\. Introduction and main result

In this paper, we first study the second-order elliptic equations in
divergence form with discontinuous coefficients in two dimensions

$L_{\varepsilon;r_{1},r_{2}}u:=D_{i}(a(x)D_{i}u)=D_{i}f_{i}\quad\mbox{in}~{}~{}\mathcal{D},$
(1.1)

where $\mathcal{D}$ is a bounded open subset of $\mathbb{R}^{2}$,

$a(x)=k_{1}\chi_{\mathcal{B}_{1}}+k_{2}\chi_{\mathcal{B}_{2}}+k_{0}\chi_{\mathcal{B}_{0}},$

$k_{0}=1$, $k_{1},k_{2},r_{1},r_{2}\in(0,\infty)$ are constants,

$\mathcal{B}_{1}:=B_{r_{1}}(\varepsilon/2+r_{1},0),\quad\mathcal{B}_{2}:=B_{r_{2}}(-\varepsilon/2-r_{2},0),\quad\mathcal{B}_{0}:=\mathbb{R}^{2}\setminus(\overline{\mathcal{B}_{1}\cup\mathcal{B}_{2}}),$

and $\chi$ is the indicator function. The equation models the conductivity
problem in composite material. The gradient of the voltage potential $u$
represents the electric field, and $a(x)$ is the conductivity which is a
constant on each inclusion, and a different constant on the background matrix.
It is significant from an engineering point of view to estimate the
derivatives of the solutions.

In [BASL], Babuška et al. analyzed an analogous elliptic system, and
numerically showed that, when the ellipticity constants are away from $0$ and
infinity, the gradient of solutions remains bounded independent of
$\varepsilon$, the distance between inclusions. When $\varepsilon=0$,
Bonnetier and Vogelius [BV] proved that $|Du|$ is bounded for a fixed
$k=k_{1}=k_{2}$ away from $0$ and infinity. This result was extended by Li and
Vogelius [LV] to general second order elliptic equations in divergence form
with piecewise Hölder coefficients and general shape of inclusions in any
dimension. Furthermore, they established a stronger piecewise $C^{1,\alpha}$
control of $u$, which is independent of $\varepsilon$. Li and Nirenberg [LN]
further extended this global Lipschitz and piecewise $C^{1,\alpha}$ result to
general second order elliptic systems in divergence form, including the linear
system of elasticity. Some higher order derivative estimates in dimension
$n=2$ were obtained in [DZ, DL, JiKang].

On the other hand, if $k_{1},k_{2}$ are allowed to be $0$ or $\infty$, it was
shown in [Kel, BudCar, Mar] that the gradient of solutions generally becomes
unbounded as $\varepsilon\to 0$. For the perfect conductivity problem
($k_{1}=k_{2}=\infty$), it has been proved that the generic blow-up rate of
$|Du|$ is $\varepsilon^{-1/2}$ in two dimensions,
$|\varepsilon\log\varepsilon|^{-1}$ in three dimensions, and
$\varepsilon^{-1}$ in dimensions greater than three; see [AKLLL, AKL, Y1, Y2,
BLY1, BLY2]. These bounds were shown to be optimal and are independent of the
shape of inclusions, as long as the inclusions are relatively strictly convex.
Moreover, more detailed characterizations of the singular behavior of $\nabla
u$ have been obtained. For further works on the perfect conductivity problem
and closely related works, see e.g. [ACKLY, BT1, BT2, DL, AKKY, KLY1, KLY2, L,
LLY, LWX, BLL, BLL2, DZ, KL, CY, ADY, Gor, LimYun, Kang] and the references
therein.

For the insulated conductivity problem ($k_{1}=k_{2}=0$), it was shown in
[AKLLL, AKL] that the optimal blow-up rate is $\varepsilon^{-1/2}$ in two
dimensions. The proof uses a harmonic conjugate argument to link the insulated
conductivity case to the perfect conductivity case, which fails in dimensions
greater than two. For the higher dimensional case, Bao, Li, and Yin in [BLY2]
established an upper bound of order $\varepsilon^{-1/2}$. Yun in [Y3] proved
the optimal blow-up rate on the shortest line segment connecting two spherical
inclusions in three dimensions is $\varepsilon^{\frac{\sqrt{2}-2}{2}}$. Later,
the upper bound $\varepsilon^{-1/2}$ was improved by Li and Yang in [LY2] to
be $\varepsilon^{-1/2+\beta}$, for some $\beta>0$. See also [LY]. Weinkove in
[We] used a Bernstein-type argument to obtain a more explicit upper bound of
$\beta$ in dimensions greater than three. Dong, Li, and Yang in recent works
[DLY, DLY2] identified the optimal blow-up rate. They proved the optimal
gradient estimate for a class of inclusions including balls and “almost”
optimal gradient estimate for general strictly convex inclusions. Unlike the
perfect conductivity case, the optimal blow-up rate is related to the
principal curvature of the inclusions.

Recently, among other results, Ji and Kang in [JiKang] used some spectral
properties of Neumann-Poincaré operators to study the problem (1.1) for the
case when $0<k_{1}<1$ and $k_{2}>1$, and proved that

$|D^{m}u|\leq
C\left(-\frac{(k_{1}+1)(k_{2}+1)}{(k_{1}-1)(k_{2}-1)}-1+\sqrt{\frac{2(r_{1}+r_{2})\varepsilon}{r_{1}r_{2}}}\right)^{-m+1},\quad
m=1,2,\ldots.$

In particular, when $k_{1}\to 0$ and $k_{2}\to\infty$, this implies

$|D^{m}u|\leq C\varepsilon^{-(m-1)/2},\quad m=1,2,\ldots.$

In the first part of this paper, we apply the Green function method developed
in [DL] to show that $|D^{m}u|$ is in fact bounded independent of
$\varepsilon$ for any $m=1,2,\ldots$. Compared to the method in [JiKang], we
use $m$-th order finite differences to estimate the $m$-th order derivatives
more precisely.

Let

$\alpha=\frac{k_{1}-1}{k_{1}+1},\quad\beta=\frac{k_{2}-1}{k_{2}+1},\quad\mbox{and}\quad\gamma=-\alpha\beta.$

In particular, we are interested in the case when $k_{1}\to 0,k_{2}\to+\infty$
(or $\alpha\to-1,\beta\to 1$). Therefore, we may restrict
$\gamma>\frac{1}{2}$. First we assume that $r_{1}=r_{2}=1$.

######  Theorem 1.1.

Let $\varepsilon\in(0,1/2)$ and $\mu\in(0,1)$ be constants. Assume that $u$ is
a weak solution of (1.1) in $B_{1}:=B_{1}(0)$ with $r_{1}=r_{2}=1$,
$k_{1}\in(0,1)$, $k_{2}\in(1,\infty)$, and $\gamma\in(1/2,1)$. For any
$m\in\mathbb{N}$, if $f$ is piecewise $C^{2m-1,\mu}$ in $B_{1}$, and for some
constant $C_{m}>0$,

$\|u\|_{L^{2}(B_{1})}\leq
C_{m},\quad\|f\|_{C^{2m-1,\mu}(B_{1}\cap\mathcal{B}_{j})}\leq C_{m}k_{j},\quad
j=0,1,2,$

then we have

$|D^{m}u(x)|\leq CC_{m}\quad\mbox{in}~{}~{}B_{1/2},$ (1.2)

where $C>0$ is a positive constant depending only on $m$ and $\mu$, and in
particular is independent of $\varepsilon$, $k_{1}$, and $k_{2}$.

When $u$ satisfies (1.1) in a domain that contains $\mathcal{B}_{1}$ and
$\mathcal{B}_{2}$, we have more precise estimates as follows.

######  Theorem 1.2.

Let $\varepsilon\in(0,1/2)$ and $\mu\in(0,1)$ be constants. Assume that
$\mathcal{B}_{1}\cup\mathcal{B}_{2}\Subset\mathcal{D}_{1}\Subset\mathcal{D}_{2}\Subset\mathcal{D}$
for some domains $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$, $u$ is a weak
solution of (1.1) in $\mathcal{D}$ with $r_{1}=r_{2}=1$, $0<k_{1}<1$,
$k_{2}>1$, and $\gamma\in(1/2,1)$. For any $m\in\mathbb{N}$, if $f$ is
piecewise $C^{2m-1,\mu}$ in $\mathcal{D}$, and for some constant $C_{m}>0$,

$\|u\|_{L^{2}(\mathcal{D})}\leq
C_{m},\quad\|f\|_{C^{2m-1,\mu}(B_{1}\cap\mathcal{B}_{j})}\leq
C_{m}\min\\{1,k_{j}\\},\quad j=0,1,2,$

then we have

$|D^{m}u(x)|\leq\left\\{\begin{aligned}
&CC_{m}&&\mbox{in}~{}~{}\mathcal{D}_{1}\cap\mathcal{B}_{0},\\\
&\frac{CC_{m}}{k_{1}+1}&&\mbox{in}~{}~{}\mathcal{B}_{1},\\\
&\frac{CC_{m}}{k_{2}+1}&&\mbox{in}~{}~{}\mathcal{B}_{2},\end{aligned}\right.$

where $C>0$ is a constant depending only on $m$, $\mu$, $\mathcal{D}_{1}$,
$\mathcal{D}_{2}$, and $\mathcal{D}$.

For the general case when $r_{1}$ and $r_{2}$ are not necessarily equal to
$1$, we have the following theorem.

######  Theorem 1.3.

Let $\varepsilon\in(0,1/2)$, $\mu\in(0,1)$ be constants, and
$1/2<r_{1},r_{2}<10$. Then there exist domains
$\mathcal{D}_{1}\Subset\mathcal{D}_{2}\Subset\mathcal{D}$ that depend on
$r_{1},r_{2}$, such that if
$\mathcal{B}_{1}\cup\mathcal{B}_{2}\Subset\mathcal{D}_{1}$, and $u$ is a weak
solution of (1.1) in $\mathcal{D}$ with $0<k_{1}<1$, $k_{2}>1$, and
$\gamma\in(1/2,1)$, we have, for any $m\in\mathcal{N}$,

$|D^{m}u(x)|\leq\left\\{\begin{aligned}
&CC_{m}&&\mbox{in}~{}~{}\mathcal{D}_{1}\cap\mathcal{B}_{0},\\\
&\frac{CC_{m}}{k_{1}+1}&&\mbox{in}~{}~{}\mathcal{B}_{1},\\\
&\frac{CC_{m}}{k_{2}+1}&&\mbox{in}~{}~{}\mathcal{B}_{2},\end{aligned}\right.$

where $C>0$ is a constant depending only on $m$, $\mu$, $r_{1}$, and $r_{2}$.

In the final part of this paper, we partially answer a question raised by Kang
in [Kang], where in the conclusion section, he mentioned that the extensions
to general shape of inclusions and higher dimensions for the case
$(k_{1}-1)(k_{2}-1)<0$ are quite challenging. We prove the gradient estimates
for the extreme case when $k_{1}=0$, $k_{2}=\infty$, and $f=0$ for general
strictly convex inclusions in dimensions $n\geq 2$.

The setting of this problem is as follows. Let
$\mathcal{D}\subset\mathbb{R}^{n}$ be a bounded domain containing two smooth
relatively strictly convex open sets $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$
so that $\mathcal{B}_{1}\cup\mathcal{B}_{2}\Subset\mathcal{D}$,
$\operatorname{dist}(\mathcal{B}_{1}\cap\mathcal{B}_{2})=\varepsilon$, and
$\operatorname{dist}(\mathcal{B}_{1}\cup\mathcal{B}_{2},\mathcal{D})>0$. It is
known that when $k_{1}=0$, $k_{2}=\infty$, and $f=0$, (1.1) is reduced to

$\begin{cases}\Delta{u}=0&\mbox{in}~{}\mathcal{D}\setminus\overline{(\mathcal{B}_{1}\cup\mathcal{B}_{2})},\\\
\frac{\partial u}{\partial\nu}=0&\mbox{on}~{}\partial{\mathcal{B}}_{1},\\\
u=C\mbox{ (Constant)}&\mbox{on}~{}\partial{\mathcal{B}}_{2},\\\
\int_{\partial{\mathcal{B}}_{2}}\frac{\partial{u}}{\partial\nu}=0.\\\
\end{cases}$

We use the notation $x=(x^{\prime},x_{n})$, where
$x^{\prime}\in\mathbb{R}^{n-1}$. After choosing a coordinate system properly,
we can assume that near the origin, the part of $\partial\mathcal{B}_{1}$ and
$\partial\mathcal{B}_{2}$, denoted by $\Gamma_{+}$ and $\Gamma_{-}$, are
respectively the graphs of two smooth functions in terms of $x^{\prime}$. That
is,

$\displaystyle\Gamma_{+}=\left\\{x_{n}=\frac{\varepsilon}{2}+h_{1}(x^{\prime}),~{}|x^{\prime}|<1\right\\}~{}~{}\mbox{and}~{}~{}\Gamma_{-}=\left\\{x_{n}=-\frac{\varepsilon}{2}+h_{2}(x^{\prime}),~{}|x^{\prime}|<1\right\\},$

where $h_{1}$ and $h_{2}$ are sufficiently smooth functions satisfying

$h_{1}(x^{\prime})>h_{2}(x^{\prime})\quad\mbox{for}~{}~{}0<|x^{\prime}|<1,$
$h_{1}(0^{\prime})=h_{2}(0^{\prime})=0,\quad
D_{x^{\prime}}h_{1}(0^{\prime})=D_{x^{\prime}}h_{2}(0^{\prime})=0,\quad
D^{2}(h_{1}-h_{2})(0^{\prime})>0.$

For $0<r\leq 1$, we denote

$\displaystyle\Omega_{r}:=\left\\{(x^{\prime},x_{n})\in\mathcal{D}\setminus\overline{(\mathcal{B}_{1}\cup\mathcal{B}_{2})}~{}\big{|}~{}-\frac{\varepsilon}{2}+h_{2}(x^{\prime})<x_{n}<\frac{\varepsilon}{2}+h_{1}(x^{\prime}),~{}|x^{\prime}|<r\right\\}.$

We focus on the following problem near the origin:

$\left\\{\begin{aligned} -D_{i}(a^{ij}(x)D_{j}u(x))=0\quad&\mbox{in
}\Omega_{1},\\\ a^{ij}(x)D_{j}u(x)\nu_{i}=0\quad&\mbox{on }\Gamma_{+},\\\
u=C\mbox{ (Constant)}\quad&\mbox{on}~{}\Gamma_{-},\\\ \end{aligned}\right.$
(1.3)

where $(a^{ij}(x))$ satisfies, for some constants $\sigma\in(0,1)$ and any
$x\in\Omega_{1},\xi\in\mathbb{R}^{n}$,

$\sigma|\xi|^{2}\leq
a^{ij}(x)\xi_{i}\xi_{j},\quad|a^{ij}(x)|\leq\frac{1}{\sigma}.$

######  Theorem 1.4.

Assume the above and let $u\in H^{1}(\Omega_{1})$ be a weak solution of (1.3).
For $m\in\mathbb{N}$, some constants $\alpha\in(0,1)$, and $C_{m,\alpha}>0$,
if

$\|a\|_{C^{m-1,\alpha}(\Omega_{1})}+\|h_{1}\|_{C^{m,\alpha}(\\{|x^{\prime}|<1\\})}+\|h_{2}\|_{C^{m,\alpha}(\\{|x^{\prime}|<1\\})}\leq
C_{m,\alpha},$

then there exist constants $\mu\in(0,1)$ and $C$, depending only on $n$,
$\sigma$, $m$, $\alpha$, and $C_{m,\alpha}$ such that

$|D^{m}u(x)|\leq
C\mu^{\frac{1}{\sqrt{\varepsilon}+|x^{\prime}|}}\|u\|_{L^{2}(\Omega_{1})}\quad\mbox{for}~{}~{}x\in\Omega_{1/2}.$

The rest of this paper is organized as follows. First, we review the Green
function of the operator $L_{\varepsilon;1,1}$ constructed in [DL], and derive
some preliminary estimates in Section 2. In Section 3, we prove Theorem 1.1
with $m=1$ to illustrate the main idea without getting into too much
technicalities. Then we prove Theorem 1.1 with general $m\in\mathbb{N}$ in
Section 4, and Theorem 1.2 in Section 5. In Section 6, we prove Theorem 1.3 by
introducing a conformal map to reduce the problem to the case considered in
Theorem 1.2. Finally, Theorem 1.4 is proved in Section 7.

##  2\. Preliminary

In this section, we first review the Green function of the operator
$L_{\varepsilon;1,1}$ constructed in [DL], and then derive some preliminary
estimates.

Let $\Phi_{1}(x),\Phi_{2}(x)$ denote the inversion maps of a point
$x\in\mathbb{R}^{2}$ with respect to $\partial\mathcal{B}_{1}$ and
$\partial\mathcal{B}_{2}$, respectively, that is

$\Phi_{1}(x_{1},x_{2}):=\left(\frac{x_{1}-(1+\varepsilon/2)}{(x_{1}-1-\varepsilon/2)^{2}+x_{2}^{2}}+1+\varepsilon/2,\frac{x_{2}}{(x_{1}-1-\varepsilon/2)^{2}+x_{2}^{2}}\right)$

and

$\Phi_{2}(x_{1},x_{2}):=\left(\frac{x_{1}+1+\varepsilon/2}{(x_{1}+1+\varepsilon/2)^{2}+x_{2}^{2}}-1-\varepsilon/2,\frac{x_{2}}{(x_{1}+1+\varepsilon/2)^{2}+x_{2}^{2}}\right).$

The auxiliary function $\mathcal{G}(x,y)$ is given as follows:

  1. (1)

When $y\in\mathcal{B}_{0}$, $\mathcal{G}(x,y)$ equals

$\displaystyle\frac{2}{k_{1}+1}\sum_{k=0}^{\infty}(\alpha\beta)^{k}\Big{(}\log|(\Phi_{1}\Phi_{2})^{k}(x)-y|-\beta\log|(\Phi_{2}\Phi_{1})^{k}\Phi_{2}(x)-y|\Big{)}$
$\displaystyle\hskip
252.94499pt\mbox{for}~{}~{}x\in\overline{\mathcal{B}}_{1};$
$\displaystyle\log|x-y|+\sum_{k=0}^{\infty}\Big{[}(\alpha\beta)^{k}\Big{(}\log|(\Phi_{1}\Phi_{2})^{k}(x)-y|+\log|(\Phi_{2}\Phi_{1})^{k}(x)-y|\Big{)}$
$\displaystyle-(\alpha\beta)^{k-1}\Big{(}\beta\log|(\Phi_{2}\Phi_{1})^{k}\Phi_{2}(x)-y|+\alpha\log|(\Phi_{1}\Phi_{2})^{k}\Phi_{1}(x)-y|\Big{)}\Big{]}$
$\displaystyle\hskip 252.94499pt\mbox{for}~{}~{}x\in\mathcal{B}_{0};$
$\displaystyle\frac{2}{k_{2}+1}\sum_{l=0}^{\infty}(\alpha\beta)^{k}\Big{(}\log|(\Phi_{2}\Phi_{1})^{k}(x)-y|-\alpha\log|(\Phi_{1}\Phi_{2})^{k}\Phi_{1}(x)-y|\Big{)}$
$\displaystyle\hskip
252.94499pt\mbox{for}~{}~{}x\in\overline{\mathcal{B}}_{2};$

  2. (2)

When $y\in\mathcal{B}_{1}$, $\mathcal{G}(x,y)$ equals

$\displaystyle\frac{1}{k_{1}}(\log|x-y|+\alpha\log|\Phi_{1}(x)-y|)-\frac{4\beta}{(k_{1}+1)^{2}}\sum_{k=0}^{\infty}(\alpha\beta)^{k}\log|(\Phi_{2}\Phi_{1})^{k}\Phi_{2}(x)-y|$
$\displaystyle\hskip
216.81pt\mbox{for}~{}~{}x\in\overline{\mathcal{B}}_{1}\setminus\\{(1+\varepsilon/2,0)\\};$
$\displaystyle\frac{2}{k_{1}+1}\sum_{k=0}^{\infty}(\alpha\beta)^{k}\Big{(}\log|(\Phi_{2}\Phi_{1})^{k}(x)-y|-\beta\log|(\Phi_{2}\Phi_{1})^{k}\Phi_{2}(x)-y|\Big{)}$
$\displaystyle\hskip 216.81pt\mbox{for}~{}~{}x\in\mathcal{B}_{0};$
$\displaystyle\frac{4}{(k_{1}+1)(k_{2}+1)}\sum_{k=0}^{\infty}(\alpha\beta)^{k}\log|(\Phi_{2}\Phi_{1})^{k}(x)-y|$
$\displaystyle\hskip 216.81pt\mbox{for}~{}~{}x\in\overline{\mathcal{B}}_{2};$

  3. (3)

When $y\in\mathcal{B}_{2}$, $\mathcal{G}(x,y)$ equals

$\displaystyle\frac{4}{(k_{1}+1)(k_{2}+1)}\sum_{k=0}^{\infty}(\alpha\beta)^{k}\log|(\Phi_{1}\Phi_{2})^{k}(x)-y|$
$\displaystyle\hskip 216.81pt\mbox{for}~{}~{}x\in\overline{\mathcal{B}}_{1};$
$\displaystyle\frac{2}{k_{2}+1}\sum_{k=0}^{\infty}(\alpha\beta)^{k}\Big{(}\log|(\Phi_{1}\Phi_{2})^{k}(x)-y|-\alpha\log|(\Phi_{1}\Phi_{2})^{k}\Phi_{1}(x)-y|\Big{)}$
$\displaystyle\hskip 216.81pt\mbox{for}~{}~{}x\in\mathcal{B}_{0};$
$\displaystyle\frac{1}{k_{2}}(\log|x-y|+\beta\log|\Phi_{2}(x)-y|)-\frac{4\alpha}{(k_{2}+1)^{2}}\sum_{k=0}^{\infty}(\alpha\beta)^{k}\log|(\Phi_{1}\Phi_{2})^{k}\Phi_{1}(x)-y|$
$\displaystyle\hskip
216.81pt\mbox{for}~{}~{}x\in\overline{\mathcal{B}}_{2}\setminus\\{(-1-\varepsilon/2,0)\\};$

Define

$G(x,y)=\left\\{\begin{aligned}
&\mathcal{G}(x,y)&&\mbox{for}~{}~{}y\in\mathcal{B}_{0},\\\
&\mathcal{G}(x,y)+\frac{\alpha}{1-\alpha}\mathcal{G}(x,(1+\varepsilon/2,0))&&\mbox{for}~{}~{}y\in\mathcal{B}_{1},\\\
&\mathcal{G}(x,y)+\frac{\beta}{1-\beta}\mathcal{G}(x,(-1-\varepsilon/2,0))&&\mbox{for}~{}~{}y\in\mathcal{B}_{2}.\end{aligned}\right.$
(2.1)

By [DL]*Proposition 2.3, $G$ is a Green function of $L_{\varepsilon;1,1}$ in
the sense that

$a(x)\Delta_{x}G(x,y)=\delta(x-y)\quad\mbox{for}~{}~{}x\not\in\partial\mathcal{B}_{1}\cup\partial\mathcal{B}_{2},$

and $\mathcal{G}(\cdot,y)$, $aD_{\nu}\mathcal{G}(\cdot,y)$ are continuous
across $\partial\mathcal{B}_{1}$ and $\partial\mathcal{B}_{2}$. Let $\eta\in
C^{\infty}_{0}(B_{3/4})$ be a cutoff function such that $\eta=1$ in $B_{1/2}$.
Let $v=u\eta$, where $u$ is a solution of (1.1). Then $v$ satisfies

$D_{i}(a(x)D_{i}v)=D_{i}\tilde{f}_{i}+\tilde{f}_{3}\quad\mbox{in}~{}~{}\mathbb{R}^{2},$

where

$\tilde{f}_{i}=f_{i}\eta+auD_{i}\eta,\quad\tilde{f}_{3}=-f_{i}D_{i}\eta+aD_{i}uD_{i}\eta.$
(2.2)

We define

$\displaystyle\tilde{u}(x)=$
$\displaystyle-\int_{\mathcal{B}_{1}}D_{y_{i}}G(x,y)\tilde{f}_{i}(y)\,dy-\int_{\mathcal{B}_{2}}D_{y_{i}}G(x,y)\tilde{f}_{i}(y)\,dy$
$\displaystyle-\int_{\mathcal{B}_{0}}D_{y_{i}}G(x,y)\tilde{f}_{i}(y)\,dy+\int_{B_{1}}G(x,y)\tilde{f}_{3}(y)\,dy$
$\displaystyle:=$ $\displaystyle-w_{1}(x)-w_{2}(x)-w_{0}(x)+w_{3}(x).$ (2.3)

We know from [DL]*pp. 1447 that $u=\tilde{u}+C_{0}$ for some constant $C_{0}$.
We define for $j=0,1,2,$

$h_{j}(x)=\int_{\mathcal{B}_{j}}D_{y_{i}}\log|x-y|\tilde{f}_{i}(y)\,dy$ (2.4)

and

$g_{j}(x)=\int_{\mathcal{B}_{j}}\log|x-y|\tilde{f}_{3}(y)\,dy.$ (2.5)

Since supp$(D\eta)\subset B_{3/4}\setminus B_{1/2}$, by [DL]*Lemma 3.2 and
[DZ]*Lemma 2.1, we have for $m\in\mathbb{N}$,

$\|u\|_{C^{2m,\mu}(\mathcal{B}_{j}\cap\operatorname{supp}(D\eta))}\leq
CC_{m},\,\,\|\tilde{f}_{i}\|_{C^{2m-1,\mu}(\mathcal{B}_{j})}\leq
CC_{m}k_{j}\,\,\mbox{for}~{}~{}i=1,2,3,~{}~{}j=0,1,2,$ (2.6)

and

$\displaystyle\|h_{j}\|_{C^{2m,\mu}(B_{3}\cap\mathcal{B}_{i})}+\|g_{j}\|_{C^{2m,\mu}(B_{3}\cap\mathcal{B}_{i})}\leq
CC_{m}k_{j}\quad\mbox{for}~{}~{}i,j=0,1,2.$ (2.7)

From now on, we identify a point $x=(x_{1},x_{2})\in\mathbb{R}^{2}$ with a
complex number $z=x_{1}+ix_{2}\in\mathbb{C}$. We will derive some derivative
estimates of the maps $(\Phi_{2}\Phi_{1})^{k}(z)$ with respect to $k$ and
$z\in\mathcal{B}_{2}$. Estimates of $(\Phi_{1}\Phi_{2})^{k}(z)$ with respect
to $k$ and $z$ for $z\in\mathcal{B}_{1}$ will follow similarly.

We denote $a=1+\varepsilon/2$ for convenience, after a change of variable

$2az-(2a^{2}-1)\rightarrow z,$

we have

$(\Phi_{2}\Phi_{1})(z)=-1/z-2(2a^{2}-1),$

and the two fixed points of $\Phi_{2}\Phi_{1}$ are given by

$\displaystyle\lambda_{1}:$
$\displaystyle=-(2a^{2}-1)+2a\sqrt{a^{2}-1}\sim-1+2\sqrt{\varepsilon},$
$\displaystyle\lambda_{2}:$
$\displaystyle=-(2a^{2}-1)-2a\sqrt{a^{2}-1}\sim-1-2\sqrt{\varepsilon}.$

We denote $\psi=\Phi_{2}\Phi_{1}$ for simplicity. For $z\in\mathcal{B}_{2}$,
we denote $r:=|z-\lambda_{2}|,$ and

$I_{k}:=(z-\lambda_{2}^{-1})-|\lambda_{2}|^{-k}(z-\lambda_{2})=(z-\lambda_{2}^{-1})(1-|\lambda_{2}|^{-k})+(\lambda_{2}-\lambda_{2}^{-1})|\lambda_{2}|^{-k}.$

It is easy to see that $|z-\lambda_{2}^{-1}|\sim r+\sqrt{\varepsilon}$, Re
$(z-\lambda_{2}^{-1})<0$, and
$\lambda_{2}-\lambda_{2}^{-1}\sim-\sqrt{\varepsilon}$. Hence

$|I_{k}|\sim(r+\sqrt{\varepsilon})(1-\lambda_{2}^{-k})+\sqrt{\varepsilon}.$

Following the iteration argument from Section 3 of [DL], we have, for any
$\alpha\in\mathbb{N}$,

$\displaystyle\psi^{k}(z)$
$\displaystyle=\lambda_{2}+(\lambda_{2}^{2}-1)\lambda_{2}^{-2k-1}(z-\lambda_{2})I_{2k}^{-1},$
(2.8) $\displaystyle D^{\alpha}(\psi^{k}(z))$
$\displaystyle=\frac{(\lambda_{2}-\lambda_{2}^{-1})^{2}}{\lambda_{2}^{2k}}(-1)^{\alpha-1}\alpha!(1-\lambda_{2}^{-2k})^{\alpha-1}I_{2k}^{-(\alpha+1)},$
(2.9)

and in particular, since
$|\lambda_{2}-\lambda_{2}^{-1}|\lesssim\sqrt{\varepsilon}$ and
$|I_{2}|\gtrsim\sqrt{\varepsilon}$,

$|D^{\alpha}(\psi(z))|\leq C,$ (2.10)

where $C$ is a positive constant depending only on $\alpha$. By (2.8), for any
$\beta\geq 1$,

$\displaystyle D_{k}^{\beta}(\psi^{k})(z)$
$\displaystyle=D_{k}^{\beta}\Big{[}(\lambda_{2}^{2}-1)\lambda_{2}^{-2k-1}(z-\lambda_{2})I_{2k}^{-1}\Big{]}$
$\displaystyle=\sum_{\beta_{1}+\beta_{2}=\beta}C_{\beta_{1},\beta_{2}}(\lambda_{2}^{2}-1)(z-\lambda_{2})[D_{k}^{\beta_{1}}\lambda_{2}^{-2k-1}][D_{k}^{\beta_{2}}I_{2k}^{-1}].$

Since

$D_{k}\lambda_{2}^{-2k}=-2\log|\lambda_{2}|\lambda_{2}^{-2k}\sim-\sqrt{\varepsilon}\lambda_{2}^{-2k}$

and

$D_{k}^{\beta_{2}}I_{2k}^{-1}=\sum_{j=0}^{\beta_{2}}C_{j}(\log|\lambda_{2}|)^{\beta_{2}}\lambda_{2}^{-2jk}(z-\lambda_{2})^{j}I_{2k}^{-j-1},$

where $C_{j}$ is some constant independent of $k$ and $\varepsilon$, we obtain

$|D_{k}^{\beta}(\psi^{k})(z)|\lesssim\frac{\varepsilon^{(1+\beta)/2}}{\lambda_{2}^{2k}}\sum_{j=0}^{\beta}\frac{r^{j+1}}{|I_{2k}|^{j+1}}.$
(2.11)

For any $\beta\geq 0$ and $\alpha\in\mathbb{N}$, by (2.9), we have

$\displaystyle
D^{\beta}_{k}D^{\alpha}(\psi^{k}(z))=\sum_{\beta_{1}+\beta_{2}+\beta_{3}=\beta}$
$\displaystyle
C_{\beta_{1},\beta_{2},\beta_{3}}(\lambda_{2}-\lambda_{2}^{-1})^{2}(-1)^{\alpha-1}\alpha![D_{k}^{\beta_{1}}\lambda_{2}^{-2k}]\times$
$\displaystyle\times[D_{k}^{\beta_{2}}(1-\lambda_{2}^{-2k})^{\alpha-1}][D_{k}^{\beta_{3}}I_{2k}^{-(\alpha+1)}].$

Observe that

$\displaystyle\left|D_{k}^{\beta_{2}}(1-\lambda_{2}^{-2k})^{\alpha-1}\right|$
$\displaystyle=\left|\sum_{j=0}^{\min(\beta_{2},\alpha-1)}C_{j}(1-\lambda_{2}^{-2k})^{\alpha-j-1}\lambda_{2}^{-2jk}(\log|\lambda_{2}|)^{\beta_{2}}\right|$
$\displaystyle\lesssim\varepsilon^{\beta_{2}/2}\sum_{j=0}^{\min(\beta_{2},\alpha-1)}|1-\lambda_{2}^{-2k}|^{\alpha-j-1},$

and

$\displaystyle|D_{k}^{\beta_{3}}I_{2k}^{-(\alpha+1)}|$
$\displaystyle=\left|\sum_{j=0}^{\beta_{3}}C_{j}(\log|\lambda_{2}|)^{\beta_{3}}\lambda_{2}^{-2jk}(z-\lambda_{2})^{j}I_{2k}^{-(\alpha+j+1)}\right|$
$\displaystyle\lesssim\varepsilon^{\beta_{3}/2}\sum_{j=0}^{\beta_{3}}\frac{r^{j}}{|I_{2k}|^{\alpha+j+1}},$

where $C_{j}$ is some constant and can be different from line to line. Thus we
have

$\displaystyle|D^{\beta}_{k}D^{\alpha}(\psi^{k}(z))|\lesssim\frac{\varepsilon^{1+\beta/2}}{\lambda_{2}^{2k}}\sum_{\begin{subarray}{c}\beta_{2}+\beta_{3}\leq\beta\\\
\beta_{2}\leq\alpha-1\end{subarray}}|1-\lambda_{2}^{-2k}|^{\alpha-\beta_{2}-1}\frac{r^{\beta_{3}}}{|I_{2k}|^{\alpha+\beta_{3}+1}}.$
(2.12)

Note that $\beta_{2},\beta_{3}$ in (2.12) might be different from the ones
above.

##  3\. First order derivative

In this section, we prove Theorem 1.1 when $m=1$. Without loss of generality,
we may assume that $C_{1}=1$.

Case 1: $x\in\mathcal{B}_{0}\cap B_{1/2}$. In this case,
$\psi^{k}(x)\in\mathcal{B}_{2}$ for $k\in\mathbb{N}$. By (2.1) and (2.3), we
have

$\displaystyle Dw_{1}(x)=$
$\displaystyle\frac{2}{k_{1}+1}\sum_{k=0}^{\infty}(-1)^{k}\gamma^{k}\Big{(}D[h_{1}(\psi^{k}(x)]-\beta
D[h_{1}(\psi^{k}\Phi_{2}(x))]\Big{)}.$

We denote

$\Theta_{k}:=\gamma^{k}D[h_{1}(\psi^{k}(x)]=\gamma^{k}Dh_{1}[\psi^{k}(x)]D(\psi^{k-1})[\psi(x)]D\psi(x).$

Then

$\sum_{k=0}^{\infty}(-1)^{k}\Theta_{k}=\frac{1}{2}\sum_{k=1}^{\infty}(-1)^{k}(\Theta_{k}-\Theta_{k+1})+\Theta_{0}-\frac{1}{2}\Theta_{1}.$

By (2.6) and (2.7), it is clear that $|\Theta_{0}-\frac{1}{2}\Theta_{1}|\leq
Ck_{1}$, where $C$ is a positive constant independent of $\varepsilon$ and
$\gamma$. Therefore, the goal is to estimate $|\Theta_{k}-\Theta_{k+1}|$. By
the mean value theorem,

$\Theta_{k}-\Theta_{k+1}=D_{k}\Theta_{k}\Big{|}_{k=\bar{k}}$

for some $\bar{k}\in(k,k+1)$. We claim that

$\sum_{k=1}^{\infty}|\Theta_{k}-\Theta_{k+1}|\lesssim\sum_{k=1}^{\infty}|D_{k}\Theta_{k}|\lesssim
k_{1}.$ (3.1)

By the chain rule and the product rule, we have

$\displaystyle D_{k}\Theta_{k}=$
$\displaystyle(\log\gamma)\gamma^{k}Dh_{1}[\psi^{k}(x)]D(\psi^{k-1})[\psi(x)]D\psi(x)$
$\displaystyle+\gamma^{k}D^{2}h_{1}[\psi^{k}(x)]D_{k}(\psi^{k-1})[\psi(x)]D(\psi^{k-1})[\psi(x)]D\psi(x)$
$\displaystyle+\gamma^{k}Dh_{1}[\psi^{k}(x)]D_{k}D(\psi^{k-1})(x)D\psi(x)$
$\displaystyle=$ $\displaystyle:J_{k}^{1}+J_{k}^{2}+J_{k}^{3}.$

Recall that $|\lambda_{2}|>1$ and $|\log\gamma|\sim 1-\gamma$, by (2.7),
(2.10), (2.11), and (2.12), we have

$\displaystyle|J_{k}^{1}|\lesssim$ $\displaystyle
k_{1}|\log\gamma|\gamma^{k}\frac{\varepsilon}{\lambda_{2}^{2k}|I_{2k}|^{2}},$
$\displaystyle|J_{k}^{2}|\lesssim$ $\displaystyle
k_{1}\gamma^{k}\frac{\varepsilon}{\lambda_{2}^{2k}}\left(\frac{r}{|I_{2k}|}+\frac{r^{2}}{|I_{2k}|^{2}}\right)\frac{\varepsilon}{\lambda_{2}^{2k}}\frac{1}{|I_{2k}|^{2}}\lesssim
k_{1}\frac{\varepsilon^{2}}{\lambda_{2}^{2k}}\left(\frac{r}{|I_{2k}|^{3}}+\frac{r^{2}}{|I_{2k}|^{4}}\right),$
$\displaystyle|J_{k}^{3}|\lesssim$ $\displaystyle
k_{1}\gamma^{k}\frac{\varepsilon^{3/2}}{\lambda_{2}^{2k}}\left(\frac{1}{|I_{2k}|^{2}}+\frac{r}{|I_{2k}|^{3}}\right).$

When $k<\frac{1}{r+\sqrt{\varepsilon}}$, we have

$|I_{2k}|\gtrsim\sqrt{\varepsilon}.$

Therefore,

$\displaystyle\Big{|}\sum_{k<\frac{1}{r+\sqrt{\varepsilon}}}J_{k}^{1}\Big{|}$
$\displaystyle\lesssim
k_{1}(1-\gamma)\sum_{k<\frac{1}{r+\sqrt{\varepsilon}}}\gamma^{k}\frac{\varepsilon}{\lambda_{2}^{2k}|I_{2k}|^{2}}$
$\displaystyle\lesssim
k_{1}(1-\gamma)\sum_{k<\frac{1}{r+\sqrt{\varepsilon}}}\gamma^{k}\lesssim
k_{1},$
$\displaystyle\Big{|}\sum_{k<\frac{1}{r+\sqrt{\varepsilon}}}J_{k}^{2}\Big{|}$
$\displaystyle\lesssim
k_{1}\sum_{k<\frac{1}{r+\sqrt{\varepsilon}}}\frac{\varepsilon^{2}}{\lambda_{2}^{2k}}\left(\frac{r}{|I_{2k}|^{3}}+\frac{r^{2}}{|I_{2k}|^{4}}\right)$
$\displaystyle\lesssim
k_{1}\sum_{k<\frac{1}{r+\sqrt{\varepsilon}}}\varepsilon^{2}\left(\frac{r}{\varepsilon^{3/2}}+\frac{r^{2}}{\varepsilon^{2}}\right)$
$\displaystyle\lesssim
k_{1}\frac{\varepsilon^{2}}{r+\sqrt{\varepsilon}}\left(\frac{r}{\varepsilon^{3/2}}+\frac{r^{2}}{\varepsilon^{2}}\right)\lesssim
k_{1}(\sqrt{\varepsilon}+r),$

and

$\displaystyle\Big{|}\sum_{k<\frac{1}{r+\sqrt{\varepsilon}}}J_{k}^{3}\Big{|}$
$\displaystyle\lesssim
k_{1}\sum_{k<\frac{1}{r+\sqrt{\varepsilon}}}\gamma^{k}\frac{\varepsilon^{3/2}}{\lambda_{2}^{2k}}\left(\frac{1}{|I_{2k}|^{2}}+\frac{r}{|I_{2k}|^{3}}\right)$
$\displaystyle\lesssim
k_{1}\sum_{k<\frac{1}{r+\sqrt{\varepsilon}}}\varepsilon^{3/2}\left(\frac{1}{\varepsilon}+\frac{r}{\varepsilon^{3/2}}\right)$
$\displaystyle\lesssim
k_{1}\frac{\varepsilon^{3/2}}{r+\sqrt{\varepsilon}}\left(\frac{1}{\varepsilon}+\frac{r}{\varepsilon^{3/2}}\right)\lesssim
k_{1}.$

When $\frac{1}{r+\sqrt{\varepsilon}}\leq k<\frac{1}{\sqrt{\varepsilon}}$, we
have

$|I_{2k}|\gtrsim(r+\sqrt{\varepsilon})k\sqrt{\varepsilon}.$

Therefore,

$\displaystyle\Big{|}\sum_{\frac{1}{r+\sqrt{\varepsilon}}\leq
k<\frac{1}{\sqrt{\varepsilon}}}J_{k}^{1}\Big{|}$ $\displaystyle\lesssim
k_{1}\sum_{\frac{1}{r+\sqrt{\varepsilon}}\leq
k<\frac{1}{\sqrt{\varepsilon}}}|\log\gamma|\gamma^{k}\frac{\varepsilon}{\lambda_{2}^{2k}|I_{2k}|^{2}}$
$\displaystyle\lesssim k_{1}\sum_{\frac{1}{r+\sqrt{\varepsilon}}\leq
k<\frac{1}{\sqrt{\varepsilon}}}|\log\gamma|\gamma^{k}\frac{\varepsilon}{\lambda_{2}^{2k}(r+\sqrt{\varepsilon})^{2}k^{2}\varepsilon}$
$\displaystyle\lesssim
k_{1}\gamma^{\frac{1}{r+\sqrt{\varepsilon}}}|\log\gamma|\frac{1}{(r+\sqrt{\varepsilon})^{2}}\sum_{\frac{1}{r+\sqrt{\varepsilon}}\leq
k<\frac{1}{\sqrt{\varepsilon}}}\frac{1}{k^{2}}$ $\displaystyle\lesssim
k_{1}e^{\frac{\log\gamma}{r+\sqrt{\varepsilon}}}\frac{|\log\gamma|}{r+\sqrt{\varepsilon}}\lesssim
k_{1},$

where we used the fact that $|e^{-x}x|\leq C$ for any $x>0$,

$\displaystyle\Big{|}\sum_{\frac{1}{r+\sqrt{\varepsilon}}\leq
k<\frac{1}{\sqrt{\varepsilon}}}J_{k}^{2}\Big{|}$ $\displaystyle\lesssim
k_{1}\sum_{\frac{1}{r+\sqrt{\varepsilon}}\leq
k<\frac{1}{\sqrt{\varepsilon}}}\gamma^{k}\frac{\varepsilon^{2}}{\lambda_{2}^{2k}}\left(\frac{r}{|I_{2k}|^{3}}+\frac{r^{2}}{|I_{2k}|^{4}}\right)$
$\displaystyle\lesssim k_{1}\sum_{\frac{1}{r+\sqrt{\varepsilon}}\leq
k<\frac{1}{\sqrt{\varepsilon}}}\varepsilon^{2}\left(\frac{r}{(r+\sqrt{\varepsilon})^{3}k^{3}\varepsilon^{3/2}}+\frac{r^{2}}{(r+\sqrt{\varepsilon})^{4}k^{4}\varepsilon^{2}}\right)$
$\displaystyle\lesssim k_{1}\sum_{\frac{1}{r+\sqrt{\varepsilon}}\leq
k<\frac{1}{\sqrt{\varepsilon}}}\left(\frac{1}{(r+\sqrt{\varepsilon})k^{3}}+\frac{1}{(r+\sqrt{\varepsilon})^{2}k^{4}}\right)$
$\displaystyle\lesssim k_{1}(r+\sqrt{\varepsilon}),$

and

$\displaystyle\Big{|}\sum_{\frac{1}{r+\sqrt{\varepsilon}}\leq
k<\frac{1}{\sqrt{\varepsilon}}}J_{k}^{3}\Big{|}$ $\displaystyle\lesssim
k_{1}\sum_{\frac{1}{r+\sqrt{\varepsilon}}\leq
k<\frac{1}{\sqrt{\varepsilon}}}\gamma^{k}\frac{\varepsilon^{3/2}}{\lambda_{2}^{2k}}\left(\frac{1}{|I_{2k}|^{2}}+\frac{r}{|I_{2k}|^{3}}\right)$
$\displaystyle\lesssim k_{1}\sum_{\frac{1}{r+\sqrt{\varepsilon}}\leq
k<\frac{1}{\sqrt{\varepsilon}}}\varepsilon^{3/2}\left(\frac{1}{(r+\sqrt{\varepsilon})^{2}k^{2}\varepsilon}+\frac{r}{(r+\sqrt{\varepsilon})^{3}k^{3}\varepsilon^{3/2}}\right)$
$\displaystyle\lesssim k_{1}\sum_{\frac{1}{r+\sqrt{\varepsilon}}\leq
k<\frac{1}{\sqrt{\varepsilon}}}\left(\frac{1}{(r+\sqrt{\varepsilon})k^{2}}+\frac{1}{(r+\sqrt{\varepsilon})^{2}k^{3}}\right)\lesssim
k_{1}.$

Finally, when $k\geq\frac{1}{\sqrt{\varepsilon}}$, we have

$|I_{2k}|\gtrsim r+\sqrt{\varepsilon}.$

Therefore,

$\displaystyle\Big{|}\sum_{k\geq\frac{1}{\sqrt{\varepsilon}}}J_{k}^{1}\Big{|}$
$\displaystyle\lesssim
k_{1}\sum_{k\geq\frac{1}{\sqrt{\varepsilon}}}|\log\gamma|\gamma^{k}\frac{\varepsilon}{\lambda_{2}^{2k}|I_{2k}|^{2}}$
$\displaystyle\lesssim
k_{1}(1-\gamma)\sum_{k\geq\frac{1}{\sqrt{\varepsilon}}}\gamma^{k}\lesssim
k_{1}\gamma^{\frac{1}{\sqrt{\varepsilon}}},$
$\displaystyle\Big{|}\sum_{k\geq\frac{1}{\sqrt{\varepsilon}}}J_{k}^{2}\Big{|}$
$\displaystyle\lesssim
k_{1}\sum_{k\geq\frac{1}{\sqrt{\varepsilon}}}\gamma^{k}\frac{\varepsilon^{2}}{\lambda_{2}^{2k}}\left(\frac{r}{|I_{2k}|^{3}}+\frac{r^{2}}{|I_{2k}|^{4}}\right)$
$\displaystyle\lesssim
k_{1}\sum_{k\geq\frac{1}{\sqrt{\varepsilon}}}\frac{\gamma^{k}}{\lambda_{2}^{2k}}\varepsilon^{2}\left(\frac{r}{(r+\sqrt{\varepsilon})^{3}}+\frac{r^{2}}{(r+\sqrt{\varepsilon})^{4}}\right)$
$\displaystyle\lesssim
k_{1}\frac{\varepsilon(\gamma/\lambda_{2}^{2})^{\frac{1}{\sqrt{\varepsilon}}}}{1-\gamma/\lambda_{2}^{2}}\lesssim
k_{1}\sqrt{\varepsilon}(\gamma/\lambda_{2}^{2})^{\frac{1}{\sqrt{\varepsilon}}},$

where we used $1-\gamma/\lambda_{2}^{2}\gtrsim\sqrt{\varepsilon}$, and

$\displaystyle\Big{|}\sum_{k\geq\frac{1}{\sqrt{\varepsilon}}}J_{k}^{3}\Big{|}$
$\displaystyle\lesssim
k_{1}\sum_{k\geq\frac{1}{\sqrt{\varepsilon}}}\gamma^{k}\frac{\varepsilon^{3/2}}{\lambda_{2}^{2k}}\left(\frac{1}{|I_{2k}|^{2}}+\frac{r}{|I_{2k}|^{3}}\right)$
$\displaystyle\lesssim
k_{1}\sum_{k\geq\frac{1}{\sqrt{\varepsilon}}}\frac{\gamma^{k}}{\lambda_{2}^{2k}}\varepsilon^{3/2}\left(\frac{1}{(r+\sqrt{\varepsilon})^{2}}+\frac{r}{(r+\sqrt{\varepsilon})^{3}}\right)$
$\displaystyle\lesssim
k_{1}\frac{\sqrt{\varepsilon}(\gamma/\lambda_{2}^{2})^{\frac{1}{\sqrt{\varepsilon}}}}{1-\gamma/\lambda_{2}^{2}}\lesssim
k_{1}(\gamma/\lambda_{2}^{2})^{\frac{1}{\sqrt{\varepsilon}}}.$

Therefore, (3.1) follows from the estimates above, and hence

$\left|\sum_{k=0}^{\infty}(-1)^{k}\Theta_{k}\right|\leq Ck_{1},$

where $C$ is a positive constant independent of $\varepsilon$ and $\gamma$.

By the same argument, we can estimate

$\left|\sum_{k=0}^{\infty}(-1)^{k}\gamma^{k}D[h_{1}(\psi^{k}\Phi_{2}(x))]\right|\leq
Ck_{1}.$

Therefore,

$|Dw_{1}(x)|\leq\frac{Ck_{1}}{k_{1}+1}\quad\mbox{for}~{}~{}x\in\mathcal{B}_{0}\cap
B_{1/2}.$

By (2.1) and (2.3) again, we have, for $x\in\mathcal{B}_{0}\cap B_{1/2}$,

$\displaystyle Dw_{2}(x)=$
$\displaystyle\frac{2}{k_{2}+1}\sum_{k=0}^{\infty}(-1)^{k}\gamma^{k}\Big{(}D[h_{2}((\Phi_{1}\Phi_{2})^{k}(x)]-\alpha
D[h_{2}((\Phi_{1}\Phi_{2})^{k}\Phi_{1}(x))]\Big{)},$ $\displaystyle
Dw_{0}(x)=$ $\displaystyle
Dh_{0}(x)+\sum_{k=0}^{\infty}\Big{[}(-1)^{k}\gamma^{k}\Big{(}D[h_{0}((\Phi_{1}\Phi_{2})^{k}(x)]+D[h_{0}((\Phi_{2}\Phi_{1})^{k}(x))]\Big{)}$
$\displaystyle-(-1)^{k-1}\gamma^{k-1}\Big{(}\beta
D[h_{0}((\Phi_{2}\Phi_{1})^{k-1}\Phi_{2}(x)]+\alpha
D[h_{0}((\Phi_{1}\Phi_{2})^{k-1}\Phi_{1}(x))]\Big{)}\Big{]}.$

Therefore, in the same way we can estimate

$|Dw_{2}(x)|\leq\frac{Ck_{2}}{k_{2}+1}\quad\mbox{and}\quad|Dw_{0}(x)|\leq
C\quad\mbox{for}~{}~{}x\in\mathcal{B}_{0}\cap B_{1/2},$

where $C$ is a positive constant independent of $\varepsilon$, $k_{1}$, and
$k_{2}$. To estimate $w_{3}$, since supp$(\tilde{f}_{3})\subset B_{1}$, we can
write

$\displaystyle w_{3}(x)$
$\displaystyle=\int_{\mathcal{B}_{1}}G(x,y)\tilde{f}_{3}(y)\,dy+\int_{\mathcal{B}_{2}}G(x,y)\tilde{f}_{3}(y)\,dy+\int_{\mathcal{B}_{0}}G(x,y)\tilde{f}_{3}(y)\,dy$
$\displaystyle=\frac{2}{k_{1}+1}\sum_{k=0}^{\infty}(-1)^{k}\gamma^{k}\Big{(}g_{1}((\Phi_{2}\Phi_{1})^{k}(x))-\beta
g_{1}((\Phi_{2}\Phi_{1})^{k}\Phi_{2}(x))\Big{)}$
$\displaystyle\quad+\alpha\sum_{k=0}^{\infty}(-1)^{k}\gamma^{k}\Big{(}\log|(\Phi_{2}\Phi_{1})^{k}(x)-(a,0)|$
$\displaystyle\quad-\beta\log|(\Phi_{2}\Phi_{1})^{k}\Phi_{2}(x)-(a,0)|\Big{)}\int_{\mathcal{B}_{1}}\tilde{f}_{3}(y)\,dy$
$\displaystyle\quad+\frac{2}{k_{2}+1}\sum_{k=0}^{\infty}(-1)^{k}\gamma^{k}\Big{(}g_{2}((\Phi_{1}\Phi_{2})^{k}(x)-\alpha
g_{2}((\Phi_{1}\Phi_{2})^{k}\Phi_{1}(x))\Big{)}$
$\displaystyle\quad+\beta\sum_{k=0}^{\infty}(-1)^{k}\gamma^{k}\Big{(}\log|(\Phi_{1}\Phi_{2})^{k}(x)+(a,0)|$
$\displaystyle\quad-\alpha\log|(\Phi_{1}\Phi_{2})^{k}\Phi_{1}(x)+(a,0)|\Big{)}\int_{\mathcal{B}_{2}}\tilde{f}_{3}(y)\,dy$
$\displaystyle\quad+g_{0}(x)+\sum_{k=0}^{\infty}\Big{[}(-1)^{k}\gamma^{k}\Big{(}g_{0}((\Phi_{1}\Phi_{2})^{k}(x)+g_{0}((\Phi_{2}\Phi_{1})^{k}(x))\Big{)}$
$\displaystyle\quad-(-1)^{k-1}\gamma^{k-1}\Big{(}\beta
g_{0}((\Phi_{2}\Phi_{1})^{k-1}\Phi_{2}(x))+\alpha
g_{0}((\Phi_{1}\Phi_{2})^{k-1}\Phi_{1}(x))\Big{)}\Big{]}.$

Note that for any $x\in\mathcal{B}_{0}\cap B_{1/2}$,

$\displaystyle|(\Phi_{2}\Phi_{1})^{k}(x)-(a,0)|\geq 1,$
$\displaystyle|(\Phi_{2}\Phi_{1})^{k}\Phi_{2}(x)-(a,0)|\geq 1,$
$\displaystyle|(\Phi_{1}\Phi_{2})^{k}(x)+(a,0)|\geq 1,$
$\displaystyle|(\Phi_{1}\Phi_{2})^{k}\Phi_{1}(x)+(a,0)|\geq 1.$

By (2.2) and (1.1),

$\int_{B_{1}}\tilde{f}_{3}=\int_{B_{1}}-f_{i}D_{i}\eta+a(x)D_{i}uD_{i}\eta=0.$

Therefore, by (2.6),

$\left|\int_{\mathcal{B}_{2}}\tilde{f}_{3}\right|=\left|\int_{\mathcal{B}_{1}\cup\mathcal{B}_{0}}-f_{i}D_{i}\eta+a(x)D_{i}uD_{i}\eta\right|\leq
CC_{m},$

where $C$ is some positive constant independent of $\varepsilon$, $k_{1}$, and
$k_{2}$. Then we can estimate $Dw_{3}$ similarly as above to obtain

$|Dw_{3}(x)|\leq C\quad\mbox{for}~{}~{}x\in\mathcal{B}_{0}\cap B_{1/2}.$

Therefore,

$|Du(x)|\leq C\quad\mbox{for}~{}~{}x\in\mathcal{B}_{0}\cap B_{1/2},$

where $C$ is a positive constant independent of $\varepsilon$, $k_{1}$, and
$k_{2}$.

Case 2: $x\in\mathcal{B}_{1}\cap B_{1/2}$. By (2.1) and (2.3), we have

$\displaystyle w_{1}(x)=$
$\displaystyle\frac{1}{k_{1}}h_{1}(x)+\frac{\alpha}{k_{1}}h_{1}(\Phi_{1}(x))-\frac{4\beta}{(k_{1}+1)^{2}}\sum_{k=0}^{\infty}(-1)^{k}\gamma^{k}h_{1}((\Phi_{2}\Phi_{1})^{k}\Phi_{2}(x)),$
$\displaystyle w_{2}(x)=$
$\displaystyle\frac{4}{(k_{1}+1)(k_{2}+1)}\sum_{k=0}^{\infty}(-1)^{k}\gamma^{k}h_{2}((\Phi_{1}\Phi_{2})^{k}(x)),$
$\displaystyle w_{0}(x)=$
$\displaystyle\frac{2}{k_{1}+1}\sum_{k=0}^{\infty}(-1)^{k}\gamma^{k}\Big{(}h_{0}(\Phi_{1}\Phi_{2})^{k}(x)-\beta
h_{0}((\Phi_{2}\Phi_{1})^{k}\Phi_{2}(x))\Big{)},$

and

$\displaystyle w_{3}(x)$
$\displaystyle=\int_{\mathcal{B}_{1}}G(x,y)\tilde{f}_{3}(y)\,dy+\int_{\mathcal{B}_{2}}G(x,y)\tilde{f}_{3}(y)\,dy+\int_{\mathcal{B}_{0}}G(x,y)\tilde{f}_{3}(y)\,dy$
$\displaystyle=\frac{1}{k_{1}}g_{1}(x)+\frac{\alpha}{k_{1}}g_{1}(\Phi_{1}(x))-\frac{4\beta}{(k_{1}+1)^{2}}\sum_{k=0}^{\infty}(-1)^{k}\gamma^{k}g_{1}((\Phi_{2}\Phi_{1})^{k}\Phi_{2}(x))$
$\displaystyle\quad+\frac{\alpha}{k_{1}}\log|x-(a,0)|\int_{\mathcal{B}_{1}}\tilde{f}_{3}(y)\,dy$
$\displaystyle\quad-\frac{2\alpha\beta}{k_{1}+1}\sum_{k=0}^{\infty}(-1)^{k}\gamma^{k}\log|(\Phi_{2}\Phi_{1})^{k}\Phi_{2}(x)-(a,0)|\int_{\mathcal{B}_{1}}\tilde{f}_{3}(y)\,dy$
$\displaystyle\quad+\frac{4}{(k_{1}+1)(k_{2}+1)}\sum_{k=0}^{\infty}(-1)^{k}\gamma^{k}g_{2}((\Phi_{1}\Phi_{2})^{k}(x))$
$\displaystyle\quad+\frac{2\beta}{k_{1}+1}\sum_{k=0}^{\infty}(-1)^{k}\gamma^{k}\log|(\Phi_{1}\Phi_{2})^{k}(x)+(a,0)|\int_{\mathcal{B}_{2}}\tilde{f}_{3}(y)\,dy$
$\displaystyle\quad+\frac{2}{k_{1}+1}\sum_{k=0}^{\infty}(-1)^{k}\gamma^{k}\Big{(}g_{0}(\Phi_{1}\Phi_{2})^{k}(x)-\beta
g_{0}((\Phi_{2}\Phi_{1})^{k}\Phi_{2}(x))\Big{)},$

where we used $\log|x-(a,0)|=-\log|\Phi_{1}(x)-(a,0)|$ for
$x\in\mathcal{B}_{1}\cap B_{1/2}$. Note that

$\displaystyle|x-(a,0)|\geq
1/2,\quad|(\Phi_{2}\Phi_{1})^{k}\Phi_{2}(x)-(a,0)|\geq
1,\quad|(\Phi_{1}\Phi_{2})^{k}(x)+(a,0)|\geq 1.$

We can estimate $Dw_{i},i=0,1,2,3,$ as above to obtain

$\displaystyle|Dw_{1}(x)|$ $\displaystyle\leq
C,\quad|Dw_{2}(x)|\leq\frac{Ck_{2}}{(k_{1}+1)(k_{2}+1)},$
$\displaystyle|Dw_{0}(x)|$
$\displaystyle\leq\frac{C}{k_{1}+1},\quad|Dw_{3}(x)|\leq C.$

Therefore,

$|Du(x)|\leq C\quad\mbox{for}~{}~{}x\in\mathcal{B}_{1}\cap B_{1/2},$

where $C$ is a positive constant independent of $\varepsilon$, $k_{1}$, and
$k_{2}$.

Case 3: $x\in\mathcal{B}_{2}\cap B_{1/2}$. By (2.1) and (2.3) again, we have

$\displaystyle w_{1}(x)=$
$\displaystyle\frac{4}{(k_{1}+1)(k_{2}+1)}\sum_{k=0}^{\infty}(-1)^{k}\gamma^{k}h_{1}((\Phi_{2}\Phi_{1})^{k}(x)),$
$\displaystyle w_{2}(x)=$
$\displaystyle\frac{1}{k_{2}}h_{2}(x)+\frac{\beta}{k_{2}}h_{2}(\Phi_{2}(x))-\frac{4\alpha}{(k_{2}+1)^{2}}\sum_{k=0}^{\infty}(-1)^{k}\gamma^{k}h_{2}((\Phi_{1}\Phi_{2})^{k}\Phi_{1}(x)),$
$\displaystyle w_{0}(x)=$
$\displaystyle\frac{2}{k_{2}+1}\sum_{k=0}^{\infty}(-1)^{k}\gamma^{k}\Big{(}h_{0}(\Phi_{2}\Phi_{1})^{k}(x)-\alpha
h_{0}((\Phi_{1}\Phi_{2})^{k}\Phi_{1}(x))\Big{)},$

and

$\displaystyle w_{3}(x)$
$\displaystyle=\int_{\mathcal{B}_{1}}G(x,y)\tilde{f}_{3}(y)\,dy+\int_{\mathcal{B}_{2}}G(x,y)\tilde{f}_{3}(y)\,dy+\int_{\mathcal{B}_{0}}G(x,y)\tilde{f}_{3}(y)\,dy$
$\displaystyle=\frac{4}{(k_{1}+1)(k_{2}+1)}\sum_{k=0}^{\infty}(-1)^{k}\gamma^{k}g_{1}((\Phi_{2}\Phi_{1})^{k}(x))$
$\displaystyle\quad+\frac{2\alpha}{k_{2}+1}\sum_{k=0}^{\infty}(-1)^{k}\gamma^{k}\log|(\Phi_{2}\Phi_{1})^{k}(x)-(a,0)|\int_{\mathcal{B}_{1}}\tilde{f}_{3}(y)\,dy$
$\displaystyle\quad+\frac{1}{k_{2}}g_{2}(x)+\frac{\beta}{k_{2}}g_{2}(\Phi_{2}(x))-\frac{4\alpha}{(k_{2}+1)^{2}}\sum_{k=0}^{\infty}(-1)^{k}\gamma^{k}g_{2}((\Phi_{1}\Phi_{2})^{k}\Phi_{1}(x))$
$\displaystyle\quad+\frac{\beta}{k_{2}}\log|x+(a,0)|\int_{\mathcal{B}_{2}}\tilde{f}_{3}(y)\,dy$
$\displaystyle\quad-\frac{2\alpha\beta}{k_{2}+1}\sum_{k=0}^{\infty}(-1)^{k}\gamma^{k}\log|(\Phi_{1}\Phi_{2})^{k}\Phi_{1}(x)+(a,0)|\int_{\mathcal{B}_{2}}\tilde{f}_{3}(y)\,dy$
$\displaystyle\quad+\frac{2}{k_{2}+1}\sum_{k=0}^{\infty}(-1)^{k}\gamma^{k}\Big{(}g_{0}(\Phi_{2}\Phi_{1})^{k}(x)-\alpha
g_{0}((\Phi_{1}\Phi_{2})^{k}\Phi_{1}(x))\Big{)}.$

where we used $\log|x+(a,0)|=-\log|\Phi_{2}(x)+(a,0)|$ for
$x\in\mathcal{B}_{2}\cap B_{1/2}$. Note that

$\displaystyle|x+(a,0)|\geq
1/2,\quad|(\Phi_{1}\Phi_{2})^{k}\Phi_{1}(x)+(a,0)|\geq
1,\quad|(\Phi_{2}\Phi_{1})^{k}(x)-(a,0)|\geq 1.$

We can estimate $Dw_{i},i=0,1,2,3,$ as above to obtain

$\displaystyle|Dw_{1}(x)|\leq\frac{Ck_{1}}{(k_{1}+1)(k_{2}+1)},\quad
Dw_{2}(x)|\leq C,$
$\displaystyle|Dw_{0}(x)|\leq\frac{C}{k_{2}+1},\quad|Dw_{3}(x)|\leq C.$

Therefore,

$|Du(x)|\leq C\quad\mbox{for}~{}~{}x\in\mathcal{B}_{2}\cap B_{1/2},$

where $C$ is a positive constant independent of $\varepsilon$, $k_{1}$ and
$k_{2}$. Hence (1.2) is proved for $m=1$.

##  4\. Higher order derivatives

In this section, we prove Theorem 1.1 for $m\geq 2$. The idea is essentially
the same as the case when $m=1$. Without loss of generality, we assume that
$C_{m}=1$.

For $x\in\mathcal{B}_{0}\cap B_{1/2}$, by (2.1) and (2.3), we have

$\displaystyle
D^{m}w_{1}(x)=\frac{2}{k_{1}+1}\sum_{k=0}^{\infty}(-1)^{k}\gamma^{k}\Big{(}D^{m}[h_{1}(\psi^{k}(x))]-\beta
D^{m}[h_{1}(\psi^{k}\Phi_{2}(x))]\Big{)}$

We denote

$\Theta_{m,k}:=\gamma^{k}D^{m}[h_{1}(\psi^{k}(x))].$

Since $\sum_{i=0}^{m}\binom{m}{i}=2^{m}$, we have

$\displaystyle\sum_{k=0}^{\infty}(-1)^{k}\Theta_{m,k}=$
$\displaystyle\frac{1}{2^{m}}\sum_{k=1}^{\infty}(-1)^{k}\left[\sum_{i=0}^{m}(-1)^{i}\binom{m}{i}\Theta_{m,k+i}\right]$
$\displaystyle+\Theta_{m,0}+\frac{1}{2^{m}}\sum_{i=1}^{m}\left[(-1)^{i}\Theta_{m,i}\sum_{j=i}^{m}\binom{m}{j}\right].$
(4.1)

The second line of (4) is clearly bounded independent of $\varepsilon$ and
$\gamma$. The main goal is to estimate the first term on the right-hand side
of (4). By the mean value theorem,

$\sum_{i=0}^{m}(-1)^{i}\binom{m}{i}\Theta_{m,k+i}=D_{k}^{m}\Theta_{m,k}\Big{|}_{k=\bar{k}}$

for some $\bar{k}\in(k,k+m)$. We claim that

$\left|\sum_{k=1}^{\infty}\sum_{i=0}^{m}(-1)^{i}\binom{m}{i}\Theta_{m,k+i}\right|\lesssim\sum_{k=1}^{\infty}|D_{k}^{m}\Theta_{m,k}|\lesssim
k_{1}.$ (4.2)

By the chain rule and the product rule,

$D^{m}[h_{1}(\psi^{k}(x))]=\sum_{n=1}^{m}\left(D^{n}h_{1}(\psi^{k}(x))\sum_{\sum_{j=1}^{n}\alpha_{n,j}=m}\prod_{j=1}^{n}D^{\alpha_{n,j}}(\psi^{k})(x)\right),$

where $\alpha_{n,j}\geq 1$. For each $n$ and $\\{\alpha_{n,j}\\}_{j=1}^{n}$
satisfying $\sum_{j=1}^{n}\alpha_{n,j}=m$,

$D^{\alpha_{n,j}}(\psi^{k})(x)=\sum_{\tilde{\alpha}_{n,j}=1}^{\alpha_{n,j}}\left(D^{\tilde{\alpha}_{n,j}}\psi^{k-1}[\psi(x)]\sum_{\sum_{p=1}^{\tilde{\alpha}_{n,j}}\alpha_{n,j,p}={\alpha_{n,j}}}\prod_{p=1}^{\tilde{\alpha}_{n,j}}D^{\alpha_{n,j,p}}\psi(x)\right),$

where $\alpha_{n,j,p}\geq 1$. Then by (2.10),

$\displaystyle\Big{|}D_{k}^{m}\gamma^{k}D^{m}[h_{1}(\psi^{k}(x))]\Big{|}$
$\displaystyle\lesssim\sum_{\begin{subarray}{c}\tau+\beta_{0}=0\\\
\tau,\beta_{0}\geq
0\end{subarray}}^{m}\sum_{n=1}^{m}\bigg{(}\gamma^{k}|\log\gamma|^{\tau}\big{|}D_{k}^{\beta_{0}}[D^{n}h_{1}(\psi^{k}(x))]\big{|}\times$
$\displaystyle\quad\times\sum_{\sum_{j=1}^{n}\beta_{j}=m-\tau-\beta_{0}}\prod_{j=1}^{n}\sum_{\tilde{\alpha}_{n,j}=1}^{\alpha_{n,j}}\big{|}D_{k}^{\beta_{j}}D^{\tilde{\alpha}_{n,j}}(\psi^{k-1})[\psi(x)]\big{|}\bigg{)}.$
(4.3)

When $\beta_{0}\geq 1$, by the chain rule and the product rule,

$\displaystyle
D_{k}^{\beta_{0}}[D^{n}h_{1}(\psi^{k}(x))]=\sum_{s=1}^{\beta_{0}}\left(D^{n+s}h_{1}(\psi^{k}(x))\sum_{\sum_{i=1}^{s}\gamma_{s,i}=\beta_{0}}\prod_{i=1}^{s}D_{k}^{\gamma_{s,i}}\psi^{k-1}[\psi(x)]\right),$

where $\gamma_{s,i}\geq 1$. For each $s$ and $\\{\gamma_{s,i}\\}_{i=1}^{s}$
satisfying $\gamma_{s,i}\geq 1$ and $\sum_{i=1}^{s}\gamma_{s,i}=\beta_{0}$, by
(2.11)

$\displaystyle\left|\prod_{i=1}^{s}D_{k}^{\gamma_{s,i}}\psi^{k-1}[\psi(x)]\right|$
$\displaystyle\lesssim\prod_{i=1}^{s}\left(\frac{\varepsilon^{(1+\gamma_{s,i})/2}}{\lambda_{2}^{2k}}\sum_{j=0}^{\gamma_{s,i}}\frac{r^{j+1}}{|I_{2k}|^{j+1}}\right)$
$\displaystyle\lesssim\frac{\varepsilon^{(s+\beta_{0})/2}}{\lambda_{2}^{2k}}\sum_{j=0}^{\beta_{0}}\frac{r^{j+s}}{|I_{2k}|^{j+s}}.$
(4.4)

To prove the claim, we consider the following two cases:

Case 1: $\beta_{0}\geq 1$. For each $n$, by (4), (2.7), and (2.12), the right-
hand side of (4) can be estimated by (up to a positive constant independent of
$k$)

$\displaystyle
k_{1}\gamma^{k}|\log\gamma|^{\tau}\sum_{s=1}^{\beta_{0}}\left(\frac{\varepsilon^{(s+\beta_{0})/2}}{\lambda_{2}^{2k}}\sum_{j=0}^{\beta_{0}}\frac{r^{j+s}}{|I_{2k}|^{j+s}}\right)\frac{\varepsilon^{n+(m-\tau-\beta_{0})/2}}{\lambda_{2}^{2k}}\times$
$\displaystyle\quad\times\sum_{\begin{subarray}{c}0\leq\tilde{\beta}_{2}+\tilde{\beta}_{3}\leq
m-\tau-\beta_{0}\\\
0\leq\tilde{\beta}_{2}\leq\tilde{m}-n\end{subarray}}|1-\lambda_{2}^{-2k}|^{\tilde{m}-n-\tilde{\beta}_{2}}\frac{r^{\tilde{\beta}_{3}}}{|I_{2k}|^{\tilde{m}+n+\tilde{\beta}_{3}}}$
$\displaystyle\lesssim
k_{1}\sum_{s=1}^{\beta_{0}}\gamma^{k}|\log\gamma|^{\tau}\frac{\varepsilon^{n+(m-\tau+s)/2}}{\lambda_{2}^{2k}}\times$
$\displaystyle\quad\times\sum_{\begin{subarray}{c}0\leq\tilde{\beta}_{2}+\tilde{\beta}_{3}\leq
m-\tau-\beta_{0}\\\ 0\leq\tilde{\beta}_{2}\leq\tilde{m}-n\\\ 0\leq
j\leq\beta_{0}\end{subarray}}|1-\lambda_{2}^{-2k}|^{\tilde{m}-n-\tilde{\beta}_{2}}\frac{r^{\tilde{\beta}_{3}+j+s}}{|I_{2k}|^{\tilde{m}+n+\tilde{\beta}_{3}+j+s}},$

where $n\leq\tilde{m}\leq m$. For every $s$, $\tau$, $j$, $\tilde{\beta}_{2}$,
and $\tilde{\beta}_{3}$ that satisfy $1\leq s\leq\beta_{0}$, $0\leq
j\leq\beta_{0}$, $0\leq\tilde{\beta}_{2}\leq\tilde{m}-n$, $\tau\geq 0$, and
$0\leq\tilde{\beta}_{2}+\tilde{\beta}_{3}\leq m-\tau-\beta_{0}$, we consider
the following three cases:

When $k<\frac{1}{r+\sqrt{\varepsilon}}$, we have

$|1-\lambda_{2}^{-2k}|\lesssim
k\sqrt{\varepsilon}\quad\mbox{and}\quad|I_{2k}|\gtrsim\sqrt{\varepsilon}.$

Since

$\displaystyle\sum_{k<\frac{1}{r+\sqrt{\varepsilon}}}\left(\frac{\gamma}{\lambda_{2}^{2}}\right)^{k}k^{\tilde{m}-n-\tilde{\beta}_{2}}$
$\displaystyle\lesssim\min\left\\{\frac{1}{(1-\gamma/\lambda_{2}^{2})^{\tilde{m}-n-\tilde{\beta}_{2}+1}},\frac{1}{(r+\sqrt{\varepsilon})^{\tilde{m}-n-\tilde{\beta}_{2}+1}}\right\\}$
$\displaystyle\lesssim\frac{1}{(1-\gamma+r+\sqrt{\varepsilon})^{\tilde{m}-n-\tilde{\beta}_{2}+1}},$

we have

$\displaystyle\sum_{k<\frac{1}{r+\sqrt{\varepsilon}}}\gamma^{k}|\log\gamma|^{\tau}\frac{\varepsilon^{n+(m-\tau+s)/2}}{\lambda_{2}^{2k}}|1-\lambda_{2}^{-2k}|^{\tilde{m}-n-\tilde{\beta}_{2}}\frac{r^{\tilde{\beta}_{3}+j+s}}{|I_{2k}|^{\tilde{m}+n+\tilde{\beta}_{3}+j+s}}$
$\displaystyle\lesssim\sum_{k<\frac{1}{r+\sqrt{\varepsilon}}}\left(\frac{\gamma}{\lambda_{2}^{2}}\right)^{k}k^{\tilde{m}-n-\tilde{\beta}_{2}}|\log\gamma|^{\tau}\varepsilon^{(m+\tilde{m}+n-\tau-\tilde{\beta}_{2}+s)/2}\frac{r^{\tilde{\beta}_{3}+j+s}}{\varepsilon^{(\tilde{m}+n+\tilde{\beta}_{3}+j+s)/2}}$
$\displaystyle\lesssim\frac{1}{(1-\gamma+r+\sqrt{\varepsilon})^{\tilde{m}-n-\tilde{\beta}_{2}+1}}(1-\gamma)^{\tau}\varepsilon^{(m-\tau-\tilde{\beta}_{2}-\tilde{\beta}_{3}-j)/2}r^{\tilde{\beta}_{3}+j+s}$
$\displaystyle\lesssim\frac{1}{(1-\gamma+r+\sqrt{\varepsilon})^{\tilde{m}-n-\tilde{\beta}_{2}+1}}(1-\gamma+r+\sqrt{\varepsilon})^{m-\tilde{\beta}_{2}+s}\lesssim
1,$

as $\tilde{m}\leq m$ and $n-1+s\geq 1>0$. When
$\frac{1}{r+\sqrt{\varepsilon}}\leq k<\frac{1}{\sqrt{\varepsilon}}$, we have

$|1-\lambda_{2}^{-2k}|\lesssim
k\sqrt{\varepsilon}\quad\mbox{and}\quad|I_{2k}|\gtrsim(r+\sqrt{\varepsilon})k\sqrt{\varepsilon}.$

Therefore,

$\displaystyle\sum_{\frac{1}{r+\sqrt{\varepsilon}}\leq
k<\frac{1}{\sqrt{\varepsilon}}}\gamma^{k}|\log\gamma|^{\tau}\frac{\varepsilon^{n+(m-\tau+s)/2}}{\lambda_{2}^{2k}}|1-\lambda_{2}^{-2k}|^{\tilde{m}-n-\tilde{\beta}_{2}}\frac{r^{\tilde{\beta}_{3}+j+s}}{|I_{2k}|^{\tilde{m}+n+\tilde{\beta}_{3}+j+s}}$
$\displaystyle\lesssim\sum_{\frac{1}{r+\sqrt{\varepsilon}}\leq
k<\frac{1}{\sqrt{\varepsilon}}}\gamma^{\frac{1}{r+\sqrt{\varepsilon}}}k^{\tilde{m}-n-\tilde{\beta}_{2}}|\log\gamma|^{\tau}\varepsilon^{(m+\tilde{m}+n-\tau-\tilde{\beta}_{2}+s)/2}\frac{r^{\tilde{\beta}_{3}+j+s}}{[(r+\sqrt{\varepsilon})k\sqrt{\varepsilon}]^{\tilde{m}+n+\tilde{\beta}_{3}+j+s}}$
$\displaystyle\lesssim\gamma^{\frac{1}{r+\sqrt{\varepsilon}}}|\log\gamma|^{\tau}\varepsilon^{(m-\tau-\tilde{\beta}_{2}-\tilde{\beta}_{3}-j)/2}\frac{1}{(r+\sqrt{\varepsilon})^{\tilde{m}+n}}\sum_{\frac{1}{r+\sqrt{\varepsilon}}\leq
k<\frac{1}{\sqrt{\varepsilon}}}\frac{1}{k^{2n+\tilde{\beta}_{2}+\tilde{\beta}_{3}+s+j}}$
$\displaystyle\lesssim
e^{\frac{\log\gamma}{r+\sqrt{\varepsilon}}}|\log\gamma|^{\tau}\varepsilon^{(m-\tau-\tilde{\beta}_{2}-\tilde{\beta}_{3}-j)/2}\frac{1}{(r+\sqrt{\varepsilon})^{\tilde{m}+n}}(r+\sqrt{\varepsilon})^{2n+\tilde{\beta}_{2}+\tilde{\beta}_{3}+s+j-1}$
$\displaystyle\lesssim
e^{\frac{\log\gamma}{r+\sqrt{\varepsilon}}}\left|\frac{\log\gamma}{r+\sqrt{\varepsilon}}\right|^{\tau}(r+\sqrt{\varepsilon})^{m-\tilde{m}+n+s-1}\lesssim
1,$

where we used the fact that $|e^{-x}x^{\tau}|\leq C_{\tau}$ for any $x>0$,
$m-\tau-\tilde{\beta}_{2}-\tilde{\beta}_{3}-j\geq 0$, $\tilde{m}\leq m$, and
$n+s-1\geq 1>0$.

When $k\geq\frac{1}{\sqrt{\varepsilon}}$, we have

$|1-\lambda_{2}^{-2k}|\lesssim 1\quad\mbox{and}\quad|I_{2k}|\gtrsim
r+\sqrt{\varepsilon}.$

Therefore,

$\displaystyle\sum_{k\geq\frac{1}{\sqrt{\varepsilon}}}\gamma^{k}|\log\gamma|^{\tau}\frac{\varepsilon^{n+(m-\tau+s)/2}}{\lambda_{2}^{2k}}|1-\lambda_{2}^{-2k}|^{\tilde{m}-n-\tilde{\beta}_{2}}\frac{r^{\tilde{\beta}_{3}+j+s}}{|I_{2k}|^{\tilde{m}+n+\tilde{\beta}_{3}+j+s}}$
$\displaystyle\lesssim\sum_{k\geq\frac{1}{\sqrt{\varepsilon}}}\left(\frac{\gamma}{\lambda_{2}^{2}}\right)^{k}|\log\gamma|^{\tau}\varepsilon^{n+(m-\tau+s)/2}\frac{1}{(r+\sqrt{\varepsilon})^{m+n}}$
$\displaystyle\lesssim\frac{\gamma^{\frac{1}{\sqrt{\varepsilon}}}}{1-\gamma/\lambda_{2}^{2}}\left|\frac{\log\gamma}{\sqrt{\varepsilon}}\right|^{\tau}\varepsilon^{(n+s)/2}$
$\displaystyle\lesssim
e^{\frac{\log\gamma}{\sqrt{\varepsilon}}}\left|\frac{\log\gamma}{\sqrt{\varepsilon}}\right|^{\tau}\varepsilon^{(n+s-1)/2}\lesssim
1,$

where we used the fact that $|e^{-x}x^{\tau}|\leq C_{\tau}$ for any $x>0$,
$\tilde{m}-n-\tilde{\beta}_{2}\geq 0$, $n+s-1\geq 1>0$, and
$1-\gamma/\lambda_{2}^{2}\gtrsim\sqrt{\varepsilon}$. Therefore, (4.2) follows
from the estimates above.

Case 2: $\beta_{0}=0$. For each $n$, by (2.12), the right-hand side of (4) can
be estimated by (up to a positive constant independent of $k$)

$\displaystyle
k_{1}\gamma^{k}|\log\gamma|^{\tau}\frac{\varepsilon^{n+(m-\tau)/2}}{\lambda_{2}^{2k}}\sum_{\begin{subarray}{c}0\leq\tilde{\beta}_{2}+\tilde{\beta}_{3}\leq
m-\tau\\\
0\leq\tilde{\beta}_{2}\leq\tilde{m}-n\end{subarray}}|1-\lambda_{2}^{-2k}|^{\tilde{m}-n-\tilde{\beta}_{2}}\frac{r^{\tilde{\beta}_{3}}}{|I_{2k}|^{\tilde{m}+n+\tilde{\beta}_{3}}},$

where $n\leq\tilde{m}\leq m$. Then (4.2) follows by repeating the exact same
argument as in Case 1 with $s=0$ and $j=0$.

Using the same argument but replacing $\psi^{k-1}(\psi(x))$ with
$\psi^{k}\Phi_{2}(x)$, we have

$\left|\sum_{k=0}^{\infty}(-1)^{k}\gamma^{k}D^{m}[h_{1}(\psi^{k}\Phi_{2}(x))]\right|\leq
Ck_{1},$

where $C$ is a positive constant independent of $\varepsilon$ and $\gamma$.
Therefore,

$|D^{m}w_{1}(x)|\leq\frac{Ck_{1}}{k_{1}+1}\quad\text{for}\,x\in\mathcal{B}_{0}\cap
B_{1/2}.$

As in Section 3, we can estimate $|D^{m}w_{1}(x)|$, $|D^{m}w_{2}(x)|$,
$|D^{m}w_{3}(x)|$, and $|D^{m}w_{0}(x)|$ in all three regions. Therefore,
(1.2) is proved.

##  5\. Proof of Theorem 1.2

In this section, we prove Theorem 1.2. The proof is similar to that of Theorem
1.1. Without loss of generality, we assume $C_{m}=1$. Take a cutoff function
$\eta\in C_{0}^{\infty}(\mathcal{D}_{2})$ such that $\eta=1$ on
$\mathcal{D}_{1}$. Then $v:=u\eta$ satisfies

$D_{i}(a(x)D_{i}v)=D_{i}\tilde{f}_{i}+\tilde{f}_{3}\quad\mbox{in}~{}~{}\mathbb{R}^{2},$

where

$\tilde{f}_{i}=f_{i}\eta+uD_{i}\eta,\quad\tilde{f}_{3}=-f_{i}D_{i}\eta+D_{i}uD_{i}\eta.$

For $i=0,1,2,3$, $j=1,2,3$, we define $\tilde{u}$, $w_{i}$, $h_{j}$, and
$g_{j}$ as in (2.3), (2.4), and (2.5). Instead of (2.6) and (2.7), we have

$\|\tilde{f}_{i}\|_{C^{2m-1,\mu}(\mathcal{B}_{j})}\leq
C\min\\{1,k_{j}\\}\quad\mbox{for}~{}~{}i=1,2,3,~{}~{}j=0,1,2,$

and

$\displaystyle\|h_{j}\|_{C^{2m,\mu}(B_{3}\cap\mathcal{B}_{i})}+\|g_{j}\|_{C^{2m,\mu}(B_{3}\cap\mathcal{B}_{i})}\leq
C\min\\{1,k_{j}\\}\quad\mbox{for}~{}~{}i,j=0,1,2.$ (5.1)

As in the proof of Theorem 1.1, for $x\in\mathcal{B}_{0}\cap\mathcal{D}_{1}$,
we have, by (5.1),

$\displaystyle|D^{m}w_{1}(x)|\leq\frac{Ck_{1}}{k_{1}+1},\quad|D^{m}w_{2}(x)|\leq\frac{C}{k_{2}+1},\quad\mbox{and}~{}~{}|D^{m}w_{0}(x)|+|D^{m}w_{3}(x)|\leq
C.$

For $x\in\mathcal{B}_{1}$, we have

$\displaystyle|D^{m}w_{1}(x)|$ $\displaystyle\leq
C+\frac{Ck_{1}}{(k_{1}+1)^{2}},$ $\displaystyle|D^{m}w_{2}(x)|$
$\displaystyle\leq\frac{C}{(k_{1}+1)(k_{2}+1)},$
$\displaystyle|D^{m}w_{0}(x)|$ $\displaystyle\leq\frac{C}{k_{1}+1}.$

To estimate $|D^{m}w_{3}(x)|$, we note that

$\displaystyle\frac{\alpha}{k_{1}}g_{1}(\Phi_{1}(x))+\frac{\alpha}{k_{1}}\log|x-(a,0)|\int_{\mathcal{B}_{1}}\tilde{f}_{3}(y)\,dy$
$\displaystyle=\frac{\alpha}{k_{1}}\int_{\mathcal{B}_{1}}(\log|\Phi_{1}(x)-y|+\log|x-(a,0)|)\tilde{f}_{3}(y)\,dy,$

and for fixed $y\in\mathcal{B}_{1}$, $x=(a,0)$ is a removable singular point
for $\log|\Phi_{1}(x)-y|+\log|x-(a,0)|$. Therefore
$\log|\Phi_{1}(x)-y|+\log|x-(a,0)|$ is harmonic in $\mathcal{B}_{1}$, and

$\left|D^{m}\left(\frac{\alpha}{k_{1}}g_{1}(\Phi_{1}(x))+\frac{\alpha}{k_{1}}\log|x-(a,0)|\int_{\mathcal{B}_{1}}\tilde{f}_{3}(y)\,dy\right)\right|\leq
C.$

The rest of the terms can be estimated similarly as before, and hence we
obtain

$|D^{m}w_{3}(x)|\leq C,$

which yields

$|D\tilde{u}(x)|\leq\frac{C}{k_{1}+1}\quad\mbox{for}~{}~{}x\in\mathcal{B}_{1}.$

Finally, for $x\in\mathcal{B}_{2}$, we have

$\displaystyle|D^{m}w_{1}(x)|$
$\displaystyle\leq\frac{Ck_{1}}{(k_{1}+1)(k_{2}+1)},$
$\displaystyle|D^{m}w_{2}(x)|$
$\displaystyle\leq\frac{C}{k_{2}}+\frac{C}{(k_{2}+1)^{2}},$
$\displaystyle|D^{m}w_{0}(x)|$ $\displaystyle\leq\frac{C}{k_{2}+1},$
$\displaystyle|D^{m}w_{3}(x)|$
$\displaystyle\leq\frac{Ck_{1}}{(k_{1}+1)(k_{2}+1)}+\frac{Ck_{1}}{k_{2}+1}+\frac{C}{k_{2}}+\frac{C}{(k_{2}+1)^{2}}+\frac{C}{k_{2}+1},$

which yields

$|D\tilde{u}(x)|\leq\frac{C}{k_{2}+1}\quad\mbox{for}~{}~{}x\in\mathcal{B}_{2}.$

Theorem 1.2 is proved.

##  6\. Proof of Theorem 1.3

When $r_{1}=r_{2}$, Theorem 1.3 follows from Theorem 1.2 after scaling. When
$r_{1}\neq r_{2}$, we will find a conformal map $T:\mathbb{C}\to\mathbb{C}$
that maps $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ to circles of the same
radius. Without loss of generality, we may assume $r_{2}>r_{1}$.

Let $T(z)=\frac{1}{z-z_{0}}$, where $z_{0}\in\mathbb{C}$ and
$z_{0}=z_{1}+z_{2}i$. It is well known that if
$z_{0}\not\in\overline{\mathcal{B}_{1}\cup\mathcal{B}_{2}}$, $T$ maps
$\mathcal{B}_{1}\cup\mathcal{B}_{2}$ to two disks. After a direct computation,
we know that $T$ maps $\mathcal{B}_{1}$ to the disk of center
$\frac{\bar{z}_{0}-\varepsilon/2-r_{1}}{r_{1}^{2}-|z_{0}-\varepsilon/2-r_{1}|^{2}}$,
radius $\frac{r_{1}}{|r_{1}^{2}-|z_{0}-\varepsilon/2-r_{1}|^{2}|}$, and maps
$\mathcal{B}_{2}$ to the disk of center
$\frac{\bar{z}_{0}+\varepsilon/2+r_{2}}{r_{2}^{2}-|z_{0}+\varepsilon/2+r_{2}|^{2}}$,
radius $\frac{r_{2}}{|r_{2}^{2}-|z_{0}+\varepsilon/2+r_{2}|^{2}|}$. We only
need to find $z_{0}=z_{1}+z_{2}i$ such that

$r_{1}(r_{2}^{2}-|z_{0}+\varepsilon/2+r_{2}|^{2})=r_{2}(r_{1}^{2}-|z_{0}-\varepsilon/2-r_{1}|^{2}).$

This is equivalent to

$\displaystyle\left(z_{1}-\Big{(}\frac{\varepsilon}{2}\frac{r_{1}+r_{2}}{r_{2}-r_{1}}+\frac{2r_{1}r_{2}}{r_{2}-r_{1}}\Big{)}\right)^{2}+z_{2}^{2}$
$\displaystyle=\Big{(}\frac{2r_{1}r_{2}}{r_{2}-r_{1}}\Big{)}^{2}+\frac{2\varepsilon
r_{1}r_{2}(r_{1}+r_{2})}{(r_{2}-r_{1})^{2}}+\frac{\varepsilon^{2}}{4}\left[\Big{(}\frac{r_{1}+r_{2}}{r_{2}-r_{1}}\Big{)}^{2}-1\right].$

We take

$\displaystyle z_{1}$
$\displaystyle=\sqrt{\Big{(}\frac{2r_{1}r_{2}}{r_{2}-r_{1}}\Big{)}^{2}+\frac{2\varepsilon
r_{1}r_{2}(r_{1}+r_{2})}{(r_{2}-r_{1})^{2}}+\frac{\varepsilon^{2}}{4}\left[\Big{(}\frac{r_{1}+r_{2}}{r_{2}-r_{1}}\Big{)}^{2}-1\right]}$
$\displaystyle\quad+\frac{\varepsilon}{2}\frac{r_{1}+r_{2}}{r_{2}-r_{1}}+\frac{2r_{1}r_{2}}{r_{2}-r_{1}},$
$\displaystyle z_{2}$ $\displaystyle=0.$

It is easy to see that $z_{1}>\varepsilon/2+4r_{1}$. Therefore
$z_{0}\not\in\overline{\mathcal{B}_{1}\cup\mathcal{B}_{2}}$, and we can choose
domains $D_{1},D_{2}$, and $D$ such that
$\mathcal{B}_{1}\cup\mathcal{B}_{2}\Subset D_{1}\Subset D_{2}\Subset D$, and
$z_{0}\not\in\overline{D}$. Hence $T$ is smooth in $D$ and is the desired
conformal map.

##  7\. The extreme case

In this section, we prove Theorem 1.4. The proof essentially follows that of
[BLLY]*Theorem 1.1, with some modifications.

By considering $u-C$ instead of $u$, we may assume the constant $C=0$ in
(1.3). For any $0<t<s<1$, let $\eta\in C_{c}^{\infty}(\Omega_{s})$ be a cutoff
function such that $\eta=1$ in $\Omega_{t}$ and $|D\eta|\leq C(s-t)^{-1}$.
Multiplying $u\eta^{2}$ on both sides of (1.3) and integrating by parts, we
have

$\int_{\Omega_{1}}a^{ij}D_{j}uD_{i}u\eta^{2}+2a^{ij}D_{j}uD_{i}\eta u\eta=0.$

By Young’s inequality,

$\int_{\Omega_{t}}|Du|^{2}\leq\frac{C}{(s-t)^{2}}\int_{\Omega_{s}\setminus\Omega_{t}}u^{2}.$

Since $u=0$ on $\Gamma_{-}$, by the Poincaré inequality in the $x_{n}$
direction, we have

$\int_{\Omega_{s}\setminus\Omega_{t}}u^{2}\leq
C(\varepsilon+s^{2})^{2}\int_{\Omega_{s}\setminus\Omega_{t}}|Du|^{2}.$

Therefore,

$\int_{\Omega_{t}}|Du|^{2}\leq
C_{0}\left(\frac{\varepsilon+s^{2}}{s-t}\right)^{2}\int_{\Omega_{s}\setminus\Omega_{t}}|Du|^{2}.$
(7.1)

Let $t_{0}=r\in(\sqrt{\varepsilon},1/2)$ and $t_{j}=(1-jr)r$ for
$j\in\mathbb{N}$. Taking $s=t_{j},t=t_{j+1}$ in (7.1), we have

$\int_{\Omega_{t_{j+1}}}|Du|^{2}\leq
4C_{0}\int_{\Omega_{t_{j}}\setminus\Omega_{t_{j+1}}}|Du|^{2}.$

Adding both sides by $4C_{0}\int_{\Omega_{t_{j+1}}}|Du|^{2}$ and dividing both
sides by $1+4C_{0}$, we have

$\int_{\Omega_{t_{j+1}}}|Du|^{2}\leq\frac{4C_{0}}{1+4C_{0}}\int_{\Omega_{t_{j}}}|Du|^{2}.$

Let $k=\lfloor\frac{1}{2r}\rfloor$ and iterate the above inequality $k$ times.
We have

$\int_{\Omega_{r/2}}|Du|^{2}\leq\left(\frac{4C_{0}}{1+4C_{0}}\right)^{k}\int_{\Omega_{r}}|Du|^{2}\leq
C\mu^{\frac{1}{r}}\int_{\Omega_{1}}|u|^{2},$ (7.2)

where $\mu\in(0,1)$ and $C$ are constants depending only on $n$, $\sigma$,
$\|h_{1}\|_{C^{2}}$, and $\|h_{2}\|_{C^{2}}$. For any
$\bar{x}=(\bar{x}^{\prime},\bar{x}_{n})\in\Omega_{1/2}$, let
$R=\varepsilon+h_{1}(\bar{x}^{\prime})-h_{2}(\bar{x}^{\prime})$. We make the
change of variables by setting

$\left\\{\begin{aligned} &y^{\prime}=x^{\prime}-\bar{x}^{\prime},\\\
&y_{n}=2R\left(\frac{x_{n}-h_{2}(x^{\prime})+\varepsilon/2}{\varepsilon+h_{1}(x^{\prime})-h_{2}(x^{\prime})}-\frac{1}{2}\right)\end{aligned}\right.$

for
$x=(x^{\prime},x_{n})\in\\{|x^{\prime}-\bar{x}^{\prime}|<R,-\varepsilon/2+h_{2}(x^{\prime})<x_{n}<\varepsilon/2+h_{1}(x^{\prime})\\}$.
This change of variables maps the domain above to $Q_{R}$, where

$Q_{s}:=\\{(y^{\prime},y_{n})\in\mathbb{R}^{n}~{}\big{|}~{}|y^{\prime}|<s,|y_{n}|<s\\}$

for $s>0$. Let

$(b^{ij}(y))=\frac{(\partial_{x}y)\big{(}a^{ij}(x(y))\big{)}(\partial_{x}y)^{t}}{\det(\partial_{x}y)},$

$\tilde{b}^{ij}(y)={b}^{ij}(Ry)$, and $\tilde{u}(y)=u(Rx)$. Then $\tilde{u}$
satisfies

$\left\\{\begin{aligned}
-\partial_{i}(\tilde{b}^{ij}(y)\partial_{j}\tilde{u}(y))&=0\quad\mbox{in
}Q_{1},\\\ \tilde{b}^{nj}(y)\partial_{j}\tilde{u}(y)&=0\quad\mbox{on
}\\{y_{n}=1\\},\\\ \tilde{u}(y)&=0\quad\mbox{on
}\\{y_{n}=-1\\}.\end{aligned}\right.$

It is straightforward to verify that

$\frac{I}{C}\leq\tilde{b}\leq
CI\quad\mbox{and}\quad\|\tilde{b}\|_{C^{m-1,\alpha}(Q_{1})}\leq C,$

where $C$ is a positive constant depending only on $n$, $\sigma$, $m$,
$\alpha$, and $C_{m,\alpha}$. Then by the Schauder estimate, we have

$\max_{-1\leq y_{n}\leq 1}|D^{m}\tilde{u}(0^{\prime},y_{n})|\leq
C\|D\tilde{u}\|_{L^{2}(Q_{1})},$

which gives, after reversing the change of variables,

$|D^{m}u(\bar{x})|\leq
CR^{1-m-n/2}\|Du\|_{L^{2}(\Omega_{|\bar{x}^{\prime}|+R})}\quad\forall\bar{x}\in\Omega_{1/2},$
(7.3)

where $C$ is a positive constant depending only on $n$, $\sigma$, $m$,
$\alpha$, and $C_{m,\alpha}$. Then by using (7.2) and (7.3) with
$r=2\max(\sqrt{\varepsilon},|x^{\prime}|+R)$, we conclude the proof.

## References