# Diffusive limits of the steady state radiative heat transfer system:
Curvature effects

Mohamed Ghattassi  Department of Mathematics, New York University in Abu
Dhabi, Saadiyat Island, P.O. Box 129188, Abu Dhabi, United Arab Emirates,
<EMAIL_ADDRESS>,  Xiaokai Huo  Department of Mathematics, Iowa State
University, 411 Morrill Rd, Ames, 50011, IA, USA<EMAIL_ADDRESS>and  Nader
Masmoudi  Department of Mathematics, New York University in Abu Dhabi,
Saadiyat Island, P.O. Box 129188, Abu Dhabi, United Arab Emirates– Courant
Institute of Mathematical Sciences, New York University, 251 Mercer Street,
New York, NY 10012, USA<EMAIL_ADDRESS>
###### Abstract.

This paper is devoted to the diffusive limit of the nonlinear radiative heat
transfer system with curved boundary domain (two dimensional disk). The
solution constructed in [5] by the leading order interior solution and the
boundary layer corrections fails here to approximate the solutions in
$L^{\infty}$ sense for the diffusive limit. The present paper aims to
construct a geometric correction to the boundary layer problem and obtain a
valid approximate solution in $L^{\infty}$ sense. The main tools to overcome
the convergence problem, are to use matched asymptotic expansion techniques,
fixed-point theorems, linear and nonlinear stability analysis of the boundary
layer problem. In particular, the spectral assumption on the leading order
interior solution, which was proposed for the flat case in [6], is shown to be
still valid which guarantee the stability of the boundary layer expansion with
geometric corrections. Moreover, the convergence result established in [5,
Lemma 10] remain applicable for the approximate solution with geometric
corrections.

###### Key words and phrases:

Radiative transfer system, Diffusive limits, Boundary layer, Milne problem,
geometric correction

###### Contents

  1. 1 Introduction
    1. 1.1 Main results
    2. 1.2 Contributions
    3. 1.3 Organizations
  2. 2 Approximate solutions
    1. 2.1 Interior expansions
    2. 2.2 Boundary layer corrections
    3. 2.3 Construction of approximate solutions
    4. 2.4 Geometric corrections
    5. 2.5 Approximate solution with geometric corrections
  3. 3 Nonlinear Milne problem with geometric correction
    1. 3.1 Existence on the bounded interval
    2. 3.2 Weighted estimates on the solutions on bounded interval
    3. 3.3 Existence on the half line
  4. 4 Linearized Milne problem
    1. 4.1 Spectral assumption (A)
    2. 4.2 Existence on the bounded interval
    3. 4.3 Weighted estimate
    4. 4.4 Existence on the half line
    5. 4.5 Decay properties of the solutions
    6. 4.6 Uniqueness
  5. 5 Convergence of the diffusive limit
    1. 5.1 Errors of the approximation
    2. 5.2 Estimation on the approximation errors
    3. 5.3 Proof of Theorem 1
  6. A Transport equation with geometric correction in half-space
  7. B Existence for the linearized Milne problem without geometric correction

##  1\. Introduction

We consider the following radiative heat transfer system

$\displaystyle\varepsilon^{2}\Delta
T^{\varepsilon}+\langle\psi^{\varepsilon}-(T^{\varepsilon})^{4}\rangle=0,$
(1.1)
$\displaystyle\varepsilon\beta\cdot\nabla\psi^{\varepsilon}+\psi^{\varepsilon}-(T^{\varepsilon})^{4}=0,$
(1.2)

on the two dimensional disk $\Omega=\\{x=(x_{1},x_{2}):|x|\leq 1\\}$ with
radiative velocity $\beta\in\mathbb{S}^{1}$. Here
$T^{\varepsilon}=T^{\varepsilon}(x)$ is the temperature and
$\psi^{\varepsilon}=\psi^{\varepsilon}(x,\beta)$ is the radiative intensity
and $\langle\cdot\rangle$ denotes the integration over $\mathbb{S}^{1}$, i.e.
$\langle f\rangle:=\int_{\mathbb{S}^{1}}f(\beta)d\beta$. We denote by
$\varepsilon>0$ a small parameter, describing the ratio of a typical photon
mean free path to a typical length scale of the problem, for more details we
refer to [4]. The system (1.1)-(1.2) is supplemented with the non-homogenous
Dirichlet boundary conditions

$\displaystyle T^{\varepsilon}(x)=T_{b}(x),\quad\text{for any
}x\in\partial\Omega,$ (1.3)
$\displaystyle\psi^{\varepsilon}(x,\beta)=\psi_{b}(x,\beta),\quad\text{for any
}(x,\beta)\in\Gamma_{-},$ (1.4)

where $\partial\Omega$ is the boundary of $\Omega$ and, the inflow boundary
$\Gamma_{-}$ and out flow boundary $\Gamma_{+}$ are defined respectively by

$\displaystyle\Gamma_{-}:=\\{0(x,\beta)\in\partial\Omega\times\mathbb{S}^{1}:n(x)\cdot\beta<0\\},$
$\displaystyle\Gamma_{+}:=\\{(x,\beta)\in\partial\Omega\times\mathbb{S}^{1}:n(x)\cdot\beta>0\\},$

where $n(x)$ is the exterior normal vector on the boundary. Similarly, the
out-flow boundary and $\Gamma:=\Gamma_{+}\cup\Gamma_{-}\cup\Gamma_{0}$ with
$\Gamma_{0}:=\\{(x,\beta)\partial\Omega\times\mathbb{S}^{1}:n(x)\cdot\beta=0\\}$.

However, when $\psi_{b}=T_{b}^{4}$, there is no boundary layer for (1.1)-(1.2)
and one can prove the solution to (1.1)-(1.2) converges to the solution of the
nonlinear elliptic equation

$\displaystyle\Delta T_{0}+\pi\Delta T_{0}^{4}=0,\quad\psi_{0}=T_{0}^{4},$
(1.5)

with Dirichlet boundary conditions $T_{0}(x)=T_{b}(x)$ for
$x\in\partial\Omega$. However, for general boundary data, boundary layer
exists and when $\varepsilon\to 0$, $T^{\varepsilon}$ no longer converges to
$T_{0}$ solution to the above nonlinear elliptic equation (1.5). A boundary
layer correction $(\bar{T}_{0},\bar{\psi}_{0})$ should be added to
$(T_{0},\psi_{0})$ to get the approximate solution. For the flat boundary, it
was proved in [5] that

$\displaystyle\|T^{\varepsilon}-T_{0}-\bar{T}_{0}\|_{L^{\infty}(\Omega)}=O(\varepsilon),\quad\|\psi^{\varepsilon}-T_{0}^{4}-\bar{\psi}_{0}\|_{L^{\infty}(\Omega)}=O(\varepsilon).$
(1.6)

The boundary layer $\bar{T}_{0}$ is defined by a suitable cut-off of the
corresponding half-space boundary Milne problem of system (1.1)-(1.2). Let
$(\tilde{T}_{0},\tilde{\psi}_{0})$ be the leading order of the half-space
Milne problem. The above inequalities were proved under the condition that
$\tilde{T}_{0}$ satisfy a spectral assumption [6, (A)], which reads as

  1. (A)

A function $h\in C^{1}(\mathbb{R}_{+})$ satisfies the spectral assumption if
there exists a constant $\beta>0$, such that

$\displaystyle
M\int_{0}^{\infty}e^{2\beta\eta}(2h^{\frac{3}{2}})^{2}|\partial_{\eta}f|^{2}d\eta\geq
4\int_{0}^{\infty}e^{2\beta\eta}|\partial_{\eta}(2h^{\frac{3}{2}})|^{2}f^{2}d\eta,$
(1.7)

holds for some constant $M<1$ and for any measurable function $f\in
C^{1}(\mathbb{R}_{+})$ with $f(0)=0$.

The spectral assumption implies the linear stability of the boundary layer
problem which is crucial for establishing the convergence result (1.6).
However, for the not-flat boundary case, curvature effects brings
singularities of the boundary layer problem at the boundary and the
inequalities (1.6) fail to hold. We address this problem by constructing a
correction $(\bar{T}_{0}^{\varepsilon},\bar{\psi}^{\varepsilon})$ of the
boundary layer to improve the approximations near the boundary and erase the
singularities. Moreover, it is expected that the spectral assumption (A) still
implies the linear stability of the boundary layer problem with geometric
correction.

###  1.1. Main results

######  Theorem 1.

Let $T_{b}\in C^{2}(\partial\Omega)$, $\psi_{b}\in C^{1}(\Gamma_{-})$ be non-
negative on the boundary. Let $(T_{0},\psi_{0})$ be the solution to (2.33) and
$(\bar{T}_{0}^{\varepsilon},\bar{\psi}_{0}^{\varepsilon})$ be the solution to
(2.56). $\tilde{T}_{0}^{\varepsilon}$ is the solution to the geometric
correction in the half-space Milne problem (2.56). Assume
$\tilde{T}^{\varepsilon}_{0}$ satisfy the spectral assumption (A) and
$\tilde{T}^{\varepsilon}_{0}\geq a$ for some constant $a>0$. Then for
$\varepsilon$ sufficiently small, the solution to system (1.1)-(1.2) with
boundary conditions (1.3)-(1.4) is unique and satisfies

$\displaystyle\|T^{\varepsilon}-T_{0}-\bar{T}_{0}^{\varepsilon}\|_{L^{\infty}(\Omega)}=O(\varepsilon),\quad\|\psi^{\varepsilon}-\psi_{0}-\bar{\psi}_{0}^{\varepsilon}\|_{L^{\infty}(\Omega\times\mathbb{S}^{1})}=O(\varepsilon).$
(1.8)

The above theorem shows that by adding the geometric correction boundary layer
solution $(\bar{T}_{0}^{\varepsilon},\bar{\psi}_{0}^{\varepsilon})$, (1.6)
still holds for the non-flat boundary and the sum of the interior solution
$(T_{0},\psi_{0})$ and the geometric correction
$(\bar{T}_{0}^{\varepsilon},\bar{\psi}_{0}^{\varepsilon})$ give an approximate
solution to system (1.1)-(1.2) valid in the diffusive limit.

###  1.2. Contributions

When $T^{4}=\langle\psi\rangle/2\pi$, system (1.1)-(1.2) reduce to the
classical transport equation
$\varepsilon\beta\cdot\nabla\psi^{\varepsilon}+\psi^{\varepsilon}-\langle\psi^{\varepsilon}\rangle/2\pi=0$.
The diffusive limit was first studied in [1], where an approximate solution is
constructed by adding interior solution and boundary layer corrections. In
[10], a counter-example shows that such an approximation can fail when the
boundary is not flat. By introducing a geometric correction to the boundary
layer problem, a valid approximate solution was constructed and can be proved
to satisfy
$\|\psi^{\varepsilon}-\psi_{0}-\bar{\psi}_{0}\|_{L^{\infty}(\Omega\times\mathbb{S}^{1})}=O(\varepsilon)$,
where $\psi_{0}\in L^{\infty}(\Omega)$ satisfy $\Delta\psi_{0}=0$ and
$\bar{\psi}_{0}^{\varepsilon}$ is the solution of the geometric corrected
boundary correction. The aim of this paper is to find a similar geometric
correction to the approximate solution for system (1.1)-(1.2). However, the
above discussion is not comprehensive: we refer the reader to the review
articles, [7, 8] and references therein for a more thorough review of the
boundary layer analysis of some singular perturbation problems.

We now state the difference between our work and [10]. First, our system
(1.1)-(1.2) is nonlinear which generates a coupling between the interior
solution and the boundary layer corrections. Indeed, to construct the boundary
layer expansions, we need to approximate the interior solutions by their
Taylor’s expansion. Moreover, the corresponding boundary half-space Milne
problems associated to system (1.1)-(1.2) are nonlinear for the leading order
and linear for higher orders, different from [10] where the Milne problems are
linear for any order of expansions. The coupling between the elliptic and
transport equation brings additional difficulties for proving the existence,
uniqueness and stability of the Milne problems. In particular, the spectral
assumption (A) is needed to show the linear stability of the Milne problems in
our problem. We use techniques from our previous work [6] but accounting for
the geometric corrections. We can show the spectral assumption (A) also
implies the linear stability of the geometric corrected Milne problem.
Furthermore, due to the nonlniearity of our problem it’s not possible to
derive the estimates (1.8) by using the Maximum principle, since the
difference
$(T^{\varepsilon}-T_{0}-\bar{T}_{0}^{\varepsilon},\psi^{\varepsilon}-\psi_{0}-\bar{\psi}^{\varepsilon})$
satisfies a different system from (1.1)-(1.2). Instead, the Banach fixed point
theorem is used. We here show that such a technique used in [5] for the flat
case, can be extended to the curvature boundary case by considering the
geometric corrections.

###  1.3. Organizations

Our paper is organized as follows: In the next section, we use asymptotic
analysis to construct the approximate interior solution and boundary layer
corrections with geometric corrections. Section 3 is devoted to the analysis
of leading order boundary layer Milne problem. Section 4 is devoted to the
analysis of linear Milne problem of the higher order boundary expansions. In
section 5, the proof of Theorem 1 is given.

Notations : Throughout the paper, we use $T_{k}$, $k\geq 0$ to denote the
interior solution and $\bar{T}_{k}$, $k\geq 0$ for the boundary layer
correction, $\bar{T}_{k}^{\varepsilon}$, $k\geq 0$ for the geometric corrected
boundary layer correction., and $\tilde{T}_{k}$, $k\geq 0$ for the solution of
the half-space Milne problem. We use $C$ to denote the generic constant.

##  2\. Approximate solutions

In this section, we construct approximation solutions to system (1.1)-(1.2)
using asymptotic expansions.

###  2.1. Interior expansions

We take the ansatz

$\displaystyle
T^{\varepsilon}\sim\sum_{k=0}^{\infty}\varepsilon^{k}T_{k},\quad\psi^{\varepsilon}\sim\sum_{k=0}^{\infty}\varepsilon^{k}\psi_{k}$

into system (1.1)-(1.2). Collecting terms with the same order gives

$\displaystyle\left\\{\begin{matrix}\langle\psi_{0}-T_{0}^{4}\rangle=0,\\\
\psi_{0}-T_{0}^{4}=0,\end{matrix}\right.$
$\displaystyle\left\\{\begin{matrix}\langle\psi_{1}-4T_{0}^{3}T_{1}\rangle=0,\\\
\beta\cdot\nabla\psi_{0}+\psi_{1}-4T_{0}^{3}T_{1}=0,\end{matrix}\right.$
$\displaystyle\left\\{\begin{matrix}\Delta
T_{0}+\langle\psi_{2}-4T_{0}^{3}T_{2}-6T_{0}^{2}T_{1}^{2}\rangle=0,\\\
\beta\cdot\nabla\psi_{1}+\psi_{2}-4T_{0}^{3}T_{2}-6T_{0}^{2}T_{1}^{2}=0,\end{matrix}\right.$
$\displaystyle\qquad\vdots$ $\displaystyle\left\\{\begin{matrix}\Delta
T_{k-2}+\langle\psi_{k}-\mathcal{C}(T,k)\rangle=0,\\\
\beta\cdot\nabla\psi_{k-1}+\psi_{k}-\mathcal{C}(T,k)=0,\end{matrix}\right.$
(2.1)

where $\mathcal{C}(T,k)$ is defined by

$\displaystyle\mathcal{C}(T,k)=\sum_{\begin{subarray}{c}i+j+l+m=k\\\
i,j,l,m\geq 0\end{subarray}}T_{i}T_{j}T_{l}T_{m}.$

Comparing the two equations in (2.1) gives

$\displaystyle\Delta T_{k-2}=\nabla\cdot\langle\beta\psi_{k-1}\rangle.$

Using $\psi_{k-1}=\mathcal{C}(T,k-1)-\beta\cdot\nabla\psi_{k-2}$ in the above
equation leads to

$\displaystyle\Delta
T_{k-2}=-\nabla\cdot\langle\beta\beta\cdot\nabla\psi_{k-2}\rangle.$

Using $\psi_{k-2}=\mathcal{C}(T,k-2)-\beta\cdot\nabla\psi_{k-3}$ gives

$\displaystyle\Delta
T_{k-2}=-\nabla\cdot\langle\beta\beta\cdot\nabla\mathcal{C}(T,k-2)\rangle+\langle(\beta\cdot\nabla)^{3}\psi_{k-3}\rangle,$

which is

$\displaystyle\Delta
T_{k-2}+\pi\Delta\mathcal{C}(T,k-2)=\langle(\beta\cdot\nabla)^{3}\psi_{k-3}\rangle.$

Thus we get $(T_{k},\psi_{k})$ satisfy for any $k\geq 0$,

$\displaystyle\Delta
T_{k}+\pi\Delta\mathcal{C}(T,k)=\langle(\beta\cdot\nabla)^{3}\psi_{k-1}\rangle,$
$\displaystyle\psi_{k}=\mathcal{C}(T,k)-\beta\cdot\nabla\psi_{k-1}.$

Note here $T_{k},\psi_{k}$ are zero for any index $k<0$. Considering
$\mathcal{C}(T,0)=T_{0}^{4}$ and for $k\geq 1$,
$\mathcal{C}(T,k)=4T_{0}T_{k}+\mathcal{E}(T,k-1)$ with

$\displaystyle\mathcal{E}(T,k-1)=\sum_{\begin{subarray}{c}i+j+l+m=k\\\ 1\leq
i,j,l,m\leq k-1\end{subarray}}T_{i}T_{j}T_{l}T_{m},$

which only depends on $T_{s}$, $s\geq k-1$, we can rewrite the previous
equations as for $k=0$,

$\displaystyle\Delta T_{0}+\pi\Delta T_{0}^{4}=0,\quad\psi_{0}=T_{0}^{4},$

and for $k\geq 1$,

$\displaystyle\Delta
T_{k}+\pi\Delta(4T_{0}^{4}T_{k})=\langle(\beta\cdot\nabla)^{3}\psi_{k-1}\rangle-\pi\Delta\mathcal{E}(T,k-1),\quad\psi_{k}=4T_{0}^{3}T_{k}+\mathcal{E}(T,k-1)-\beta\cdot\nabla\psi_{k-1}.$

Therefore, the leading order system is a nonlinear elliptic equation while the
higher order system is a linear elliptic equation.

###  2.2. Boundary layer corrections

The interior expansions $T^{o}:=\sum_{k=0}^{\infty}\varepsilon^{k}T_{k}$ and
$\psi^{o}:=\sum_{k=0}^{\infty}\varepsilon^{k}\psi_{k}$ provide a good
approximation to system (1.1)-(1.2) inside the domain. Near the boundary we
need to introduce a correction term $(T^{i},\psi^{i})$ describing the effects
of boundary layers. Suppose $T^{\varepsilon}=T^{o}+T^{i}$,
$\psi^{\varepsilon}=\psi^{o}+\psi^{i}$, then system (1.1)-(1.2) becomes

$\displaystyle\varepsilon^{2}\Delta(T^{o}+T^{i})+\langle\psi^{o}+\psi^{i}-(T^{o}+T^{i})^{4}\rangle=0,$
$\displaystyle\varepsilon\beta\cdot\nabla(\psi^{o}+\psi^{i})+\psi^{o}+\psi^{i}-(T^{o}+T^{i})^{4}=0.$

Using the fact that $T^{o}$, $\psi^{o}$ solves system (1.1)-(1.2), the above
system becomes

$\displaystyle\varepsilon^{2}\Delta
T^{i}+\langle\psi^{i}-(T^{o}+T^{i})^{4}+(T^{o})^{4}\rangle=0,$ (2.2)
$\displaystyle\varepsilon\beta\cdot\nabla\psi^{i}+\psi^{i}-(T^{o}+T^{i})^{4}+(T^{o})^{4}=0.$
(2.3)

We now follow [10] by introducing the transformation of variables
$(x_{1},x_{2})\to(r,\theta)$ given by

$\displaystyle\left\\{\begin{array}[]{ll}x_{1}=(1-r)\cos\theta,\\\
x_{2}=(1-r)\sin\theta,\end{array}\right.$

with $r$ denotes the distance to the boundary $\partial\Omega$. With the
coordinate $(r,\theta)$ and $\beta=(\beta_{1},\beta_{2})$, system (2.2)-(2.3)
can be rewritten as

$\displaystyle\varepsilon^{2}\partial_{r}^{2}T^{i}-\frac{\varepsilon^{2}}{1-r}\partial_{r}T^{i}+\frac{\varepsilon^{2}}{(1-r)^{2}}\partial^{2}_{\theta}T^{i}+\langle\psi^{i}-(T^{o}+T^{i})^{4}+(T^{o})^{4}\rangle=0,$
$\displaystyle-\varepsilon(\beta_{1}\cos\theta+\beta_{2}\sin\theta)\partial_{r}\psi^{i}-\frac{\varepsilon}{1-r}(\beta_{1}\sin\theta-\beta_{2}\cos\theta)\partial_{\theta}\psi^{i}$
$\displaystyle\quad+\psi^{i}-(T^{o}+T^{i})^{4}+(T^{o})^{4}=0,$

and the boundary conditions are

$\displaystyle T^{i}(0,\theta)+T^{o}(0,\theta)=T_{b}(\theta),$
$\displaystyle\psi^{i}(0,\theta,\beta)+\psi^{o}(0,\theta)=\psi_{b}(\theta,\beta),\quad\text{for
}\beta_{1}\cos\theta+\beta_{2}\sin\theta<0.$

We introduce a new coordinate $\eta=r/\varepsilon$ and rewrite the above
system as

$\displaystyle\partial_{\eta}^{2}T^{i}-\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}T^{i}+\frac{\varepsilon^{2}}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}T^{i}+\langle\psi^{i}-(T^{o}+T^{i})^{4}+(T^{o})^{4}\rangle=0,$
$\displaystyle-(\beta_{1}\cos\theta+\beta_{2}\sin\theta)\partial_{\eta}\psi^{i}-\frac{\varepsilon}{1-\varepsilon\eta}(\beta_{1}\sin\theta-\beta_{2}\cos\theta)\partial_{\theta}\psi^{i}$
$\displaystyle\quad+\psi^{i}-(T^{o}+T^{i})^{4}+(T^{o})^{4}=0.$

By introducing the transformation of variable

$\displaystyle\left\\{\begin{array}[]{ll}\beta_{1}=-\sin\xi,\\\
\beta_{2}=-\cos\xi,\end{array}\right.$

we can rewrite the system as

$\displaystyle\partial_{\eta}^{2}T^{i}-\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}T^{i}+\frac{\varepsilon^{2}}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}T^{i}+\langle\psi^{i}-(T^{o}+T^{i})^{4}+(T^{o})^{4}\rangle=0,$
(2.4)
$\displaystyle\sin(\theta+\xi)\partial_{\eta}\psi^{i}-\frac{\varepsilon}{1-\varepsilon\eta}\cos(\theta+\xi)\partial_{\theta}\psi^{i}+\psi^{i}-(T^{o}+T^{i})^{4}+(T^{o})^{4}=0.$
(2.5)

Since we are considering boundary layers, we may take Taylor’s expansion of
$(T^{o}=T^{o}(\varepsilon\eta,r),\psi^{o}=\psi^{o}(\varepsilon\eta,r))$ around
the boundary $\eta=0$ and get

$\displaystyle
T^{o}=\sum_{k=0}^{\infty}\varepsilon^{k}P_{k},\quad\psi^{o}=\sum_{k=0}^{\infty}\varepsilon^{k}Q_{k},$

where

$\displaystyle
P_{k}(\eta,\theta)=\sum_{l=0}^{k}\frac{\eta^{l}}{l!}\frac{\partial^{l}}{\partial_{r}^{l}}T_{k-l}(0,\theta),\quad
Q_{k}(\eta,\theta,\xi)=\sum_{l=0}^{k}\frac{\eta^{l}}{l!}\frac{\partial^{l}}{\partial_{r}^{l}}\psi_{k-l}(0,\theta)$

Taking the ansatz

$\displaystyle
T^{i}=\sum_{k=1}^{\infty}\varepsilon^{k}\bar{T}_{k},\quad\psi^{i}=\sum_{k=1}^{\infty}\varepsilon^{k}\bar{\psi}_{k}$
(2.6)

into the system (2.4)-(2.5) and collecting terms with the same order, we get

$\displaystyle\left\\{\begin{array}[]{ll}\partial_{\eta}^{2}\bar{T}_{0}+\langle\bar{\psi}_{0}-(\bar{T}_{0}+P_{0})^{4}+P_{0}^{4}\rangle=0,\\\
\sin(\theta+\xi)\partial_{\eta}\bar{\psi}_{0}+\bar{\psi}_{0}-(\bar{T}_{0}+P_{0})^{4}+P_{0}^{4}=0,\end{array}\right.$
(2.9)
$\displaystyle\left\\{\begin{array}[]{ll}\partial_{\eta}^{2}\bar{T}_{1}+\langle\bar{\psi}_{1}-4(P_{0}+\bar{T}_{0})^{3}(P_{1}+\bar{T}_{1})+4P_{0}^{3}P_{1}\rangle=\frac{1}{1-\varepsilon\eta}\partial_{\eta}\bar{T}_{0},\\\
\sin(\theta+\xi)\partial_{\eta}\bar{\psi}_{1}+\bar{\psi}_{1}-4(P_{0}+\bar{T}_{0})^{3}(P_{1}+\bar{T}_{1})+4P_{0}^{3}P_{1}=\frac{1}{1-\varepsilon\eta}\cos(\theta+\xi)\partial_{\theta}\bar{\psi}_{0},\\\
\end{array}\right.$ (2.12) $\displaystyle\qquad\vdots$
$\displaystyle\left\\{\begin{array}[]{ll}\partial_{\eta}^{2}\bar{T}_{k}+\langle\bar{\psi}_{k}-4(P_{0}+\bar{T}_{0})^{3}(P_{k}+\bar{T}_{k})+4P_{0}^{3}P_{k}\rangle=\frac{1}{1-\varepsilon\eta}\partial_{\eta}\bar{T}_{k-1}-\frac{1}{(1-\varepsilon\eta)^{2}}\partial_{\theta}\bar{T}_{k-2}\\\
\qquad+\langle\mathcal{E}(P+\bar{T},k-1)-\mathcal{E}(P,k-1)\rangle,\\\
\sin(\theta+\xi)\partial_{\eta}\bar{\psi}_{k}+\bar{\psi}_{k}-4(P_{0}+\bar{T}_{0})^{3}(P_{k}+\bar{T}_{k})+4P_{0}^{3}P_{k}=\frac{1}{1-\varepsilon\eta}\cos(\theta+\xi)\partial_{\theta}\bar{\psi}_{k-1}\\\
\quad+\mathcal{E}(P+\bar{T},k-1)-\mathcal{E}(P,k-1),\end{array}\right.$ (2.17)

with the boundary conditions

$\displaystyle\bar{T}_{0}(0,\theta)+T_{0}(0,\theta)=T_{b}(\theta),$
$\displaystyle\bar{\psi}_{0}(0,\theta,\xi)+\psi_{0}(0,\theta,\xi)=\psi_{b}(\theta,\xi),\quad\text{for
}\sin(\theta+\xi)>0,$

and for $k\geq 1$,

$\displaystyle\bar{T}_{k}(0,\theta)+T_{k}(0,\theta)=0,$
$\displaystyle\bar{\psi}_{k}(0,\theta,\xi)+\psi_{k}(0,\theta,\xi)=0,\quad\text{for
}\sin(\theta+\xi)>0,$

However, the boundary condition $T_{k}(0,\theta)$, $\psi_{k}(0,\theta)$ are
not known. Hence we introduce the variables

$\displaystyle\tilde{T}_{k}=\bar{T}_{k}+T_{k}(\varepsilon\eta=0)=\bar{T}_{k}+P_{k}(\eta=0),$
$\displaystyle\tilde{\psi}_{k}=\bar{\psi}_{k}+4P_{0}^{3}P_{k}(\eta=0).$

We denote $P_{k}(0)=P_{k}(\eta=0)$, $Q_{k}(0)=Q_{k}(\eta=0)$. Equations
(2.9)-(2.17) can be written as

$\displaystyle\left\\{\begin{array}[]{ll}\partial_{\eta}^{2}\tilde{T}_{0}+\langle\tilde{\psi}_{0}-\tilde{T}_{0}^{4}\rangle=0,\\\
\sin(\theta+\xi)\partial_{\eta}\tilde{\psi}_{0}+\tilde{\psi}_{0}-\tilde{T}_{0}^{4}=0,\end{array}\right.$
(2.20)
$\displaystyle\left\\{\begin{array}[]{ll}\partial_{\eta}^{2}\tilde{T}_{1}+\tilde{\psi}_{1}-4\tilde{T}_{0}^{3}\tilde{T}_{1}=\frac{1}{1-\varepsilon\eta}\partial_{\eta}\bar{T}_{0}+\langle
4(\tilde{T}_{0}^{3}-P_{0}^{3})(P_{1}-P_{1}(0))\rangle,\\\
\sin(\theta+\xi)\partial_{\eta}\tilde{\psi}_{1}+\tilde{\psi}_{1}-4\tilde{T}_{0}^{3}\tilde{T}_{1}=\frac{1}{1-\varepsilon\eta}\cos(\theta+\xi)\partial_{\theta}\bar{\psi}_{0}+4(\tilde{T}_{0}^{3}-P_{0}^{3})(P_{1}-P_{1}(0))\end{array}\right.$
(2.23) $\displaystyle\qquad\vdots$
$\displaystyle\left\\{\begin{array}[]{ll}\partial_{\eta}^{2}\tilde{T}_{k}+\tilde{\psi}_{k}-4\tilde{T}_{0}^{3}\tilde{T}_{k}=\frac{1}{1-\varepsilon\eta}\partial_{\eta}\tilde{T}_{k-1}+\frac{1}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\tilde{T}_{k-2}+\langle
4(\tilde{T}_{0}^{3}-P_{0}^{3})(P_{k}-P_{k}(0))\rangle\\\
\quad+\langle\mathcal{E}(\bar{T}+P,k-1)-\mathcal{E}(P,k-1),\rangle\\\
\sin(\theta+\xi)\partial_{\eta}\tilde{\psi}_{k}+\tilde{\psi}_{k}-4\tilde{T}_{0}^{3}\tilde{T}_{k}=\frac{1}{1-\varepsilon\eta}\cos(\theta+\xi)\partial_{\theta}\bar{\psi}_{k-1}+4(\tilde{T}_{0}^{3}-P_{0}^{3})(P_{k}-P_{k}(0))\\\
\quad+\mathcal{E}(\bar{T}+P,k-1)-\mathcal{E}(P,k-1).\end{array}\right.$ (2.28)

The boundary conditions are

$\displaystyle\tilde{T}_{0}(0,\theta)=T_{b}(\theta),\quad\tilde{\psi}_{0}(0,\theta,\xi)=\psi_{b}(\theta,\xi),\quad\text{for
}\sin(\theta+\xi)>0,$

and

$\displaystyle\tilde{T}_{k}(0,\theta)=0,\quad\tilde{\psi}_{k}(0,\theta,\xi)=\beta\cdot\nabla\psi_{k-1}(0)-\mathcal{E}(T(0),k-1),\quad\text{for
}\sin(\theta+\xi)>0.$

The reason why the boundary condition for $\tilde{\psi}_{k}$ is not zero is
because we define
$\tilde{\psi}_{k}=\bar{\psi}_{k}+4P_{0}^{3}P_{k}(0)=\bar{\psi}_{k}+4P_{0}^{3}T_{k}(0)$
so that

$\displaystyle\bar{\psi}_{k}(\eta=0)+\psi_{k}(\varepsilon\eta=0)$
$\displaystyle=\tilde{\psi}_{k}(0)-4P_{0}^{3}T_{k}(0)+\psi_{k}(0)$
$\displaystyle=\beta\cdot\nabla\psi_{k-1}(0)-\mathcal{E}(T(0),k-1)-4P_{0}^{3}T_{k}(0)+\psi_{k}(0)=0,$

due to the relation
$\psi_{k}=-\beta\cdot\nabla\psi_{k-1}+4T_{0}^{3}T_{k}+\mathcal{E}(T,k-1)$.
Here we call (2.20)-(2.28) the _Milne problems_ corresponding to (1.1)-(1.2)
and (2.9)-(2.17) the _boundary layer correction_.

###  2.3. Construction of approximate solutions

Let $\delta>0$ be a small constant. Define the cut-off function

$\displaystyle\chi_{0}(r)=\left\\{\begin{array}[]{ll}1,&\text{for }0\leq
r\leq\tfrac{1}{2}\delta,\\\ 0,&\text{for }r\geq\tfrac{3}{4}\delta,\\\
\in(0,1),&\text{otherwise},\end{array}\right.$

and

$\displaystyle\chi(r)=\left\\{\begin{array}[]{ll}1,&\text{for }0\leq
r\leq\tfrac{1}{4}\delta,\\\ 0,&\text{for }r\geq\tfrac{3}{8}\delta,\\\
\in(0,1),&\text{otherwise}.\end{array}\right.$

We begin by constructing the leading order approximations. The zeroth-order
boundary layer correction $(\bar{T}_{0},\bar{\psi}_{0})$ is provided by
solving

$\displaystyle\left\\{\begin{array}[]{ll}\bar{T}_{0}(\eta,\theta)=\chi(\varepsilon\eta)(\tilde{T}_{0}(\eta,\theta)-\tilde{T}_{0,\infty}(\infty,\theta)),~{}\bar{\psi}_{0}(\eta,\theta,\xi)=\chi(\varepsilon\eta)(\tilde{\psi}_{0}(\eta,\theta,\xi)-\tilde{\psi}_{0,\infty}(\infty,\theta,\xi))\\\
\partial_{\eta}^{2}\tilde{T}_{0}+\langle\tilde{\psi}_{0}-\tilde{T}_{0}^{4}\rangle=0,\\\
\sin(\theta+\xi)\partial_{\eta}\tilde{\psi}_{0}+\tilde{\psi}_{0}-\tilde{T}_{0}^{4}=0,\end{array}\right.$
(2.32)

combined to the following boundary condition

$\tilde{T}_{0}(0,\theta)=T_{b}(\theta),\quad\tilde{\psi}_{0}(0,\theta,\xi)=\psi_{b}(\theta,\xi),\text{
for }\sin(\theta+\xi)>0.$

Assuming sufficient regularity on the boundary data, one can prove as $\eta\to
0$, there exists a constant function
$\tilde{T}_{\infty}=\tilde{T}_{0,\infty}(\theta)$ independent of $\eta$ such
that $\lim_{\eta\to\infty}\tilde{T}_{0}(\eta)=\tilde{T}_{0,\infty}$ and
$\lim_{\eta\to\infty}\tilde{\psi}_{0}(\eta)=\tilde{T}^{4}_{0,\infty}$. This
provides the boundary condition for the interior expansions. The leading order
of the interior expansion is

$\displaystyle\Delta T_{0}+\pi\Delta T_{0}^{4}=0,\quad\text{in }\Omega$ (2.33)
$\displaystyle\psi=T_{0}^{4}\qquad\text{in }\Omega\times\mathbb{S}^{1}$
$\displaystyle T_{0}(x)=\tilde{T}_{0,\infty},\quad\text{for
}x\in\partial\Omega.$

We next construct the first order approximations. The boundary layer
correction $(\bar{T}_{1},\bar{\psi}_{1})$ is given by solving

$\displaystyle\left\\{\begin{array}[]{ll}\bar{T}_{1}(\eta,\theta)=\chi(\varepsilon\eta)(\tilde{T}_{1}(\eta,\theta)-\tilde{T}_{1,\infty}(\infty,\theta)),~{}\bar{\psi}_{1}(\eta,\theta,\xi)=\chi(\varepsilon\eta)(\tilde{\psi}_{1}(\eta,\theta,\xi)-\tilde{\psi}_{1,\infty}(\infty,\theta,\xi))\\\
\\\
\partial_{\eta}^{2}\tilde{T}_{1}+\tilde{\psi}_{1}-4\tilde{T}_{0}^{3}\tilde{T}_{1}=\frac{\chi_{0}(\varepsilon\eta)}{1-\varepsilon\eta}\partial_{\eta}\tilde{T}_{0}\langle
4(\tilde{T}_{0}^{3}-P_{0}^{3})(P_{1}-P_{1}(0))\rangle,\\\ \\\
\sin(\theta+\xi)\partial_{\eta}\tilde{\psi}_{1}+\tilde{\psi}_{1}-4\tilde{T}_{0}^{3}\tilde{T}_{1}=\frac{\chi_{0}(\varepsilon\eta)}{1-\varepsilon\eta}\cos(\theta+\xi)\partial_{\theta}\bar{\psi}_{0}+4(\tilde{T}_{0}^{3}-P_{0}^{3})(P_{1}-P_{1}(0)),\\\
\\\ \text{B.C.:
}\tilde{T}_{1}(0,\theta)=0,\quad\tilde{\psi}_{1}(0,\theta,\xi)=\beta\cdot\nabla\psi_{0}(0),\quad\text{for
}\sin(\theta+\xi)>0.\end{array}\right.$ (2.41)

The interior expansion $(T_{0},\psi_{0})$ is given by

$\displaystyle\Delta
T_{1}+\pi\Delta(4T_{0}^{3}T_{1})=0,\quad\psi_{1}=4T_{0}^{3}T_{1}-\beta\cdot\nabla\psi_{0},\quad\text{in
}\Omega\times\mathbb{S}^{1},$ $\displaystyle
T_{1}(x)=\tilde{T}_{1,\infty},\quad\text{for }x\in\partial\Omega.$

The boundary layer corrections $(\bar{T}_{k},\bar{\psi}_{k})$, $k\geq 1$ are
given by

$\displaystyle\left\\{\begin{array}[]{ll}\bar{T}_{k}(\eta,\theta)=\chi(\varepsilon\eta)(\tilde{T}_{k}(\eta,\theta)-\tilde{T}_{k,\infty}(\infty,\theta)),~{}\bar{\psi}_{k}(\eta,\theta,\xi)=\chi(\varepsilon\eta)(\tilde{\psi}_{k}(\eta,\theta,\xi)-\tilde{\psi}_{k,\infty}(\infty,\theta,\xi))\\\
\\\
\partial_{\eta}^{2}\tilde{T}_{k}+\tilde{\psi}_{k}-4\tilde{T}_{0}^{3}\tilde{T}_{k}=\frac{\chi_{0}(\varepsilon\eta)}{1-\varepsilon\eta}\partial_{\eta}\tilde{T}_{k-1}+\frac{\chi_{0}(\varepsilon\eta)}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\tilde{T}_{k-2}+\langle
4(\tilde{T}_{0}^{3}-P_{0}^{3})(P_{k}-P_{k}(0))\rangle\\\ \\\
\qquad+\langle\mathcal{E}(\bar{T}+P,k-1)-\mathcal{E}(P,k-1),\rangle\\\ \\\
\sin(\theta+\xi)\partial_{\eta}\tilde{\psi}_{k}+\tilde{\psi}_{k}-4\tilde{T}_{0}^{3}\tilde{T}_{k}=\frac{\chi_{0}(\varepsilon\eta)}{1-\varepsilon\eta}\cos(\theta+\xi)\partial_{\theta}\bar{\psi}_{k-1}+4(\tilde{T}_{0}^{3}-P_{0}^{3})(P_{k}-P_{k}(0))\\\
\\\ \qquad+\mathcal{E}(\bar{T}+P,k-1)-\mathcal{E}(P,k-1),\\\ \\\ \text{B.C.:
}\tilde{T}_{k}(0,\theta)=0,\quad\tilde{\psi}_{k}(0,\theta,\xi)=\beta\cdot\nabla\psi_{k-1}(0)-\mathcal{E}(T(0),k-1),\quad\text{for
}\sin(\theta+\xi)>0.\end{array}\right.$

The interior expansions $(T_{k},\psi_{k})$, $k\geq 1$ are solved by

$\displaystyle\Delta
T_{k}+\pi\Delta(4T_{0}^{3}T_{k})=-\pi\Delta\mathcal{E}(T,k-1)+\langle(\beta\cdot\nabla)^{3}\psi_{k-1}\rangle,$
$\displaystyle\psi_{k}=4T_{0}^{3}T_{k}+\mathcal{E}(T,k-1)-\beta\cdot\nabla\psi_{k-1},$
$\displaystyle\text{B.C.: }T_{k}(x)=\tilde{T}_{k,\infty},\quad\text{for
}x\in\partial\Omega.$

It is expected that when $\varepsilon$ is small, the addition of the boundary
layer corrects well describes the boundary layer effects and we should have
for example

$\displaystyle\|T^{\varepsilon}-T_{0}-\bar{T}_{0}\|_{L^{\infty}(\Omega)}=O(\varepsilon),\quad\|\psi^{\varepsilon}-\psi_{0}-\bar{\psi}_{0}\|_{L^{\infty}(\Omega\times\mathbb{S}^{1})}=O(\varepsilon).$

For the flat boundary case, this has been proved in [5]. However, for the unit
disk we consider here, a similar problem arises as the case of linear
transport equation in [10]. When solving (2.41), the term
$\cos(\theta+\xi)\partial_{\theta}\tilde{\psi}_{0}$ appears on the right of
the equation of $\tilde{\psi}_{1}$. In order for $\tilde{\psi}_{1}\in
L^{\infty}(\mathbb{R}_{+}\times[-\pi,\pi)\times[-\pi,\pi))$, we need

$\displaystyle\frac{\chi_{0}(\varepsilon\eta)}{1-\varepsilon\eta}\cos(\theta+\xi)\partial_{\theta}\tilde{\psi}_{0}\in
L^{\infty}(\mathbb{R}_{+}\times[-\pi,\pi)\times[-\pi,\pi)).$ (2.42)

Considering that the support of $\chi_{0}$ depends on $\varepsilon$, we need
$\partial_{\theta}\tilde{\psi}_{0}\in L^{\infty}$. Note that
$\partial_{\theta}\tilde{\psi}_{0}$ is a solution to the equations

$\displaystyle\partial_{\eta}^{2}(\partial_{\theta}\tilde{T}_{0})+\langle\partial_{\theta}\tilde{\psi}_{0}-4\tilde{T}_{0}^{3}\partial_{\theta}\tilde{T}_{0}\rangle=0,$
$\displaystyle\sin(\theta+\xi)\partial_{\eta}(\partial_{\theta}\tilde{\psi}_{0})+\partial_{\theta}\tilde{\psi}_{0}-4\tilde{T}_{0}^{3}\partial_{\theta}\tilde{T}_{0}=-\cos(\theta+\xi)\partial_{\eta}\tilde{\psi}_{0}.$

Thus in order for $\partial_{\theta}\tilde{\psi}_{0}\in L^{\infty}$ we need
$\partial_{\eta}\tilde{\psi}_{0}\in L^{\infty}$. However, this fails to be
hold, according to the following lemma.

######  Lemma 2.

For the Milne problem

$\displaystyle\left\\{\begin{array}[]{lll}&\partial_{\eta}^{2}\tilde{T}_{0}+\langle\tilde{\psi}_{0}-\tilde{T}_{0}^{4}\rangle=0,\\\
&\sin(\theta+\xi)\partial_{\eta}\tilde{\psi}_{0}+\tilde{\psi}_{0}-\tilde{T}_{0}^{4}=0,\\\
&\mbox{B.C.:}~{}\tilde{T}_{0}(0,\theta)=T_{b}(\theta),\quad\tilde{\psi}_{0}(0,\theta,\xi)=\psi_{b}(\theta,\xi),\quad\text{for
}\sin(\theta+\xi)>0,\end{array}\right.$ (2.46)

if $T_{b}=0$, $\psi_{b}(\theta,\xi)=\cos(2(\theta+\xi))$, then

$\displaystyle\partial_{\eta}\tilde{\psi}_{0}\not\in
L^{\infty}([0,\infty)\times[-\pi,\pi)\times[-\pi,\pi)).$

###### Proof.

For $\eta=0$, according to (2.46)

$\displaystyle\partial_{\eta}\tilde{\psi}_{0}(0,\theta,\xi)=-\frac{\tilde{\psi}_{0}(0,\theta,\xi)-\tilde{T}_{0}^{4}(0,\theta)}{\sin(\theta+\xi)}.$

If we take $\theta+\xi=0$, then

$\displaystyle\lim_{\xi\to-\theta^{+}}\partial_{\eta}\tilde{\psi}_{0}(0,\theta,\xi)=-\lim_{\xi\to-\theta^{+}}\frac{\tilde{\psi}_{0}(0,\theta,\xi)-\tilde{T}_{0}^{4}(0,\theta)}{\sin(\theta+\xi)}=-\frac{1-0}{0}=-\infty,$

hence $\partial_{\eta}\tilde{\psi}_{0}\not\in
L^{\infty}([0,\infty)\times[-\pi,\pi)\times[-\pi,\pi))$. ∎

###  2.4. Geometric corrections

As can be seen from above analysis, the term (2.42) does not lie in
$L^{\infty}$. In order to overcome this difficulty, we follow the work [10]
and introduce a geometric correction to the boundary layer corrections. First
we introduce the transformation of variables $(\theta,\xi)\to(\theta,\phi)$
defined by

$\displaystyle\left\\{\begin{array}[]{ll}\theta=\theta,\\\
\phi=\theta+\xi.\end{array}\right.$

With the new coordinate, $\partial_{\theta}f(\theta,\xi)$ becomes

$\displaystyle\partial_{\theta}f(\theta,\xi)=\partial_{\theta}f(\theta,\phi)+\partial_{\phi}f(\theta,\phi),$

and $\partial_{\theta}^{2}f$ becomes

$\displaystyle\partial_{\theta}^{2}f=\partial_{\theta}^{2}f(\theta,\phi)+\partial_{\phi}^{2}f(\theta,\phi)+2\partial_{\theta\phi}f(\theta,\phi).$

Note that any function $h=h(\theta)$ remains the same in the new coordinate.

Equations (2.4)-(2.5) become

$\displaystyle\partial_{\eta}^{2}T^{i}-\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}T^{i}+\frac{\varepsilon^{2}}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}T^{i}+\langle\psi^{i}-(T^{o}+T^{i})^{4}+(T^{o})^{4}\rangle=0,$
$\displaystyle\sin\phi\partial_{\eta}\psi^{i}-\frac{\varepsilon}{1-\varepsilon\eta}\cos\phi(\partial_{\theta}\psi^{i}+\partial_{\phi}\psi^{i})+\psi^{i}-(T^{o}+T^{i})^{4}+(T^{o})^{4}=0.$

Taking the ansatz (2.6) into the above equations and collect the terms with
the same order while treating the terms
$\varepsilon^{2}\partial_{\phi}^{2}\bar{T}_{k}$,
$\varepsilon\partial_{\phi}\bar{\psi}_{k}$ to be $O(1)$ gives

$\displaystyle\left\\{\begin{array}[]{ll}\partial_{\eta}^{2}\bar{T}_{0}-\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}\bar{T}_{0}+\bar{\psi}_{0}-(\bar{T}_{0}+P_{0})^{4}+P_{0}^{4}\rangle=0,\\\
\sin\phi\partial_{\eta}\bar{\psi}_{0}-\frac{\varepsilon}{1-\varepsilon\eta}\cos\phi\partial_{\phi}\bar{\psi}_{0}+\bar{\psi}_{0}-(\bar{T}_{0}+P_{0})^{4}+P_{0}^{4}=0,\end{array}\right.$
(2.47)
$\displaystyle\left\\{\begin{array}[]{ll}\partial_{\eta}^{2}\bar{T}_{1}-\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}\bar{T}_{1}+\langle\bar{\psi}_{1}-4(P_{0}+\bar{T}_{0})^{3}(P_{1}+\bar{T}_{1})+4P_{0}^{3}P_{1}\rangle=0,\\\
\\\
\sin\phi\partial_{\eta}\bar{\psi}_{1}-\frac{\varepsilon}{1-\varepsilon\eta}\cos\phi\partial_{\phi}\bar{\psi}_{1}+\bar{\psi}_{1}-4(P_{0}+\bar{T}_{0})^{3}(P_{1}+\bar{T}_{1})+4P_{0}^{3}P_{1}=\frac{1}{1-\varepsilon\eta}\cos(\theta+\xi)\partial_{\theta}\bar{\psi}_{0},\\\
\end{array}\right.$ $\displaystyle\qquad\vdots$
$\displaystyle\left\\{\begin{array}[]{ll}\partial_{\eta}^{2}\bar{T}_{k}-\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}\bar{T}_{k}+\langle\bar{\psi}_{k}-4(P_{0}+\bar{T}_{0})^{3}(P_{k}+\bar{T}_{k})+4P_{0}^{3}P_{k}\rangle=-\frac{1}{(1-\varepsilon\eta)^{2}}\partial_{\theta}\bar{T}_{k-2}\\\
\\\ \qquad+\langle\mathcal{E}(P+\bar{T},k-1)-\mathcal{E}(P,k-1)\rangle,\\\ \\\
\sin\phi\partial_{\eta}\bar{\psi}_{k}-\frac{\varepsilon}{1-\varepsilon\eta}\cos\phi\partial_{\phi}\bar{\psi}_{k}+\bar{\psi}_{k}-4(P_{0}+\bar{T}_{0})^{3}(P_{k}+\bar{T}_{k})+4P_{0}^{3}P_{k}=\frac{1}{1-\varepsilon\eta}\cos(\theta+\xi)\partial_{\theta}\bar{\psi}_{k-1}\\\
\\\ \qquad+\mathcal{E}(P+\bar{T},k-1)-\mathcal{E}(P,k-1),\end{array}\right.$

In the above, the terms

$\displaystyle-\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}\bar{T}_{k},\quad-\frac{\varepsilon}{1-\varepsilon\eta}\cos\phi\partial_{\phi}\bar{\psi}_{k}$

are included in the boundary layer problem, which is used to overcome the
singularity of the normal derivatives of boundary layer solutions. The second
term was introduced in [10]. Here since we need to consider the elliptic
equation, the first term should also be introduced.

Following a similar procedure, we can reformulate the equations using
$\tilde{T}_{k},\tilde{\psi}_{k}$. Define $F(\varepsilon;\eta)$ by

$\displaystyle
F(\varepsilon;\eta)=-\frac{\varepsilon\chi_{0}(\varepsilon\eta)}{1-\varepsilon\eta}.$
(2.48)

The approximate solutions can be constructed as follows.

The zeroth-order boundary layer correction
$\bar{T}_{0}^{\varepsilon},\bar{\psi}_{0}^{\varepsilon}$ with geometric
correction is given by

$\displaystyle\left\\{\begin{array}[]{ll}\bar{T}^{\varepsilon}_{0}(\eta,\theta)=\chi(\varepsilon\eta)(\tilde{T}^{\varepsilon}_{0}(\eta,\theta)-\tilde{T}_{0,\infty}(\infty,\theta)),~{}\bar{\psi}^{\varepsilon}_{0}(\eta,\theta,\xi)=\chi(\varepsilon\eta)(\tilde{\psi}^{\varepsilon}_{0}(\eta,\theta,\xi)-\tilde{\psi}^{\varepsilon}_{0,\infty}(\infty,\theta,\xi))\\\
\\\
\partial_{\eta}^{2}\tilde{T}^{\varepsilon}_{0}+F(\varepsilon;\eta)\partial_{\eta}\tilde{T}^{\varepsilon}_{0}+\langle\tilde{\psi}_{0}^{\varepsilon}-(\tilde{T}^{\varepsilon}_{0})^{4}\rangle=0,\\\
\\\
\sin\phi\partial_{\eta}\tilde{\psi}_{0}^{\varepsilon}+F(\varepsilon;\eta)\cos\phi\partial_{\phi}\tilde{\psi}_{0}^{\varepsilon}+\tilde{\psi}_{0}^{\varepsilon}-(\tilde{T}_{0}^{\varepsilon})^{4}=0,\\\
\\\ \text{B.C.:
}\tilde{T}^{\varepsilon}_{0}(0,\theta)=T_{b}(\theta),\quad\tilde{\psi}_{0}^{\varepsilon}(0,\theta,\phi)=\psi_{b}(\theta,\phi),\text{
for }\sin\phi>0.\end{array}\right.$ (2.56)

The first order boundary layer correction
$\bar{T}_{1}^{\varepsilon},\bar{\psi}_{1}^{\varepsilon}$ with geometric
correction is given by

$\displaystyle\left\\{\begin{array}[]{ll}\bar{T}_{1}^{\varepsilon}(\eta,\theta)=\chi(\varepsilon\eta)(\tilde{T}_{1}^{\varepsilon}(\eta,\theta)-\tilde{T}_{1,\infty}(\infty,\theta)),~{}\bar{\psi}_{1}^{\varepsilon}(\eta,\theta,\xi)=\chi(\varepsilon\eta)(\tilde{\psi}_{1}^{\varepsilon}(\eta,\theta,\xi)-\tilde{\psi}^{\varepsilon}_{1,\infty}(\infty,\theta,\xi))\\\
\\\
\partial_{\eta}^{2}\tilde{T}_{1}^{\varepsilon}+F(\varepsilon;\eta)\partial_{\eta}\tilde{T}_{1}^{\varepsilon}+\tilde{\psi}_{1}^{\varepsilon}-4(\tilde{T}^{\varepsilon}_{0})^{3}\tilde{T}_{1}^{\varepsilon}=\langle
4((\tilde{T}^{\varepsilon}_{0})^{3}-(P_{0}^{\varepsilon})^{3})(P_{1}^{\varepsilon}-P_{1}^{\varepsilon}(0))\rangle,\\\
\\\
\sin\phi\partial_{\eta}\tilde{\psi}_{1}^{\varepsilon}+F(\varepsilon;\eta)\cos\phi\partial_{\phi}\tilde{\psi}_{1}^{\varepsilon}+\tilde{\psi}_{1}^{\varepsilon}-4(\tilde{T}_{0}^{\varepsilon})^{3}\tilde{T}_{1}^{\varepsilon}=\frac{\chi_{0}(\varepsilon\eta)}{1-\varepsilon\eta}\cos\phi\partial_{\theta}\bar{\psi}_{0}^{\varepsilon}\\\
\\\
\qquad+4((\tilde{T}^{\varepsilon}_{0})^{3}-(P_{0}^{\varepsilon})^{3})(P_{1}^{\varepsilon}-P_{1}^{\varepsilon}(0)),\\\
\\\ \text{B.C.:
}\tilde{T}^{\varepsilon}_{1}(0,\theta)=0,\quad\tilde{\psi}_{1}^{\varepsilon}(0,\theta,\phi)=\beta\cdot\nabla\psi^{\varepsilon}_{0}(0),\quad\text{for
}\sin\phi>0.\end{array}\right.$ (2.66)

The boundary layer corrections with geometric corrections
$(\bar{T}^{\varepsilon}_{k},\bar{\psi}_{k}^{\varepsilon})$, $k\geq 1$ are
given by

$\displaystyle\left\\{\begin{array}[]{ll}\bar{T}_{k}^{\varepsilon}(\eta,\theta)=\chi(\varepsilon\eta)(\tilde{T}_{k}^{\varepsilon}(\eta,\theta)-\tilde{T}^{\varepsilon}_{k,\infty}(\infty,\theta)),~{}\bar{\psi}_{k}^{\varepsilon}(\eta,\theta,\xi)=\chi(\varepsilon\eta)(\tilde{\psi}_{k}^{\varepsilon}(\eta,\theta,\xi)-\tilde{\psi}^{\varepsilon}_{k,\infty}(\infty,\theta,\xi))\\\
\\\
\partial_{\eta}^{2}\tilde{T}^{\varepsilon}_{k}+F(\varepsilon;\eta)\partial_{\eta}\tilde{T}^{\varepsilon}_{k}+\tilde{\psi}_{k}^{\varepsilon}-4(\tilde{T}_{0}^{\varepsilon})^{3}\tilde{T}_{k}^{\varepsilon}=+\frac{\chi_{0}(\varepsilon\eta)}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\tilde{T}_{k-2}^{\varepsilon}+\langle
4((\tilde{T}^{\varepsilon}_{0})^{3}-(P_{0}^{\varepsilon})^{3})(P^{\varepsilon}_{k}-P^{\varepsilon}_{k}(0))\rangle\\\
\\\
\qquad+\langle\mathcal{E}(\bar{T}^{\varepsilon}+P^{\varepsilon},k-1)-\mathcal{E}(P^{\varepsilon},k-1)\rangle\\\
\\\
\sin\phi\partial_{\eta}\tilde{\psi}_{k}^{\varepsilon}+F(\varepsilon;\eta)\cos\phi\partial_{\phi}\tilde{\psi}^{\varepsilon}_{k}+\tilde{\psi}_{k}^{\varepsilon}-4(\tilde{T}_{0}^{\varepsilon})^{3}\tilde{T}_{k}^{\varepsilon}=\frac{\chi_{0}(\varepsilon\eta)}{1-\varepsilon\eta}\cos\phi\partial_{\theta}\bar{\psi}_{k-1}\\\
\\\
\qquad+4((\tilde{T}^{\varepsilon}_{0})^{3}-(P_{0}^{\varepsilon})^{3})(P^{\varepsilon}_{k}-P^{\varepsilon}_{k}(0))+\mathcal{E}(\bar{T}^{\varepsilon}+P^{\varepsilon},k-1)-\mathcal{E}(P^{\varepsilon},k-1),\\\
\\\ \text{B.C.:
}\tilde{T}^{\varepsilon}_{k}(0,\theta)=0,\quad\tilde{\psi}_{k}^{\varepsilon}(0,\theta,\phi)=\beta\cdot\nabla\psi_{k-1}(0)-\mathcal{E}(T(0),k-1),\quad\text{for
}\sin\phi>0.\end{array}\right.$ (2.67)

We will show later that

$\lim_{\eta\to\infty}\tilde{T}_{k}^{\varepsilon}(\eta,\theta)=\tilde{T}_{k,\infty}^{\varepsilon},\quad\text{and}\,\,\,\lim_{\eta\to\infty}\tilde{\psi}_{k}^{\varepsilon}(\eta,\theta,\phi)=\tilde{\psi}^{\varepsilon}_{k,\infty}\quad\text{exist.}$

The leading order interior expansion
$({T}_{0}^{\varepsilon},{\psi}_{0}^{\varepsilon})$ is given by

$\displaystyle\left\\{\begin{array}[]{lll}\Delta
T^{\varepsilon}_{0}+\pi\Delta(T^{\varepsilon}_{0})^{4}=0,\quad\text{in
}\Omega,\\\ \psi_{0}^{\varepsilon}=(T_{0}^{\varepsilon})^{4}\quad\text{in
}\Omega\times\mathbb{S}^{1}\\\ \text{B.C.:
}T_{0}^{\varepsilon}(x)=\tilde{T}^{\varepsilon}_{0,\infty},\quad\text{for
}x\in\partial\Omega.\end{array}\right.$ (2.71)

The first order interior expansion
$(T_{1}^{\varepsilon},\psi_{1}^{\varepsilon})$ is given by

$\displaystyle\left\\{\begin{array}[]{lll}\Delta
T_{1}^{\varepsilon}+\pi\Delta(4(T_{0}^{\varepsilon})^{3}T_{1}^{\varepsilon})=0,\quad\text{in
}\Omega\times\mathbb{S}^{1},\\\
\psi_{1}^{\varepsilon}=4(T_{0}^{\varepsilon})^{3}T_{1}^{\varepsilon}-\beta\cdot\nabla\psi_{0}^{\varepsilon},\quad\text{in
}\Omega\times\mathbb{S}^{1},\\\ \text{B.C.:
}T_{1}^{\varepsilon}(x)=\tilde{T}^{\varepsilon}_{1,\infty},\quad\text{for
}x\in\partial\Omega.\end{array}\right.$ (2.75)

The higher order interior expansion
$(T_{k}^{\varepsilon},\psi_{k}^{\varepsilon})$, $k\geq 1$ is given by

$\displaystyle\left\\{\begin{array}[]{lll}\Delta
T_{k}^{\varepsilon}+\pi\Delta(4(T_{0}^{\varepsilon})^{4}T_{k}^{\varepsilon})=\langle(\beta\cdot\nabla)^{3}\psi_{k-1}\rangle-\pi\Delta\mathcal{E}(T^{\varepsilon},k-1),\\\
\psi_{k}^{\varepsilon}=4(T_{0}^{\varepsilon})^{3}T_{k}^{\varepsilon}+\mathcal{E}(T^{\varepsilon},k-1)-\beta\cdot\nabla\psi_{k-1}^{\varepsilon},\\\
\text{B.C.:
}T_{k}^{\varepsilon}(x)=\tilde{T}^{\varepsilon}_{k,\infty},\quad\text{for
}x\in\partial\Omega.\end{array}\right.$ (2.79)

Here we call the equations of
$(\tilde{T}_{k}^{\varepsilon},\tilde{\psi}_{k}^{\varepsilon})$ the _Milne
problem with geometric corrections_ and
$(\bar{T}^{\varepsilon},\bar{\psi}^{\varepsilon})$ the _boundary layer
solution with geometric corrections_ , and
$(T_{k}^{\varepsilon},\psi_{k}^{\varepsilon})$ the interior solutions.

###  2.5. Approximate solution with geometric corrections

We thus obtain the approximate solution with geometric corrections by adding
the interior solution and the boundary layer solution with geometric
corrections as

$\displaystyle
T^{a,\varepsilon}=\sum_{k=0}^{N}\varepsilon^{k}(T_{k}^{\varepsilon}+\bar{T}_{k}^{\varepsilon}),\quad\psi^{a,\varepsilon}=\sum_{k=0}^{N}\varepsilon^{k}(\psi_{k}^{\varepsilon}+\bar{\psi}_{k}^{\varepsilon}),$
(2.80)

where $T_{k},\psi_{k}$ are given by (2.71)-(2.75) and
$\bar{T}_{k}^{\varepsilon},\bar{\psi}_{k}^{\varepsilon}$ are given by
(2.56)-(2.67).

One can show that $(T^{a,\varepsilon},\psi^{a,\varepsilon})$ satisfies system
(1.1)-(1.2) with boundary conditions (1.3)-(1.4) in the perturbative sense.

######  Lemma 3.

Let $(T^{a,\varepsilon},\psi^{a,\varepsilon})$ be the pair of functions
constructed above. Then formally, it satisfy

$\displaystyle\varepsilon^{2}\Delta
T^{a,\varepsilon}+\langle\psi^{a,\varepsilon}-(T^{a,\varepsilon})^{4}\rangle=\mathcal{R}_{1}(T^{a,\varepsilon},\psi^{a,\varepsilon}),$
$\displaystyle\varepsilon\beta\cdot\nabla\psi^{a,\varepsilon}+\psi^{a,\varepsilon}-(T^{a,\varepsilon})^{4}=\mathcal{R}_{2}(T^{a,\varepsilon},\psi^{a,\varepsilon}),$

with boundary conditions

$\displaystyle T^{a,\varepsilon}(x)=T_{b}(x),\quad\text{for any
}x\in\partial\Omega,$
$\displaystyle\psi^{a,\varepsilon}(x,\beta)=\psi_{b}(x,\beta),\quad\text{for
any }(x,\beta)\in\Gamma_{-},$

where
$\mathcal{R}_{1}(T^{a,\varepsilon},\psi^{a,\varepsilon})=O(\varepsilon^{N+1})$,
$\mathcal{R}_{1}(T^{a,\varepsilon},\psi^{a,\varepsilon})=O(\varepsilon^{N+1})$.

The lemma will be proved later in section 5 after we show the property of the
Milne problems with geometric corrections, which is needed to obtain the error
estimates.

##  3\. Nonlinear Milne problem with geometric correction

One clearly notes that the leading order of the Milne problem with geometric
corrections (2.56) is nonlinear and the higher orders problem (2.66), (2.67)
are linear, we will consider these two problems separately.

In this section, we first study the nonlinear Milne problem (2.56). We rewrite
the system as follows

$\displaystyle\partial_{\eta}^{2}G^{\varepsilon}+F(\varepsilon;\eta)\partial_{\eta}G^{\varepsilon}+\langle\varphi^{\varepsilon}-(G^{\varepsilon})^{4}\rangle=0,$
(3.1)
$\displaystyle\sin\phi\partial_{\eta}\varphi^{\varepsilon}+F(\varepsilon;\eta)\cos\phi\partial_{\phi}\varphi^{\varepsilon}+\varphi^{\varepsilon}-(G^{\varepsilon})^{4}=0,$
(3.2)

with boundary conditions

$\displaystyle G^{\varepsilon}(0)=G_{b},$ (3.3)
$\displaystyle\varphi^{\varepsilon}(0,\phi)=\varphi_{b}(\phi),\quad\text{for
}\sin\phi>0.$ (3.4)

Note that here we drop the dependence of variables on $\theta$ since it is
irrelevant here. This section is devoted to the proof of the existence for the
above nonlinear system (3.1)-(3.4).

######  Theorem 4.

Given $0\leq G_{b}\leq\gamma$, $0\leq\varphi_{b}(\phi)\leq\gamma^{4}$ for
$\sin\phi>0$. Then there exists a weak solution
$(\varphi^{\varepsilon},G^{\varepsilon})\in L^{2}_{\rm loc}([0,\infty))\times
L^{2}_{\rm loc}([0,\infty)\times[-\pi,\pi))$ to system (3.1)-(3.2) with
boundary conditions (3.3)-(3.4). Moreover, the solutions satisfy the following
estimates

$\displaystyle\int_{0}^{\infty}4(G^{\varepsilon})^{3}|\partial_{\eta}{G^{\varepsilon}}|^{2}d\eta+\frac{1}{2}\int_{0}^{\infty}\int_{-\pi}^{\pi}(\varphi^{\varepsilon}-(G^{\varepsilon})^{4})^{2}d\phi
d\eta+\frac{1}{2}\int_{\sin\phi<0}|\sin\phi|(\varphi^{\varepsilon}(0,\phi)-G_{b}^{4})^{2}d\phi$
$\displaystyle\quad\leq\frac{1}{2}\int_{\sin\phi>0}\sin\phi(\varphi_{b}(\phi)-G_{b}^{4})^{2}d\phi,$
(3.5)

and for any $0<\alpha<1/4$,

$\displaystyle\int_{0}^{\infty}e^{2\alpha\eta}4(G^{\varepsilon})^{3}|\partial_{\eta}G^{\varepsilon}|^{2}d\eta+\frac{1}{4}\int_{0}^{\infty}\int_{-\pi}^{\pi}e^{2\alpha\eta}(\varphi^{\varepsilon}-(G^{\varepsilon})^{4})^{2}d\phi
d\eta+\frac{1}{2}\int_{\sin\phi<0}|\sin\phi|(\varphi^{\varepsilon}_{B}(0,\phi)-G^{4}_{b})^{2}d\phi$
$\displaystyle\qquad=\frac{1}{2}\int_{\sin\phi>0}\sin\phi(\varphi_{b}-G_{b}^{4})^{2}d\phi.$
(3.6)

Furthermore, the limit
$G^{\varepsilon}_{\infty}=\lim_{\eta\to\infty}G^{\varepsilon}(\eta)$ exists
and

$\displaystyle|G^{\varepsilon}(\eta)-G^{\varepsilon}_{\infty}|\leq
Ce^{-\alpha\eta},$ (3.7)

where $C>0$ is a constant only depending on $G_{b}$ and $\varphi_{b}$.

The proof of the above theorem follows the similar procedure as the case for
flat boundary without geometric corrections in [6]. We first establish the
existence in a bounded domain and then use uniform weighted estimates to carry
out the solutions to half-space. However, due to the additional terms by
geometric corrections, we need to use properties of the function
$F(\varepsilon;\eta)$ defined by (2.48) to obtain the existence proofs.

The following property of $F(\varepsilon;\eta)$ was proved in [10, Lemma 4.1].

######  Lemma 5 ([10]).

There exists a potential function $V(\varepsilon;\eta)$ satisfying
$V(\varepsilon;0)=0$ and
$F(\varepsilon;\eta)=-\partial_{\eta}V(\varepsilon;\eta)$. Moreover,
$V(\varepsilon;\eta)$ is a monotonically increasing function of $\eta$
satisfying

$\displaystyle 0\leq V(\varepsilon;\eta)\leq-\log(1-\tfrac{3}{4}\delta),$

and that $V_{\infty}(\varepsilon):=\lim_{\eta\to\infty}V(\varepsilon;\eta)$
exists and is independent of $\varepsilon$. For any $\sigma>0$, $\eta>0$,

$\displaystyle e^{V(\eta+\sigma)}-e^{V(\eta)}\leq
1+\frac{4}{4-3\delta}\varepsilon\sigma.$

For $F(\varepsilon;\eta)$, the following inequalities hold:

$\displaystyle-\log(1-\tfrac{3}{4}\delta)\leq\int_{y}^{z}F(\varepsilon;\eta)d\eta\leq
0,\quad\text{for any }0\leq y\leq z\leq\infty,$ (3.8)
$\displaystyle\int_{0}^{\infty}\int_{\eta}^{\infty}|F(y)^{2}|dyd\eta\leq\frac{3\delta}{4-3\delta}+\log(1-\tfrac{3}{4}\delta),$
(3.9)
$\displaystyle\int_{0}^{\infty}|F(\varepsilon;\eta)|^{2}d\eta\leq\frac{3\delta}{4-3\delta}\varepsilon,$
(3.10)
$\displaystyle\|F(\varepsilon;\eta)\|_{L^{\infty}(\mathbb{R}_{+})}\leq\frac{4}{4-3\delta}\varepsilon.$
(3.11)

###  3.1. Existence on the bounded interval

We first consider the nonlinear Milne problem on the bounded domain with
$\eta\in[0,B]$, $\phi\in[-\pi,\pi)$.

$\displaystyle\partial_{\eta}^{2}G^{\varepsilon}_{B}+F(\varepsilon;\eta)\partial_{\eta}G^{\varepsilon}_{B}+\langle\varphi^{\varepsilon}_{B}-(G^{\varepsilon}_{B})^{4}\rangle=0,$
(3.12)
$\displaystyle\sin\phi\partial_{\eta}\varphi^{\varepsilon}_{B}+F(\varepsilon;\eta)\cos\phi\partial_{\phi}\varphi^{\varepsilon}_{B}+\varphi^{\varepsilon}_{B}-(G^{\varepsilon}_{B})^{4}=0,$
(3.13)

subject to the boundary conditons

$\displaystyle
G_{B}^{\varepsilon}(0)=G_{b},\quad\partial_{\eta}G^{\varepsilon}_{B}(B)=0,$
(3.14)
$\displaystyle\varphi^{\varepsilon}_{B}(0,\phi)=\varphi_{b}(\phi),\quad\varphi^{\varepsilon}_{B}(B,\phi)=\varphi^{\varepsilon}_{B}(B,-\phi),\quad\text{for
}\sin\phi>0.$ (3.15)

The existence of solutions to the above problem is guaranteed by the following
lemma.

######  Lemma 6.

Assume $0\leq G_{b}\leq\gamma$ and $0\leq\varphi_{b}\leq\gamma^{4}$ for some
constant $\gamma>0$. Then there exists a unique solution
$(G_{B}^{\varepsilon},\varphi_{B}^{\varepsilon})\in C^{2}([0,B])\times
C^{1}([0,B]\times[-\pi,\pi))$ to system (3.12)-(3.13) with boundary conditions
(3.14)-(3.15), and the solution satisfies

$\displaystyle 0\leq G_{B}^{\varepsilon}(\eta)\leq\gamma,\quad
0\leq\varphi^{\varepsilon}_{B}(\eta,\phi)\leq\gamma^{4},\quad\text{for
}\eta\in[0,B],\,\phi\in[-\pi,\pi)$

In order to show the Lemma 6, we follow our previous work on the Milne problem
[6]. The proof for the flat case in [6] relys on the monotonicity of solutions
for the elliptic equation and transport equation. Here we show that the
geometric correction does not alter the monotonicity of the solutions. We
start by proving the monotonicity of the elliptic equation (3.12).

######  Lemma 7.

Suppose $0\leq G_{b}\leq\gamma$ for some constant $\gamma>0$. Suppose
$\zeta=\zeta(x)$ is an increasing monotonic function defined on
$x\in\mathbb{R}$ and $g\in C([0,B])$ is a continuous bounded function
satisfying $0\leq g\leq\zeta(\gamma)$. Then there exists a unique bounded
solution $G^{\varepsilon}_{B}\in C^{2}([0,B])$ to the equation

$\displaystyle-\partial_{\eta}^{2}G+F(\varepsilon;\eta)\partial_{\eta}G+\zeta(G)=g(\eta),\quad\text{on
}\eta\in[0,B],$ (3.16)

with boundary conditions

$\displaystyle G(0)=G_{b},\quad\partial_{\eta}G(B)=0,$ (3.17)

and the solution satisfies $0\leq G(\eta)\leq\gamma$ for any $\eta\in[0,B]$.

Moreover, let $G_{1}$, $G_{2}$ be two solutions of the above equation
corresponding to source terms $g_{1}$, $g_{2}$ and boundary data $G_{b1}$,
$G_{b2}$, respectively. Suppose $0\leq g_{1}\leq g_{2}\leq\gamma$ on $[0,B]$
and $0\leq G_{b1}\leq G_{b2}\leq\gamma$, then $0\leq G_{1}\leq
G_{2}\leq\gamma$ on $[0,B]$.

###### Proof.

The proof of the lemma is based on the Leray-Schauder fixed point theorem and
the maximum principle.

_Existence._ Let $h\in C([0,B])$ be the solution to the problem

$\displaystyle-\partial_{\eta}^{2}f-F(\varepsilon;\eta)\partial_{\eta}f=h,\text{
in [0,B]},\quad f(0)=0,\quad\partial_{\eta}f(B)=0.$

Then the solution can be explicitly solved by

$\displaystyle
f(\eta)=\int_{0}^{\eta}e^{-\int_{0}^{s}F(\varepsilon;p)dp}\left(\int_{0}^{s}-e^{\int_{0}^{q}F(\varepsilon;p)dp}h(q)dq+\int_{0}^{B}e^{\int_{0}^{q}F(\varepsilon;p)dp}h(q)dq\right)ds.$
(3.18)

Due to (3.8),
$|\int_{0}^{s}F(\varepsilon;p)dp|\leq-\log(1-\tfrac{3}{4}\delta)$ is bounded,
so $e^{\pm\int_{0}^{s}F(\varepsilon;p)dp}$ is also bounded and the above
formula is well-defined. Denote the above relation by $f=\mathcal{G}h$ with
$\mathcal{G}$ being a linear operator mapping from $\mathcal{C}([0,B])$ to
$C^{2}([0,B])$. We next show the nonlinear operator $\mathcal{N}$ defined by

$\displaystyle\mathcal{N}G:=\mathcal{G}(g-\zeta(G))+G_{b},$

has a fixed point. First, the operator
$\mathcal{N}:C([0,B])\times\mathbb{C}^{2}([0,B])$ is a compact mapping from
the Banach space $(C[0,B])$ to $C([0,B])$, due to the compact embedding
$C^{2}([0,B])\subset C([0,B])$. Next we show the set

$\displaystyle\\{G\in C([0,B]):G=\sigma\mathcal{N}G,\quad\text{for some
}\sigma\in[0,1]\\}$ (3.19)

is bounded. To show this, suppose $G$ is in this set, then

$\displaystyle-\partial_{\eta}^{2}G-F(\varepsilon;\eta)\partial_{\eta}G+\sigma\zeta(G)=\sigma
g,\quad\text{ in }[0,B]$ $\displaystyle
G(0)=G_{b},\quad\partial_{\eta}G(B)=0.$

First we show $G\geq 0$. Otherwise the minimum value of $G(\eta)$ on $[0,B]$
is small than $0$. Let $\eta_{\min}$ be the minimum point, then
$G(\eta_{\min})<0$. If $\eta_{\min}=0$ then $G(\eta_{\min})=G_{b}\geq 0$. If
$\eta_{\min}\in(0,B]$, then $\partial_{\eta}G(\eta_{\min})=0$ and

$\displaystyle-\partial_{\eta}^{2}G(\eta_{\min})=\sigma
g(\eta_{\min})-\sigma\zeta(G(\eta_{\min}))>0,$ (3.20)

due to $\zeta$ being an increasing function and $G(\eta_{\min})<0$. Hence
$G({\eta})$ is locally concave around $\eta_{\min}$, which contradicts to the
assumption that $\eta_{\min}$ is the minimum point. Hence $G(\eta)\geq 0$ for
any $\eta\in[0,B]$.

We can also show $G\leq\gamma$. Otherwise the maxium value of $G(\eta)$ is
larger than $\gamma$. Let $\eta_{\max}$ be the maximum point, then
$G(\eta_{\max})>\gamma$. Since $G_{b}\leq\gamma$, $\eta_{\max}\neq 0$. For
$\eta_{\max}\in(0,B]$, we have $\partial_{\eta}G(\eta_{\max})=0$ and

$\displaystyle-\partial_{\eta}^{2}G(\eta_{\max})=\sigma
g(\eta_{\max})-\sigma\zeta(G(\eta_{\max}))\leq\sigma\zeta(\gamma)-\sigma\zeta(G(\eta_{\max}))<0,$

due to $\zeta$ being an increasing function and
$\zeta(G(\eta_{\max}))>\zeta(\gamma)$. This implies that $G(\eta)$ is locally
strictly convex near $\eta_{\max}$, which contradicts the assumption that
$\eta_{\max}$ is a maximum point. Hence $G\leq\gamma$ holds.

From $0\leq G\leq\gamma$ on $[0,B]$, we have $\|G\|_{C([0,B])}\leq\gamma$ and
the set (3.19) is bounded. Consequently, we can apply the Leray-Schauder fixed
point theorem and conclude that $\mathcal{N}$ has a fixed point in $C([0,B])$.
Let $G=G(\eta)\in C([0,B])$ be a fixed point, then $\mathcal{N}G=G$ and by the
definition of $\mathcal{N}$, $G\in C^{2}([0,B])$ and solves equation (3.16)
with boundary condition (3.17). By the same argument as above we also have
$0\leq G\leq\gamma$ in $[0,B]$.

_Uniqueness._ Note that the Leray-Schauder fixed point theorem only guarantees
the existence for equation (3.16). To show the uniqueness of solutions, we use
the maximum principle. Suppose $G_{1}\neq G_{2}$ be two solutions to (3.16).
Let $\ell:=G_{1}-G_{2}$, suppose $\ell(\eta)$ attains its maximum at
$\eta_{\max}$ and its minimum at $\eta_{\min}$. First we show
$\ell(\eta_{\max})\leq 0$. Otherwise suppose $\ell(\eta_{\max})>0$, then
$\eta_{\max}\neq 0$ since $G_{1}(0)=G_{2}(0)=G_{b}.$ For
$\eta_{\max}\in(0,B]$, we have $\partial_{\eta}\ell(\eta_{\max})=0$, hence

$\displaystyle-\partial_{\eta}^{2}\ell(\eta_{\max})=-(\zeta(G_{1}(\eta_{\max}))-\zeta(G_{2}(\eta_{\max})))<0,$

due to $G_{1}(\eta_{\max})>G_{2}(\eta_{\max})$ and $\zeta$ being an increasing
function. This implies $\ell$ is locally convex near $\eta_{\max}$, which
contradicts the assumption that $\eta_{\max}$ is a maximum point. Therefore,
$\ell(\eta)\leq\ell(\eta_{\max})\leq 0$ for any $\eta\in[0,B]$. By a similar
argument, one can show $\ell(\eta)\geq\ell(\eta_{\min})\geq 0$ and we conclude
that $\ell(\eta)=0$ for $\eta\in[0,B]$. Therefore, $G_{1}(\eta)=G_{2}(\eta)$
on $\eta\in[0,B]$ and the solution to (3.16) is unique.

_Monotonicity._ Let $G_{1}$, $G_{2}$ be two solutions as described in Lemma 7.
Let $\ell:=G_{1}-G_{2}$. We first show the maximum value of $\ell$ cannot be
bigger than zero. Otherwise, suppose $\ell(\eta)$ attains its maximum at
$\eta_{\max}$ with $\ell(\eta_{\max})>0.$ Then due to
$G_{1}(0)-G_{2}(0)=G_{b1}-G_{b2}\leq 0$, $\eta_{\max}\neq 0$. For
$\eta_{\max}\in(0,B]$, we have $\partial_{\eta}\ell(\eta_{\max})=0$ and

$\displaystyle-\partial_{\eta}^{2}\ell(\eta_{\max})=g_{1}(\eta_{\max})-g_{2}(\eta_{\max})-(\zeta(G_{1}(\eta_{\max}))-\zeta(G_{2}(\eta_{\max})))<0,$

due to the assumption $g_{1}\leq g_{2}$ and the monotonicity of $\zeta$. Hence
$\ell(\eta)$ is strictly convex near $\eta_{\max}$, which contradicts the
assumption that $\eta_{\max}$ is a maximum point. Therefore,
$\ell(\eta)\leq\ell(\eta_{\max})\leq 0$ for any $\eta\in[0,B]$. That is,
$G_{1}(\eta)\leq G_{2}(\eta)$ for any $\eta\in[0,B]$. Thus we finish the proof
of the lemma. ∎

######  Remark 8.

Compared to the proof for the flat case in [6], we can see here that adding
$F(\varepsilon;\eta)\partial_{\eta}G$ does not change the maximum principle
since the stationary point $\partial_{\eta}G=0$ vanishes.

Next, we show the monotonicity of the transport equation (3.13).

######  Lemma 9.

Given $\varphi_{b}=\varphi_{b}(\phi)\in L^{\infty}((0,\pi))$. Let
$h=h(\eta,\phi)\in C([0,B]\times[-\pi,\pi))$. Then there exists a unique
solution $\varphi\in C^{1}([0,B]\times[-\pi,\pi))$ to the equation

$\displaystyle\sin\phi\partial_{\eta}\varphi+F(\varepsilon;\eta)\cos\phi\partial_{\phi}\varphi+\varphi=h,\text{
in }[0,B]\times[-\pi,\pi),$ (3.21)

with boundary conditions

$\displaystyle\varphi(0,\phi)=\varphi_{b}(\phi),\quad\varphi(B,\phi)=\varphi(B,-\phi),\quad\text{for
}\sin\phi>0,$ (3.22)

and the solution satisfies

$\displaystyle\|\varphi\|_{L^{\infty}([0,B])}\leq\|h\|_{L^{\infty}([0,B]\times[-\pi,\pi))}+\|\varphi_{b}\|_{L^{\infty}((0,\pi))}.$
(3.23)

Moreover, if $0\leq\varphi_{b}\leq\gamma$ and $0\leq h\leq\gamma$, then the
solution $\varphi\geq 0$ on $[0,B]\times[-\pi,\pi)$. Furthermore, let
$\varphi_{1},\varphi_{2}$ be two solutions of the above equation corresponding
to the source term $h_{1}$, $h_{2}$ and boundary data $\varphi_{b1}$,
$\varphi_{b2}$, respectively. Suppose $0\leq h_{1}\leq h_{2}\leq\gamma$ on
$[0,B]$ and $0\leq\varphi_{b1}\leq\varphi_{b2}\leq\gamma$ on $(0,\pi)$, then
$0\leq\varphi_{1}\leq\varphi_{2}\leq\gamma$ on $[0,B]\times[-\pi,\pi)$.

###### Proof.

To solve (3.21), we use the method of characteristics. Suppose
$\phi=\phi(\eta)$ is the characteristic line given by

$\displaystyle\frac{d}{d\eta}\phi=F(\varepsilon;\eta)\cos\phi=-\cos\phi\partial_{\eta}V(\varepsilon;\eta),\quad\phi(\eta=0)=\phi_{0}$

whose solution is given by

$\displaystyle\cos\phi e^{-V(\varepsilon;\eta)}=\cos\phi_{0}.$

Define the energy $E(\eta,\phi)=\cos\phi e^{-V(\eta)}$. The characteristics
lines for (3.21) is the curves with constant energy. Let the curves are
parameterized by $|E|=e^{-V(\eta_{+})}$. We define
$\phi^{\prime}=\phi^{\prime}(\phi,\eta,\eta^{\prime})$ to denote the solution
of $\cos\phi
e^{-V(\eta)}=\cos\phi^{\prime}e^{-V(\eta^{\prime})}=e^{-V(\eta_{+})}$, which
is for $0\leq\eta^{\prime}\leq\eta_{+}$,

$\displaystyle\phi^{\prime}(\phi,\eta,\eta^{\prime})=\arccos\left(\cos\phi
e^{V(\eta^{\prime})-V(\eta)}\right),$

where $\arccos$ is defined in $[0,\pi)$ as a single-valued function. In order
to extend the above relation to $[-\pi,\pi]$, we use the symmetric property of
$\cos\phi$ and define for $\phi<0$,

$\displaystyle\phi^{\prime}(-\phi,\eta,\eta^{\prime})=-\phi^{\prime}(\phi,\eta,\eta^{\prime}).$

Thus $\phi^{\prime}=\phi^{\prime}(\phi,\eta,\eta^{\prime})$ defines a function
on $[-\pi,\pi]\times[0,B]\times[0,B]$.

Equation (3.21) becomes

$\displaystyle\sin\phi(\eta)\frac{d}{d\eta}\varphi(\eta,\phi(\eta))+\varphi=h,$
(3.24)

with boundary condition

$\displaystyle\varphi(0,\phi_{0})=\varphi_{b}(\phi_{0}),\quad\text{for
}\sin\phi_{0}>0,\quad\varphi(B,\phi(B))=\varphi(B,-\phi(B)),\quad\text{ for
}\sin\phi_{B}>0.$

Based on the values of $\sin\phi$ and $\eta$, we divide into three different
cases.

_Case 1. $\sin\phi>0$._ We integrate (3.24) over $[0,\eta]$ and get

$\displaystyle\varphi(\eta,\phi(\eta))=\varphi_{b}(\phi_{0})e^{-\int_{0}^{\eta}\frac{1}{\sin\phi(\xi)}d\xi}+\int_{0}^{\eta}\frac{h(\xi,\phi^{\prime}(\phi,\eta,\xi))}{\sin\phi(\xi)}e^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi(\rho)}d\rho}d\xi,$

which using $\phi_{0}=\phi^{\prime}(\phi,\eta,0)$,
$\phi(\xi)=\phi^{\prime}(\phi,\eta,\xi)$, can be written as

$\displaystyle\varphi(\eta,\phi)=\varphi_{b}(\phi^{\prime}(\phi,\eta,0))e^{-\int_{0}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\xi)}d\xi}+\int_{0}^{\eta}\frac{h(\xi,\phi^{\prime}(\phi,\eta,\xi))}{\sin\phi^{\prime}(\phi,\eta,\xi)}e^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}d\xi.$
(3.25)

_Case 2. $\sin\phi<0$ and $|E(\phi,\eta)|\leq e^{-V(\varepsilon;B)}$._ In this
situation, we can integrate (3.24) over $[\eta,B]$ to get

$\displaystyle\varphi(B,\phi(B))e^{\int_{0}^{L}\frac{1}{\sin\phi(\xi)}d\xi}-\varphi(\eta,\phi(\eta))e^{\int_{0}^{\eta}\frac{1}{\sin\phi(\xi)}d\xi}=\int_{\eta}^{B}\frac{h(\xi,\phi^{\prime}(\phi,\eta,\xi))}{\sin\phi(\xi)}e^{\int_{0}^{\xi}\frac{1}{\sin\phi(\rho)}d\rho}d\xi.$

Since $\sin\phi<0$, we have
$\phi(\xi)=\phi^{\prime}(\phi,\eta,\xi)=-\phi^{\prime}(-\phi,\eta,\xi)$ and
$\phi(B)=\phi^{\prime}(\phi,B,\xi)=-\phi^{\prime}(-\phi,\eta,B)$. The above
relationship can be rewritten as

$\displaystyle\varphi(\eta,\phi)=\varphi(B,-\phi^{\prime}(-\phi,\eta,L))e^{\int_{\eta}^{B}\frac{1}{-\sin\phi^{\prime}(-\phi,\eta,\xi)}d\xi}-\int_{\eta}^{B}\frac{h(\xi,-\phi^{\prime}(-\phi,\eta,\xi))}{-\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{-\int_{\xi}^{\eta}\frac{1}{-\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi.$

Since $\sin\phi(B)<0$, we have
$\varphi(B,\phi(B))=\varphi(B,-\phi(B))=\varphi(B,\phi^{\prime}(-\phi,\eta,B))$
due to the boundary condition (3.22). By (3.25), we have

$\displaystyle\varphi(B,\phi^{\prime}(-\phi,\eta,B))=\varphi_{b}(\phi^{\prime}(-\phi,\eta,0))e^{-\int_{0}^{B}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\xi)}d\xi}+\int_{0}^{B}\frac{h(\xi,\phi^{\prime}(-\phi,\eta,\xi))}{\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{-\int_{\xi}^{B}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi.$

Taking this into the previous equation leads to

$\displaystyle\varphi(\eta,\phi)$
$\displaystyle=\varphi_{b}(\phi^{\prime}(-\phi,\eta,0))e^{-\int_{0}^{B}+\int_{\eta}^{B}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\xi)}d\xi}+\int_{0}^{B}\frac{h(\xi,\phi^{\prime}(-\phi,\eta,\xi))}{\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{-\int_{\xi}^{B}+\int_{\eta}^{B}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi$
$\displaystyle\quad+\int_{\eta}^{B}\frac{h(\xi,-\phi^{\prime}(-\phi,\eta,\xi))}{\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi.$
(3.26)

Here we use the notation $\int_{0}^{B}+\int_{\eta}^{B}\cdot
d\xi=\int_{0}^{B}\cdot d\xi+\int_{\eta}^{B}\cdot d\xi$.

_Case 3. $\sin\phi<0$ and $|E(\phi,\eta)|>e^{-V(\varepsilon;B)}$._ In this
case, the characteristic curve does not touch the boundary $\eta=B$ but
reaches $(\phi=0,\eta=\eta_{+})$. We integrate (3.24) over $[\eta,\eta_{+}]$
and get

$\displaystyle\varphi(\eta,\phi)=\varphi(\eta_{+},0)e^{\int_{\eta}^{\eta_{+}}\frac{1}{-\sin\phi^{\prime}(-\phi,\eta,\xi)}d\xi}-\int_{\eta}^{\eta_{+}}\frac{h(\xi,-\phi^{\prime}(-\phi,\eta,\xi))}{-\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{-\int_{\xi}^{\eta}\frac{1}{-\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi.$

Since $(\eta_{+},0)$ also lies on the right characteristic lines with
$\phi^{\prime}(-\phi,\eta,\eta_{+})=0$, we get from (3.25),

$\displaystyle\varphi(\eta,\phi^{\prime}(-\phi,\eta,\eta_{+}))$
$\displaystyle=\varphi_{b}(\phi^{\prime}(-\phi,\eta,0))e^{-\int_{0}^{\eta_{+}}\frac{1}{-\sin\phi^{\prime}(-\phi,\eta,\xi)}d\xi}$
$\displaystyle\quad+\int_{0}^{\eta_{+}}\frac{h(\xi,\phi^{\prime}(-\phi,\eta,\xi))}{-\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{-\int_{\xi}^{\eta}\frac{1}{-\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi.$

Taking it into the previous equation leads to

$\displaystyle\varphi(\eta,\phi)$
$\displaystyle=\varphi_{b}(\phi^{\prime}(-\phi,\eta,0))e^{-\int_{0}^{\eta_{+}}+\int_{\eta}^{\eta_{+}}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\xi)}d\xi}+\int_{0}^{\eta_{+}}\frac{h(\xi,\phi^{\prime}(-\phi,\eta,\xi))}{\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{-\int_{\xi}^{\eta_{+}}+\int_{\eta}^{\eta_{+}}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi$
$\displaystyle\quad+\int_{\eta}^{\eta_{+}}\frac{h(\xi,-\phi^{\prime}(-\phi,\eta,\xi))}{\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi.$
(3.27)

In any case, we can show that (3.25), (3.1) and (3.1) is a solution to (3.21)
with boundary condition (3.22). Moreover, inequality (3.23) holds. To show
this, we have in case 1, from (3.25),

$\displaystyle\|\varphi(\eta,\phi)\|_{L^{\infty}([0,B]\times[-\pi,\pi))}$
$\displaystyle\leq\|\varphi_{b}\|_{L^{\infty}((0,\pi))}+\|h\|_{L^{\infty}([0,B]\times[-\pi,\pi)}\int_{0}^{\eta}\frac{1}{\sin\phi(\xi)}e^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi(\rho)}d\rho}d\xi$
$\displaystyle=\|\varphi_{b}\|_{L^{\infty}((0,\pi))}+\|h\|_{L^{\infty}([0,B]\times[-\pi,\pi)}\left(1-e^{-\int_{0}^{\eta}\frac{1}{\sin\phi(\rho)}d\rho}\right)$
$\displaystyle\leq\|\varphi_{b}\|_{L^{\infty}((0,\pi))}+\|h\|_{L^{\infty}([0,B]\times[-\pi,\pi)},$

and in case 2 from (3.1),

$\displaystyle\|\varphi(\eta,\phi)\|_{L^{\infty}([0,B]\times[-\pi,\pi))}$
$\displaystyle\quad\leq\|\varphi_{b}\|_{L^{\infty}((0,\pi))}+\|h\|_{L^{\infty}([0,B]\times[-\pi,\pi)}(1-e^{-\int_{0}^{B}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho})e^{-\int_{\eta}^{B}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}$
$\displaystyle\qquad+\|h\|_{L^{\infty}([0,B]\times[-\pi,\pi)}(1-e^{-\int_{\eta}^{B}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho})$
$\displaystyle\quad\leq\|\varphi_{b}\|_{L^{\infty}((0,\pi))}+\|h\|_{L^{\infty}([0,B]\times[-\pi,\pi)},$

and in case 3 from (3.1),

$\displaystyle\|\varphi(\eta,\phi)\|_{L^{\infty}([0,B]\times[-\pi,\pi))}$
$\displaystyle\quad\leq\|\varphi_{b}\|_{L^{\infty}((0,\pi))}+\|h\|_{L^{\infty}([0,B]\times[-\pi,\pi)}(1-e^{-\int_{0}^{\eta_{+}}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho})e^{-\int_{\eta}^{\eta_{+}}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}$
$\displaystyle\qquad+\|h\|_{L^{\infty}([0,B]\times[-\pi,\pi)}(1-e^{-\int_{\eta}^{\eta_{+}}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho})$
$\displaystyle\quad\leq\|\varphi_{b}\|_{L^{\infty}((0,\pi))}+\|h\|_{L^{\infty}([0,B]\times[-\pi,\pi)},$

Next assume $0\leq\varphi_{b}\leq\gamma$, $0\leq h\leq\gamma$, we show the
boundness of solutions, we can see obviously from (3.25), (3.1) and (3.1) that
as long as $\varphi_{b}\geq 0$ and $h\geq 0$, $\varphi(\eta,\phi)\geq 0$ for
all $\eta\in[0,B]$ and $\phi\in[-\pi,\pi)$. Next we show if
$\varphi_{b}\leq\gamma$, $h\leq\gamma$, then $\varphi\leq\gamma$. We show this
for case 1 and leave the other cases to the reader. For case 1, from (3.25),

$\displaystyle\varphi(\eta,\phi)$ $\displaystyle\leq\gamma
e^{-\int_{0}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\xi)}d\xi}+\gamma\int_{0}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\xi)}e^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}d\xi$
$\displaystyle=\gamma
e^{-\int_{0}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\xi)}d\xi}+\gamma\int_{0}^{\eta}de^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}$
$\displaystyle=\gamma
e^{-\int_{0}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\xi)}d\xi}+\gamma
e^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}\big{|}_{\xi=0}^{\eta}=\gamma.$

Therefore, $0\leq\varphi(\eta,\phi)\leq\gamma$ for any $\eta\in[0,B]$,
$\phi\in[-\pi,\pi)$.

Finally, the monotonicity of solutions for equation (3.24) can be seen from
the formulas (3.25), (3.1) and (3.1) directly. For example, if
$\varphi_{b1}\leq\varphi_{b2}$ and $h_{1}\leq h_{2}$, then for the case 1,

$\displaystyle(\varphi_{1}-\varphi_{2})(\eta,\phi)$
$\displaystyle=(\varphi_{b1}-\varphi_{b2})(\phi^{\prime}(\phi,\eta,0))e^{-\int_{0}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\xi)}d\xi}$
$\displaystyle\quad+\int_{0}^{\eta}\frac{(h_{1}-h_{2})(\xi,\phi^{\prime}(\phi,\eta,\xi))}{\sin\phi^{\prime}(\phi,\eta,\xi)}e^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}d\xi\leq
0,$

hence $\varphi_{1}\leq\varphi_{2}$ on $[0,B]\times[-\pi,\pi)$ and thus finish
the proof. ∎

######  Remark 10.

Compared to the proof of the flat case in [6], the additional variable $\phi$
could be integrated in the characteristics and the monotonicity of solution
remains true. Moreover, the estimate (3.23) has already been proved in [10,
equations (4.52)-(4.54)] where the monotonicity property is new which is
crucial in our approach.

We now introduce the concept of _sub-solutions_. Our definition is consistent
from the theory in the [3, 2] or also in [6].

######  Definition 1.

A pair of functions $(G,\varphi)\in C^{2}([0,B])\times
C^{1}([0,B]\times[-\pi,\pi))$ is called a _sub-solution_ to system
(3.12)-(3.13) if it satisfies the inequality

$\displaystyle-\partial_{\eta}^{2}G-F(\varepsilon;\eta)\partial_{\eta}G-\langle\varphi-G^{4}\rangle\leq
0,$
$\displaystyle\sin\phi\partial_{\eta}\varphi+F(\varepsilon;\eta)\cos\phi\partial_{\phi}\varphi+\varphi-G^{4}\leq
0,$

and on the left boundary $G(0)\leq G_{b}$,
$\varphi(0,\phi)\leq\varphi_{b}(\phi)$ for $\sin\phi>0$ and
$\partial_{\eta}G(B)=0$, $\varphi(B,\phi)=\varphi(B,-\phi)$ on the right
boundary.

Next we proceed for the proof of Lemma 6.

###### Proof of Lemma 6.

The proof of the lemma follows the same way as the proof of [6, Theorem 4].
Here we give a brief description of the proof. For the details, we refer the
reader to [6].

First we introduce the function $\zeta$ by

$\displaystyle\zeta(G)=\left\\{\begin{array}[]{ll}\frac{G}{1-G},&\text{for
}G<0,\\\ G^{4},&\text{for }0\leq G\leq\gamma,\\\
G^{4}+\frac{G-\gamma}{1+G-\gamma},&\text{for }G>\gamma,\end{array}\right.$

which extends the power function $G^{4}$ to $\mathbb{R}$ and $\zeta$ is an
increasing function. Starting with any sub-solution $G^{0},\varphi^{0}$ (e.g.
$G^{0}=0,\varphi^{0}=0$), we construct a sequence of functions
$\\{G^{k},\varphi^{k}\\}$ by solving iteratively

$\displaystyle-\partial_{\eta}^{2}G^{k}-F(\varepsilon;\eta)\partial_{\eta}G^{k}+\langle\zeta(G^{4})\rangle=\langle\varphi^{k-1}\rangle,$
(3.28)
$\displaystyle\sin\phi\partial_{\eta}\varphi^{k}+F(\varepsilon;\eta)\cos\phi\partial_{\phi}\varphi^{k}+\varphi^{k}=(G^{k-1})^{4}.$
(3.29)

We can use Lemma 7 and Lemma 9 and show by induction that $\gamma\geq
G^{k}(\eta)\geq G^{k-1}(\eta)\geq 0$ and
$\gamma\geq\varphi^{k}(\eta,\phi)\geq\varphi^{k-1}(\eta,\phi)\geq 0$ for any
$\eta\in[0,B]$ and $\phi\in[-\pi,\pi)$. Therefore, $\\{G^{k},\varphi^{k}\\}$
are bounded increasing sequences. Hence there existence bounded functions
$(G,\varphi)$ such that $(G^{k},\varphi^{k})\to(G,\varphi)$ pointwise as
$k\to\infty$. Moreover, since the solutions to (3.28)-(3.29) could be
expressed using integral formulas using Lemma 7 and Lemma 9, we can pass to
the limit in these formulas and use Beppo Levi’s lemma to conclude that
$(G,\varphi)\in C^{2}([0,B])\times C^{2}([0,B]\times[-\pi,\pi])$. Therefore,
by Dini’s theorem, $\\{G^{k},\varphi^{k}\\}$ converge uniformly to
$(G,\varphi)$ and $(G,\varphi)$ solves (3.12)-(3.13) with boundary conditions
(3.14)-(3.15).

The proof of uniqueness is slightly different than that of [6]. Suppose
$(G_{1},\varphi_{1})$ and $(G_{2},\varphi_{2})$ be two solutions to system
(3.12)-(3.13). Let

$\displaystyle
G^{0}=\max\\{G_{1},G_{2}\\},\quad\varphi^{0}=\max\\{\varphi_{1},\varphi_{2}\\},$

and we can construct sequence of functions $\\{G^{k},\varphi^{k}\\}$ by the
above precedure. Following the above arguments, $(G^{k},\psi^{k})$ converges
to $(G,\varphi)$ which is a solution to (3.12)-(3.13) and $G\geq
G^{0},\varphi\geq\varphi^{0}$. Thus $G-G_{1}$, $\varphi-\varphi_{1}$ is a non-
negative solution to the system

$\displaystyle-\partial_{\eta}^{2}(G-G_{1})-F(\varepsilon;\eta)\partial_{\eta}(G-G_{1})+\langle
G^{4}-(G_{1})^{4}\rangle=\langle\varphi-\varphi_{1}\rangle,$ (3.30)
$\displaystyle\sin\phi\partial_{\eta}(\varphi-\varphi_{1})+F(\varepsilon;\eta)\cos\phi\partial_{\phi}(\varphi-\varphi_{1})+\varphi-\varphi_{1}=G^{4}-(G_{1})^{4},$
(3.31)

with

$\displaystyle G(0)-G_{1}(0)=0,\quad\partial_{x}(G-G_{1})(B)=0,$
$\displaystyle\varphi(0,\phi)-\varphi_{1}(0,\phi)=0,\quad\varphi(B,\phi)-\varphi_{1}(B,\phi)=\varphi(B,-\phi)-\varphi_{1}(B,-\phi),\quad\text{for
}\sin\phi>0.$

Comparing (3.30) with (3.31),

$\displaystyle\partial_{\eta}^{2}(G-G_{1})+F(\varepsilon;\eta)\partial_{\eta}(G-G_{1})$
$\displaystyle=\partial_{\eta}\langle\sin\phi(\varphi-\varphi_{1})\rangle+F(\varepsilon;\eta)\langle\cos\phi\partial_{\phi}(\varphi-\varphi_{1})\rangle$
$\displaystyle=\partial_{\eta}\langle\sin\phi(\varphi-\varphi_{1})\rangle-F(\varepsilon;\eta)\langle\partial_{\phi}\cos\phi(\varphi-\varphi_{1})\rangle$
$\displaystyle=\partial_{\eta}\langle\sin\phi(\varphi-\varphi_{1})\rangle+F(\varepsilon;\eta)\langle\sin\phi(\varphi-\varphi_{1})\rangle.$

That is

$\displaystyle\partial_{\eta}\left(\partial_{\eta}(G-G_{1})-\langle\sin\phi(\varphi-\varphi_{1})\right)=-F(\varepsilon;\eta)\left(\partial_{\eta}(G-G_{1})-\langle\sin\phi(\varphi-\varphi_{1})\right).$

Due to $F(\varepsilon;\eta)\geq 0$ on $[0,B]$ and at $\eta=B$,

$\displaystyle\partial_{\eta}(G-G_{1})(B)-\langle\sin\phi(\varphi-\varphi_{1})(B,\phi)\rangle=0,$

the previous equation implies

$\displaystyle\partial_{\eta}(G-G_{1})(\eta)-\langle\sin\phi(\varphi-\varphi_{1})(\eta,\phi)\rangle=0,\quad\text{for
any }\eta\in[0,B].$ (3.32)

When $|E(\phi,\eta)|\geq e^{-V(\varepsilon;B)}$, for $\sin\phi<0$, the
solution $\varphi-\varphi_{1}$ can be written using (3.1) as

$\displaystyle\varphi(\eta,\phi)$
$\displaystyle=\int_{0}^{B}\frac{G(\xi)-G_{1}(\xi)}{\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{-\int_{\xi}^{B}+\int_{\eta}^{B}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi+\int_{\eta}^{B}\frac{G(\xi)-G_{1}(\xi)}{\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{\int_{\xi}^{B}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi.$

If $G(\xi)-G_{1}(\xi)>0$ on some interval $\xi\in(a,b)\in[0,B]$, then
$\varphi(\eta,\phi)>0$. Therefore,

$\displaystyle\langle\sin\phi(\varphi-\varphi_{1})(0,\phi)\rangle$
$\displaystyle=\int_{\sin\phi>0}\sin\phi(\varphi-\varphi_{1})(\eta,\phi)d\phi+\int_{\sin\phi<0}\sin\phi(\varphi-\varphi_{1})(\eta,\phi)d\phi<0,$

which together with (3.32) implies

$\displaystyle\partial_{\eta}(G-G_{1})(0)<0,$

which contradicts the fact that $G(0)-G_{1}(0)=0$ and $G-G_{1}$ is non-
negative on $[0,B]$. Similarly, when $|E(\phi,\eta)|<e^{-V(\varepsilon;B)}$,
we can use (3.1) and get the same contradiction argument. Thus
$G(\eta)=G_{1}(\eta)$ for any $\eta\in[0,B]$. The uniqueness for (3.31) then
implies $\varphi-\varphi_{1}=0$. Therefore, $G=G_{1}$ and
$\varphi=\varphi_{1}$. By the same argument, we can show $G=G_{2}$ and
$\varphi=\varphi_{2}$. Hence $G_{1}=G_{2}$ and $\varphi_{1}=\varphi_{2}$ and
the solution to (3.12)-(3.13) is unique. ∎

###  3.2. Weighted estimates on the solutions on bounded interval

Next we show a weighted estimate on the solution to system (3.12)-(3.13).

######  Lemma 11.

Assuming the assumption of Lemma 6 holds and
$(G_{B}^{\varepsilon},\varphi_{B}^{\varepsilon})$ be the unique solution to
system (3.12)-(3.13) with boundary conditions (3.14)-(3.15). Then the
following inequalities hold for $0\leq\varepsilon\leq 1/4$:

$\displaystyle\int_{0}^{B}4(G_{B}^{\varepsilon})^{3}|\partial_{\eta}{G_{B}^{\varepsilon}}|^{2}d\eta+\frac{1}{2}\int_{0}^{B}\int_{-\pi}^{\pi}(\varphi_{B}^{\varepsilon}-(G_{B}^{\varepsilon})^{4})^{2}d\phi
d\eta+\frac{1}{2}\int_{\sin\phi<0}|\sin\phi|(\varphi_{B}^{\varepsilon}(0,\phi)-G_{b}^{4})^{2}d\phi$
$\displaystyle\quad\leq\frac{1}{2}\int_{\sin\phi>0}\sin\phi(\varphi_{b}(\phi)-G_{b}^{4})^{2}d\phi,$
(3.33)

and for $0\leq\varepsilon\leq 1/4$, $0\leq\alpha\leq 1/4$,

$\displaystyle\int_{0}^{B}e^{2\alpha\eta}4(G_{B}^{\varepsilon})^{3}|\partial_{\eta}G_{B}^{\varepsilon}|^{2}d\eta+\frac{1}{4}\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\alpha\eta}(\varphi^{\varepsilon}_{B}-(G_{B}^{\varepsilon})^{4})^{2}d\phi
d\eta+\frac{1}{2}\int_{\sin\phi<0}|\sin\phi|(\varphi^{\varepsilon}_{B}(0,\phi)-G^{4}_{b})^{2}d\phi$
$\displaystyle\qquad=\frac{1}{2}\int_{\sin\phi>0}\sin\phi(\varphi_{b}-G_{b}^{4})^{2}d\phi.$
(3.34)

Moreover, setting

$\displaystyle\omega_{B}^{\varepsilon}=\varphi_{B}^{\varepsilon}-\frac{1}{2\pi}\int_{-\pi}^{\pi}\varphi_{B}^{\varepsilon}d\phi,\quad
q_{B}^{\varepsilon}=\varphi_{B}^{\varepsilon}-\omega_{B}^{\varepsilon},$

then the following inequalities hold:

$\displaystyle\|\partial_{\eta}G^{\varepsilon}_{B}\|_{L^{2}([0,B])},\quad\|\omega_{B}^{\varepsilon}\|_{L^{2}([0,B]\times[-\pi,\pi))}\leq\left(\int_{\sin\phi>0}\sin\phi(\varphi_{b}-G_{b}^{4})^{2}d\phi\right)^{1/2},\quad$
(3.35) $\displaystyle|q_{B}^{\varepsilon}(\eta)|\leq
C(\delta,\varphi_{b},G_{b})(1+\sqrt{\eta}+\|\omega_{B}^{\varepsilon}(\eta,\cdot)\|_{L^{2}([-\pi,\pi))}),~{}|G(\eta)|\leq
C(G_{b},\varphi_{b})(1+\sqrt{\eta}),\text{for any }\eta\in[0,B].$ (3.36)

###### Proof.

For simplicity of notations, we drop the superscript $\varepsilon$ and the
subscript $B$.

_The energy estimate._ We first multiply (3.12) by $G^{4}$ and (3.13) by
$\varphi$, and integrate to get

$\displaystyle-\int_{0}^{B}G^{4}\partial_{\eta}^{2}Gd\eta+\int_{0}^{B}\int_{-\pi}^{\pi}\sin\phi\partial_{\eta}\frac{\varphi^{2}}{2}d\phi
d\eta+\int_{0}^{B}\int_{-\pi}^{\pi}(\varphi-G^{4})^{2}d\phi d\eta$
$\displaystyle\quad-\int_{0}^{B}F(\varepsilon;\eta)G^{4}\partial_{\eta}Gd\eta+\int_{0}^{B}\int_{-\pi}^{\pi}F(\varepsilon;\eta)\cos\phi\varphi\partial_{\phi}\varphi
d\phi d\eta=0.$ (3.37)

Using integration-by-parts and the boundary condition (3.14), we obtain

$\displaystyle-\int_{0}^{B}G^{4}\partial_{\eta}^{2}Gd\eta=-G^{4}\partial_{\eta}G\bigg{|}_{0}^{B}+\int_{0}^{B}4G^{3}|\partial_{\eta}G|^{2}d\eta=G^{4}(0)\partial_{\eta}G(0)+\int_{0}^{B}4G^{3}|\partial_{\eta}G|^{2}d\eta.$
(3.38)

Using integration-by-parts and the boundary condition (3.15), we obtain

$\displaystyle\int_{0}^{B}\int_{-\pi}^{\pi}\sin\phi\partial_{\eta}\frac{\varphi^{2}}{2}d\phi
d\eta=\int_{-\pi}^{\pi}\sin\phi\frac{\varphi^{2}}{2}d\phi\bigg{|}_{0}^{B}=-\frac{1}{2}\langle\sin\phi\varphi^{2}(0,\phi)\rangle.$
(3.39)

Comparing (3.12) and $\langle\eqref{eq:B2}\rangle$, it holds that

$\displaystyle\partial_{\eta}(\partial_{\eta}G-\langle\sin\phi\varphi\rangle)=-F(\varepsilon;\eta)(\partial_{\eta}G-\langle\sin\phi\varphi\rangle),$

which with $\partial_{\eta}G(B)-\langle\sin\phi\varphi(B,\phi)\rangle=0$,
implies that

$\displaystyle\partial_{\eta}G(\eta)-\langle\sin\phi\varphi(\eta,\phi)\rangle=0,\quad\text{
for any }\eta\in[0,B].$ (3.40)

Therefore,

$\displaystyle\partial_{\eta}G(0)=\langle\sin\phi\varphi(0,\phi)\rangle,$

and

$\displaystyle
G^{4}(0)\partial_{\eta}G(0)-\frac{1}{2}\langle\sin\phi\varphi^{2}(0,\phi)\rangle$
$\displaystyle=G^{4}(0)\sin\phi\varphi(0,\phi)\rangle-\frac{1}{2}\langle\sin\phi\varphi^{2}(0,\phi)\rangle$
$\displaystyle=-\frac{1}{2}\langle\sin\phi(\varphi(0,\phi)-G^{4}(0))^{2}\rangle,$
(3.41)

due to $\langle\sin\phi G^{4}(0)\rangle=0.$

Using (3.40), we also have

$\displaystyle-\int_{0}^{B}F(\varepsilon;\eta)G^{4}\partial_{\eta}Gd\eta+\int_{0}^{B}\int_{-\pi}^{\pi}F(\varepsilon;\eta)\cos\phi\varphi\partial_{\phi}\varphi
d\phi d\eta$
$\displaystyle\quad=-\int_{0}^{B}F(\varepsilon;\eta)G^{4}\partial_{\eta}Gd\eta+\int_{0}^{B}\int_{-\pi}^{\pi}F(\varepsilon;\eta)\sin\phi\frac{\varphi^{2}}{2}d\phi
d\eta$
$\displaystyle\quad=-\int_{0}^{B}\int_{-\pi}^{\pi}F(\varepsilon;\eta)G^{4}\sin\phi\varphi
d\phi
d\eta+\int_{0}^{B}\int_{-\pi}^{\pi}F(\varepsilon;\eta)\sin\phi\frac{\varphi^{2}}{2}d\phi
d\eta$
$\displaystyle\quad=\frac{1}{2}\int_{0}^{B}\int_{-\pi}^{\pi}F(\varepsilon;\eta)\sin\phi(\varphi-G^{4})^{2}d\phi
d\eta.$

Combing this above relation with (3.2), (3.38), (3.39), (3.2) gives

$\displaystyle\int_{0}^{B}4G^{3}|\partial_{\eta}G|^{2}d\eta+\int_{0}^{B}\int_{-\pi}^{\pi}(\varphi-G^{4})^{2}d\phi
d\eta+\frac{1}{2}\int_{0}^{B}\int_{-\pi}^{\pi}F(\varepsilon;\eta)\sin\phi(\varphi-G^{4})^{2}d\phi
d\eta$
$\displaystyle\quad+\frac{1}{2}\langle\sin\phi(\varphi(0,\phi)-G^{4}(0))^{2}\rangle=0.$

The last term on the left side of the above equation can be rewritten using
the boundary conditon (3.15) as

$\displaystyle\frac{1}{2}\langle\sin\phi(\varphi(0,\phi)-G^{4}(0))^{2}\rangle$
$\displaystyle=\frac{1}{2}\int_{\sin\phi>0}\sin\phi(\varphi(0,\phi)-G^{4}(0))^{2}d\phi+\frac{1}{2}\int_{\sin\phi<0}\sin\phi(\varphi(0,\phi)-G^{4}(0))^{2}d\phi$
$\displaystyle=\frac{1}{2}\int_{\sin\phi>0}\sin\phi(\varphi_{b}(\phi)-G^{4}(0))^{2}d\phi-\frac{1}{2}\int_{\sin\phi<0}|\sin\phi|(\varphi(0,\phi)-G^{4}(0))^{2}d\phi.$

Taking this into the previous equation, we arrive at the following energy
equality:

$\displaystyle\int_{0}^{B}4G^{3}|\partial_{\eta}G|^{2}d\eta+\int_{0}^{B}\int_{-\pi}^{\pi}(\varphi-G^{4})^{2}d\phi
d\eta+\frac{1}{2}\int_{0}^{B}\int_{-\pi}^{\pi}F(\varepsilon;\eta)\sin\phi(\varphi-G^{4})^{2}d\phi
d\eta$
$\displaystyle\quad+\frac{1}{2}\int_{\sin\phi<0}|\sin\phi|(\varphi(0,\phi)-G^{4}(0))^{2}d\phi=\frac{1}{2}\int_{\sin\phi>0}\sin\phi(\varphi_{b}(\phi)-G^{4}(0))^{2}d\phi.$
(3.42)

Note due to (3.11), $\|F\|_{L^{\infty}}\leq\tfrac{4}{4-3\delta}\varepsilon$,
hence for $\delta\leq 1$,

$\displaystyle|F(\varepsilon;\eta)\sin\phi|\leq\frac{4}{4-3\delta}\varepsilon\leq
4\varepsilon,$ (3.43)

and for $\varepsilon\leq 1/4$,

$\displaystyle 1+\frac{1}{2}F(\varepsilon;\eta)\sin\phi\geq
1-2\varepsilon\geq\frac{1}{2}.$

The energy equality (3.2) thus implies (11).

_The weighted estimate._ Let $\alpha>0$ be a positive constant. We multiply
(3.12) by $e^{2\alpha\eta}G^{4}$ and (3.13) by $e^{2\alpha\eta}\varphi$ and
integrate to get

$\displaystyle-\int_{0}^{B}e^{2\alpha\eta}G^{4}\partial_{\eta}^{2}Gd\eta+\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\alpha\eta}\sin\phi\partial_{\eta}\frac{\varphi^{2}}{2}d\phi
d\eta+\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\alpha\eta}(\varphi-G^{4})^{2}d\phi
d\eta$
$\displaystyle\quad-\int_{0}^{B}e^{2\alpha\eta}F(\varepsilon;\eta)G^{4}\partial_{\eta}Gd\eta+\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\alpha\eta}F(\varepsilon;\eta)\cos\phi\varphi\partial_{\phi}\varphi
d\phi d\eta=0.$ (3.44)

Using integration-by-parts and the boundary conditions (3.14)-(3.15), we
obtain

$\displaystyle-\int_{0}^{B}e^{2\alpha\eta}G^{4}\partial_{\eta}^{2}Gd\eta+\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\alpha\eta}\sin\phi\partial_{\eta}\frac{\varphi^{2}}{2}d\phi
d\eta$
$\displaystyle\quad=-e^{2\alpha\eta}G^{4}\partial_{\eta}G\bigg{|}_{0}^{B}+\int_{0}^{B}e^{2\alpha\eta}4G^{3}|\partial_{\eta}G|^{2}d\eta+\int_{0}^{B}2\alpha
e^{2\alpha\eta}G^{4}\partial_{\eta}Gd\eta+\frac{1}{2}\int_{-\pi}^{\pi}e^{2\alpha\eta}\sin\phi\varphi^{2}d\phi\bigg{|}_{0}^{B}$
$\displaystyle\qquad-\frac{1}{2}\int_{0}^{B}\int_{-\pi}^{\pi}2\alpha
e^{2\alpha\eta}\sin\phi\varphi^{2}d\phi d\eta$
$\displaystyle\quad=G^{4}(0)\partial_{\eta}G(0)+\int_{0}^{B}e^{2\alpha\eta}4G^{3}|\partial_{\eta}G|^{2}d\eta+\int_{0}^{B}2\alpha
e^{2\alpha\eta}G^{4}\partial_{\eta}Gd\eta-\frac{1}{2}\int_{-\pi}^{\pi}\sin\phi\varphi^{2}(0,\phi)d\phi$
$\displaystyle\qquad-\frac{1}{2}\int_{0}^{B}\int_{-\pi}^{\pi}2\alpha
e^{2\alpha\eta}\sin\phi\varphi^{2}d\phi d\eta$
$\displaystyle\quad=-\frac{1}{2}\langle\sin\phi(\varphi(0,\phi)-G_{b}^{4})^{2}\rangle+\int_{0}^{B}e^{2\alpha\eta}4G^{3}|\partial_{\eta}G|^{2}d\eta-\alpha\int_{0}^{B}\int_{-\pi}^{\pi}\alpha
e^{2\alpha\eta}\sin\phi(\varphi-G^{4})^{2}\phi d\eta$
$\displaystyle\quad=-\frac{1}{2}\int_{\sin\phi>0}\sin\phi(\varphi_{b}(\phi)-G_{b}^{4})^{2}d\phi+\frac{1}{2}\int_{\sin\phi<0}|\sin\phi|(\varphi(0,\phi)-G_{b}^{4})^{2}d\phi+\int_{0}^{B}e^{2\alpha\eta}4G^{3}|\partial_{\eta}G|^{2}d\eta$
$\displaystyle\qquad-\alpha\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\alpha\eta}\sin\phi(\varphi-G^{4})^{2}\phi
d\eta.$

Using integration-by-parts and the relation (3.40), we get

$\displaystyle-\int_{0}^{B}e^{2\alpha\eta}F(\varepsilon;\eta)G^{4}\partial_{\eta}Gd\eta+\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\alpha\eta}F(\varepsilon;\eta)\cos\phi\varphi\partial_{\phi}\varphi
d\phi d\eta$
$\displaystyle\quad=-\int_{0}^{B}e^{2\alpha\eta}F(\varepsilon;\eta)G^{4}\partial_{\eta}Gd\eta+\frac{1}{2}\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\alpha\eta}F(\varepsilon;\eta)\sin\phi\varphi^{2}d\phi
d\eta$
$\displaystyle\quad=\frac{1}{2}\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\alpha\eta}F(\varepsilon;\eta)\sin\phi(\varphi-G^{4})^{2}d\phi
d\eta.$

Taking the above two equations into (3.2) gives

$\displaystyle\int_{0}^{B}e^{2\alpha\eta}4G^{3}|\partial_{\eta}G|^{2}d\eta+\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\alpha\eta}(\varphi-G^{4})^{2}d\phi
d\eta+\frac{1}{2}\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\alpha\eta}F(\varepsilon;\eta)\sin\phi(\varphi-G^{4})^{2}d\phi
d\eta$
$\displaystyle\quad-\alpha\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\alpha\eta}\sin\phi(\varphi-G^{4})^{2}\phi
d\eta+\frac{1}{2}\int_{\sin\phi<0}|\sin\phi|(\varphi(0,\phi)-G^{4}_{b})^{2}d\phi$
$\displaystyle\qquad=\frac{1}{2}\int_{\sin\phi>0}\sin\phi(\varphi_{b}-G_{b}^{4})^{2}d\phi.$

Using (3.43) and for $\varepsilon\leq 1/4$ and $0<\alpha\leq\tfrac{1}{4}$
small,

$\displaystyle\left|1+\frac{1}{2}F(\varepsilon;\eta)\sin\phi-\alpha\right|\geq
1-2\varepsilon-\alpha\geq\frac{1}{4}.$ (3.45)

Taking this into the previous inequality leads to the estimate (11).

_Boundness of $\varphi$._ Note that from the estimates (11) and (11),
$\varphi-G^{4}$ is in $L^{2}$ but we lack an estimate on $\varphi$ or $G$.
However, from the definition of $\omega$, $\langle\omega\rangle=0$ and then

$\displaystyle\int_{-\pi}^{\pi}(\varphi-G^{4})^{2}d\phi=\int_{-\pi}^{\pi}\left(\omega+q-G^{4}\right)^{2}d\phi=\int_{-\pi}^{\pi}\omega^{2}d\phi+2\pi(q-G^{4})^{2}.$
(3.46)

Hence we get from (11) and (11) that

$\displaystyle\|\omega\|_{L^{2}([0,B]\times[-\pi,\pi))}^{2}=\int_{0}^{B}\int_{-\pi}^{\pi}\omega^{2}d\phi
d\eta\leq\int_{\sin\phi>0}(\varphi_{b}-G_{b}^{4})^{2}d\phi,$

and

$\displaystyle\|e^{\alpha\eta}\omega\|_{L^{2}([0,B]\times[-\pi,\pi))}=\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\alpha\eta}\omega^{2}d\phi
d\eta\leq 2\int_{\sin\phi>0}(\varphi_{b}-G_{b}^{4})^{2}d\phi.$

Next we derive estimate on $q$. Multiplying (3.13) by $\sin\phi$ and integrate
over $[-\pi,\pi)$ gives

$\displaystyle\frac{d}{d\eta}\langle\sin^{2}\phi\varphi\rangle$
$\displaystyle=-F(\varepsilon;\eta)\langle\cos\phi\sin\phi\partial_{\phi}\varphi\rangle-\langle(\varphi-G^{4})\sin\phi\rangle$
$\displaystyle=\frac{1}{2}F(\varepsilon;\eta)\langle\partial_{\phi}\sin(2\phi)\varphi\rangle-\langle\sin\phi\varphi\rangle$
$\displaystyle=F(\varepsilon;\eta)\langle\cos(2\phi)\omega\rangle-\langle\sin\phi\omega\rangle.$

Integrating over $[0,\eta]$ and using (3.10) gives

$\displaystyle|\langle\sin^{2}\phi\varphi(\eta,\cdot)\rangle-\langle\sin^{2}\phi\varphi(0,\cdot)\rangle|$
$\displaystyle\quad=\int_{0}^{\eta}(F(\varepsilon;\tau)\langle\cos(2\phi)\omega(\tau,\cdot)\rangle-\langle\sin\phi\omega(\tau,\cdot)\rangle)d\tau$
$\displaystyle\quad\leq\left(\int_{0}^{\eta}\int_{-\pi}^{\pi}\cos^{2}(2\phi)F^{2}(\varepsilon;\tau)d\phi
d\eta\right)^{1/2}\left(\int_{0}^{\eta}\int_{-\pi}^{\pi}\omega^{2}d\phi
d\eta\right)^{1/2}$
$\displaystyle\qquad+\left(\int_{0}^{\eta}\int_{-\pi}^{\pi}\sin^{2}\phi d\phi
d\eta\right)^{1/2}\left(\int_{0}^{\eta}\int_{-\pi}^{\pi}\omega^{2}d\phi
d\eta\right)^{1/2}$
$\displaystyle\leq\sqrt{\pi}\|F\|_{L^{2}([0,\eta])}\|\omega\|_{L^{2}([0,B]\times[-\pi,\pi))}+\sqrt{\pi}\sqrt{\eta}\|\omega\|_{L^{2}([0,B]\times[-\pi,\pi))}$
$\displaystyle\leq(\frac{3\delta}{4-3\delta}\sqrt{\pi}+\sqrt{\pi}\sqrt{\eta})\|\omega\|_{L^{2}([0,B]\times[-\pi,\pi))}$
$\displaystyle\leq
C(\delta)(1+\sqrt{\eta})\left(\int_{\sin\phi>0}(\varphi_{b}-G_{b}^{4})^{2}d\phi\right)^{1/2}.$

Since

$\displaystyle|\langle\sin^{2}\phi\varphi(0,\cdot)\rangle|$
$\displaystyle=\left|\int_{\sin\phi>0}\sin^{2}\phi\varphi_{b}d\phi+\int_{\sin\phi<0}\sin^{2}\phi\varphi(0,\phi)d\phi\right|$
$\displaystyle\leq\int_{\sin\phi>0}\sin^{2}\phi\varphi_{b}d\phi+\left(\int_{\sin\phi<0}|\sin\phi|^{\tfrac{9}{4}}d\phi\right)^{1/2}\left(\int_{\sin\phi<0}|\sin\phi|\varphi^{2}(0,\phi)d\phi\right)^{1/2}$
$\displaystyle\leq\int_{\sin\phi>0}\sin^{2}\phi\varphi_{b}d\phi+C\left(\int_{\sin\phi>0}(\varphi_{b}-G_{b}^{4})^{2}d\phi\right)^{1/2},$

we get from the previous inequality that

$\displaystyle|\langle\sin^{2}\phi\varphi(\eta,\cdot)\rangle|$
$\displaystyle\leq|\langle\sin^{2}\phi\varphi(0,\cdot)\rangle|+C(\delta)(1+\sqrt{\eta})\left(\int_{\sin\phi>0}(\varphi_{b}-G_{b}^{4})^{2}d\phi\right)^{1/2}$
$\displaystyle\leq C(\delta,\varphi_{b},G_{b})(1+\sqrt{\eta}),$

with $C(\delta,\varphi_{b},G_{b})>0$ is a positive constant depending on
$\delta,\varphi_{b},G_{b}$. Since

$\langle\sin^{2}\phi\varphi(\eta,\cdot)\rangle=\langle\sin^{2}\phi
q(\eta)\rangle+\langle\sin^{2}\phi\omega(\eta,\cdot)\rangle=\pi
q(\eta)+\langle\sin^{2}\phi\omega(\eta,\cdot)\rangle$

and

$|\langle\sin^{2}\phi\omega(\eta,\cdot)\rangle|\leq\|\sin^{2}\phi\|_{L^{2}([-\pi,\pi))}\|\omega(\eta,\cdot)\|_{L^{2}([-\pi,\pi))}\leq
C\|\omega(\eta,\cdot)\|_{L^{2}([-\pi,\pi))},$

we obtain

$\displaystyle|q(\eta)|\leq
C(\delta,\varphi_{b},G_{b})(1+\sqrt{\eta}+\|\omega(\eta,\cdot)\|_{L^{2}([-\pi,\pi))})$

for any $\eta\in[0,B]$. Similarly, using the relation (3.40), we can get

$\displaystyle|G(\eta)|$
$\displaystyle=\left|G_{b}+\int_{0}^{\eta}\partial_{\eta}Gd\eta\right|$
$\displaystyle\leq G_{b}+\int_{0}^{\eta}|\langle\sin\phi\varphi\rangle|d\eta$
$\displaystyle\leq G_{b}+\int_{0}^{\eta}|\langle\sin\phi\omega\rangle|d\eta$
$\displaystyle\leq G_{b}+\left(\int_{0}^{\eta}\int_{-\pi}^{\pi}\sin^{2}\phi
d\phi d\eta\right)^{1/2}\|\omega\|_{L^{2}([0,\eta]\times[-\pi,\pi))}$
$\displaystyle\leq
G_{b}+C\sqrt{\eta}\left(\int_{\sin\phi>0}(\varphi_{b}-G_{b}^{4})^{2}d\phi\right)^{1/2}$
$\displaystyle\leq C(G_{b},\varphi_{b})(1+\sqrt{\eta}),$

which finishes the proof. ∎

###  3.3. Existence on the half line

Next we show the existence of solutions for system (3.1)-(3.2).

###### Proof of Theorem 4.

From (3.35) and (3.36),
$\varphi_{B}^{\varepsilon}=q_{B}^{\varepsilon}+\omega_{B}^{\varepsilon}\in
L^{2}_{\rm loc}([0,\infty)\times[-\pi,\pi))$ and $G_{B}^{\varepsilon}\in
L^{2}_{\rm loc}([0,\infty))$, thus there exists a subsequence
$\\{B_{k}\\}_{k=1}^{\infty}$ such that

$\displaystyle G_{B_{k}}^{\varepsilon}\rightharpoonup
G^{\varepsilon},\quad\text{weakly in }L^{2}_{\rm loc}([0,\infty)),$
$\displaystyle\varphi^{\varepsilon}_{B_{k}}\rightharpoonup\varphi^{\varepsilon},\quad\text{weakly
in }L^{2}_{\rm loc}([0,\infty)\times[-\pi,\pi)).$

By the estimate (3.35),
$\|\partial_{\eta}G_{B}^{\varepsilon}\|_{L^{2}([0,B])}$ is uniformly bounded,
hence by trace theorem we get
$G^{\varepsilon}(0)=G_{B}^{\varepsilon}(0)=G_{b}$ holds. Moreover, from (11),

$\displaystyle\sin\phi\partial_{\eta}\varphi_{B}^{\varepsilon}+F(\varepsilon;\eta)\cos\phi\partial_{\phi}\varphi_{B}^{\varepsilon}\in
L^{2}([0,B]\times[-\pi,\pi)),$

by the the Ukai’s trace theorem [9, Theorem 5.1], $\varphi^{\varepsilon}$ is
well defined on the boundary and

$\displaystyle\varphi^{\varepsilon}(0,\phi)=\varphi_{b}(\phi),\quad\text{for
}\sin\phi>0.$

Since $\|G^{\varepsilon}_{B}\|_{H^{1}([0,B])}$ is bounded for any finite $B$,
the Sobolev embedding theorem implies that
$\|G^{\varepsilon}_{B}\|_{C([0,B])}$ is bounded and thus as $B\to 0$,

$\displaystyle(G_{B}^{\varepsilon})^{4}\to(G^{\varepsilon})^{4},\quad\text{uniformly
in }C([0,B]).$

Therefore, $(G^{\varepsilon},\phi^{\varepsilon})$ solves system (3.1)-(3.2)
with boundary conditions (3.3)-(3.4). Moreover, due to $0\leq
G^{\varepsilon}_{B}\leq\gamma$ and
$0\leq\varphi^{\varepsilon}_{B}\leq\gamma^{4}$, $0\leq
G^{\varepsilon}\leq\gamma$ and $0\leq\varphi^{\varepsilon}\leq\gamma^{4}$ also
hold.

Next we show the uniform boundness of the solutions. First, by the lower semi-
continuity of the $L^{2}$ norm, (11) and (11) implies (4) and (4).

From the relation (3.40),
$\partial_{\eta}G^{\varepsilon}_{B}=\langle\sin\phi\varphi_{B}^{\varepsilon}\rangle$,
we can pass to the weak limit $B\to\infty$ and get that

$\displaystyle\partial_{\eta}G^{\varepsilon}=\langle\sin\phi\varphi^{\varepsilon}\rangle$

holds in the weak sense in $L^{2}_{\rm loc}([0,\infty))$. Therefore,

$\displaystyle\int_{0}^{\infty}|\partial_{\eta}G^{\varepsilon}|^{2}d\eta$
$\displaystyle=\int_{0}^{\infty}|\langle\sin\phi\varphi^{\varepsilon}\rangle|^{2}d\eta=\int_{0}^{\infty}|\langle\sin\phi(\varphi^{\varepsilon}-(G^{\varepsilon})^{4})\rangle|^{2}d\eta$
$\displaystyle\leq\int_{0}^{\infty}\left(\int_{-\pi}^{\pi}\sin^{2}\phi
d\phi\int_{-\pi}^{\pi}(\varphi^{\varepsilon}-(G^{\varepsilon})^{4})^{2}d\phi\right)d\eta\leq\pi\int_{\sin\phi>0}\sin\phi(\varphi_{b}(\phi)-G_{b}^{4})^{2}d\phi,$

and

$\displaystyle\int_{0}^{\infty}e^{2\alpha\eta}|\partial_{\eta}G^{\varepsilon}|^{2}d\eta=\int_{0}^{\infty}e^{2\alpha\eta}|\langle\sin\phi\varphi^{\varepsilon}\rangle|^{2}d\eta=\int_{0}^{\infty}e^{2\alpha\eta}|\langle\sin\phi(\varphi^{\varepsilon}-(G^{\varepsilon})^{4})\rangle|^{2}d\eta$
$\displaystyle\quad\leq\pi\int_{0}^{\infty}\int_{-\pi}^{\pi}e^{2\alpha\eta}(\varphi^{\varepsilon}-(G^{\varepsilon})^{4})^{2}d\phi
d\eta\leq
2\pi\int_{\sin\phi>0}\sin\phi(\varphi_{b}(\phi)-G_{b}^{4})^{2}d\phi.$

Therefore,

$\displaystyle G^{\varepsilon}(\eta)$
$\displaystyle=G_{b}+\int_{0}^{\eta}\partial_{\eta}G^{\varepsilon}d\eta$
$\displaystyle=G_{b}+\int_{0}^{\eta}e^{-\alpha\eta}e^{\alpha\eta}\partial_{\eta}G^{\varepsilon}d\eta\leq
G_{b}+\left(\int_{0}^{\eta}e^{-2\alpha\eta}d\eta\right)^{1/2}\left(\int_{0}^{\eta}e^{2\alpha\eta}|\partial_{\eta}G^{\varepsilon}|^{2}d\eta\right)^{1/2}$
$\displaystyle\leq
G_{b}+\frac{\sqrt{1-e^{-2\alpha\eta}}}{\sqrt{2\alpha}}\sqrt{2\pi}\left(\int_{\sin\phi>0}\sin\phi(\varphi_{b}(\phi)-G_{b}^{4})^{2}d\phi\right)^{1/2}.$

Hence as $\eta\to\infty$, $G^{\varepsilon}(\eta)$ is uniformly bounded, and so
$\lim_{\eta\to\infty}G^{\varepsilon}(\eta)=:G^{\varepsilon}_{\infty}$ exists.
Moreover,

$\displaystyle|G^{\varepsilon}(\eta)-G_{\infty}^{\varepsilon}|$
$\displaystyle=\left|-\int_{\eta}^{\infty}\partial_{\eta}G^{\varepsilon}d\eta\right|=\left|\int_{\eta}^{\infty}e^{-\alpha\eta}e^{\alpha\eta}\partial_{\eta}G^{\varepsilon}d\eta\right|$
$\displaystyle\leq\left(\int_{\eta}^{\infty}e^{-2\alpha\eta}d\eta\right)^{1/2}\left(\int_{\eta}^{\infty}e^{2\alpha\eta}|\partial_{\eta}G^{\varepsilon}|^{2}d\eta\right)^{1/2}$
$\displaystyle\leq\frac{e^{-\alpha\eta}}{\sqrt{2\alpha}}\left(\int_{0}^{\infty}e^{2\alpha\eta}|\partial_{\eta}G^{\varepsilon}|^{2}d\eta\right)^{1/2}$
$\displaystyle\leq
e^{-\alpha\eta}\frac{\sqrt{\pi}}{\sqrt{\alpha}}\left(\int_{\sin\phi>0}\sin\phi(\varphi_{b}(\phi)-G_{b}^{4})^{2}d\phi\right)^{1/2}.$

Finally, we show $\varphi^{\varepsilon}-(G^{\varepsilon}_{\infty})^{4}$ decays
exponentially to $0$ as $\eta\to 0$. We use Lemma 20 in Appendix A to get
this. Note that from the exponential decay of $G^{\varepsilon}$,

$\displaystyle|(G^{\varepsilon}(\eta))^{4}-(G^{\varepsilon}_{\infty})^{4}|$
$\displaystyle\leq|G^{\varepsilon}(\eta)-G^{\varepsilon}_{\infty}|\cdot|(G^{\varepsilon}(\eta))^{3}-(G^{\varepsilon}(\eta))^{2}G^{\varepsilon}_{\infty}+G^{\varepsilon}(\eta)(G^{\varepsilon}_{\infty})^{2}-(G^{\varepsilon}_{\infty})^{3}|$
$\displaystyle\quad\leq
e^{-\alpha\eta}\frac{\sqrt{\pi}}{\sqrt{\alpha}}\left(\int_{\sin\phi>0}\sin\phi(\varphi_{b}(\phi)-G_{b}^{4})^{2}d\phi\right)^{1/2}\cdot
4\gamma^{3}.$

Let
$C_{b}:=\frac{\sqrt{\pi}}{\sqrt{\alpha}}\left(\int_{\sin\phi>0}\sin\phi(\varphi_{b}(\phi)-G_{b}^{4})^{2}d\phi\right)^{1/2}\cdot
4\gamma^{3}$, then
$|(G^{\varepsilon}(\eta))^{4}-(G^{\varepsilon}_{\infty})^{4}|\leq
C_{b}e^{-\alpha\eta}.$ First, for $\sin\phi>0$, from (A.4),

$\displaystyle|\varphi^{\varepsilon}(\eta,\phi)-(G^{\varepsilon}_{\infty})^{4}|$
$\displaystyle\quad=\left|\varphi_{b}(\phi^{\prime}(\phi,\eta,0))e^{-\int_{0}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\xi)}d\xi}+\int_{0}^{\eta}\frac{(G^{\varepsilon}(\xi))^{4}}{\sin\phi^{\prime}(\phi,\eta,\xi)}e^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}d\xi-(G^{\varepsilon}_{\infty})^{4}\right|$
$\displaystyle\quad=\left|(\varphi_{b}(\phi^{\prime}(\phi,\eta,0))-(G_{\infty}^{\varepsilon})^{4})e^{-\int_{0}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\xi)}d\xi}+\int_{0}^{\infty}\frac{(G^{\varepsilon}(\xi))^{4}-(G_{\infty}^{\varepsilon})^{4}}{\sin\phi^{\prime}(\phi,\eta,\xi)}e^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}d\xi\right|$
$\displaystyle\quad\leq|\varphi_{b}(\phi^{\prime}(\phi,\eta,0))-(G_{\infty}^{\varepsilon})^{4}|e^{-\eta}$
$\displaystyle\qquad+\frac{\sqrt{\pi}}{\sqrt{\alpha}}\left(\int_{\sin\phi>0}\sin\phi(\varphi_{b}(\phi)-G_{b}^{4})^{2}d\phi\right)^{1/2}\cdot
4\gamma^{3}\int_{0}^{\eta}\frac{e^{-\alpha\xi}}{\sin\phi^{\prime}(\phi,\eta,\xi)}e^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}d\xi.$

Using integration-by-parts, we obtain

$\displaystyle\int_{0}^{\eta}\frac{e^{-\alpha\xi}}{\sin\phi^{\prime}(\phi,\eta,\xi)}e^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}d\xi$
$\displaystyle=\int_{0}^{\eta}e^{-\alpha\xi}de^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}$
$\displaystyle=e^{-\alpha\xi}e^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}\big{|}_{0}^{\eta}+\alpha\int_{0}^{\eta}e^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}e^{-\alpha\xi}d\xi$
$\displaystyle\leq
e^{-\alpha\eta}-e^{-\int_{0}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}+\alpha\int_{0}^{\eta}e^{-(\eta-\xi)}e^{-\alpha\xi}d\xi$
$\displaystyle\leq
e^{-\alpha\eta}+\frac{\alpha}{1-\alpha}(e^{-\alpha\eta}-e^{-\eta})\leq\frac{1}{1-\alpha}e^{-\alpha\eta}.$

Taking this into the previous inequality leads to

$\displaystyle|\varphi^{\varepsilon}(\eta,\phi)-(G^{\varepsilon}_{\infty})^{4}|$
$\displaystyle\quad\leq|\varphi_{b}(\phi^{\prime}(\phi,\eta,0))-(G_{\infty}^{\varepsilon})^{4}|e^{-\eta}+e^{-\alpha\eta}\frac{4\gamma^{3}\sqrt{\pi}}{\sqrt{\alpha}\left(\int_{\sin\phi>0}\sin\phi(\varphi_{b}(\phi)-G_{b}^{4})^{2}d\phi\right)^{1/2}(1-\alpha)},$

for any $\eta\geq 0$ and $\sin\phi>0$.

Similarly, for $\sin\phi<0$ and $|E(\phi,\eta)|\leq e^{-V_{\infty}}$, we use
(A.5) and get

$\displaystyle\varphi^{\varepsilon}(\eta,\phi)$
$\displaystyle=\varphi_{b}(\phi^{\prime}(-\phi,\eta,0))e^{-\int_{0}^{\infty}+\int_{\eta}^{\infty}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\xi)}d\xi}+\int_{0}^{\infty}\frac{(G^{\varepsilon}(\xi))^{4}}{\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{-\int_{\xi}^{\infty}+\int_{\eta}^{\infty}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi$
$\displaystyle\quad+\int_{\eta}^{\infty}\frac{(G^{\varepsilon}(\xi))^{4}}{\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi$
$\displaystyle=\int_{\eta}^{\infty}\frac{(G^{\varepsilon}(\xi))^{4}}{\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi.$

Hence

$\displaystyle|\varphi^{\varepsilon}(\eta,\phi)-(G^{\varepsilon}_{\infty})^{4}|$
$\displaystyle\quad=\bigg{|}\int_{\eta}^{\infty}\frac{(G^{\varepsilon}(\xi))^{4}}{\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi-(G^{\varepsilon}_{\infty})^{4}\bigg{|}$
$\displaystyle\quad=\bigg{|}\int_{\eta}^{\infty}\frac{(G^{\varepsilon}(\xi))^{4}-(G^{\varepsilon}_{\infty})^{4}}{\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi\bigg{|}$
$\displaystyle\quad\leq\int_{\eta}^{\infty}\frac{|(G^{\varepsilon}(\xi))^{4}-(G^{\varepsilon}_{\infty})^{4}|}{\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi$
$\displaystyle\quad\leq\frac{\sqrt{\pi}}{\sqrt{\alpha}}\left(\int_{\sin\phi>0}\sin\phi(\varphi_{b}(\phi)-G_{b}^{4})^{2}d\phi\right)^{1/2}\cdot
4\gamma^{3}\int_{\eta}^{\infty}\frac{e^{-\alpha\xi}}{\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi$
$\displaystyle\quad\leq\frac{4\gamma^{3}\sqrt{\pi}}{\sqrt{\alpha}}\left(\int_{\sin\phi>0}\sin\phi(\varphi_{b}(\phi)-G_{b}^{4})^{2}d\phi\right)^{1/2}\int_{\eta}^{\infty}-e^{-\alpha\xi}de^{\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}$
$\displaystyle\quad\leq\frac{4\gamma^{3}\sqrt{\pi}}{\sqrt{\alpha}}\left(\int_{\sin\phi>0}\sin\phi(\varphi_{b}(\phi)-G_{b}^{4})^{2}d\phi\right)^{1/2}$
$\displaystyle\qquad\cdot\bigg{(}-e^{-\alpha\xi}e^{\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}\big{|}_{\eta}^{\infty}+\alpha\int_{\eta}^{\infty}e^{-\alpha\xi}e^{\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi\bigg{)}$
$\displaystyle\quad\leq\frac{4\gamma^{3}\sqrt{\pi}}{\sqrt{\alpha}}\left(\int_{\sin\phi>0}\sin\phi(\varphi_{b}(\phi)-G_{b}^{4})^{2}d\phi\right)^{1/2}(e^{-\alpha\eta}+\alpha\int_{\eta}^{\infty}e^{-\alpha\xi}e^{\eta-\xi}d\xi)$
$\displaystyle\quad\leq\frac{4\gamma^{3}\sqrt{\pi}}{\sqrt{\alpha}}\left(\int_{\sin\phi>0}\sin\phi(\varphi_{b}(\phi)-G_{b}^{4})^{2}d\phi\right)^{1/2}(e^{-\alpha\eta}+\frac{\alpha}{\alpha+1}e^{-\alpha\eta})$
$\displaystyle\quad\leq
e^{-\alpha\eta}\frac{4\gamma^{3}\sqrt{\pi}(2\alpha+1)}{\sqrt{\alpha}(\alpha+1)}\left(\int_{\sin\phi>0}\sin\phi(\varphi_{b}(\phi)-G_{b}^{4})^{2}d\phi\right)^{1/2}.$

For $\sin\phi<0$ and $E(\phi,\eta)>e^{-V_{\infty}}$, we use (20) and get

$\displaystyle|\varphi^{\varepsilon}(\eta,\phi)-(G_{\infty}^{\varepsilon})^{4}|$
$\displaystyle\quad=\Big{|}\varphi_{b}(\phi^{\prime}(-\phi,\eta,0))e^{-\int_{0}^{\eta_{+}}+\int_{\eta}^{\eta_{+}}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\xi)}d\xi}+\int_{0}^{\eta_{+}}\frac{(G^{\varepsilon}(\xi))^{4}}{\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{-\int_{\xi}^{\eta_{+}}+\int_{\eta}^{\eta_{+}}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi$
$\displaystyle\qquad+\int_{\eta}^{\eta_{+}}\frac{(G^{\varepsilon}(\xi))^{4}}{\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi-(G_{\infty}^{\varepsilon})^{4}\Big{|}$
$\displaystyle\quad=\Big{|}(\varphi_{b}(\phi^{\prime}(-\phi,\eta,0))-(G^{\varepsilon}_{\infty})^{4})e^{-\int_{0}^{\eta_{+}}+\int_{\eta}^{\eta_{+}}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\xi)}d\xi}$
$\displaystyle\qquad+\int_{0}^{\eta_{+}}\frac{(G^{\varepsilon}(\xi))^{4}-(G_{\infty}^{\varepsilon})^{4}}{\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{-\int_{\xi}^{\eta_{+}}+\int_{\eta}^{\eta_{+}}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi$
$\displaystyle\qquad+\int_{\eta}^{\eta_{+}}\frac{(G^{\varepsilon}(\xi))^{4}-(G^{\varepsilon}_{\infty})^{4}}{\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi\Big{|}$
$\displaystyle\quad\leq|(\varphi_{b}(\phi^{\prime}(-\phi,\eta,0))-(G^{\varepsilon}_{\infty})^{4})|e^{-\int_{0}^{\eta_{+}}+\int_{\eta}^{\eta_{+}}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\xi)}d\xi}$
$\displaystyle\qquad+C_{b}\int_{0}^{\eta_{+}}\frac{e^{-\alpha\xi}}{\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{-\int_{\xi}^{\eta_{+}}+\int_{\eta}^{\eta_{+}}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi+C_{b}\int_{\eta}^{\eta_{+}}\frac{e^{-\alpha\xi}}{\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi$
$\displaystyle\quad\leq|(\varphi_{b}(\phi^{\prime}(-\phi,\eta,0))-(G^{\varepsilon}_{\infty})^{4})|e^{-(2\eta_{+}-\eta)}+C_{b}\int_{0}^{\eta_{+}}e^{-\alpha\xi}de^{-\int_{\xi}^{\eta_{+}}+\int_{\eta}^{\eta_{+}}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}$
$\displaystyle\qquad-
C_{b}\int_{\eta}^{\eta_{+}}e^{-\alpha\xi}de^{\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}$
$\displaystyle\quad\leq|(\varphi_{b}(\phi^{\prime}(-\phi,\eta,0))-(G^{\varepsilon}_{\infty})^{4})|e^{-(2\eta_{+}-\eta)}+C_{b}e^{-\alpha\xi}e^{-\int_{\xi}^{\eta_{+}}+\int_{\eta}^{\eta_{+}}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}\Big{|}_{\xi=\eta}^{\eta_{+}}$
$\displaystyle\qquad+\alpha
C_{b}\int_{0}^{\eta_{+}}e^{-\alpha\xi}e^{-\int_{\xi}^{\eta_{+}}+\int_{\eta}^{\eta_{+}}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi-
C_{b}e^{-\alpha\xi}e^{\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}\Big{|}_{\xi=\eta}^{\eta_{+}}$
$\displaystyle\qquad-\alpha
C_{b}\int_{\eta}^{\eta_{+}}e^{-\alpha\xi}e^{\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi$
$\displaystyle\quad\leq|(\varphi_{b}(\phi^{\prime}(-\phi,\eta,0))-(G^{\varepsilon}_{\infty})^{4})|e^{-(2\eta_{+}-\eta)}+C_{b}e^{-\alpha\eta_{+}}e^{-(\eta_{+}-\eta)}-C_{b}e^{-\alpha\eta}e^{-2\int_{\eta}^{\eta_{+}}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}$
$\displaystyle\qquad+\alpha
C_{b}\int_{0}^{\eta_{+}}e^{-\alpha\xi}e^{-(2\eta_{+}-\eta-\xi)}d\xi-
C_{b}e^{-\alpha\eta_{+}}e^{\int_{\eta_{+}}^{\eta}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}+C_{b}e^{-\alpha\eta}$
$\displaystyle\qquad-\alpha
C_{b}\int_{\eta}^{\eta_{+}}e^{-\alpha\xi}e^{\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi$
$\displaystyle\quad\leq|(\varphi_{b}(\phi^{\prime}(-\phi,\eta,0))-(G^{\varepsilon}_{\infty})^{4})|e^{-(2\eta_{+}-\eta)}+C_{b}e^{-(\alpha+1)\eta_{+}+\eta}+\alpha
C_{b}\frac{e^{-(2\eta_{+}-\eta)}}{1-\alpha}(1-e^{\eta_{+}(1-\alpha)})+C_{b}e^{-\alpha\eta}$
$\displaystyle\leq|(\varphi_{b}(\phi^{\prime}(-\phi,\eta,0))-(G^{\varepsilon}_{\infty})^{4})|e^{-\eta}+C_{b}e^{-\alpha\eta}+\frac{\alpha}{1-\alpha}C_{b}e^{-\eta}+C_{b}e^{-\alpha\eta},$

where in the last inequality we use $\eta\leq\eta_{+}$, which implies
$e^{-(2\eta_{+}-\eta)}\leq e^{-\eta}$.

Therefore, we conclude that in any of the three cases,

$\displaystyle|\varphi^{\varepsilon}(\eta,\phi)-(G_{\infty}^{\varepsilon})^{4}|\leq
Ce^{-\alpha\eta},$ (3.47)

where $C$ depends on $G_{b}$, $\varphi_{b}$, $G_{\infty}^{\varepsilon}$,
$C_{b}$ and $\alpha$. ∎

######  Remark 12.

We can see from the above proof that the geometric correction gives a slightly
weaker estimate on the convergence rate. By (3.45), $\alpha\leq
1-\frac{1}{2}\|F\|_{L^{\infty}}\leq 1-2\varepsilon$. Hence the convergence
rate $\alpha$ should be strictly less than $1$. In the flat case without
geometric correction, $\alpha$ can be $1$ [6].

##  4\. Linearized Milne problem

For the higher order boundary layer expansions (2.66) and (2.67), the function
$\tilde{T}_{k}$, $\tilde{\psi}_{k}$, $k\geq 1$ satisfies the linear Milne
problem of the form

$\displaystyle\partial_{\eta}^{2}G^{\varepsilon}+F(\varepsilon;\eta)\partial_{\eta}G^{\varepsilon}+\langle\varphi^{\varepsilon}-4h^{3}G^{\varepsilon}\rangle=S_{1},$
(4.1)
$\displaystyle\sin\phi\partial_{\eta}\varphi^{\varepsilon}+F(\varepsilon;\eta)\cos\phi\partial_{\phi}\varphi^{\varepsilon}+\varphi^{\varepsilon}-4h^{3}G^{\varepsilon}=S_{2},$
(4.2)

with boundary conditions

$\displaystyle G^{\varepsilon}(0)=0,$ (4.3)
$\displaystyle\varphi^{\varepsilon}(0,\phi)=\varphi_{b}(\phi),\quad\text{for
}\sin\phi>0,$ (4.4)

where $S_{1}$, $S_{2}$ are known functions. This section is devoted to the
proof of the existence for the above linear Milne problem. The boundary data
$\varphi_{b}$ and the source term $S_{1}$, $S_{2}$ are assumed to satisfy

$\displaystyle|\varphi_{b}(\phi)|\leq
C_{b},\quad|S_{1}(\eta)|,\,|S_{2}(\eta,\phi)|\leq
C_{S}e^{-K\eta},\quad\text{for any
}\eta\in\mathbb{R}_{+},\,\phi\in[-\pi,\pi).$ (4.5)

In this section, we focus on proving the existence and uniqueness of solutions
for the above linear Milne problem, stated in the following theorem.

######  Theorem 13.

Given $h=h(\eta)\in C^{1}_{\rm\operatorname{loc}}([0,\infty))$ satisfying the
spectral assumption (A). Given functions $S_{1}=S_{1}(\eta)\in L^{2}_{\rm
loc}([0,\infty))$, $S_{2}=S_{2}(\eta,\phi)\in L^{2}_{\rm
loc}([0,\infty)\times[-\pi,\pi))$ satisfying (4.5). Then there exists a unique
solution $(G^{\varepsilon},\varphi^{\varepsilon})\in L^{2}_{\rm
loc}(\mathbb{R}_{+})\times L^{2}_{\rm loc}(\mathbb{R}_{+}\times[-\pi,\pi))$ to
system (4.1)-(4.2) with boundary conditions (4.3)-(4.4). Moreover, the
solution satisfies

$\displaystyle\frac{1}{8}\int_{0}^{\infty}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}(\varphi^{\varepsilon}-4h^{3}G^{\varepsilon})^{2}d\phi
d\eta+\frac{1}{2}\int_{\sin\phi<0}|\sin\phi|(\varphi^{\varepsilon})^{2}(0,\cdot)d\phi$
$\displaystyle\quad\leq\frac{1}{2}\int_{\sin\phi>0}\sin\phi\varphi_{b}^{2}d\phi+\|e^{\beta^{\prime}\eta}S_{2}\|_{L^{2}([0,\infty)\times[-\pi,\pi))}^{2},$
(4.6)

and there exists constants $G^{\varepsilon}_{\infty}$, such that

$\displaystyle|G^{\varepsilon}(\eta)-G^{\varepsilon}_{\infty}|\leq
Ce^{-\beta^{\prime}\eta},\quad|G^{\varepsilon}(\eta,\phi)-4h_{\infty}^{3}G_{\infty}^{\varepsilon}|\leq
Ce^{-\beta^{\prime}\eta},$

for any $0\leq\beta^{\prime}\leq\min\\{1/2,\beta,K\\}$, where $C>0$ is a
constant depending on $\varphi_{b}$.

###  4.1. Spectral assumption (A)

The spectral assumption is motivated by the corresponding eigenvalue problem
of system (4.1)-(4.2) without the geometric correction terms. The assumption
was proposed in [6] to study system (4.1)-(4.2) without the geometric
correction terms. Here we will show that under the spectral assumption, the
geometric corrections does not affect the linear stability of the system. We
first recall the following lemma that was proved in [6].

######  Lemma 14 ([6]).

Assume $h\in C^{1}_{\rm loc}(\mathbb{R}_{+})$ atisfies the spectral assumption
(A). Let $F=F(\varepsilon;\eta)$ be given by (2.48). Then it holds the
following inequality

$\displaystyle\int_{0}^{\infty}e^{2\beta^{\prime}\eta}4h^{3}|\partial_{\eta}f|^{2}d\eta+\int_{0}^{\infty}e^{2\beta^{\prime}\eta}\partial_{\eta}(4h^{3})f\partial_{\eta}fd\eta\geq
0,$ (4.7)

holds for any $0\leq\beta^{\prime}\leq\beta$ and for any $f\in C^{1}_{\rm
loc}(\mathbb{R}_{+})$ satisfying $f(0)=0$.

It has been proved in [6] that without geometric correction, system
(4.1)-(4.2) has a unique solution under the assumption that $h$ satisfies the
spectral assumption (A), see Appendix B for details.

###  4.2. Existence on the bounded interval

We first consider the existence for system (4.1)-(4.2) on the bounded interval
$\eta\in[0,B]$:

$\displaystyle\partial_{\eta}^{2}G_{B}^{\varepsilon}+F(\varepsilon;\eta)\partial_{\eta}G^{\varepsilon}_{B}+\langle\varphi^{\varepsilon}_{B}-4h^{3}G^{\varepsilon}_{B}\rangle=S_{1},$
(4.8)
$\displaystyle\sin\phi\partial_{\eta}\varphi^{\varepsilon}_{B}+F(\varepsilon;\eta)\cos\phi\partial_{\phi}\varphi^{\varepsilon}_{B}+\varphi^{\varepsilon}_{B}-4h^{3}G^{\varepsilon}_{B}=S_{2},$
(4.9)

with boundary conditions

$\displaystyle
G^{\varepsilon}_{B}(0)=0,\quad\partial_{\eta}G^{\varepsilon}(B)=0,$ (4.10)
$\displaystyle\varphi^{\varepsilon}_{B}(0,\phi)=\varphi_{b}(\phi),\quad\varphi^{\varepsilon}_{B}(B,\phi)=\varphi^{\varepsilon}_{B}(B,-\phi),\quad\text{for
}\sin\phi>0.$ (4.11)

We will consider the case when $\langle S_{2}\rangle=S_{1}$ and prove the
following lemma.

######  Lemma 15.

Let $h$ satisfy the spectral assumption (A). Assume $\varphi_{b}$, $S_{1}$,
$S_{2}$ satisfy (4.5) and $S_{1}(\eta)=\langle S_{2}(\eta,\cdot)\rangle$ for
any $\eta\in\mathbb{R}_{+}$ and $\varphi_{b}\in C^{\infty}(0,\pi)$. Then there
exists a solution $(G_{B}^{\varepsilon},\varphi_{B}^{\varepsilon})\in
L^{2}([0,B])\times L^{2}([0,B]\times[-\pi,\pi))$ to system (4.8)-(4.9) with
boundary conditions (4.10)-(4.11), and the solution verifies

$\displaystyle\frac{1}{4}\int_{0}^{B}\int_{-\pi}^{\pi}(\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon})^{2}d\phi
d\eta+\frac{1}{2}\int_{\sin\phi<0}|\sin\phi|(\varphi_{B}^{\varepsilon})^{2}(0,\phi)d\phi$
$\displaystyle\quad\leq\frac{1}{2}\int_{\sin\phi>0}|\sin\phi|(\varphi_{b})^{2}d\phi+\|S_{2}\|_{L^{2}([0,B]\times[-\pi,\pi))}^{2},$
(4.12)

for $\varepsilon$ sufficiently small.

###### Proof.

The proof is based on Theorem 22 for existence of the linear Milne problem
without geometric correction and the Leray-Schauder fixed point theorem.

First we construct a sequence of functions
$\\{(G^{k},\varphi^{k})\\}_{k=0}^{\infty}$ by $G^{0}=0,\varphi^{0}=0$ and

$\displaystyle\partial_{\eta}^{2}G^{k}+F(\varepsilon;\eta)\partial_{\eta}G^{k-1}+\langle\varphi^{k}-4h^{3}G^{k}\rangle=S_{1},$
(4.13)
$\displaystyle\sin\phi\partial_{\eta}\varphi^{k}+F(\varepsilon;\eta)\cos\phi\partial_{\phi}\varphi^{k-1}+\varphi^{k}-4h^{3}G^{k}=S_{2},$
(4.14)

with boundary conditions

$\displaystyle G^{k}(0)=0,\quad\partial_{\eta}G^{k}(B)=0,$ (4.15)
$\displaystyle\varphi^{k}(0,\phi)=\varphi_{b}(\phi),\quad\varphi^{k}(B,\phi)=\varphi^{k}(B,-\phi),\quad\text{for
}\sin\phi>0.$ (4.16)

Given bounded functions $(G^{k-1},\varphi^{k-1})$, according to Theorem 22 in
Appendix B, there exists unique solution $(G^{k},\varphi^{k})$ for the above
system.

Next we show the boundness of $(G^{k},\varphi^{k})$ in $L^{2}([0,B])\times
L^{2}([0,B]\times[-\pi,\pi])$. First, the relation
$\partial_{\eta}G^{k}=\langle\sin\phi\varphi^{k}\rangle$ on $[0,B]$ is
preserved during the iterations. Suppose
$\partial_{\eta}G^{k-1}=\langle\sin\phi\varphi^{k-1}\rangle$ on $[0,B]$, then
comparing (4.13) to $\langle$(4.14)$\rangle$ and using $S_{1}=\langle
S_{2}\rangle$ yields

$\displaystyle\partial_{\eta}(\partial_{\eta}G^{k}-\langle\sin\phi\varphi^{k}\rangle)$
$\displaystyle=-F(\varepsilon;\eta)(\partial_{\eta}G^{k-1}-\langle\cos\phi\partial_{\phi}\varphi^{k-1}\rangle)$
$\displaystyle=-F(\varepsilon;\eta)(\partial_{\eta}G^{k-1}-\langle\sin\phi\varphi^{k-1}\rangle)=0.$

This, together with the boundary condition
$\partial_{\eta}G^{k}(B)=\langle\varphi^{k}(B,\cdot)\rangle=0$, implies that
$\partial_{\eta}G^{k}(\eta)=\langle\sin\phi\varphi^{k}(\eta,\cdot)\rangle$ for
any $\eta\in[0,B]$.

We multiply (4.13) by $4h^{3}G^{k}$ and (4.14) by $\varphi^{k}$ and integrate
to get

$\displaystyle-\int_{0}^{B}4h^{3}G^{k}\partial_{\eta}^{2}G^{k}d\eta+\frac{1}{2}\int_{0}^{B}\int_{-\pi}^{\pi}\sin\phi\partial_{\eta}(\varphi^{k})^{2}d\phi
d\eta+\int_{0}^{B}\int_{-\pi}^{\pi}(\varphi^{k}-4h^{3}G^{k})^{2}d\phi d\eta$
$\displaystyle\quad+\int_{0}^{B}\int_{-\pi}^{\pi}F(\varepsilon;\eta)\cos\phi\partial_{\phi}\varphi^{k-1}\cdot\varphi^{k}d\phi
d\eta-\int_{0}^{B}F(\varepsilon;\eta)4h^{3}G^{k}\partial_{\eta}G^{k-1}d\eta$
$\displaystyle\qquad=\int_{0}^{B}\int_{-\pi}^{\pi}S_{2}(\varphi^{k}-4h^{3}G^{k})d\phi
d\eta.$ (4.17)

The inequality (4.7) from the spectral assumption implies

$\displaystyle-\int_{0}^{B}4h^{3}G^{k}\partial_{\eta}^{2}G^{k}=\int_{0}^{B}4h^{3}|\partial_{\eta}G^{k}|^{2}d\eta+\int_{0}^{B}\partial_{\eta}(4h^{3})G^{k}\partial_{\eta}G^{k}d\eta\geq
0.$ (4.18)

Boundary conditions (4.16) implies

$\displaystyle\frac{1}{2}\int_{0}^{B}\int_{-\pi}^{\pi}\sin\phi\partial_{\eta}(\varphi^{k})^{2}d\phi
d\eta$
$\displaystyle=\frac{1}{2}\int_{-\pi}^{\pi}\sin\phi(\varphi^{k})^{2}d\phi\bigg{|}_{0}^{B}$
$\displaystyle=-\frac{1}{2}\int_{\sin\phi>0}|\sin\phi|(\varphi_{b})^{2}d\phi+\frac{1}{2}\int_{\sin\phi<0}|\sin\phi|(\varphi^{k})^{2}(0,\phi)d\phi.$
(4.19)

The relation $\partial_{\eta}G^{k-1}=\langle\sin\phi\varphi^{k-1}\rangle$
implies

$\displaystyle\int_{0}^{B}\int_{-\pi}^{\pi}F(\varepsilon;\eta)\cos\phi\partial_{\phi}\varphi^{k-1}\cdot\varphi^{k}d\phi
d\eta-\int_{0}^{B}4h^{3}G^{k}\partial_{\eta}G^{k-1}d\eta$
$\displaystyle\quad=\int_{0}^{B}\int_{-\pi}^{\pi}F(\varepsilon;\eta)\cos\phi\partial_{\phi}\varphi^{k-1}\cdot\varphi^{k}d\phi
d\eta-\int_{0}^{B}\int_{-\pi}^{\pi}\sin\phi\varphi^{k-1}4h^{3}G^{k}d\phi
d\eta$
$\displaystyle\quad=\int_{0}^{B}\int_{-\pi}^{\pi}F(\varepsilon;\eta)\cos\phi\partial_{\phi}\varphi^{k-1}\cdot\varphi^{k}d\phi
d\eta-\int_{0}^{B}\int_{-\pi}^{\pi}\cos\phi\partial_{\phi}\varphi^{k-1}\cdot
4h^{3}G^{k}d\phi d\eta$
$\displaystyle\quad=\int_{0}^{B}\int_{-\pi}^{\pi}F(\varepsilon;\eta)\cos\phi\partial_{\phi}\varphi^{k-1}\cdot(\varphi^{k}-4h^{3}G^{k})d\phi
d\eta$
$\displaystyle\quad\leq\frac{1}{2}\int_{0}^{B}\int_{-\pi}^{\pi}(\varphi^{k}-4h^{3}G^{k})^{2}d\phi
d\eta+\frac{1}{2}\int_{0}^{B}\int_{-\pi}^{\pi}|F(\varepsilon;\eta)|^{2}|\partial_{\phi}\varphi^{k-1}|^{2}d\phi
d\eta$
$\displaystyle\quad\leq\frac{1}{2}\int_{0}^{B}\int_{-\pi}^{\pi}(\varphi^{k}-4h^{3}G^{k})^{2}d\phi
d\eta+\frac{1}{2}\|F\|_{L^{\infty}(\mathbb{R}_{+})}\|^{2}\|\partial_{\phi}\varphi^{k-1}\|_{L^{2}([0,B]\times[-\pi,\pi))}^{2}$
$\displaystyle\quad\leq\frac{1}{2}\int_{0}^{B}\int_{-\pi}^{\pi}(\varphi^{k}-4h^{3}G^{k})^{2}d\phi
d\eta+\frac{2}{4-3\delta}\varepsilon\|\partial_{\phi}\varphi^{k-1}\|_{L^{2}([0,B]\times[-\pi,\pi))}^{2},$
(4.20)

where the last inequality is due to (3.11).

Using the Young’s inequality, the last term in (4.2) can be estimated by

$\displaystyle\int_{0}^{B}\int_{-\pi}^{\pi}S_{2}(\varphi^{k}-4h^{3}G^{k})d\phi
d\eta\leq\frac{1}{4}\int_{0}^{B}\int_{-\pi}^{\pi}(\varphi^{k}-4h^{3}G^{k})^{2}d\phi
d\eta+\|S_{2}\|_{L^{2}([0,B]\times[-\pi,\pi))}^{2}.$

Taking the above inequality with (4.18)-(4.2) into (4.2), we obtain

$\displaystyle\frac{1}{4}\int_{0}^{B}\int_{-\pi}^{\pi}(\varphi^{k}-4h^{3}G^{k})^{2}d\phi
d\eta+\frac{1}{2}\int_{\sin\phi<0}|\sin\phi|(\varphi^{k})^{2}(0,\phi)d\phi$
$\displaystyle\quad\leq\frac{1}{2}\int_{\sin\phi>0}|\sin\phi|(\varphi_{b})^{2}d\phi+\|S_{2}\|_{L^{2}([0,B]\times[-\pi,\pi))}^{2}+\frac{2}{4-3\delta}\varepsilon\|\partial_{\phi}\varphi^{k-1}\|_{L^{2}([0,B]\times[-\pi,\pi))}^{2}.$
(4.21)

Hence, assume $\partial_{\phi}\varphi^{k-1}\in L^{2}([0,B]))$, one can get
$\varphi^{k}-4h^{3}G^{k}\in L^{2}([0,B]\times[-\pi,\pi))$. By equation (4.13),
$\partial_{\eta}^{2}G^{k}\in L^{2}([0,B])$ and so

$\displaystyle
G^{k}(\eta)=-\int_{0}^{\eta}\partial_{s}G^{k}(s)ds=\int_{0}^{\eta}\int_{s}^{B}\partial_{\ell}^{2}G^{k}(\ell)d\ell
ds\leq\int_{0}^{B}\int_{0}^{B}|\partial_{\eta}^{2}G^{k}|d\eta ds\leq
B\sqrt{B}\|\partial^{2}_{\eta}G\|_{L^{2}([0,B])},$ (4.22)

and thus is bounded for any $\eta\in[0,B]$. Moreover, by the boundness of $h$,
$\varphi^{k}=\varphi^{k}-4h^{3}G^{k}+4h^{3}G^{k}\in
L^{2}([0,B]\times[-\pi,\pi))$. Note here since $\varphi^{k-1}(\eta,\cdot)\in
C^{\infty}([-\pi,\pi))$, $\varphi^{k}(\eta,\cdot)\in C^{\infty}([-\pi,\pi))$,
since the regularity on $\phi$ is preserved by system (4.13)-(4.14).

Therefore, we have shown that there exists a solution to the problem
(4.13)-(4.16) satisfying $(G^{k},\varphi^{k})\in L^{2}([0,B])\times
L^{2}([0,B]\times[-\pi,\pi))$. Define the operator
$\mathcal{L}:L^{2}([0,B])\times L^{2}([0,B]\times[-\pi,\pi))\mapsto
L^{2}([0,B])\times L^{2}([0,B]\times[-\pi,\pi))$ given by (4.13)-(4.16) with
$(G^{k},\varphi^{k})=\mathcal{L}((G^{k-1},\varphi^{k-1}))$. We can also get
from the above derivation that $G^{k}\in H^{2}([0,B])$ and $\varphi^{k}\in
H^{1}([0,B]\times[-\pi,\pi))$. Hence the operator $\mathcal{L}$ is continous
and compact. In order to show $\mathcal{L}$ has a fixed point, we need to show
the set

$\displaystyle\\{(G,\varphi)\in L^{2}([0,B])\times
L^{2}([0,B]\times[-\pi,\pi)):(G,\varphi)=\sigma\mathcal{L}((G,\varphi))\text{
for some }0\leq\sigma\leq 1\\}$ (4.23)

is bounded. Then by the Schauder fixed-point theorem, there exists a fixed
point of $\mathcal{L}$. To show the above set is bounded, let
$(G^{,}\varphi)=\sigma\mathcal{L}((G,\varphi))$, then $(G,\varphi)$ solves

$\displaystyle\partial_{\eta}^{2}G+\sigma
F(\varepsilon;\eta)\partial_{\eta}G+\langle\varphi-4h^{3}G\rangle=S_{1},$
(4.24) $\displaystyle\sin\phi\partial_{\eta}\varphi+\sigma
F(\varepsilon;\eta)\cos\phi\partial_{\phi}\varphi+\varphi-4h^{3}G=S_{2},$
(4.25)

with boundary conditions

$\displaystyle G(0)=0,\quad\partial_{\eta}G(B)=0,$
$\displaystyle\varphi(0,\phi)=\varphi_{b}(\phi),\quad\varphi(B,\phi)=\varphi(B,-\phi),\quad\text{for
}\sin\phi>0.$

Note that $\partial_{\eta}\varphi=\langle\sin\phi\varphi\rangle$ also hold on
$[0,B]$. Comparing equation (4.24) to (4.25) gives

$\displaystyle\partial_{\eta}(\partial_{\eta}G-\langle\sin\phi\varphi)\rangle)=-\sigma
F(\varepsilon;\eta)(\partial_{\eta}G-\langle\cos\phi\partial_{\phi}\varphi\rangle)=-\sigma
F(\varepsilon;\eta)(\partial_{\eta}G-\langle\sin\phi\varphi)\rangle),$

which together with the boundary condition
$\partial_{\eta}G(B)-\langle\sin\phi\varphi(B,\cdot)\rangle=0$, implies

$\displaystyle\partial_{\eta}G=\langle\sin\phi\varphi\rangle.$ (4.26)

We can estimate the above system similarly as the estimates of system
(4.13)-(4.14) and the result is (see the estimate of (4.2))

$\displaystyle\frac{3}{4}\int_{0}^{B}\int_{-\pi}^{\pi}(\varphi-4h^{3}G)^{2}d\phi
d\eta+\frac{1}{2}\int_{\sin\phi<0}|\sin\phi|\varphi^{2}(0,\phi)d\phi$
$\displaystyle\quad\leq\frac{1}{2}\int_{\sin\phi>0}|\sin\phi|(\varphi_{b})^{2}d\phi+\|S_{2}\|_{L^{2}([0,B]\times[-\pi,\pi))}^{2}+\int_{0}^{B}\int_{-\pi}^{\pi}F(\varepsilon;\eta)\cos\phi\partial_{\phi}\varphi(\varphi-4h^{3}G)d\phi
d\eta.$

Integration by parts in $\phi$ on the last term gives

$\displaystyle\int_{0}^{B}\int_{-\pi}^{\pi}F(\varepsilon;\eta)\cos\phi\partial_{\phi}\varphi(\varphi-4h^{3}G)d\phi
d\eta$
$\displaystyle\quad=\int_{0}^{B}\int_{-\pi}^{\pi}F(\varepsilon;\eta)\sin\phi\varphi(\varphi-4h^{3}G)d\phi
d\eta$
$\displaystyle\quad=\frac{1}{2}\int_{0}^{B}\int_{-\pi}^{\pi}F(\varepsilon;\eta)\sin\phi(\varphi-4h^{3}G)^{2}d\eta$
$\displaystyle\quad\leq\frac{1}{2}\|F\|_{L^{\infty}(\mathbb{R}_{+})}\int_{0}^{B}\int_{-\pi}^{\pi}(\varphi-4h^{3}G)^{2}d\phi
d\eta$
$\displaystyle\quad\leq\frac{2}{4-3\delta}\varepsilon\int_{0}^{B}\int_{-\pi}^{\pi}(\varphi-4h^{3}G)^{2}d\phi
d\eta.$

Taking it into the previous inequality leads to

$\displaystyle\left(\frac{3}{4}-\frac{2}{4-3\delta}\varepsilon\right)\int_{0}^{B}\int_{-\pi}^{\pi}(\varphi-4h^{3}G)^{2}d\phi
d\eta+\frac{1}{2}\int_{\sin\phi<0}|\sin\phi|\varphi^{2}(0,\phi)d\phi$
$\displaystyle\quad\leq\frac{1}{2}\int_{\sin\phi>0}|\sin\phi|(\varphi_{b})^{2}d\phi+\|S_{2}\|_{L^{2}([0,B]\times[-\pi,\pi))}^{2}.$

For $\varepsilon$ small, for example $\varepsilon\leq(4-3\delta)/8$,
$\left(\frac{3}{4}-\frac{2}{4-3\delta}\varepsilon\right)\geq\frac{1}{4}$, and
the above estimate becomes

$\displaystyle\frac{1}{4}\int_{0}^{B}\int_{-\pi}^{\pi}(\varphi-4h^{3}G)^{2}d\phi
d\eta+\frac{1}{2}\int_{\sin\phi<0}|\sin\phi|\varphi^{2}(0,\phi)d\phi$
$\displaystyle\quad\leq\frac{1}{2}\int_{\sin\phi>0}|\sin\phi|(\varphi_{b})^{2}d\phi+\|S_{2}\|_{L^{2}([0,B]\times[-\pi,\pi))}^{2}.$
(4.27)

Moreover, by equation (4.24),

$\displaystyle\|\partial_{\eta}^{2}G\|_{L^{2}([0,B])}$
$\displaystyle\leq\sigma\|F\partial_{\eta}G\|_{L^{2}([0,B])}+\|\langle(\varphi-4h^{3}G)\rangle\|_{L^{2}([0,B])}+\|S_{1}\|_{L^{2}([0,B])}$
$\displaystyle\leq\sigma\|F\|_{L^{\infty}(\mathbb{R}_{+})}\|\langle\sin\phi(\varphi-4h^{3}G)\rangle\|_{L^{2}[0,B]}+\|\langle(\varphi-4h^{3}G)\rangle\|_{L^{2}([0,B])}+\|S_{1}\|_{L^{2}([0,B])}$
$\displaystyle\leq\left(\frac{4\sigma}{4-3\delta}\varepsilon+1\right)\sqrt{2\pi}\|(\varphi-4h^{3}G)\|_{L^{2}([0,B]\times[-\pi,\pi))}+\|S_{1}\|_{L^{2}([0,B])}$
$\displaystyle\leq
C\left(\int_{\sin\phi>0}|\sin\phi|(\varphi_{b})^{2}d\phi\right)^{\frac{1}{2}}+C\|S_{2}\|_{L^{2}([0,B]\times[-\pi,\pi))}+\|S_{1}\|_{L^{2}([0,B])}.$

Hence $\|G\|_{H^{2}([0,B])}$ is bounded. Moreover
$\varphi=(\varphi-4h^{3}G)+4h^{3}G\in L^{2}([0,B]\times[-\pi,\pi))$ is
bounded. Therefore, the set (4.24) is bounded. By the Schauder fixed point
theorem, there exists a fixed point
$(G^{\varepsilon}_{B},\varphi^{\varepsilon}_{B})\in L^{2}([0,B])\times
L^{2}([0,B]\times[-\pi,\pi))$ such that
$(G^{\varepsilon}_{B},\varphi^{\varepsilon}_{B})=\mathcal{L}((G^{\varepsilon}_{B},\varphi^{\varepsilon}_{B}))$.
By the definition of $\mathcal{L}$,
$(G^{\varepsilon}_{B},\varphi^{\varepsilon}_{B})$ solves system (4.8)-(4.9)
with boundary conditions (4.10)-(4.11).

Taking $\sigma=1$ in the derivation of the estimate (4.2) implies that
$(G^{\varepsilon}_{B},\varphi_{B}^{\varepsilon})$ satisfy the estimate (15)
and finishes the proof. ∎

###  4.3. Weighted estimate

Next we derive the uniform weighted estimate on the solutions of system
(4.8)-(4.9). We have the following lemma.

######  Lemma 16.

Assuming the hypothesis of Lemma 15 hold and
$(G_{B}^{\varepsilon},\varphi_{B}^{\varepsilon})$ is solution obtained from
Lemma 15. Then for $0\leq\beta^{\prime}\leq\min\\{1/2,\beta,K\\}$, the
following estimate holds

$\displaystyle\frac{1}{8}\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}(\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon})^{2}d\phi
d\eta+\frac{1}{2}\int_{\sin\phi<0}|\sin\phi|(\varphi_{B}^{\varepsilon})^{2}(0,\cdot)d\phi$
$\displaystyle\quad\leq\frac{1}{2}\int_{\sin\phi>0}\sin\phi\varphi_{b}^{2}d\phi+\|e^{\beta^{\prime}\eta}S_{2}\|_{L^{2}([0,B]\times[-\pi,\pi))}^{2},$
(4.28)

and

$\displaystyle|G_{B}^{\varepsilon}(\eta)|\leq C$ (4.29)

for some constant $C>0$ dependent on $\varphi_{b},S_{1},S_{2}$ but independent
of $\varepsilon$. Moreover, the solution satisfies

$\displaystyle\partial_{\eta}G_{B}^{\varepsilon}(\eta)=\langle\sin\phi\varphi^{\varepsilon}_{B}(\eta,\cdot)\rangle,$
(4.30)

for any $\eta\in[0,B]$.

###### Proof.

First, we have
$\partial_{\eta}G_{B}^{\varepsilon}=\langle\sin\phi\varphi_{B}^{\varepsilon}\rangle$
which is due to the same reason as (4.26). We multiply (4.8) by
$e^{2\beta^{\prime}\eta}4hG_{B}^{\varepsilon}$ and (4.9) by
$\varphi_{B}^{\varepsilon}$ and integrate to obtain

$\displaystyle-\int_{0}^{B}e^{2\beta^{\prime}\eta}4h^{3}G^{\varepsilon}_{B}\partial_{\eta}^{2}G^{\varepsilon}_{B}d\eta+\frac{1}{2}\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}\sin\phi\partial_{\eta}(\varphi^{\varepsilon}_{B})^{2}d\phi
d\eta+\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}(\varphi^{\varepsilon}_{B}-4h^{3}G^{\varepsilon}_{B})^{2}d\phi
d\eta$
$\displaystyle\quad+\frac{1}{2}\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}F(\varepsilon;\eta)\cos\phi\partial_{\phi}(\varphi^{\varepsilon}_{B})^{2}d\phi
d\eta-\int_{0}^{B}e^{2\beta^{\prime}\eta}F(\varepsilon;\eta)4h^{3}G^{\varepsilon}_{B}\partial_{\eta}G^{\varepsilon}_{B}d\eta$
$\displaystyle\qquad=\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}S_{2}(\varphi^{\varepsilon}_{B}-4h^{3}G^{\varepsilon}_{B})d\phi
d\eta.$ (4.31)

Using integration by parts and the inequality (4.7) from the spectral
assumption (A), we can get

$\displaystyle-\int_{0}^{B}e^{2\beta^{\prime}\eta}4h^{3}G^{\varepsilon}_{B}\partial_{\eta}^{2}G^{\varepsilon}_{B}d\eta$
$\displaystyle\quad=\int_{0}^{B}e^{2\beta^{\prime}\eta}4h^{3}|\partial_{\eta}G_{B}^{\varepsilon}|^{2}d\eta+\int_{0}^{B}e^{2\beta^{\prime}\eta}\partial_{\eta}(4h^{3})G_{B}^{\varepsilon}\partial_{\eta}G_{B}^{\varepsilon}d\eta+2\beta^{\prime}\int_{0}^{B}e^{2\beta^{\prime}\eta}4h^{3}G_{B}^{\varepsilon}\partial_{\eta}G_{B}^{\varepsilon}d\eta$
$\displaystyle\quad\geq
2\beta^{\prime}\int_{0}^{B}e^{\beta^{\prime}\eta}4h^{3}G_{B}^{\varepsilon}\partial_{\eta}G_{B}^{\varepsilon}d\eta.$

Using integration by parts and the boundary conditions (4.11), we have

$\displaystyle\frac{1}{2}\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}\sin\phi\partial_{\eta}(\varphi^{\varepsilon}_{B})^{2}d\phi
d\eta$
$\displaystyle\quad=-\beta^{\prime}\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}\sin\phi(\varphi_{B}^{\varepsilon})^{2}d\beta
d\eta+\frac{1}{2}e^{2\beta^{\prime}\eta}\int_{-\pi}^{\pi}\sin\phi(\varphi_{B}^{\varepsilon})^{2}(\eta,\cdot)d\phi\bigg{|}_{0}^{B}$
$\displaystyle\quad=-\beta^{\prime}\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}\sin\phi(\varphi_{B}^{\varepsilon})^{2}d\beta
d\eta-\frac{1}{2}\int_{\sin\phi>0}\sin\phi\varphi_{b}^{2}d\phi+\frac{1}{2}\int_{\sin\phi<0}|\sin\phi|(\varphi_{B}^{\varepsilon})^{2}(0,\cdot)d\phi.$

The relation
$\partial_{\eta}G_{B}^{\varepsilon}=\langle\sin\phi\varphi_{B}^{\varepsilon}\rangle$
gives

$\displaystyle\frac{1}{2}\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}F(\varepsilon;\eta)\cos\phi\partial_{\phi}(\varphi^{\varepsilon}_{B})^{2}d\phi
d\eta-\int_{0}^{B}e^{2\beta^{\prime}\eta}F(\varepsilon;\eta)4h^{3}G^{\varepsilon}_{B}\partial_{\eta}G^{\varepsilon}_{B}d\eta$
$\displaystyle\quad=\frac{1}{2}\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}F(\varepsilon;\eta)\sin\phi(\varphi_{B}^{\varepsilon})^{2}d\phi
d\eta-\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}F(\varepsilon;\eta)4h^{3}G^{\varepsilon}_{B}\sin\phi\varphi_{B}^{\varepsilon}d\phi
d\eta$
$\displaystyle\quad=\frac{1}{2}\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}F(\varepsilon;\eta)\sin\phi(\varphi_{B}^{\varepsilon}-G_{B}^{\varepsilon})^{2}d\phi
d\eta,$

and also

$\displaystyle-\beta^{\prime}\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}\sin\phi(\varphi_{B}^{\varepsilon})^{2}d\beta