# Truncation Error Analysis for an Accurate Nonlocal Approximation to Manifold
Poisson Models with Dirichlet Boundary ††thanks: This work was supported by
NSFC grant 12071244, 11671005.

Yajie Zhang Department of Statistics and Mathematics, Zhongnan University of
Economics and Laws, Wuhan, China, 430000 Email<EMAIL_ADDRESS>Zuoqiang
Shi Corresponding Author, Department of Mathematical Sciences, Tsinghua
University, Beijing, China, 100084\. Email<EMAIL_ADDRESS>
###### Abstract

In this work, we introduced a class of nonlocal models to accurately
approximate the Poisson model on manifolds that are embedded in high
dimensional Euclid spaces with Dirichlet boundary. In comparison to the
existing nonlocal approximation to Poisson models, instead of utilizing
volumetric boundary constraint to reduce the truncation error to its local
counterpart, we rely on the Poisson equation itself along the boundary to
explicitly express the second order normal derivative by some geometry-based
terms, so that to create a new model with $\mathcal{O}(\delta)$ truncation
error along the $2\delta-$boundary layer and $\mathcal{O}(\delta^{2})$ at
interior, with $\delta$ be the nonlocal interaction horizon. Our concentration
is on the construction and the truncation error analysis of such nonlocal
model. The control on the truncation error is currently optimal among all
nonlocal models, and is sufficient to attain second order localization rate
that will be derived in our subsequent work.

###### keywords:

Manifold Poisson equation, Dirichlet boundary, nonlocal approximation,
truncation error analysis, second order convergence.

45P05; 45A05; 35A15; 46E35

##  1 Introduction.

Partial differential equations on manifolds have never failed to attract
researchers in the past decades. Its application includes material science [9]
[18] , fluid flow [20] [22], and biology physics [6] [19] [30]. In addition,
many recent work on machine learning [7] [11] [25] [28] [34] and image
processing [10] [21] [23] [24] [31] [33] [42] relates their model with
manifold PDEs. One frequently used manifold model is the Poisson equation.
Apart from its broad application, such equation is mathematically interesting
by itself since it reveals much information of the manifold. Among all the
recent study on the numerical analysis of the Poisson models, one approach is
its nonlocal approximation. The main advantage for nonlocal model is that the
use of explicit spatial differential operators is avoided, hence new numerical
schemes can be explored, for instance, the point integral method(PIM). In
spite of the large quantity of articles on the nonlocal approximation of
Poisson model on Euclid spaces, less work have been done on its extension from
Euclid spaces to manifolds. Due to the demand of numerical methods for
manifold PDEs, it is necessary to construct one nonlocal model that
approximates the manifold Poisson equation with high accuracy, hence to be
able to solve it by proper numerical scheme.

In this paper, our purpose is to seek for a nonlocal model that approximates
the following Poisson problem with Dirichlet boundary with high accuracy:

$\begin{cases}-\Delta_{\mathcal{M}}u({x})=f({x})&{x}\in\mathcal{M},\\\
u=0&{x}\in\partial\mathcal{M}.\end{cases}$ (1.1)

Here $\mathcal{M}$ is a compact, smooth $m$ dimensional manifold embedded in
$\mathbb{R}^{d}$, with $\partial\mathcal{M}$ a smooth $(m-1)$ dimensional
curve with bounded curvature. $f$ is an $H^{2}$ function on $\mathcal{M}$,
$\Delta_{\mathcal{M}}$ is the Laplace-Beltrami operator on $\mathcal{M}$. We
next state the definition of $\Delta_{\mathcal{M}}$: Let
$\Phi:\Omega\subset\mathcal{R}^{m}\to\mathcal{M}\subset\mathcal{R}^{d}$ be a
local parametrization of $\mathcal{M}$ and $\theta\in\Omega$, then
$\Delta_{\mathcal{M}}$ is defined as

$\Delta_{\mathcal{M}}u(\Phi(\theta))=div\
\nabla_{\mathcal{M}}u(\Phi(\theta)),$ (1.2)

where the gradient $\nabla_{\mathcal{M}}$ on $\mathcal{M}$ is

$\nabla_{\mathcal{M}}f(\Phi(\theta))=\sum\limits_{i,j=1}^{m}g^{ij}(\theta)\frac{\partial\Phi}{\partial\theta_{i}}(\theta)\frac{\partial
f(\Phi(\theta))}{\partial\theta_{j}}(\theta)$ (1.3)

for any differentiable function $f:\mathcal{M}\to\mathcal{R}$; and the
divergence operator on $\mathcal{M}$ is defined as

$div(F)=\frac{1}{\sqrt{det\
G}}\sum\limits_{k=1}^{d}\sum\limits_{i,j=1}^{m}\frac{\partial}{\partial\theta_{i}}(\sqrt{det\
G}g^{ij}F^{k}(\Phi(\theta))\frac{\partial\phi^{k}}{\partial\theta_{j}})$ (1.4)

for any vector field $F:\mathcal{M}\to\mathcal{T}_{x}\mathcal{M}$ on
$\mathcal{M}$. Here $\mathcal{T}_{{x}}\mathcal{M}$ is the tangent space of
$\mathcal{M}$ at ${x}=\Phi(\theta)\in\mathcal{M}$,
$(F^{1}({x}),...,F^{d}({x}))^{t}$ is the representation of $F$ in the
embedding coordinates, and $G(\theta)=(g_{ij})_{i,j=1,2,...,m}$ is the first
fundamental form which is defined by

$g_{ij}(\theta)=\sum\limits_{k=1}^{d}\frac{\partial\phi^{k}}{\partial\theta_{i}}(\theta)\frac{\partial\phi^{k}}{\partial\theta_{j}}(\theta),\qquad
i,j=1,...,m.$ (1.5)

And $(g^{ij})_{i,j=1,...,k}=G^{-1}$, $det\ G$ is the determinant of matrix
$G$.

In the past decade, researchers devoted on the construction of nonlocal
Poisson models on Euclid domains, among which the most commonly studied
equation is

$\frac{1}{\delta^{2}}\int_{\Omega}(u_{\delta}({x})-u_{\delta}({y}))R_{\delta}({x},{y})d{y}=f({x}),\qquad{x}\in\Omega.$
(1.6)

Here $\Omega\subset\mathbb{R}^{k}$ is a bounded Euclid domain with smooth
boundary, $f\in H^{2}(\Omega)$, $\delta$ is the nonlocal horizon parameter
that describes the range of nonlocal interaction,
$R_{\delta}({x},{y})=C_{\delta}R\big{(}\frac{|{x}-{y}|^{2}}{4\delta^{2}}\big{)}$
is the nonlocal kernel function, where $R\in C^{2}(\mathbb{R}^{+})\cap
L^{1}[0,\infty)$ is a properly-chosen positive function with compact support,
and $C_{\delta}=\frac{1}{(4\pi\delta^{2})^{k/2}}$ is the normalization factor.
Such equation usually appeared in the discussion of peridynamics models [3]
[8] [13] [32] [38] [40]. The truncation error of (1.6) to its local
counterpart $\Delta u=f$ has been proved to be $\mathcal{O}(\delta^{2})$ in
the interior region away from boundary, and $\mathcal{O}(\delta^{-1})$ in the
$2\delta-$layer along the boundary. Efforts have been made to reduce the order
of truncation error by making modification of (1.6) along the boundary layer.
See [4] [5] [12] [14] for Neumann boundary condition and [1] [2] [15] [29]
[45] for other types of boundary conditions. Other related modification of
(1.6) involves fractional nonlocal kernel [17] or nonlocal gradient terms
[16]. Those studies yield to at most $\mathcal{O}(\delta)$ convergence rate
from $u_{\delta}$ to $u$.

As a breakthrough, in one dimensional [39] and two dimensional [43] cases, the
nonlocal models with $\mathcal{O}(\delta^{2})$ convergence rate to its local
counterpart were successfully constructed under Neumann boundary condition.
One year later, Lee H. and Du Q. in [26] introduced a nonlocal model under
Dirichlet boundary condition by imposing a special volumetric constraint along
the boundary layer, which assures $\mathcal{O}(\delta^{2})$ convergence rate
in 1d segment and 2d plain disk.

Nevertheless, due to the demand of massive calculation, less work have been
done on the extension of nonlocal Poisson model from Euclid space into
manifold. This restricted the exploration of new numerical methods for
manifold PDEs. In 2018, nonlocal Poisson model on manifold was first studied
in [37] in the case of Neumann boundary. Its subsequent works describe the
manifold nonlocal models with Dirichlet boundary [35] [41] , with interface
[44] , and with anisotropic coefficients [36]. In [37], we constructed a
nonlocal Poisson model by utilizing the following approximation:

$\\!\\!\\!\\!\\!\\!\\!\\!-\int_{\mathcal{M}}\Delta_{\mathcal{M}}u({y})\bar{R}_{\delta}({x},{y})d\mu_{{y}}\approx\int_{\mathcal{M}}\frac{1}{\delta^{2}}R_{\delta}({x},{y})(u({x})-u({y}))d\mu_{{y}}-2\int_{\partial\mathcal{M}}\bar{R}_{\delta}({x},{y})\frac{\partial
u}{\partial{n}}({y})d\tau_{y},$ (1.7)

Here $\mathcal{M}$ is the manifold, $u\in H^{4}(\mathcal{M})$, $\frac{\partial
u}{\partial{n}}=\nabla_{\mathcal{M}}u\cdot{n}$, $d\mu_{y}$ and $d\tau_{y}$ are
the volume form of $\mathcal{M}$ and $\partial\mathcal{M}$. The kernel
functions
$R_{\delta}({x},{y})=C_{\delta}R\big{(}\frac{|{x}-{y}|^{2}}{4\delta^{2}}\big{)}$,
$\bar{R}_{\delta}({x},{y})=C_{\delta}\bar{R}\big{(}\frac{|{x}-{y}|^{2}}{4\delta^{2}}\big{)}$
and $\bar{R}(r)=\int_{r}^{+\infty}R(s)ds$. The error of (1.7) has been rawly
analyzed in [37]. Such model has convergence rate $\mathcal{O}(\delta)$ to its
local counterpart. The same rate was reached in [27], where the Dirichlet
boundary was approximated by Robin condition.

Due to the demand of high accuracy for nonlocal approximation, one natural
question to consider is the existence of nonlocal model that approaches the
manifold Poisson equation in a rate of $\mathcal{O}(\delta^{2})$,
specifically, with Dirichlet boundary (1.1) as we are studying in this work.
Since such rate has been obtained in low dimensional Euclid space, we can
expect to extend it into high dimensional manifolds by exploiting a new
approximation instead of (1.7). Our innovation is that, first, we show the
error of (1.7) in the interior region is $\mathcal{O}(\delta^{2})$ using
classical Cauchy-Lebniz formula, which agrees to the case with Euclid domains.
Secondly, at the boundary layer, by regarding the normal derivative in (1.7)
as a variable, we are able to reduce the order of error in the
$2\delta-$boundary layer in (1.7) into $\mathcal{O}(\delta)$ after adding a
term that involves the $2^{nd}$ normal derivative. According to the theory of
normal curvature function, such term can be explicitly expressed by the first
order derivative and Laplace-Beltrami operator. We can then set up an accurate
nonlocal model accordingly.

The main contribution of this work is the construction of nonlocal manifold
Poisson model with Dirichlet boundary, and the control on the truncation error
of model to its local counterpart. The well-posedness of model will be given
in our subsequent papers. Our results can be easily generalized into the case
with non-homogeneous Dirichlet boundary condition. To the author’s best
knowledge, even in the Euclid spaces, no work has ever appeared on the
construction of nonlocal Poisson model with second order convergence under
dimension $d\geq 3$, having such model will result in higher efficiency in the
numerical implementation. In addition, our model can be easily discretized and
be solved by meshless scheme like point integral method(PIM, see [27]) , which
brings more convenience on solving manifold models numerically. On the
contrary, we can not apply the finite-element method(FEM) on high dimensional
manifold due to the complication of mesh construction. Besides, this work
gives an orientation on how to handle nonlocal problems on manifolds.

The paper is organized as follows: we introduce the idea to formulate our
model in section 2. In section 3, we controlled the truncation error of model
on both the interior and the boundary. Some complicated calculation and proof
have been moved into appendix.

##  2 Nonlocal Model

In this section, we briefly introduce how our model is formulated. Recall the
Poisson model to be approximated:

$\begin{cases}-\Delta_{\mathcal{M}}u({x})=f({x})&{x}\in\mathcal{M},\\\
u=0&{x}\in\partial\mathcal{M}.\end{cases}$ (2.1)

In our nonlocal model, the kernel function
$R_{\delta}({x},{y})=C_{\delta}R\big{(}\frac{|{x}-{y}|^{2}}{4\delta^{2}}\big{)}$
appeared in (1.6) plays a crucial role. A better choice of $R$ will result in
much less work in the error analysis. To this end, we make the following
assumptions on $R(r)$:

  1. 1.

Smoothness: $\frac{d^{2}}{dr^{2}}R(r)$ is bounded, i.e., for any $r\geq 0$ we
have $\big{|}\frac{d^{2}}{dr^{2}}R(r)\big{|}\leq C$;

  2. 2.

Nonnegativity: $R(r)>0$ for any $r\geq 0$;

  3. 3.

Compact support: $R(r)=0$ for any $r>1$;

  4. 4.

Nondegenearcy: $\exists\ \delta_{0}>0$ so that $R(r)\geq\delta_{0}>0$ for
$0\leq r\leq 1/2$.

These assumptions are very mild, almost all the smooth functions in the
literature satisfy these conditions. The first condition makes $R$ not blow up
near the origin, while the fourth condition assures a non-degenerate interval
for $R$. Despite that the removal of third condition will make the problem
more interesting, we keep this condition to avoid complication in the error
analysis. A frequently used kernel function $R$ is the following:

$R(r)=\begin{cases}\frac{1}{2}(1+\cos\pi r),&0\leq r\leq 1,\\\
0,&r>1.\end{cases}$ (2.2)

Based on $R$, we write down the other two related functions
$\bar{R}(r)=\int_{r}^{+\infty}R(s)ds$,
$\overset{=}{R}(r)=\int_{r}^{+\infty}\bar{R}(s)ds$ and their corresponding
kernel functions

$\bar{R}_{\delta}({x},{y})=C_{\delta}\bar{R}\big{(}\frac{|{x}-{y}|^{2}}{4\delta^{2}}\big{)},\qquad\overset{=}{R}_{\delta}({x},{y})=C_{\delta}\overset{=}{R}\big{(}\frac{|{x}-{y}|^{2}}{4\delta^{2}}\big{)}.$
(2.3)

It is clear that $\bar{R}$ and $\overset{=}{R}$ satisfy the assumptions 1-4 as
well by replacing $\delta_{0}$ with some other positive constants in the
fourth condition.

In the next section, we will show the error of (1.7) can be improved by adding
a term involving the second order normal derivative:

$\begin{split}\\!\\!\\!\\!\\!-\int_{\mathcal{M}}\Delta_{\mathcal{M}}u({y})\bar{R}_{\delta}({x},{y})d\mu_{y}\approx\
&\frac{1}{\delta^{2}}\int_{\mathcal{M}}(u({x})-u({y}))R_{\delta}({x},{y})d\mu_{y}-2\int_{\partial\mathcal{M}}\frac{\partial
u}{\partial{n}}({y})\bar{R}_{\delta}({x},{y})d\tau_{y}\\\
&-\int_{\partial\mathcal{M}}(({x}-{y})\cdot{n}({y}))\frac{\partial^{2}u}{\partial^{2}{n}}({y})\bar{R}_{\delta}({x},{y})d\tau_{y},\qquad{x}\in\mathcal{M}.\end{split}$
(2.4)

Here
$\frac{\partial^{2}u}{\partial^{2}{n}}=\sum\limits_{i=1}^{d}\sum\limits_{j=1}^{d}n^{i}n^{j}\nabla_{\mathcal{M}}^{i}\nabla_{\mathcal{M}}^{j}u$
and $n^{i},\ n^{j}$ is the $i,j^{th}$ component of ${n}$. For simplicity, we
always write $\frac{\partial^{2}u}{\partial^{2}{n}}$ as
$n^{i}n^{j}\nabla^{i}\nabla^{j}u$ or $u_{{n}{n}}$. The last term of (2.4) is
supported in the layer adjacent to the boundary with width $\delta$.

The main difficulty to apply (2.4) into a nonlocal model is the need of
handling the term $\frac{\partial^{2}u}{\partial^{2}{n}}$, which usually keep
people away from improving the accuracy of model. Fortunately, in the case
when $u\equiv 0$ throughout the boundary, certain relation can be exploited
between $\frac{\partial^{2}u}{\partial^{2}{n}}$ and $\nabla_{\mathcal{M}}u$
after further study on the shape of boundary. In lemma 3.3 of section 3, we
will prove the following equality

$\frac{\partial^{2}u}{\partial^{2}{n}}({y})=\Delta_{\mathcal{M}}u({y})+\kappa_{{n}}({y})\frac{\partial
u}{\partial{n}}({y}),\qquad\forall\ {y}\in\partial\mathcal{M}$ (2.5)

whenever $u\in H^{4}(\mathcal{M})$ and $u\equiv 0$ on $\partial\mathcal{M}$.
This is fundamental in the geometric analysis. Here $\kappa_{{n}}$ is a
function that depends only on the shape of $\mathcal{M}$ and
$\partial\mathcal{M}$. A detailed description of $\kappa_{{n}}$ will be given
in lemma 3.3. Specially, in the two dimensional case where $m=2$,

$\kappa_{{n}}({y})=\kappa({y})({n}({y})\cdot{n}^{b}({y})),\qquad\forall\
{y}\in\partial\mathcal{M},$ (2.6)

where $\kappa({y})$ denotes the curvature of $\partial\mathcal{M}$ at ${y}$,
and ${n}^{b}({y})$ denotes the unit principal normal vector of
$\partial\mathcal{M}$ at ${y}$.

Based on (2.4) and (2.5), we are ready to set up an equation that approximates
the Poisson problem (2.1): we express $\frac{\partial^{2}u}{\partial^{2}{n}}$
in (2.4) by the equality (2.5) and assign $\Delta_{\mathcal{M}}u$ as $-f$ in
(2.4), then equalize two sides of (2.4) after replacing $u$ and
$\frac{\partial u}{\partial{n}}$ by new variables $u_{\delta}$ and
$v_{\delta}$ to write

$\mathcal{L}_{\delta}u_{\delta}({x})-\mathcal{G}_{\delta}v_{\delta}({x})=\mathcal{P}_{\delta}f({x})\qquad{x}\in\mathcal{M},$
(2.7)

where the operators in (2.7) are defined as

$\mathcal{L}_{\delta}u_{\delta}({x})=\frac{1}{\delta^{2}}\int_{\mathcal{M}}(u_{\delta}({x})-u_{\delta}({y}))\
{R}_{\delta}({x},{y})d\mu_{y},$ (2.8)
$\mathcal{G}_{\delta}v_{\delta}({x})=\int_{\partial\mathcal{M}}v_{\delta}({y})\
(2+({x}-{y})\cdot\kappa_{{n}}({y}){n}({y}))\
\bar{R}_{\delta}({x},{y})d\tau_{y},$ (2.9)
$\mathcal{P}_{\delta}f({x})=\int_{\mathcal{M}}f({y})\
\bar{R}_{\delta}({x},{y})d\mu_{y}-\int_{\partial\mathcal{M}}(({x}-{y})\cdot{n}({y}))\
f({y})\ \bar{R}_{\delta}({x},{y})d\tau_{y}.$ (2.10)

We aim to approximate the solution $u$ of (2.1) by $u_{\delta}$, and
$\frac{\partial u}{\partial{n}}$ by $v_{\delta}$. Apparently, (2.7) is an
equation for $(u_{\delta},v_{\delta})$ and has infinitely many solutions, one
more equation for $(u_{\delta},v_{\delta})$ along the boundary is required to
assure the uniqueness of solution. To this end, it is necessary to explore
additional relation between $u$ and $\frac{\partial u}{\partial{n}}$. In view
of the term $\mathcal{G}_{\delta}v_{\delta}$ in (2.9), we write down the
following two approximations

$\begin{split}2\int_{\mathcal{M}}\bar{R}_{\delta}({x},{y})(u({x})-u({y}))d\mu_{{y}}-4\delta^{2}\int_{\partial\mathcal{M}}\overset{=}{R}_{\delta}({x},{y})\frac{\partial
u}{\partial{n}}({x})d\tau_{y}\\\
\approx-2\delta^{2}\int_{\mathcal{M}}\Delta_{\mathcal{M}}u({y})\overset{=}{R}_{\delta}({x},{y})d\mu_{{y}}\qquad{x}\in\partial\mathcal{M};\end{split}$
(2.11)
$\int_{\mathcal{M}}\big{(}(u({x})-u({y})-({x}-{y})\cdot{n}({x})\frac{\partial
u}{\partial{n}}({x})\big{)}\
({x}-{y})\cdot{n}({x})\kappa_{{n}}({x})\bar{R}_{\delta}({x},{y})d{y}\approx
0\qquad{x}\in\partial\mathcal{M}.$ (2.12)

Here (2.11) is a slight modification of (1.7), and (2.12) is a Taylor
expansion residue in the normal direction. The truncation error analysis will
be given in the next section. Next, similar to (2.7), we are ready to
construct our second nonlocal equation that approximates (2.1): we assign
$u({x})$ in (2.11), (2.12) as $0$ and $\Delta_{\mathcal{M}}u({y})$ as
$-f({y})$, replace $u({y})$ and $\frac{\partial u}{\partial{n}}$ by
$u_{\delta}({y})$ and $v_{\delta}$, then equalize two sides of (2.11) and
(2.12) to sum up

$\mathcal{D}_{\delta}u_{\delta}({x})+\tilde{R}_{\delta}({x})v_{\delta}({x})=\mathcal{Q}_{\delta}f({x}),\qquad{x}\in\partial\mathcal{M},$
(2.13)

where the operators are defined as

$\mathcal{D}_{\delta}u_{\delta}({x})=\int_{\mathcal{M}}u_{\delta}({y})\ (2-\
({x}-{y})\cdot\kappa_{{n}}({x}){n}({x}))\ \bar{R}_{\delta}({x},{y})d\mu_{y},$
(2.14)
$\tilde{R}_{\delta}({x})=4\delta^{2}\int_{\partial\mathcal{M}}\overset{=}{R}_{\delta}({x},{y})d\tau_{y}-\int_{\mathcal{M}}\kappa_{{n}}({x})\
(({x}-{y})\cdot n({x}))^{2}\ \bar{R}_{\delta}({x},{y})d\mu_{y},$ (2.15)
$\mathcal{Q}_{\delta}f({x})=-2\delta^{2}\int_{\mathcal{M}}f({y})\
\overset{=}{R}_{\delta}({x},{y})d\mu_{y}.$ (2.16)

Here the term $\mathcal{D}_{\delta}u_{\delta}$ in (2.14) is constructed
accordingly with $\mathcal{G}_{\delta}v_{\delta}$ in (2.9) to assure the
elimination of cross terms in the weak formulation of model. Consequently, we
set up our nonlocal model by combining (2.7) and (2.13):

$\begin{cases}\mathcal{L}_{\delta}u_{\delta}({x})-\mathcal{G}_{\delta}v_{\delta}({x})=\mathcal{P}_{\delta}f({x}),&{x}\in\mathcal{M},\\\
\mathcal{D}_{\delta}u_{\delta}({x})+\tilde{R}_{\delta}({x})v_{\delta}({x})=\mathcal{Q}_{\delta}f({x}),&{x}\in\partial\mathcal{M}.\end{cases}$
(2.17)

Our final destination is to show that (2.17) has a unique solution pair
$(u_{\delta},v_{\delta})$ and $\left\lVert
u-u_{\delta}\right\rVert_{H^{1}(\mathcal{M})}\leq C\delta^{2}\left\lVert
f\right\rVert_{H^{2}(\mathcal{M})}$ where $u$ solves (2.1). Due to its
complication in the coercivity analysis, we remain to solve such problem in
our subsequent work. In the next section, we will concentrate on the control
of the truncation error between (2.17) and (2.1).

##  3 Analysis of the Truncation Errors

In this section, we state the main theorem of our paper, which is crucial for
our subsequential papers.

######  Theorem 3.1.

Let $u\in H^{4}(\mathcal{M})$ be the solution of $\eqref{b01}$, and

$r_{in}({x})=\mathcal{L}_{\delta}u({x})-\mathcal{G}_{\delta}\frac{\partial
u}{\partial{{n}}}({x})-\mathcal{P}_{\delta}f({x}),\qquad{x}\in\mathcal{M},$
(3.1)
$r_{bd}({x})=\mathcal{D}_{\delta}u({x})+\tilde{R}_{\delta}({x})\frac{\partial
u}{\partial{{n}}}({x})-\mathcal{Q}_{\delta}f({x}),\qquad{x}\in\partial\mathcal{M}.$
(3.2)

then we can decompose $r_{in}$ into $r_{in}=r_{it}+r_{bl}$, where $r_{it}$ is
supported in the whole domain $\mathcal{M}$, with the following bound

$\frac{1}{\delta}\left\lVert
r_{it}\right\rVert_{L^{2}(\mathcal{M})}+\left\lVert\nabla
r_{it}\right\rVert_{L^{2}(\mathcal{M})}\leq C\delta\left\lVert
u\right\rVert_{H^{4}(\mathcal{M})};$ (3.3)

and $r_{bl}$ is supported in the layer adjacent to the boundary
$\partial\mathcal{M}$ with width $2\delta$:

$supp(r_{bl})\subset\\{{x}\ \big{|}\ {x}\in\mathcal{M},\
dist({x},\partial\mathcal{M})\leq 2\delta\ \\},$ (3.4)

and satisfies the following two estimates

$\frac{1}{\delta}\left\lVert
r_{bl}\right\rVert_{L^{2}(\mathcal{M})}+\left\lVert\nabla
r_{bl}\right\rVert_{L^{2}(\mathcal{M})}\leq C\delta^{\frac{1}{2}}\left\lVert
u\right\rVert_{H^{4}(\mathcal{M})};$ (3.5)
$\begin{split}\int_{\mathcal{M}}r_{bl}({x})\
f_{1}({x})d\mu_{x}\leq&C\delta^{2}\left\lVert
u\right\rVert_{H^{4}(\mathcal{M})}(\left\lVert
f_{1}\right\rVert_{H^{1}(\mathcal{M})}+\left\lVert\bar{f}_{1}\right\rVert_{H^{1}(\mathcal{M})}+\left\lVert\overset{=}{f}_{1}\right\rVert_{H^{1}(\mathcal{M})}),\\\
&\forall\ f_{1}\in H^{1}(\mathcal{M});\end{split}$ (3.6)

where the notations $\nabla=\nabla_{\mathcal{M}}$, and

$\bar{f}_{1}({x})=\frac{1}{\bar{\omega}_{\delta}({x})}\int_{\mathcal{M}}f_{1}({y})\bar{R}_{\delta}({x},{y})d\mu_{y},\qquad\overset{=}{f}_{1}({x})=\frac{1}{\overset{=}{\omega}_{\delta}({x})}\int_{\mathcal{M}}f_{1}({x})\overset{=}{R}_{\delta}({x},{y})d\mu_{y}$
(3.7)

are the weighted average of $f_{1}$ in ${B}_{2\delta}({x})$ with respect to
$\bar{R}$ and $\overset{=}{R}$, and

$\bar{\omega}_{\delta}({x})=\int_{\mathcal{M}}\bar{R}_{\delta}({x},{y})d\mu_{y},\
\overset{=}{\omega}_{\delta}({x})=\int_{\mathcal{M}}\overset{=}{R}_{\delta}({x},{y})d\mu_{y},\qquad\forall\
{x}\in\mathcal{M}.$

In addition, we have the following estimate for $r_{bd}$:

$\left\lVert r_{bd}\right\rVert_{L^{2}(\partial\mathcal{M})}\leq
C\delta^{\frac{5}{2}}\left\lVert u\right\rVert_{H^{4}(\mathcal{M})}.$ (3.8)

The constant $C$ in (3.5), (3.6) and (3.8) depend only on $\mathcal{M}$.

In general, this theorem states that $r_{in}$ is $\mathcal{O}(\delta^{2})$ in
the interior region, and is $\mathcal{O}(\delta)$ in the boundary layer, while
$r_{bd}$ is $\mathcal{O}(\delta^{\frac{5}{2}})$. In fact, $r_{in}$ is
essentially the error of the approximation (2.4). Compared with $r$ in [37]
that denotes the error of (1.7), $r_{in}$ is smaller in all aspects.
Correspondingly, it brings more complication in the proof of theorem 3.1 since
we need to further analyze the second order truncation terms.

To prove theorem (3.1), we introduce a special local parametrization of
$\mathcal{M}$. We let $\delta$ be sufficiently small so that it is less than
$0.1$ times the minimum reaches of $\mathcal{M}$ and $\partial\mathcal{M}$.
For the definition of the reach of the manifold, one can refer to the first
part of section 5 of [37]. Under such assumption, for each
${x}\in\mathcal{M}$, its neighborhood $B_{{x}}^{4\delta}$ is holomorphic to
the Euclid space $\mathbb{R}^{m}$. Since $\mathcal{M}$ is compact, there
exists a $2\delta$-net,
$\mathcal{N}_{\delta}=\\{{q}_{i}\in\mathcal{M},i=1,...,N\\}$, such that

$\mathcal{M}\subset\bigcup\limits_{i=1}^{N}B_{{q}_{i}}^{2\delta},$ (3.9)

and there exists a partition of $\mathcal{M}$, $\\{\mathcal{O}_{i},\
i=1,...,N\\}$, such that $\mathcal{O}_{i}\cap\mathcal{O}_{j}=\emptyset,\
\forall\ i\neq j$ and

$\mathcal{M}=\bigcup\limits_{i=1}^{N}\mathcal{O}_{i},\ \mathcal{O}_{i}\subset
B_{{q}_{i}}^{2\delta},\ i=1,2,...,N.$ (3.10)

######  Lemma 3.2.

There exist a parametrization
$\bm{\phi}_{i}:\Omega_{i}\subset\mathbb{R}^{m}\to U_{i}\subset\mathcal{M},\
i=1,2,...,N$, such that

  1. 1.

(Convexity) $B_{{q}_{i}}^{4\delta}\cap\mathcal{M}\subset U_{i}$ and
$\Omega_{i}$ is convex,

  2. 2.

(Smoothness) $\bm{\phi}_{i}\in C^{4}(\Omega_{i})$,

  3. 3.

(Local small deformation) For any points $\theta^{1},\theta^{2}\in\Omega_{i}$.

$\frac{1}{2}|\theta^{1}-\theta^{2}|\leq\left\lVert\bm{\phi}_{i}(\theta^{1})-\bm{\phi}_{i}(\theta^{2})\right\rVert\leq
2|\theta^{1}-\theta^{2}|.$ (3.11)

This lemma is a corollary of the proposition 1 of [37]. Such lemma gives a
parametrization of $\mathcal{M}$ on each of $N$ pieces. It indicates that for
each ${x}\in\mathcal{M}$, there is a unique index $J({x})\in\\{1,2,...,N\\}$
such that ${x}\in\mathcal{O}_{J({x})}$, while $B_{{x}}^{2\delta}\in
U_{J({x})}$. For any ${y}\in B_{{x}}^{2\delta}$, we define the function

$\bm{\xi}({x},{y})=\phi_{J({x})}^{-1}({y})-\phi_{J({x})}^{-1}({x}),$ (3.12)

and the auxiliary function $\eta({x},{y})=<\eta_{1},\eta_{2},...,\eta_{d}>$,
where

$\eta^{j}({x},{y})=\sum\limits_{k=1}^{m}\xi^{i}({x},{y})\partial_{k}\phi^{j}_{J({x})}({y}),\qquad
j\in\\{1,2,...,d\\}.$ (3.13)

Here $\bm{\eta}=\bm{\eta}({x},{y})$ is an auxiliary function that approximates
the vector $({y}-{x})$ in the tangential plane $T_{\mathcal{M}}({x})$. In the
following content, we sometimes omit the symbol of summation, for instance, we
write
$\sum\limits_{k=1}^{m}\xi^{k}({x},{y})\partial_{i}\phi^{j}_{J({x})}({y})$ as
$\xi^{k}({x},{y})\partial_{k}\phi^{j}_{J({x})}({y})$. To simplify our
notations, we always write $\bm{\alpha}$ to denote $\phi_{J({x})}^{-1}({y})$,
and $\bm{\beta}$ to denote $\phi_{J({x})}^{-1}({x})$, with
$\bm{\xi}=\bm{\beta}-\bm{\alpha}$.

Besides, the following 2 lemmas are necessary in the proof of theorem 3.1.

######  Lemma 3.3.

For each ${x}\in\partial\mathcal{M}$, we have the following equality (equation
(2.5)) hold:

$\frac{\partial^{2}u}{\partial{n}^{2}}({x})=\Delta_{\mathcal{M}}u({x})+\kappa_{{n}}({x})\
\frac{\partial u}{\partial{n}}({x}),$ (3.14)

where $\kappa_{{n}}({x})$ is $(m-1)$ times the mean curvature of the $(m-1)$
dimensional hyper-surface $\mathcal{P}_{{x}}(\partial\mathcal{M})$ at ${x}$
with normal direction ${n}({x})$, and $\mathcal{P}_{{x}}(\partial\mathcal{M})$
represents the projection of $\partial\mathcal{M}$ onto the $m$ dimensional
Euclid space $\mathcal{T}_{{x}}\mathcal{M}$. In addition, let
$\Psi:\Sigma\subset\mathbb{R}^{m-1}\to\partial\mathcal{M}\subset\mathbb{R}^{d}$
be a local parametrization of $\partial\mathcal{M}$ near ${x}$ with
$\Psi(\omega)={x}$, and $H(\omega)=(h_{ij})_{i,j=1,2,...,m-1}$ be the first
fundamental form of $\Psi$ with

$h_{ij}(\omega)=\sum\limits_{k=1}^{d}\frac{\partial\psi^{k}}{\partial\omega_{i}}(\omega)\frac{\partial\psi^{k}}{\partial\omega_{j}}(\omega),\qquad
i,j=1,...,m-1;$ (3.15)

and let $L(\omega)=(l_{ij})_{i,j=1,2,...,m-1}$ with

$l_{ij}(\omega)=\sum\limits_{k=1}^{d}\frac{\partial^{2}\psi^{k}}{\partial\omega_{i}\partial\omega_{j}}(\omega)n^{k}(\omega),\qquad
i,j=1,...,m-1;$ (3.16)

then we have the following explicit form of $\kappa_{{n}}$:

$\kappa_{{n}}({x})=\sum\limits_{i=1}^{m-1}\sum\limits_{j=1}^{m-1}h^{ij}(\omega)l_{ij}(\omega),$
(3.17)

where $h^{ij}_{i,j=1,2,...,m-1}=H^{-1}$.

The proof of such lemma is given in section 4.1 of appendix.

######  Lemma 3.4.

Let $f_{1}:\mathcal{M}\to\mathbb{R}^{1}$,
$\textbf{g}_{1}:\mathcal{M}\to\mathbb{R}^{m}$. If $f_{1}\in
H^{1}(\mathcal{M})$ and $\textbf{g}_{1}\in[H^{1}(\mathcal{M})]^{m}$, then we
have

$\begin{split}\int_{\partial\mathcal{M}}\int_{\mathcal{M}}f_{1}({x})\textbf{g}_{1}({y})\cdot&({x}-{y}-\bm{\eta}({x},{y}))R_{\delta}({x},{y})d\mu_{x}d\tau_{y}\\\
&\leq C\delta^{2}(\left\lVert
f_{1}\right\rVert_{H^{1}(\mathcal{M})}+\left\lVert\bar{f}_{1}\right\rVert_{H^{1}(\mathcal{M})})\left\lVert\textbf{g}_{1}\right\rVert_{H^{1}(\mathcal{M})},\end{split}$
(3.18)

where $\bar{f}_{1}$ is the function defined in (3.7), and $\bm{\eta}$ is
defined in (3.13).

The proof of lemma can be found in section 4.2 of appendix. Next, we are ready
to prove the main theorem of this section.

######  Proof 3.5 (Proof of Theorem 3.1).

Let us first study $r_{in}$. Recall (3.1) and the definition of
$\mathcal{L}_{\delta}$, $\mathcal{G}_{\delta}$ and $\mathcal{P}_{\delta}$ in
(2.7), we replace $f$ into $\Delta_{\mathcal{M}}u$ to write

$\begin{split}&r_{in}({x})=-\int_{\mathcal{M}}\Delta_{\mathcal{M}}u({y})\bar{R}_{\delta}({x},{y})d\mu_{y}-\frac{1}{\delta^{2}}\int_{\mathcal{M}}(u({x})-u({y}))R_{\delta}({x},{y})d\mu_{y}\\\
&-2\int_{\partial\mathcal{M}}\frac{\partial
u}{\partial{n}}({y})\bar{R}_{\delta}({x},{y})d\tau_{y}-\int_{\partial\mathcal{M}}(({x}-{y})\cdot{n}({y}))(\Delta_{\mathcal{M}}u({y})+\kappa_{{n}}({y})\
\frac{\partial
u}{\partial{n}}({y}))\bar{R}_{\delta}({x},{y})d\tau_{y}.\end{split}$ (3.19)

Next, we decompose $r_{in}$ into $r_{in}=r_{1}+r_{2}+r_{3}+r_{4}+r_{5}+r_{6}$,
where

$r_{1}({x})=\frac{1}{\delta^{2}}\int_{\mathcal{M}}(u({x})-u({y})-({x}-{y})\cdot\nabla
u({y})-\frac{1}{2}\eta^{i}\eta^{j}(\nabla^{i}\nabla^{j}u({y})))R_{\delta}({x},{y})d\mu_{y},$
(3.20)

$r_{2}({x})=\frac{1}{2\delta^{2}}\int_{\mathcal{M}}\eta^{i}\eta^{j}(\nabla^{i}\nabla^{j}u({y}))R_{\delta}({x},{y})d\mu_{y}-\int_{\mathcal{M}}\eta^{i}(\nabla^{i}\nabla^{j}u({y}))\nabla^{j}\bar{R}_{\delta}({x},{y})d\mu_{y},$
(3.21)

$\begin{split}r_{3}({x})=&\int_{\mathcal{M}}\eta^{i}(\nabla^{i}\nabla^{j}u({y}))\nabla^{j}\bar{R}_{\delta}({x},{y})d\mu_{y}+\int_{\mathcal{M}}div\
(\eta^{i}(\nabla^{i}\nabla u({y}))\bar{R}_{\delta}({x},{y})d\mu_{y}\\\
&-\int_{\partial\mathcal{M}}(({x}-{y})\cdot{n}({y}))\frac{\partial^{2}u}{\partial{n}^{2}}({y})\bar{R}_{\delta}({x},{y})d\tau_{y},\end{split}$
(3.22)

$r_{4}({x})=-\int_{\mathcal{M}}div\ (\eta^{i}(\nabla^{i}\nabla
u({y}))\bar{R}_{\delta}({x},{y})d\mu_{y}-\int_{\mathcal{M}}\Delta_{\mathcal{M}}u({y})\bar{R}_{\delta}({x},{y})d\mu_{y},$
(3.23)

$r_{5}({x})=\int_{\partial\mathcal{M}}(({x}-{y})\cdot{n}({y}))(\frac{\partial^{2}u}{\partial{n}^{2}}({y})-\Delta_{\mathcal{M}}u({y})-\kappa_{{n}}({y})\frac{\partial
u}{\partial{n}}({y}))\bar{R}_{\delta}({x},{y})d\tau_{y},$ (3.24)

$\begin{split}r_{6}({x})=&-2\int_{\partial\mathcal{M}}\frac{\partial
u}{\partial{n}}({y})\bar{R}_{\delta}({x},{y})d\tau_{y}+2\int_{\mathcal{M}}\Delta_{\mathcal{M}}u({y})\bar{R}_{\delta}({x},{y})d\mu_{y}\\\
&+\frac{1}{\delta^{2}}\int_{\mathcal{M}}({x}-{y})\cdot\nabla
u({y})R_{\delta}({x},{y})d\mu_{y}.\end{split}$ (3.25)

We have $r_{5}\equiv 0$ according to the lemma 3.3, and $r_{6}\equiv 0$ is the
equality above (22) in [37]. The terms $r_{1}$, $r_{2}$ and $r_{4}$ are
exactly $r_{1}$, $r_{2}$ and $r_{4}$ in [37]. For the term $r_{2}$, The
equation $(27)$ of [37] directly gives

$\int_{\mathcal{M}}|r_{2}({x})|^{2}d\mu_{x}\leq C\delta^{2}\left\lVert
u\right\rVert_{H^{3}(\mathcal{M})}^{2},$ (3.26) $\int_{\mathcal{M}}|\nabla
r_{2}({x})|^{2}d\mu_{x}\leq C\delta\left\lVert
u\right\rVert_{H^{3}(\mathcal{M})}^{2}.$ (3.27)

Hence it satisfies the control (3.3). What left is to control $r_{1}$, $r_{3}$
and $r_{4}$. To simplify the proof, we use $I_{1}$ to denote the set of terms
that satisfy the control (3.3), and $I_{2}$ to denote the set of terms that
satisfy the control (3.4) (3.5) and (3.6).

  1. 1.

Let us first study $r_{1}$, recall

$r_{1}({x})=\frac{1}{\delta^{2}}\int_{\mathcal{M}}(u({x})-u({y})-({x}-{y})\cdot\nabla
u({y})-\frac{1}{2}\eta^{i}\eta^{j}(\nabla^{i}\nabla^{j}u({y})))R_{\delta}({x},{y})d\mu_{y},$
(3.28)

In [37], $r_{1}({x})$ is written as the sum of 3 triple-integral terms in the
parametric domain $\Omega_{J({x})}$ using Cauchy-Leibniz formula. For this
theorem, we need to further decompose it into 3 first order terms plus 3
quadruple-integral terms, where the latter terms can be classified as $I_{2}$.
By several steps of integration by parts in $\Omega_{J({x})}$, the former
terms can be further divided into $I_{1}$ part and $I_{2}$ part. Due its
complication in the calculation, we have moved the control of $r_{1}$ into
appendix 4.3.

  2. 2.

Next, let us analyze $r_{3}$. We have

$\begin{split}&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!r_{3}=\int_{\partial\mathcal{M}}n^{j}\eta^{i}(\nabla^{i}\nabla^{j}u({y}))\bar{R}_{\delta}({x},{y})d\tau_{y}-\int_{\partial\mathcal{M}}(({x}-{y})\cdot{n}({y}))\frac{\partial^{2}u}{\partial{n}^{2}}({y})\bar{R}_{\delta}({x},{y})d\tau_{y}\\\
=&\int_{\partial\mathcal{M}}\textbf{H}_{u}:({x}-{y})\otimes{n}({y})\
\bar{R}_{\delta}({x},{y})d\tau_{y}-\int_{\partial\mathcal{M}}(({x}-{y})\cdot{n}({y}))\frac{\partial^{2}u}{\partial{n}^{2}}({y})\bar{R}_{\delta}({x},{y})d\tau_{y}\\\
&-\int_{\partial\mathcal{M}}\textbf{H}_{u}:({x}-{y}-\bm{\eta}({x},{y}))\otimes{n}({y})\
\bar{R}_{\delta}({x},{y})d\tau_{y}\\\
=&\int_{\partial\mathcal{M}}\textbf{H}_{u}:\big{(}({x}-{y})-(({x}-{y})\cdot{n}({y})){n}({y})\big{)}\otimes{n}({y})\
\bar{R}_{\delta}({x},{y})d\tau_{y}\\\
&-\int_{\partial\mathcal{M}}\textbf{H}_{u}:({x}-{y}-\bm{\eta}({x},{y}))\otimes{n}({y})\
\bar{R}_{\delta}({x},{y})d\tau_{y}.\end{split}$ (3.29)

For the second term of (3.29), we directly apply the lemma 3.4 to obtain

$\begin{split}\int_{\mathcal{M}}f_{1}({x})\int_{\partial\mathcal{M}}&\textbf{H}_{u}:{n}({y})\otimes(({x}-{y})-\bm{\eta}({x},{y}))\bar{R}_{\delta}({x},{y})d\tau_{{y}}\\\
&\leq C(\left\lVert
f_{1}\right\rVert_{H^{1}(\mathcal{M})}+\left\lVert\overset{=}{f}_{1}\right\rVert_{H^{1}(\mathcal{M})})\left\lVert
u\right\rVert_{H^{4}(\mathcal{M})},\ \forall\ f_{1}\in
H^{1}(\mathcal{M}).\end{split}$ (3.30)

In addition, since we have the following decomposition

$\begin{split}&\int_{\partial\mathcal{M}}\textbf{H}_{u}:({x}-{y}-\bm{\eta}({x},{y}))\otimes{n}({y})\
\bar{R}_{\delta}({x},{y})d\tau_{y}\\\
=&\int_{\partial\mathcal{M}}\nabla^{i}\nabla^{j}u({y})\
n^{j}\xi^{i^{\prime}}\xi^{j^{\prime}}\int_{0}^{1}\int_{0}^{1}s\partial_{j^{\prime}}\partial_{i^{\prime}}{\phi}^{i}(\bm{\alpha}+\tau
s\bm{\xi})d\tau ds\bar{R}_{\delta}({x},{y})d\tau_{y}\\\
=&\int_{\partial\mathcal{M}}\nabla^{i}\nabla^{j}u({y})\
n^{j}\xi^{i^{\prime}}\xi^{j^{\prime}}\frac{1}{2}\partial_{j^{\prime}}\partial_{i^{\prime}}{\phi}^{i}(\bm{\alpha})\bar{R}_{\delta}({x},{y})d\tau_{y}+\\\
&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\int_{\partial\mathcal{M}}\nabla^{i}\nabla^{j}u({y})\
n^{j}\xi^{i^{\prime}}\xi^{j^{\prime}}\xi^{k^{\prime}}(\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}s^{2}\tau\partial_{j^{\prime}}\partial_{i^{\prime}}\partial_{k^{\prime}}\phi^{i}(\bm{\alpha}+\tau
st\bm{\xi})dtd\tau ds){R}_{\delta}({x},{y})d\tau_{y},\end{split}$ (3.31)

we can follow the control on $d_{221}$ in (4.57) (4.58), and the control on
$d_{11}$ in (4.46) (4.47) to obtain the bound (3.5) for the above two terms.
The second term of (3.29) therefore belongs to $I_{2}$,

On the other hand, what remains to show is that the first term of (3.29)
belongs to $I_{2}$. We denote
$\mathcal{T}^{\prime}_{{y}}:\mathbb{R}^{d}\to\mathbb{R}^{d}$ as the projection
map from $\mathbb{R}^{d}$ onto the space
$\mathcal{T}_{{y}}(\partial\mathcal{M})$, while
$\mathcal{T}_{{y}}(\partial\mathcal{M})$ represents the tangent space of the
manifold $\partial\mathcal{M}$ at ${y}$. By noticing that
$({x}-{y})-(({x}-{y})\cdot{n}({y})){n}({y})=\mathcal{T}^{\prime}_{{y}}({x}-{y})$,
we can do the following integration by parts:

$\begin{split}&\int_{\partial\mathcal{M}}\textbf{H}_{u}:\big{(}({x}-{y})-(({x}-{y})\cdot{n}({y})){n}({y})\big{)}\otimes{n}({y})\
\bar{R}_{\delta}({x},{y})d\tau_{y}\\\
=&\int_{\partial\mathcal{M}}\textbf{H}_{u}:{n}({y})\otimes\mathcal{T}^{\prime}_{{y}}({x}-{y})\
\bar{R}_{\delta}({x},{y})d\tau_{y}\\\
=&\int_{\partial\mathcal{M}}\textbf{H}_{u}:{n}({y})\otimes(\nabla_{\partial\mathcal{M}}^{{y}}\overset{=}{R}_{\delta}({x},{y}))d\tau_{y}\\\
=&\int_{\partial\mathcal{M}}\nabla^{i}\nabla^{j}u({y}){n}^{j}({y})(\nabla_{\partial\mathcal{M}}^{{y}}\overset{=}{R}_{\delta}({x},{y}))^{i}d\tau_{y}\\\
=&\int_{\partial\mathcal{M}}{n}^{j}({y})\nabla(\nabla^{j}u({y}))\cdot\nabla_{\partial\mathcal{M}}^{{y}}\overset{=}{R}_{\delta}({x},{y})d\tau_{y}\\\
=&\int_{\partial\mathcal{M}}{n}^{j}({y})\nabla_{\partial\mathcal{M}}(\nabla^{j}u({y}))\cdot\nabla_{\partial\mathcal{M}}^{{y}}\overset{=}{R}_{\delta}({x},{y})d\tau_{y}\\\
=&-\int_{\partial\mathcal{M}}\nabla_{\partial\mathcal{M}}\cdot({n}^{j}({y})\nabla_{\partial\mathcal{M}}(\nabla^{j}u({y})))\overset{=}{R}_{\delta}({x},{y})d\tau_{y}\\\
=&-\int_{\partial\mathcal{M}}{n}^{j}({y})\nabla_{\partial\mathcal{M}}\cdot\nabla_{\partial\mathcal{M}}(\nabla^{j}u({y}))\
\overset{=}{R}_{\delta}({x},{y})d\tau_{y}\\\
=&-\int_{\partial\mathcal{M}}{n}^{j}({y})\Delta_{\partial\mathcal{M}}(\nabla^{j}u({y}))\
\overset{=}{R}_{\delta}({x},{y})d\tau_{y},\end{split}$ (3.32)

here we have used the fact that
$\nabla_{\partial\mathcal{M}}\cdot{n}({y})\equiv 0$ in the second last
equality, and
$\mathcal{T}^{\prime}_{{y}}\textbf{a}\cdot\textbf{b}=\mathcal{T}^{\prime}_{{y}}\textbf{a}\cdot\mathcal{T}^{\prime}_{{y}}\textbf{b}$
for any vector $\textbf{a},\textbf{b}\in\mathbb{R}^{d}$ in the fifth equality.
While in the last equality,
$\Delta_{\partial\mathcal{M}}=\nabla_{\partial\mathcal{M}}\cdot\nabla_{\partial\mathcal{M}}$
is given in the equation $(30)$ of [37].

Consequently, we can mimic the control on $d_{221}$ in (4.57), (4.58), (4.59)
in the appendix 4.3 to obtain the bound (3.5) and (3.6) for the term (3.32).

  3. 3.

for $r_{4}$, we have from (33) of [37] that

$\\!\\!\\!\\!\\!\\!r_{4}({x})=-\int_{\mathcal{M}}\frac{\xi^{l}}{\sqrt{det\
G({y})}}\partial_{i^{\prime}}\big{(}\sqrt{det\
G}g^{i^{\prime}j^{\prime}}(\partial_{j^{\prime}}\phi^{j})(\partial_{l}\phi^{i})(\nabla^{i}\nabla^{j}u({y}))\big{)}\bar{R}_{\delta}({x},{y})d\mu_{y}.$
(3.33)

We apply (4.24) to discover

$\begin{split}&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!r_{4}({x})=\int_{\mathcal{M}}\frac{1}{\sqrt{det\
G({y})}}\partial_{i^{\prime}}\big{(}\sqrt{det\
G({y})}g^{i^{\prime}j^{\prime}}(\partial_{j^{\prime}}\phi^{j})(\partial_{l}\phi^{i})(\nabla^{i}\nabla^{j}u({y}))\big{)}\xi^{l}\bar{R}_{\delta}({x},{y})d\mu_{y}\\\
&=2\delta^{2}\int_{\mathcal{M}}\frac{1}{\sqrt{det\
G({y})}}\partial_{i^{\prime}}\big{(}\sqrt{det\
G({y})}g^{i^{\prime}j^{\prime}}(\partial_{j^{\prime}}\phi^{j})(\partial_{l}\phi^{i})(\nabla^{i}\nabla^{j}u({y}))\big{)}g^{k^{\prime}l}({y})\\\
&\partial^{\bm{\alpha}}_{k^{\prime}}\overset{=}{R}_{\delta}({x},{y})d\mu_{y}+r_{40}\\\
&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=2\delta^{2}\int_{\Omega}\partial_{i^{\prime}}\big{(}\sqrt{det\
G({y})}g^{i^{\prime}j^{\prime}}(\partial_{j^{\prime}}\phi^{j})(\partial_{l}\phi^{i})(\nabla^{i}\nabla^{j}u({y}))\big{)}g^{k^{\prime}l}({y})\partial^{\bm{\alpha}}_{k^{\prime}}\overset{=}{R}_{\delta}({x},{y})d\bm{\alpha}+r_{40}\\\
&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=-2\delta^{2}\int_{\Omega}\partial_{k^{\prime}}\Big{(}\partial_{i^{\prime}}\big{(}\sqrt{det\
G({y})}g^{i^{\prime}j^{\prime}}(\partial_{j^{\prime}}\phi^{j})(\partial_{l}\phi^{i})(\nabla^{i}\nabla^{j}u({y}))\big{)}g^{k^{\prime}l}({y})\Big{)}\overset{=}{R}_{\delta}({x},{y})d\bm{\alpha}\\\
&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+2\delta^{2}\int_{\partial\Omega}\partial_{i^{\prime}}\big{(}\sqrt{det\
G({y})}g^{i^{\prime}j^{\prime}}(\partial_{j^{\prime}}\phi^{j})(\partial_{l}\phi^{i})(\nabla^{i}\nabla^{j}u({y}))\big{)}g^{k^{\prime}l}({y})n^{k^{\prime}}_{\mathcal{M}}({y})\overset{=}{R}_{\delta}({x},{y})dS_{\bm{\alpha}}\\\
&+r_{40}.\end{split}$ (3.34)

By observation, we can mimic the control (4.54) (4.55) for $d_{21}$ to obtain
the same bound for the first term of (3.34). Such bound indicates the first
term of (3.34) to be in $I_{1}$. Similarly, we mimic the control (4.57) (4.58)
and (4.59) for $d_{221}$ to carry out the same bound for the second term of
(3.34). Consequently, the second term of (3.34) is in $I_{2}$. while the third
term

$r_{40}=\int_{\mathcal{M}}\frac{1}{\sqrt{det\
G({y})}}\partial_{i^{\prime}}\big{(}\sqrt{det\
G({y})}g^{i^{\prime}j^{\prime}}(\partial_{j^{\prime}}\phi^{j})(\partial_{l}\phi^{i})(\nabla^{i}\nabla^{j}u({y}))\big{)}d_{121}^{l}d\mu_{y}$
(3.35)

is obviously a lower order term and belongs to $I_{1}$. Here $d_{121}$ is the
error function defined in (4.50) in the appendix.

So far, we have classified all the terms that composes $r_{in}$ as in set
$I_{1}$ and set $I_{2}$. Finally, what left in the theorem is the control of
$r_{bd}$. Recall the definition of $r_{bd}$ in (3.2) and refer to (2.14)
(2.15) (2.16), we write

$\begin{split}r_{bd}({x})=2\int_{\mathcal{M}}u({y})\
\bar{R}_{\delta}({x},{y})d\mu_{y}+4\delta^{2}\frac{\partial
u}{\partial{n}}({x})\int_{\partial\mathcal{M}}\overset{=}{R}_{\delta}({x},{y})d\tau_{y}+2\delta^{2}\int_{\mathcal{M}}f({y})\
\overset{=}{R}_{\delta}({x},{y})d\mu_{y}\\\
-\kappa_{{n}}({x})\big{(}\int_{\mathcal{M}}u({y})\ ({x}-{y})\cdot{n}({x})\
\bar{R}_{\delta}({x},{y})d\mu_{y}+\frac{\partial
u}{\partial{n}}({x})\int_{\mathcal{M}}\ (({x}-{y})\cdot n({x}))^{2}\
\bar{R}_{\delta}({x},{y})d\mu_{y}\big{)}.\end{split}$ (3.36)

By referring to (2.11) and (2.12) and the fact $u\equiv 0$ on
$\partial\mathcal{M}$, we then split $r_{bd}$ into
$r_{bd}=2r_{6}+2r_{7}+2\delta^{2}\bar{r}_{in}-\kappa_{{n}}r_{8}$, where

$r_{6}=\delta^{2}\int_{\partial\mathcal{M}}(({x}-{y})\cdot{n}({y}))(f({y})-\kappa_{{n}}({y})\frac{\partial
u}{\partial{n}}({y}))\overset{=}{R}_{\delta}({x},{y})d\tau_{y},$ (3.37)
$r_{7}=2\delta^{2}(\int_{\partial\mathcal{M}}\frac{\partial
u}{\partial{n}}({y})\overset{=}{R}_{\delta}({x},{y})d\tau_{y}-\int_{\partial\mathcal{M}}\frac{\partial
u}{\partial{n}}({x})\overset{=}{R}_{\delta}({x},{y})d\tau_{y}),$ (3.38)
$r_{8}=\int_{\mathcal{M}}\big{(}u({y})-u({x})+({x}-{y})\cdot\nabla
u({x})\big{)}\ (({x}-{y})\cdot{n}({x}))\ \bar{R}_{\delta}({x},{y})d\mu_{y},$
(3.39) $\begin{split}\\!\\!\\!\\!\\!\bar{r}_{in}=&\int_{\mathcal{M}}f({y})\
\overset{=}{R}_{\delta}({x},{y})d\mu_{y}-\frac{1}{\delta^{2}}\int_{\mathcal{M}}(u({x})-u({y}))\
\bar{R}_{\delta}({x},{y})d\mu_{y}+2\int_{\partial\mathcal{M}}\frac{\partial
u}{\partial{n}}({y})\overset{=}{R}_{\delta}({x},{y})d\mu_{y}\\\
&+\int_{\partial\mathcal{M}}(({x}-{y})\cdot{n}({y}))(-f({y})+\kappa_{{n}}({y})\frac{\partial
u}{\partial{n}}({y}))\overset{=}{R}_{\delta}({x},{y})d\mu_{y}.\end{split}$
(3.40)

Our next goal is to control each term of $r_{bd}$.

  1. 1.

We start with the control of $r_{6}$. In fact, due to the smoothness of
$\partial\mathcal{M}$, for ${x},{y}\in\partial\mathcal{M}$ with
$|{x}-{y}|<2\delta$, the term $({x}-{y})\cdot{n}({x})$ is
$\mathcal{O}(\delta^{2})$. We then have the following bound for $r_{6}$:

$\begin{split}&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left\lVert
r_{6}\right\rVert^{2}_{L^{2}(\partial\mathcal{M})}=\int_{\partial\mathcal{M}}(\delta^{2}\int_{\partial\mathcal{M}}(({x}-{y})\cdot{n}({y}))\
(f({y})-\kappa_{{n}}({y})\frac{\partial u}{\partial{n}}({y}))\
\overset{=}{R}_{\delta}({x},{y})d\tau_{y})^{2}d\tau_{x}\\\ &\leq
C\delta^{4}\int_{\partial\mathcal{M}}(\int_{\partial\mathcal{M}}\delta^{2}|\nabla_{\mathcal{M}}^{2}u({y})|\
\overset{=}{R}_{\delta}({x},{y})d\tau_{y})^{2}d\tau_{x}\\\ &\leq
C\delta^{8}\int_{\partial\mathcal{M}}(\int_{\partial\mathcal{M}}|\nabla_{\mathcal{M}}^{2}u({y})|^{2}\
\overset{=}{R}_{\delta}({x},{y})d\tau_{{y}})\ (\int_{\partial\mathcal{M}}\
\overset{=}{R}_{\delta}({x},{y})d\tau_{y})\ d\tau_{x}\\\ &\leq
C\delta^{6}\int_{\partial\mathcal{M}}|\nabla_{\mathcal{M}}^{2}u({y})|^{2}d\tau_{y}=C\delta^{6}\left\lVert
u\right\rVert^{2}_{H^{2}(\partial\mathcal{M})}\leq C\delta^{6}\left\lVert
u\right\rVert^{2}_{H^{3}(\mathcal{M})}.\end{split}$ (3.41)

  2. 2.

For the next term $r_{7}$, we denote

$r_{70}=\frac{1}{\delta^{2}}\int_{\partial\mathcal{M}}(\frac{\partial
u}{\partial{n}}({y})-\frac{\partial
u}{\partial{n}}({x}))\overset{=}{R}_{\delta}({x},{y})d\tau_{y}-\int_{\partial\mathcal{M}}\Delta_{\partial\mathcal{M}}(\frac{\partial
u}{\partial{n}}({y})){\overset{\equiv}{R}}_{\delta}({x},{y})d\tau_{{y}}.$
(3.42)

Here the kernel function
$\overset{\equiv}{R}_{\delta}({x},{y})=C_{\delta}\overset{\equiv}{R}\big{(}\frac{|{x}-{y}|^{2}}{4\delta^{2}}\big{)}$,
and $\overset{\equiv}{R}(r)=\int_{r}^{+\infty}\overset{=}{R}(s)ds$. We compare
(3.42) with the approximation (2.4). Due to the fact that
$\partial\mathcal{M}$ is an $(m-1)$ dimensional manifold with empty boundary,
(3.42) is exactly (2.4) after replacing $u$ by $\frac{\partial
u}{\partial{n}}$, $R$ by $\overset{=}{R}$, and $\mathcal{M}$ by
$\partial\mathcal{M}$. Consequently, $r_{70}$ is a lower order term compared
to the other terms in (3.42). We then have

$\begin{split}\left\lVert
r_{7}\right\rVert^{2}_{L^{2}(\partial\mathcal{M})}\leq&4\delta^{8}\int_{\partial\mathcal{M}}(\int_{\partial\mathcal{M}}\Delta_{\partial\mathcal{M}}\frac{\partial
u}{\partial{n}}({y}){\overset{\equiv}{R}}_{\delta}({x},{y})d\tau_{{y}})^{2}d\tau_{{x}}+4\delta^{8}\left\lVert
r_{70}\right\rVert_{L^{2}(\partial\mathcal{M})}\\\
\leq&C\delta^{8}\int_{\partial\mathcal{M}}(\int_{\partial\mathcal{M}}|\nabla_{\mathcal{M}}^{3}u({y})|{\overset{\equiv}{R}}_{\delta}({x},{y})d\tau_{{y}})^{2}d\tau_{{x}}\\\
\leq&C\delta^{8}\int_{\partial\mathcal{M}}(\int_{\partial\mathcal{M}}|\nabla_{\mathcal{M}}^{3}u({y})|^{2}\
\overset{=}{R}_{\delta}({x},{y})d\tau_{{y}})\ (\int_{\partial\mathcal{M}}\
\overset{=}{R}_{\delta}({x},{y})d\tau_{y})\ d\tau_{x}\\\
\leq&C\delta^{6}\int_{\partial\mathcal{M}}|\nabla_{\mathcal{M}}^{3}u({y})|^{2}d\tau_{y}=C\delta^{6}\left\lVert
u\right\rVert^{2}_{H^{3}(\partial\mathcal{M})}\leq C\delta^{6}\left\lVert
u\right\rVert^{2}_{H^{4}(\mathcal{M})}.\end{split}$ (3.43)

  3. 3.

Next, for the term $r_{8}$, we have

$\begin{split}|r_{8}|\leq&C\delta\int_{\mathcal{M}}\Big{|}u({y})-u({x})-({y}-{x})\cdot\nabla
u({x})\Big{|}\bar{R}_{\delta}({x},{y})d\mu_{y}\\\
\leq&C\delta\int_{\mathcal{M}}\Big{|}u({y})-u({x})-({y}-{x})\cdot\nabla
u({x})-\frac{1}{2}\eta^{i}\eta^{j}\nabla^{i}\nabla^{j}u({x})\Big{|}\bar{R}_{\delta}({x},{y})d\mu_{y}\\\
&+C\delta\int_{\mathcal{M}}\big{|}\eta^{i}\eta^{j}\nabla^{i}\nabla^{j}u({x})\big{|}\bar{R}_{\delta}({x},{y})d\mu_{y}\\\
\leq&C\delta^{3}\frac{1}{\delta^{2}}\int_{\mathcal{M}}\Big{|}u({y})-u({x})-({y}-{x})\cdot\nabla
u({x})-\frac{1}{2}\eta^{i}\eta^{j}\nabla^{i}\nabla^{j}u({x})\Big{|}\bar{R}_{\delta}({x},{y})d\mu_{y}\\\
&+C\delta^{3}|\nabla^{2}u({x})|.\end{split}$ (3.44)

For the second term of (3.44), it is clear that its boundary $L^{2}$ norm is
bounded by $\delta^{3}\left\lVert u\right\rVert^{2}_{H^{4}(\mathcal{M})}$.
What left is the control of the first term of (3.44). We denote it as
$\bar{r}_{1}$. Recall in (3.28) we have controlled the term

$r_{1}({x})=\frac{1}{\delta^{2}}\int_{\mathcal{M}}(u({y})-u({x})-({y}-{x})\cdot\nabla
u({x})-\frac{1}{2}\eta^{i}\eta^{j}\nabla^{i}\nabla^{j}u({x}))R_{\delta}({x},{y})d\mu_{{y}}$
(3.45)

with the bound

$\left\lVert r_{1}\right\rVert_{H^{1}(\mathcal{M})}\leq
C\delta^{-\frac{1}{2}}\left\lVert u\right\rVert_{H^{4}(\mathcal{M})}.$ (3.46)

We compare $r_{1}$ with $\bar{r}_{1}$ to notice that only the kernel function
has been changed from ${R}$ into $\bar{R}$, hence the same argument hold for
$\bar{r}_{1}$:

$\begin{split}\left\lVert\bar{r}_{1}\right\rVert_{H^{1}(\mathcal{M})}\leq
C\delta^{3}\ \delta^{-\frac{1}{2}}\left\lVert
u\right\rVert^{2}_{H^{4}(\mathcal{M})}=C\delta^{\frac{5}{2}}\left\lVert
u\right\rVert^{2}_{H^{4}(\mathcal{M})},\end{split}$ (3.47)

which gives

$\begin{split}\left\lVert\bar{r}_{1}\right\rVert_{L^{2}(\partial\mathcal{M})}\leq
C\left\lVert\bar{r}_{1}\right\rVert_{H^{1}(\mathcal{M})}\leq
C\delta^{\frac{5}{2}}\left\lVert
u\right\rVert^{2}_{H^{4}(\mathcal{M})},\end{split}$ (3.48)

therefore

$\left\lVert r_{8}\right\rVert_{L^{2}(\partial\mathcal{M})}\leq
C\delta^{\frac{5}{2}}\left\lVert u\right\rVert^{2}_{H^{4}(\mathcal{M})}.$
(3.49)

  4. 4.

Finally, what left is the term $\bar{r}_{in}$. By a simple observation, we see
$\bar{r}_{in}$ is exactly the term $r_{in}$ after replacing the kernel
functions $R_{\delta}$ in (3.1) by $\bar{R}_{\delta}$ in (3.40). recall our
estimate on ${r}_{in}$ in the first part of theorem:

$\begin{split}\left\rVert{r}_{in}\right\rVert_{L^{2}(\partial\mathcal{M})}\leq&\left\lVert
r_{it}\right\rVert_{L^{2}(\partial\mathcal{M})}+\left\lVert
r_{bl}\right\rVert_{L^{2}(\partial\mathcal{M})}\\\ \leq&\ C(\left\lVert
r_{it}\right\rVert_{H^{1}(\mathcal{M})}+\left\lVert
r_{bl}\right\rVert_{H^{1}(\mathcal{M})})\leq C\delta^{\frac{1}{2}}\left\lVert
u\right\rVert_{H^{4}(\mathcal{M})},\end{split}$ (3.50)

our previous calculation indicates that the same bound holds for
$\bar{r}_{in}$:

$\left\rVert\bar{r}_{in}\right\rVert_{L^{2}(\partial\mathcal{M})}\leq
C\delta^{\frac{1}{2}}\left\lVert u\right\rVert_{H^{4}(\mathcal{M})}.$

Finally, we sum up all the above 4 estimates to conclude

$\begin{split}\left\lVert
r_{bd}\right\rVert_{L^{2}(\partial\mathcal{M})}\leq&\left\lVert
r_{6}\right\rVert_{L^{2}(\partial\mathcal{M})}+\left\lVert
r_{7}\right\rVert_{L^{2}(\partial\mathcal{M})}+\left\lVert p\
r_{8}\right\rVert_{L^{2}(\partial\mathcal{M})}+2\delta^{2}\left\lVert\bar{r}_{in}\right\rVert_{L^{2}(\partial\mathcal{M})}\\\
\leq&C\delta^{\frac{5}{2}}\left\lVert
u\right\rVert_{H^{4}(\mathcal{M})}.\end{split}$ (3.51)

Hence we have completed the proof of theorem 3.1.

##  4 Appendix

###  4.1 Proof of Lemma 3.3

We start with the proof of the equality

$\frac{\partial^{2}u}{\partial{n}^{2}}({x})=\Delta_{\mathcal{M}}u({x})+\sum\limits_{i=1}^{m-1}\sum\limits_{j=1}^{m-1}h^{ij}(\omega)l_{ij}(\omega)\
\frac{\partial u}{\partial{n}}({x}).$ (4.1)

Let $\mathcal{T}_{{x}}(\partial\mathcal{M})$ be the tangent space of
$\partial\mathcal{M}$ at ${x}$. Apparently, we have
$\mathcal{T}_{{x}}(\partial\mathcal{M})\oplus{n}({x})=\mathcal{T}_{{x}}(\mathcal{M})$.
Let $\\{\bm{\tau}_{i}\\}_{i=1,2,...,m-1}$ be an orthonormal basis of
$\mathcal{T}_{{x}}(\partial\mathcal{M})$ and

$\tau_{ij}=\sum\limits_{k=1}^{m-1}h^{jk}(\frac{\partial\bm{\psi}}{\partial\omega_{k}},\bm{\tau}_{i}),$
(4.2)

then by calculation, we can decompose $\bm{\tau}_{i}$ into

$\bm{\tau}_{i}=\sum\limits_{j=1}^{m-1}\tau_{ij}\frac{\partial\bm{\psi}}{\partial\omega_{j}}.$
(4.3)

Now for each $i$, we can find a curve
$\bm{\gamma}_{i}\subset\partial\mathcal{M}$ with equation

$\bm{\gamma}_{i}(t)=\bm{\psi}(\omega_{1}+\tau_{i1}t,\omega_{2}+\tau_{i2}t,...,\omega_{m-1}+\tau_{i\
m-1}t)$ (4.4)

such that $\bm{\gamma}_{i}(0)={x}$ and

$\frac{d\bm{\gamma_{i}}}{dt}(0)=\sum\limits_{j=1}^{m-1}\tau_{ij}\frac{\partial\bm{\psi}}{\partial\omega_{j}}=\bm{\tau}_{i}.$
(4.5)

Since $u\equiv 0$ on $\bm{\gamma}_{i}$ according to the Dirichlet condition,
we have

$\begin{split}0=&\frac{d}{dt}u(\bm{\gamma}_{i}(0))=\nabla
u({x})\cdot\frac{d}{dt}\bm{\gamma}_{i}(0)=\nabla
u({x})\cdot\bm{\tau}_{i}({x}),\end{split}$ (4.6)

and

$\begin{split}0=&\frac{d^{2}}{dt^{2}}u(\bm{\gamma}_{i}(0))=\nabla
u({x})\cdot\frac{d^{2}}{dt^{2}}\bm{\gamma}_{i}(0)+\frac{d}{dt}(\nabla
u({x}))\cdot\frac{d}{dt}\bm{\gamma}_{i}(0)\\\ =&\nabla
u({x})\cdot\frac{d^{2}}{dt^{2}}\bm{\gamma}_{i}(0)+\bm{H}_{u}({x}):\bm{\tau_{i}}\otimes\bm{\tau_{i}}\\\
=&\nabla
u({x})\cdot\sum\limits_{j=1}^{m-1}\sum\limits_{k=1}^{m-1}(\frac{\partial^{2}\bm{\psi}}{\partial\omega_{j}\partial\omega_{k}}\tau_{ij}\tau_{ik})+\bm{H}_{u}({x}):\bm{\tau_{i}}\otimes\bm{\tau_{i}}.\end{split}$
(4.7)

By the definition of $\nabla$ in (1.3), since $\mathcal{T}_{{x}}\mathcal{M}$
has orthonormal basis $\\{{n},\bm{\tau}_{1},...,\bm{\tau}_{m-1}\\}$, and
$\partial_{i}\bm{\phi}\in\mathcal{T}_{{x}}(\mathcal{M})$ for each
$i\in\\{1,2,...,m\\}$, we have

$\begin{split}\nabla
u=&\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{m}(\partial_{i}\bm{\phi})g^{ij}(\partial_{j}u)=\sum\limits_{k=1}^{m-1}\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{m}(\partial_{i}\bm{\phi}\cdot\bm{\tau}_{k})\bm{\tau}_{k}g^{ij}(\partial_{j}u)\\\
&+\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{m}(\partial_{i}\bm{\phi}\cdot{n})\
{n}\ g^{ij}(\partial_{j}u)\\\ =&\sum\limits_{k=1}^{m-1}(\nabla
u\cdot\bm{\tau}_{k})\bm{\tau}_{k}+(\nabla u\cdot{n}){n}=(\nabla
u\cdot{n}){n}=\frac{\partial u}{\partial{n}}{n},\end{split}$ (4.8)

this implies

$\nabla
u({x})\cdot\frac{\partial^{2}\bm{\psi}}{\partial\omega_{j}\partial\omega_{k}}\tau_{ij}\tau_{ik}=(\frac{\partial^{2}\bm{\psi}}{\partial\omega_{j}\partial\omega_{k}},{n})\tau_{ij}\tau_{ik}\frac{\partial
u}{\partial{n}}=l_{jk}\tau_{ij}\tau_{ik}\frac{\partial u}{\partial{n}}.$ (4.9)

Using such fact, we then sum up (4.7) from $i=1$ to $m-1$ to discover

$0=\sum\limits_{i=1}^{m-1}\sum\limits_{j=1}^{m-1}\sum\limits_{k=1}^{m-1}l_{jk}\tau_{ij}\tau_{ik}\frac{\partial
u}{\partial{n}}+\sum\limits_{i=1}^{m-1}\bm{H}_{u}({x}):\bm{\tau_{i}}\otimes\bm{\tau_{i}}.$
(4.10)

For the first term of (4.10), we denote
${\Gamma}=(\tau_{ij})_{i,j=1,2,...,m-1}$. Since
$\\{\bm{\tau}_{i}\\}_{i=1,2,...,m-1}$ is orthonormal, the expansion (4.3)
gives

${\Gamma}^{T}H{\Gamma}=I,$ (4.11)

hence $\Gamma\Gamma^{T}=H^{-1}$. We then return to (4.10) to obtain

$\begin{split}\sum\limits_{i=1}^{m-1}\sum\limits_{j=1}^{m-1}\sum\limits_{k=1}^{m-1}l_{jk}\tau_{ij}\tau_{ik}=tr(\
{\Gamma}^{T}L{\Gamma})=tr(L\ {\Gamma}{\Gamma}^{T})\\\
=\sum\limits_{i=1}^{m-1}\sum\limits_{j=1}^{m-1}l_{ij}h^{ji}=\sum\limits_{i=1}^{m-1}\sum\limits_{j=1}^{m-1}l_{ij}h^{ij}.\end{split}$
(4.12)

For the last term of (4.10), as the equation (30) of [37] gives us

$\Delta_{\mathcal{M}}u=\nabla^{j}(\nabla^{j}u)=(\partial_{k^{\prime}}\phi^{j})g^{k^{\prime}l^{\prime}}\partial_{l^{\prime}}\big{(}(\partial_{m^{\prime}}\phi^{j})g^{m^{\prime}n^{\prime}}(\partial_{n^{\prime}}u)\big{)},$
(4.13)

and by the same argument as (4.8),

$\partial_{k^{\prime}}\phi^{j}=(\partial_{k^{\prime}}\bm{\phi})^{j}=\sum\limits_{i=1}^{m-1}(\partial_{k^{\prime}}\bm{\phi}\cdot\bm{\tau}_{i})\tau_{i}^{j}+(\partial_{k^{\prime}}\bm{\phi}\cdot{n})n^{j}=\sum\limits_{i=1}^{m-1}\partial_{k^{\prime}}{\phi^{l}}{\tau}_{i}^{l}\tau_{i}^{j}+\partial_{k^{\prime}}{\phi^{l}}n^{l}n^{j},$
(4.14)

this implies

$\begin{split}&\Delta_{\mathcal{M}}u({x})=(\partial_{k^{\prime}}\phi^{j})g^{k^{\prime}l^{\prime}}\partial_{l^{\prime}}\big{(}(\partial_{m^{\prime}}\phi^{j})g^{m^{\prime}n^{\prime}}(\partial_{n^{\prime}}u)\big{)}\\\
&=\sum\limits_{i=1}^{m-1}{\tau}_{i}^{l}\tau_{i}^{j}\partial_{k^{\prime}}{\phi^{l}}g^{k^{\prime}l^{\prime}}\partial_{l^{\prime}}\big{(}(\partial_{m^{\prime}}\phi^{j})g^{m^{\prime}n^{\prime}}(\partial_{n^{\prime}}u)\big{)}+n^{l}n^{j}\partial_{k^{\prime}}{\phi^{l}}g^{k^{\prime}l^{\prime}}\partial_{l^{\prime}}\big{(}(\partial_{m^{\prime}}\phi^{j})g^{m^{\prime}n^{\prime}}(\partial_{n^{\prime}}u)\big{)}\\\
&=\sum\limits_{i=1}^{m-1}{\tau}_{i}^{l}\tau_{i}^{j}\nabla^{l}\nabla^{j}u+n^{l}n^{j}\nabla^{l}\nabla^{j}u=\sum\limits_{i=1}^{m-1}\bm{H}_{u}({x}):\bm{\tau_{i}}\otimes\bm{\tau_{i}}+\frac{\partial^{2}u}{\partial{n}^{2}}({x}).\end{split}$
(4.15)

We then apply (4.12) and (4.15) into (4.10) to discover

$\frac{\partial^{2}u}{\partial{n}^{2}}({x})=\Delta_{\mathcal{M}}u({x})+\sum\limits_{i=1}^{m-1}\sum\limits_{j=1}^{m-1}l_{ij}h^{ij}\frac{\partial
u}{\partial{n}}({x}).$ (4.16)

Now we have finished the proof of (4.1). What remains to show is the equality
(3.17). Let
$\mathcal{T}_{{x}}:\partial\mathcal{M}\to\mathcal{P}_{{x}}(\partial\mathcal{M})$
be the map of projection from $\partial\mathcal{M}$ into the Euclid space
$\mathcal{T}_{{x}}(\mathcal{M})$. Apparently,
$\bm{\psi}_{P}=\mathcal{T}_{{x}}\circ\bm{\psi}:\Sigma\subset\mathbb{R}^{m-1}\to\mathcal{P}_{{x}}(\partial\mathcal{M})\subset\mathbb{R}^{d}$
is a local parametrization of $\mathcal{P}_{{x}}(\partial\mathcal{M})$ near
${x}$, with

$\frac{\partial\bm{\psi}_{P}}{\partial\omega_{j}}=\frac{\partial\bm{\psi}}{\partial\omega_{j}},\
j=1,2,...,m-1.$ (4.17)
$\frac{\partial^{2}\bm{\psi}_{P}}{\partial\omega_{j}\partial\omega_{k}}=\sum\limits_{i=1}^{m-1}(\frac{\partial^{2}\bm{\psi}}{\partial\omega_{j}\partial\omega_{k}}\cdot\bm{\tau}_{i})\bm{\tau}_{i}+(\frac{\partial^{2}\bm{\psi}}{\partial\omega_{j}\partial\omega_{k}}\cdot{n}){n},\
j,k=1,2,...,m-1.$ (4.18)

We denote $\kappa_{i},i\in\\{1,...,m-1\\}$ as the normal curvature of
$\mathcal{P}_{{x}}(\partial\mathcal{M})$ at ${x}$ in the direction of
$\bm{\tau}_{i}$. According to (4.3), $\bm{\tau}_{i}$ has the following
decomposition

$\bm{\tau}_{i}=\sum\limits_{j=1}^{m-1}\tau_{ij}\frac{\partial\bm{\psi}}{\partial\omega_{j}}=\sum\limits_{j=1}^{m-1}\tau_{ij}\frac{\partial\bm{\psi}_{P}}{\partial\omega_{j}},$
(4.19)

thereafter, we apply the normal curvature function of
$\mathcal{P}_{{x}}(\partial\mathcal{M})$ in the direction of $\bm{\tau}_{i}$
to discover

$\begin{split}\kappa_{i}=II(\bm{\tau}_{i},\bm{\tau}_{i})/I(\bm{\tau}_{i},\bm{\tau}_{i})=\frac{\sum\limits_{j=1}^{m-1}\sum\limits_{k=1}^{m-1}\tau_{ij}\tau_{ik}\frac{\partial^{2}\bm{\psi}_{P}}{\partial\omega_{j}\partial\omega_{k}}\cdot{n}}{\sum\limits_{j=1}^{m-1}\sum\limits_{k=1}^{m-1}\tau_{ij}\tau_{ik}\frac{\partial\bm{\psi}_{P}}{\partial\omega_{j}}\cdot\frac{\partial\bm{\psi}_{P}}{\partial\omega_{k}}},\end{split}$
(4.20)

by (4.17), (4.18) and $\left\lVert\bm{\tau}_{i}\right\rVert=1$, we can obtain

$\begin{split}\kappa_{i}=\frac{\sum\limits_{j=1}^{m-1}\sum\limits_{k=1}^{m-1}\tau_{ij}\tau_{ik}\frac{\partial^{2}\bm{\psi}}{\partial\omega_{j}\partial\omega_{k}}\cdot{n}}{\sum\limits_{j=1}^{m-1}\sum\limits_{k=1}^{m-1}\tau_{ij}\tau_{ik}\frac{\partial\bm{\psi}}{\partial\omega_{j}}\cdot\frac{\partial\bm{\psi}}{\partial\omega_{k}}}=\sum\limits_{j=1}^{m-1}\sum\limits_{k=1}^{m-1}\tau_{ij}\tau_{ik}l_{jk}.\end{split}$
(4.21)

By the definition of $\kappa_{{n}}$,

$\kappa_{n}=(m-1)\frac{\sum\limits_{i=1}^{m-1}\kappa_{i}}{m-1}=\sum\limits_{i=1}^{m-1}\kappa_{i}=\sum\limits_{i=1}^{m-1}\sum\limits_{j=1}^{m-1}\sum\limits_{k=1}^{m-1}\tau_{ij}\tau_{ik}l_{jk}=\sum\limits_{i=1}^{m-1}\sum\limits_{j=1}^{m-1}h^{ij}l_{ij},$
(4.22)

where the last equality comes from (4.12). Hence we have completed the proof.

###  4.2 Proof of Lemma 3.4

By the definition of $\bm{\eta}$, we can do the following expansion:

$\begin{split}&\int_{\partial\mathcal{M}}\int_{\mathcal{M}}f_{1}({x})\textbf{g}_{1}({y})\cdot({x}-{y}-\bm{\eta}({x},{y}))R_{\delta}({x},{y})d\mu_{x}d\tau_{y}\\\
=&\int_{\partial\mathcal{M}}\int_{\mathcal{M}}f_{1}({x})g_{1}^{j}({y})\xi^{i^{\prime}}\xi^{j^{\prime}}\int_{0}^{1}\int_{0}^{1}s\partial_{j^{\prime}}\partial_{i^{\prime}}\bm{\phi}^{j}(\bm{\alpha}+\tau
s\bm{\xi})d\tau ds\ R_{\delta}({x},{y})d\mu_{x}d\tau_{y}\\\
=&\int_{\partial\mathcal{M}}\int_{\mathcal{M}}f_{1}({x})g^{j}_{1}({y})\xi^{i^{\prime}}\xi^{j^{\prime}}\xi^{k^{\prime}}(\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}s^{2}\tau\partial_{j^{\prime}}\partial_{i^{\prime}}\partial_{k^{\prime}}\phi^{j}(\bm{\alpha}+\tau
st\bm{\xi})dtd\tau ds){R}_{\delta}({x},{y})\\\
&d\mu_{x}d\tau_{y}+\int_{\partial\mathcal{M}}\int_{\mathcal{M}}f_{1}({x})g^{j}_{1}({y})\frac{1}{2}\xi^{i^{\prime}}\xi^{j^{\prime}}\partial_{j^{\prime}}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha}){R}_{\delta}({x},{y})d\mu_{x}d\tau_{y}\\\
=&\int_{\partial\mathcal{M}}g^{j}_{1}({y})\int_{\mathcal{M}}f_{1}({x})\frac{1}{2}\xi^{i^{\prime}}\xi^{j^{\prime}}\partial_{j^{\prime}}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha}){R}_{\delta}({x},{y})d\mu_{x}d\tau_{y}+r_{51},\end{split}$
(4.23)

where $r_{51}$ is a lower order term and will be controlled later. Next, we
will use the fact that

$\begin{split}2\delta^{2}\partial_{l}^{\bm{\beta}}\bar{R}_{\delta}({x},{y})=&2\delta^{2}\partial_{l}\bm{\phi}(\bm{\beta})\cdot\nabla_{\mathcal{M}}^{{x}}\bar{R}_{\delta}({x},{y})=\partial_{l}\bm{\phi}(\bm{\beta})\cdot\bm{\eta}({y},{x}){R}_{\delta}({y},{x})+r_{52}\\\
=&\partial_{l}{\phi}^{k^{\prime}}(\bm{\beta}){\xi}^{i^{\prime}}({y},{x})\partial_{i^{\prime}}{\phi}^{k^{\prime}}(\bm{\beta}){R}_{\delta}({x},{y})+r_{52}\\\
=&-\partial_{l}{\phi}^{k^{\prime}}(\bm{\beta})\partial_{i^{\prime}}{\phi}^{k^{\prime}}(\bm{\beta}){\xi}^{i^{\prime}}({x},{y}){R}_{\delta}({x},{y})+r_{52}\\\
=&-g_{i^{\prime}l}({x}){\xi}^{i^{\prime}}{R}_{\delta}({x},{y})+r_{52},\end{split}$
(4.24)

where

$\begin{split}&r^{j}_{52}({x},{y})=\partial_{j}\bm{\phi}(\bm{\beta})\cdot(2\delta^{2}\nabla_{\mathcal{M}}^{{x}}\bar{R}_{\delta}({x},{y})-\bm{\eta}({y},{x}){R}_{\delta}({y},{x}))\\\
=&\partial_{j}{\phi}^{l}(\bm{\beta})(\partial_{m^{\prime}}\phi^{l}(\bm{\alpha})g^{m^{\prime}n^{\prime}}\partial_{n^{\prime}}\phi^{i}(\bm{\alpha})(x^{i}-y^{i})\bar{R}_{\delta}({x},{y})\\\
&-\partial_{m^{\prime}}\phi^{l}(\bm{\alpha})g^{m^{\prime}n^{\prime}}\partial_{n^{\prime}}\phi^{i}(\bm{\alpha})\xi^{i^{\prime}}\partial_{i^{\prime}}\phi^{i}\bar{R}_{\delta}({x},{y}))\\\
=&\partial_{j}{\phi}^{l}(\bm{\beta})\partial_{m^{\prime}}\phi^{l}(\bm{\alpha})g^{m^{\prime}n^{\prime}}\partial_{n^{\prime}}\phi^{i}(\bm{\alpha})(x^{i}-y^{i}-\xi^{i^{\prime}}\partial_{i^{\prime}}\phi^{i})\bar{R}_{\delta}({x},{y})\\\
=&\partial_{j}{\phi}^{l}(\bm{\beta})\partial_{m^{\prime}}\phi^{l}(\bm{\alpha})g^{m^{\prime}n^{\prime}}\partial_{n^{\prime}}\phi^{i}(\bm{\alpha})\bar{R}_{\delta}({x},{y})\xi^{i^{\prime}}\xi^{j^{\prime}}\int_{0}^{1}\int_{0}^{1}s\partial_{j^{\prime}}\partial_{i^{\prime}}\bm{\phi}^{i}(\bm{\alpha}+\tau
s\bm{\xi})d\tau ds,\end{split}$ (4.25)

which is obviously a lower order term comparing to the other terms in (4.24).
Next, we apply (4.24) to (4.23) to obtain

$\begin{split}\int_{\mathcal{M}}&f_{1}({x})\xi^{i^{\prime}}\xi^{j^{\prime}}\partial_{j^{\prime}}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha}){R}_{\delta}({x},{y})d\mu_{x}\\\
=&\int_{{\Omega}_{J({y})}}f_{1}({x})\xi^{i^{\prime}}\xi^{j^{\prime}}\partial_{j^{\prime}}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha}){R}_{\delta}({x},{y})\sqrt{det\
G(\bm{\beta})}\ d\bm{\beta}\\\
=&-2\delta^{2}\int_{{\Omega}_{J({y})}}f_{1}({x})\xi^{j^{\prime}}\partial_{j^{\prime}}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha})g^{i^{\prime}l}({x})\partial_{l}^{\bm{\beta}}\bar{R}_{\delta}({x},{y})\
\sqrt{det\ G(\bm{\beta})}\ d\bm{\beta}+r_{53}\\\
=&-2\delta^{2}\int_{\partial{\Omega}_{J({y})}}f_{1}({x})\xi^{j^{\prime}}\partial_{j^{\prime}}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha})g^{i^{\prime}l}({x})n_{\Omega}^{l}(\bm{\beta})\bar{R}_{\delta}({x},{y})\
\sqrt{det\ G(\bm{\beta})}\ dS_{\bm{\beta}}\\\
&+2\delta^{2}\int_{{\Omega}_{J({y})}}\partial_{l}\big{(}f_{1}({x})\xi^{j^{\prime}}\partial_{j^{\prime}}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha})g^{i^{\prime}l}({x})\
\sqrt{det\ G(\bm{\beta})}\big{)}\bar{R}_{\delta}({x},{y})\
d\bm{\beta}+r_{53}\\\
=&-2\delta^{2}\int_{\partial{{\Omega}_{J({y})}}}f_{1}({x})\xi^{j^{\prime}}\partial_{j^{\prime}}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha})g^{i^{\prime}l}({x})n_{\Omega}^{l}(\bm{\beta})\bar{R}_{\delta}({x},{y})\
\sqrt{det\ G(\bm{\beta})}\ dS_{\bm{\beta}}\\\
&+2\delta^{2}\int_{{{\Omega}_{J({y})}}}\partial_{l}\big{(}f_{1}({x})\partial_{j^{\prime}}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha})g^{i^{\prime}l}({x})\
\sqrt{det\ G(\bm{\beta})}\big{)}\xi^{j^{\prime}}\bar{R}_{\delta}({x},{y})\
d\bm{\beta}\\\
&-2\delta^{2}\int_{{{\Omega}_{J({y})}}}f_{1}({x})\partial_{l}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha})g^{i^{\prime}l}({x})\
\sqrt{det\ G(\bm{\beta})}\bar{R}_{\delta}({x},{y})\ d\bm{\beta}+r_{53}\\\
=&-2\delta^{2}\int_{\partial{{\Omega}_{J({y})}}}f_{1}({x})\xi^{j^{\prime}}\partial_{j^{\prime}}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha})g^{i^{\prime}l}({x})n_{\Omega}^{l}(\bm{\beta})\bar{R}_{\delta}({x},{y})\
\sqrt{det\ G(\bm{\beta})}\ dS_{\bm{\beta}}\\\
&-2\delta^{2}\int_{\mathcal{M}}f_{1}({x})\partial_{l}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha})g^{i^{\prime}l}({x})\bar{R}_{\delta}({x},{y})d\mu_{x}+r_{53}+r_{54}.\end{split}$
(4.26)

Here $G=(g_{ij})_{i,j=1,2,...,m}$ is the matrix defined in (1.5) such that
$d\mu_{{x}}=\sqrt{det\ G(\bm{\beta})}d\bm{\beta}$ by a calculation on the
Jacobian matrix, and ${n}_{\Omega}(\bm{\beta})$ is the unit outward normal
vector of $\Omega_{J({y})}$ at $\bm{\beta}\in\partial\Omega_{J({y})}$. To
simplify the notations, we always use $\Omega$ to denote $\Omega_{J({y})}$ in
the following content of this lemma.

For the boundary term of (4.26), we aim to show it is a lower order term.
First, we decompose it into

$\begin{split}&-2\delta^{2}\int_{\partial\Omega}f_{1}({x})\xi^{j^{\prime}}\partial_{j^{\prime}}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha})g^{i^{\prime}l}({x})n_{\Omega}^{l}(\bm{\beta})\bar{R}_{\delta}({x},{y})\
\sqrt{det\ G(\bm{\beta})}\ dS_{\bm{\beta}}\\\
=&4\delta^{4}\int_{\partial\Omega}f_{1}({x})\partial_{j^{\prime}}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha})g^{i^{\prime}l}({x})n_{\Omega}^{l}(\bm{\beta})(g^{j^{\prime}l^{\prime}}({x})\partial_{l^{\prime}}^{\bm{\beta}}\overset{=}{R}_{\delta}({x},{y}))\
\sqrt{det\ G(\bm{\beta})}\ dS_{\bm{\beta}}+r_{58}\\\
=&4\delta^{4}\int_{\partial\Omega}f_{1}({x})\partial_{j^{\prime}}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha})g^{i^{\prime}l}({x})n_{\Omega}^{l}(\bm{\beta})g^{j^{\prime}l^{\prime}}({x})(\partial^{\bm{\beta}}\overset{=}{R}_{\delta}({x},{y})\cdot{n}_{\Omega}(\bm{\beta}))n_{\Omega}^{l^{\prime}}(\bm{\beta})\
\sqrt{det\ G(\bm{\beta})}\\\
&+f_{1}({x})\partial_{j^{\prime}}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha})g^{i^{\prime}l}({x})n_{\Omega}^{l}(\bm{\beta})g^{j^{\prime}l^{\prime}}({x})\mathcal{T}^{\prime}_{\bm{\beta}}(\partial^{\bm{\beta}}\overset{=}{R}_{\delta}({x},{y}))\
\sqrt{det\ G(\bm{\beta})}\ dS_{\bm{\beta}}+r_{58}\\\
=&r_{56}+r_{57}+r_{58},\end{split}$ (4.27)

here $\mathcal{T}^{\prime}_{\bm{\beta}}:\mathbb{R}^{m}\to\mathbb{R}^{m}$ is
the projection from $\Omega$ onto $\mathcal{T}_{\bm{\beta}}(\partial\Omega)$,
while $\mathcal{T}_{\bm{\beta}}(\partial\Omega)$ denotes the tangent space of
$\partial\Omega$ at point $\bm{\alpha}$.

Let us first study $r_{56}$. Since $\partial\mathcal{M}$ is sufficiently
smooth, we have $({x}-{y})\cdot{n}({x})$ is $\mathcal{O}(\delta^{2})$ when
${x},{y}$ both lie on $\partial\mathcal{M}$ and $|{x}-{y}|\leq 2\delta$.
Consequently, the term
$\partial^{\bm{\beta}}\overset{=}{R}_{\delta}({x},{y})\cdot{n}_{\Omega}(\bm{\beta})$
is of lower order due to the fact that
$\nabla_{\mathcal{M}}^{{x}}\overset{=}{R}_{\delta}({x},{y})\cdot{n}({x})$ is a
lower order term. This observation indicates $r_{56}$ is
$\mathcal{O}(\delta^{3})$. For the term $r_{57}$, by noticing the fact that
$\mathcal{T}^{\prime}_{\bm{\beta}}(\partial^{\bm{\beta}})=\partial_{\partial\Omega}^{\bm{\beta}}$,
where $\partial_{\partial\Omega}^{\bm{\beta}}$ represents the gradient
operator defined on the manifold $\partial\Omega$ embedded in
$\mathbb{R}^{m}$, we can do the following integration by parts:

$\begin{split}r_{57}=&4\delta^{4}\int_{\partial\Omega}f_{1}({x})\partial_{j^{\prime}}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha})g^{i^{\prime}l}({x})n_{\Omega}^{l}(\bm{\beta})g^{j^{\prime}l^{\prime}}({x})\partial_{\partial\Omega}^{\bm{\beta}}\overset{=}{R}_{\delta}({x},{y})\
\sqrt{det\ G(\bm{\beta})}\ dS_{\bm{\beta}}\\\
=&-4\delta^{4}\int_{\partial\Omega}\partial_{\partial\Omega}^{\bm{\beta}}\Big{(}f_{1}({x})\partial_{j^{\prime}}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha})g^{i^{\prime}l}({x})n_{\Omega}^{l}(\bm{\beta})g^{j^{\prime}l^{\prime}}({x})\
\sqrt{det\ G(\bm{\beta})}\Big{)}\ \overset{=}{R}_{\delta}({x},{y})\
dS_{\bm{\beta}},\end{split}$ (4.28)

which indicates that $r_{57}$ is $\mathcal{O}(\delta^{3})$ as well by
continuity.

Now let us return to (4.26). For the interior term of (4.26), we multiply by
$g_{1}$ and integrate over $\partial\mathcal{M}$ to discover

$\begin{split}\int_{\partial\mathcal{M}}&g^{j}_{1}({y})\int_{\mathcal{M}}f_{1}({x})\partial_{l}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha})g^{i^{\prime}l}({x})\bar{R}_{\delta}({x},{y})d\mu_{x}d\tau_{y}\\\
=&\int_{\partial\mathcal{M}}g^{j}_{1}({y})\int_{\mathcal{M}}f_{1}({x})\partial_{l}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha})g^{i^{\prime}l}({y})\bar{R}_{\delta}({x},{y})d\mu_{x}d\tau_{y}+r_{55}\\\
=&\int_{\partial\mathcal{M}}\partial_{l}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha})g^{i^{\prime}l}({y})g^{j}_{1}({y})\big{(}\int_{\mathcal{M}}f_{1}({x})\bar{R}_{\delta}({x},{y})d\mu_{x}\big{)}d\tau_{y}+r_{55}\\\
=&\int_{\partial\mathcal{M}}\partial_{l}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha})g^{i^{\prime}l}({y})g^{j}_{1}({y})\bar{f}_{1}({y})d\tau_{y}+r_{55}\\\
\leq&C\left\lVert
g_{1}\right\rVert_{L^{2}(\partial\mathcal{M})}\left\lVert\bar{f}_{1}\right\rVert_{L^{2}(\partial\mathcal{M})}+r_{55}\\\
\leq&C\left\lVert
g_{1}\right\rVert_{H^{1}(\mathcal{M})}\left\lVert\bar{f}_{1}\right\rVert_{H^{1}(\mathcal{M})}+r_{55}\end{split}$
(4.29)

Here the lower order terms

$\begin{split}r_{51}=\int_{\partial\mathcal{M}}\int_{\mathcal{M}}f_{1}({x})g^{j}_{1}({y})\xi^{i^{\prime}}\xi^{j^{\prime}}\xi^{k^{\prime}}(\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}s^{2}\tau\partial_{j^{\prime}}\partial_{i^{\prime}}\partial_{k^{\prime}}\phi^{j}(\bm{\alpha}+\tau
st\bm{\xi})dtd\tau ds)\\\ {R}_{\delta}({x},{y})d\mu_{x}d\tau_{y},\end{split}$
(4.30)

$\begin{split}r_{54}=2\delta^{2}\int_{\Omega}\partial_{l}\big{(}f_{1}({x})\partial_{j^{\prime}}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha})g^{i^{\prime}l}({x})\
\sqrt{det\ G(\bm{\beta})}\big{)}\xi^{j^{\prime}}\bar{R}_{\delta}({x},{y})\
d\bm{\beta},\end{split}$ (4.31)

is obviously $\mathcal{O}(\delta^{3})$, for the other error terms, recall the
expression

$\begin{split}r_{53}=-2\delta^{2}\int_{\Omega}(f_{1}({x})\xi^{j^{\prime}}\partial_{j^{\prime}}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha}))r_{52}^{l}({x},{y})\
\sqrt{det\ G(\bm{\beta})}\ d\bm{\beta},\end{split}$ (4.32)
$\begin{split}r_{58}=-2\delta^{2}\int_{\partial\Omega}f_{1}({x})\partial_{j^{\prime}}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha})g^{i^{\prime}l}({x})n_{\Omega}^{l}(\bm{\beta})g^{j^{\prime}l^{\prime}}({x})r_{52}^{l^{\prime}}({x},{y})\
\sqrt{det\ G(\bm{\beta})}\ dS_{\bm{\beta}},\end{split}$ (4.33)

which indicates that they are $\mathcal{O}(\delta^{3})$ terms due to the
estimate (4.25) on $r_{52}$. For the term $r_{55}$, we have

$\begin{split}r_{55}=\int_{\partial\mathcal{M}}g^{j}_{1}({y})\int_{\mathcal{M}}f_{1}({x})\partial_{l}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha})(g^{i^{\prime}l}({x})-g^{i^{\prime}l}({y}))\bar{R}_{\delta}({x},{y})d\mu_{x}d\tau_{y},\end{split}$
(4.34)

which is of lower order by the continuity of matrix $g^{i^{\prime}l}$.

Consequently, we combine all the above estimates to conclude

$\begin{split}&\int_{\partial\mathcal{M}}\int_{\mathcal{M}}f_{1}({x})g^{j}_{1}({y})\frac{1}{2}\xi^{i^{\prime}}\xi^{j^{\prime}}\partial_{j^{\prime}}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha}){R}_{\delta}({x},{y})d\mu_{x}d\tau_{y}\\\
=&\frac{1}{2}\int_{\partial\mathcal{M}}g^{j}_{1}({y})\int_{\mathcal{M}}f_{1}({x})\xi^{i^{\prime}}\xi^{j^{\prime}}\partial_{j^{\prime}}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha}){R}_{\delta}({x},{y})d\mu_{x}d\tau_{y}\\\
=&-\delta^{2}\int_{\partial\mathcal{M}}g^{j}_{1}({y})(\int_{\mathcal{M}}f_{1}({x})\partial_{l}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha})g^{i^{\prime}l}({x})\bar{R}_{\delta}({x},{y})d\mu_{x}+r_{53}+r_{54}+r_{56}+r_{57}+r_{58})d\tau_{y}\\\
\leq&C\delta^{2}\left\lVert
g_{1}\right\rVert_{H^{1}(\mathcal{M})}(\left\lVert\bar{f}_{1}\right\rVert_{H^{1}(\mathcal{M})}+\int_{\partial\mathcal{M}}g^{j}_{1}({y})(r_{53}+r_{54}+r_{56}+r_{57}+r_{58})d\tau_{y}+r_{55}\\\
\leq&C\delta^{2}(\left\lVert
g_{1}\right\rVert_{H^{1}(\mathcal{M})}(\left\lVert\bar{f}_{1}\right\rVert_{H^{1}(\mathcal{M})}+\left\lVert{f}_{1}\right\rVert_{H^{1}(\mathcal{M})}),\end{split}$
(4.35)

and finally

$\begin{split}\int_{\partial\mathcal{M}}\int_{\mathcal{M}}&f_{1}({x})\textbf{g}_{1}({y})\cdot({x}-{y}-\bm{\eta}({x},{y}))R_{\delta}({x},{y})d\mu_{x}d\tau_{y}\\\
=\int_{\partial\mathcal{M}}&\int_{\mathcal{M}}f_{1}({x})g^{j}_{1}({y})\frac{1}{2}\xi^{i^{\prime}}\xi^{j^{\prime}}\partial_{j^{\prime}}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha}){R}_{\delta}({x},{y})d\mu_{x}d\tau_{y}+r_{51}\\\
\leq C\delta^{2}&(\left\lVert
g_{1}\right\rVert_{H^{1}(\mathcal{M})}(\left\lVert\bar{f}_{1}\right\rVert_{H^{1}(\mathcal{M})}+\left\lVert{f}_{1}\right\rVert_{H^{1}(\mathcal{M})}).\end{split}$
(4.36)

###  4.3 The Control for $r_{1}$

This subsection is about the estimate on $r_{1}$ in the proof of theorem 3.1.
We denote

$d({x},{y})=u({x})-u({y})-({x}-{y})\cdot\nabla
u({y})-\frac{1}{2}\eta^{i}\eta^{j}(\nabla^{i}\nabla^{j}u({y})),$ (4.37)

then

$r_{1}({x})=\frac{1}{\delta^{2}}\int_{\mathcal{M}}d({x},{y})R_{\delta}({x},{y})dy.$
(4.38)

Using Newton-Leibniz formula, we can discover

$\begin{split}d({x},{y})=&u({x})-u({y})-({x}-{y})\cdot\nabla
u({y})-\frac{1}{2}\eta^{i}\eta^{j}(\nabla^{i}\nabla^{j}u({y}))\\\
=&\xi^{i}\xi^{i^{\prime}}\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}s_{1}\frac{d}{ds_{3}}\Big{(}\partial_{i}{\phi}^{j}(\alpha+s_{3}s_{1}\xi)\partial_{i^{\prime}}\phi^{j^{\prime}}(\alpha+s_{3}s_{2}s_{1}\xi)\\\
&\nabla^{j^{\prime}}\nabla^{j}u(\bm{\phi}(\alpha+s_{3}s_{2}s_{1}\xi))\Big{)}ds_{3}ds_{2}ds_{1}\\\
=&\xi^{i}\xi^{i^{\prime}}\xi^{i^{\prime\prime}}\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}s_{1}^{2}s_{2}\partial_{i}\phi^{j}(\alpha+s_{3}s_{1}\xi)\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\alpha+s_{3}s_{2}s_{1}\xi)\\\
&\nabla^{j^{\prime}}\nabla^{j}u(\bm{\phi}(\alpha+s_{3}s_{2}s_{1}\xi))ds_{3}ds_{2}ds_{1}\\\
&+\xi^{i}\xi^{i^{\prime}}\xi^{i^{\prime\prime}}\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}s_{1}^{2}\partial_{i^{\prime\prime}}\partial_{i}\phi^{j}(\alpha+s_{3}s_{1}\xi)\partial_{i^{\prime}}\phi^{j^{\prime}}(\alpha+s_{3}s_{2}s_{1}\xi)\\\
&\nabla^{j^{\prime}}\nabla^{j}u(\bm{\phi}(\alpha+s_{3}s_{2}s_{1}\xi))ds_{3}ds_{2}ds_{1}\\\
&+\xi^{i}\xi^{i^{\prime}}\xi^{i^{\prime\prime}}\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}s_{1}^{2}s_{2}\partial_{i}\phi^{j}(\alpha+s_{3}s_{1}\xi)\partial_{i^{\prime}}\phi^{j^{\prime}}(\alpha+s_{3}s_{2}s_{1}\xi)\partial_{i^{\prime\prime}}\\\
&\phi^{j^{\prime\prime}}(\alpha+s_{3}s_{2}s_{1}\xi)\nabla^{j^{\prime\prime}}\nabla^{j^{\prime}}\nabla^{j}u(\bm{\phi}(\alpha+s_{3}s_{2}s_{1}\xi))ds_{3}ds_{2}ds_{1}\\\
=&d_{1}({x},{y})+d_{2}({x},{y})+d_{3}({x},{y})+d_{11}+d_{12}+d_{13},\end{split}$
(4.39)

where $\bm{\phi}=\bm{\phi}_{J({x})}$ for simplicity as we mentioned, and

$d_{1}({x},{y})=\frac{1}{6}\xi^{i}\xi^{i^{\prime}}\xi^{i^{\prime\prime}}\partial_{i}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha}),$
(4.40)
$d_{2}({x},{y})=\frac{1}{3}\xi^{i}\xi^{i^{\prime}}\xi^{i^{\prime\prime}}\partial_{i^{\prime\prime}}\partial_{i}\phi^{j}(\bm{\alpha})\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha}),$
(4.41)
$d_{3}({x},{y})=\frac{1}{6}\eta^{i}\eta^{i^{\prime}}\eta^{i^{\prime\prime}}(\nabla^{i}\nabla^{i^{\prime}}\nabla^{i^{\prime\prime}}u({y})),$
(4.42)

with the lower order terms

$\begin{split}d_{11}=&\xi^{i}\xi^{i^{\prime}}\xi^{i^{\prime\prime}}\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}s_{1}^{2}s_{2}\frac{d}{ds_{4}}(\partial_{i}\phi^{j}(\alpha+s_{4}s_{3}s_{1}\xi)\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\alpha+s_{4}s_{3}s_{2}s_{1}\xi)\\\
&\nabla^{j^{\prime}}\nabla^{j}u(\bm{\phi}(\alpha+s_{4}s_{3}s_{2}s_{1}\xi))ds_{4}ds_{3}ds_{2}ds_{1},\end{split}$
(4.43)

$\begin{split}d_{12}=&\xi^{i}\xi^{i^{\prime}}\xi^{i^{\prime\prime}}\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}s_{1}^{2}\frac{d}{ds_{4}}\partial_{i^{\prime\prime}}\partial_{i}\phi^{j}(\alpha+s_{4}s_{3}s_{1}\xi)\partial_{i^{\prime}}\phi^{j^{\prime}}(\alpha+s_{4}s_{3}s_{2}s_{1}\xi)\\\
&\nabla^{j^{\prime}}\nabla^{j}u(\bm{\phi}(\alpha+s_{4}s_{3}s_{2}s_{1}\xi))ds_{4}ds_{3}ds_{2}ds_{1},\end{split}$
(4.44)

$\begin{split}d_{13}=&\xi^{i}\xi^{i^{\prime}}\xi^{i^{\prime\prime}}\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}s_{1}^{2}s_{2}\frac{d}{ds_{4}}\partial_{i}\phi^{j}(\alpha+s_{4}s_{3}s_{1}\xi)\partial_{i^{\prime}}\phi^{j^{\prime}}(\alpha+s_{4}s_{3}s_{2}s_{1}\xi)\\\
&\partial_{i^{\prime\prime}}\phi^{j^{\prime\prime}}(\alpha+s_{4}s_{3}s_{2}s_{1}\xi)\nabla^{j^{\prime\prime}}\nabla^{j^{\prime}}\nabla^{j}u(\bm{\phi}(\alpha+s_{4}s_{3}s_{2}s_{1}\xi))ds_{4}ds_{3}ds_{2}ds_{1},\end{split}$
(4.45)

Same as the control of $d({x},{y})$ in the page 11-14 of [37], we can
calculate

$\begin{split}&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\int_{\mathcal{M}}(\int_{\mathcal{M}}d_{1i}({x},{y})R_{\delta}({x},{y})d\mu_{y})^{2}d\mu_{x}\leq
C\int_{\mathcal{M}}(\int_{\mathcal{M}}d^{2}_{1i}({x},{y})R_{\delta}({x},{y})d\mu_{{y}})(\int_{\mathcal{M}}R_{\delta}({x},{y})d\mu_{{y}})d\mu_{x}\\\
&\leq
C\int_{\mathcal{M}}\int_{\mathcal{M}}d^{2}_{1i}({x},{y})R_{\delta}({x},{y})d\mu_{{y}}d\mu_{x}=C\sum\limits_{j=1}^{N}\int_{\mathcal{O}_{j}}\int_{\mathcal{M}}d^{2}_{1i}({x},{y})R_{\delta}({x},{y})d\mu_{{y}}d\mu_{x}\\\
&\leq
C\sum\limits_{j=1}^{N}\int_{\mathcal{O}_{j}}\int_{B_{{q}_{j}}^{4\delta}}d^{2}_{1i}({x},{y})R_{\delta}({x},{y})d\mu_{{y}}d\mu_{x}\leq
C\delta^{4}\sum\limits_{j=1}^{N}\int_{B_{{q}_{j}}^{4\delta}}|D^{3,4}u({y})|^{2}d\mu_{{y}}\\\
&\leq C\delta^{4}\left\lVert u\right\rVert_{H^{4}(\mathcal{M})},\end{split}$
(4.46)

where $i=1,2,3$ and

$D^{3,4}u({y})=\sum\limits_{j,j^{\prime},j^{\prime\prime},j^{\prime\prime\prime}=1}^{d}|\nabla^{j^{\prime\prime\prime}}\nabla^{j^{\prime\prime}}\nabla^{j^{\prime}}\nabla^{j}u({x})|^{2}+\sum\limits_{j,j^{\prime},j^{\prime\prime}=1}^{d}|\nabla^{j^{\prime\prime}}\nabla^{j^{\prime}}\nabla^{j}u({x})|^{2}.$

Also

$\int_{\mathcal{M}}(\nabla_{\mathcal{M}}^{{x}}\int_{\mathcal{M}}d_{1i}({x},{y})R_{\delta}({x},{y})d\mu_{y})^{2}d\mu_{x}\leq
C\delta\left\lVert u\right\rVert_{H^{4}(\mathcal{M})},$ (4.47)

where $i=1,2,3.$ Hence we have completed the control of $d_{1i},\ i=1,2,3$.
The terms remaining is $d_{1},d_{2},d_{3}$. By a simple observation we see
$d_{2}=2d_{1}$. We now start to analyze $d_{1}$. From (4.38), the term we need
to control is

$\begin{split}\int_{\mathcal{M}}d_{1}({x},{y})R_{\delta}({x},{y})d\mu_{y}=\int_{\mathcal{M}}\xi^{i}\xi^{i^{\prime}}\xi^{i^{\prime\prime}}\partial_{i}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})R_{\delta}({x},{y})d\mu_{y}.\end{split}$
(4.48)

We aim to control it by integration by parts. The calculation of the following
term is needed:

$\begin{split}\partial_{l}^{{y}}\bar{R}_{\delta}({x},{y})=&\partial_{l}\bm{\phi}(\bm{\alpha})\cdot\nabla_{y}\bar{R}_{\delta}({x},{y})=\frac{1}{2\delta^{2}}\partial_{l}\bm{\phi}(\bm{\alpha})\cdot\bm{\eta}({x},{y}){R}_{\delta}({x},{y})+d^{l}_{121}\\\
=&\frac{1}{2\delta^{2}}\partial_{l}{\phi}^{k^{\prime}}(\bm{\alpha}){\xi}^{i^{\prime}}({x},{y})\partial_{i^{\prime}}{\phi}^{k^{\prime}}(\bm{\alpha}){R}_{\delta}({x},{y})+d^{l}_{121}\\\
=&-\frac{1}{2\delta^{2}}\partial_{l}{\phi}^{k^{\prime}}(\bm{\alpha})\partial_{i^{\prime}}{\phi}^{k^{\prime}}(\bm{\alpha}){\xi}^{i^{\prime}}{R}_{\delta}({x},{y})+d^{l}_{121}\\\
=&-\frac{1}{2\delta^{2}}g_{i^{\prime}l}(\bm{\alpha}){\xi}^{i^{\prime}}{R}_{\delta}({x},{y})+d^{l}_{121},\end{split}$
(4.49)

where $d^{l}_{121}$ is a lower order term with the form

$\begin{split}d^{l}_{121}=&\
\partial_{l}\bm{\phi}(\bm{\alpha})\cdot(\frac{1}{2\delta^{2}}\bm{\eta}({x},{y}){R}_{\delta}({x},{y})-\nabla_{y}\bar{R}_{\delta}({x},{y}))\\\
=&-\frac{1}{2\delta^{2}}\partial_{l}{\phi}^{i}(\bm{\alpha})\partial_{m^{\prime}}\phi^{i}g^{m^{\prime}n^{\prime}}\partial_{n^{\prime}}\phi^{j}(x^{j}-y^{j}-\xi^{i^{\prime}}\partial_{i^{\prime}}\phi^{j}){R}_{\delta}({x},{y})\\\
=&-\frac{1}{2\delta^{2}}\xi^{i^{\prime}}\xi^{j^{\prime}}\partial_{l}{\phi}^{i}(\bm{\alpha})\partial_{m^{\prime}}\phi^{i}g^{m^{\prime}n^{\prime}}\partial_{n^{\prime}}\phi^{j}(\int_{0}^{1}\int_{0}^{1}s\partial_{j^{\prime}}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha}+\tau
s\bm{\xi})d\tau ds){R}_{\delta}({x},{y}).\end{split}$ (4.50)

By a similar argument, we have

$\partial_{l}^{{y}}\overset{=}{R}_{\delta}({x},{y})=-\frac{1}{2\delta^{2}}g_{i^{\prime}l}(\bm{\alpha}){\xi}^{i^{\prime}}\bar{R}_{\delta}({x},{y})+d^{l}_{122},$
(4.51)

where

$d^{l}_{122}=-\frac{1}{2\delta^{2}}\xi^{i^{\prime}}\xi^{j^{\prime}}\partial_{l}{\phi}^{i}(\bm{\alpha})\partial_{m^{\prime}}\phi^{i}g^{m^{\prime}n^{\prime}}\partial_{n^{\prime}}\phi^{j}(\int_{0}^{1}\int_{0}^{1}s\partial_{j^{\prime}}\partial_{i^{\prime}}\phi^{j}(\bm{\alpha}+\tau
s\bm{\xi})d\tau ds)\bar{R}_{\delta}({x},{y}).$ (4.52)

Back to (4.48), we apply integration by parts on the parametric plane
$\Omega_{J({x})}$ of $\mathcal{M}$:

$\begin{split}&\int_{\mathcal{M}}\xi^{i}\xi^{i^{\prime}}\xi^{i^{\prime\prime}}\partial_{i}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})R_{\delta}({x},{y})d\mu_{y}\\\
=&-2\delta^{2}\int_{\mathcal{M}}\xi^{i}\xi^{i^{\prime}}\partial_{i}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})(\partial_{l}^{\bm{\alpha}}\bar{R}_{\delta}({x},{y})-r_{52})d\mu_{y}\\\
=&-2\delta^{2}\int_{\Omega_{J({x})}}\xi^{i}\xi^{i^{\prime}}\partial_{i}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})\partial_{l}^{\bm{\alpha}}\bar{R}_{\delta}({x},{y})\sqrt{Det\
G(\bm{\alpha})}d\bm{\alpha}\\\
&+2\delta^{2}\int_{\mathcal{M}}\xi^{i}\xi^{i^{\prime}}\partial_{i}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})d^{l}_{121}d\mu_{y}\\\
=&-2\delta^{2}\int_{\Omega_{J({x})}}\xi^{i}\xi^{i^{\prime}}\partial_{l}\big{(}\partial_{i}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})\sqrt{det\
G(\bm{\alpha})}\big{)}\bar{R}_{\delta}({x},{y})d\bm{\alpha}\\\
&+2\delta^{2}\int_{\Omega_{J({x})}}\xi^{i^{\prime}}\partial_{l}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})\bar{R}_{\delta}({x},{y})\sqrt{det\
G(\bm{\alpha})}d\bm{\alpha}\\\
&+2\delta^{2}\int_{\Omega_{J({x})}}\xi^{i}\partial_{i}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{l}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})\bar{R}_{\delta}({x},{y})\sqrt{det\
G(\bm{\alpha})}d\bm{\alpha}\\\
&\\!\\!\\!\\!\\!\\!\\!\\!-2\delta^{2}\int_{\partial\Omega_{J({x})}}\xi^{i}\xi^{i^{\prime}}\partial_{i}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})\sqrt{det\
G(\bm{\alpha})}n_{\Omega}^{l}(\bm{\alpha})\bar{R}_{\delta}({x},{y})d\bm{\alpha}\\\
&+2\delta^{2}\int_{\mathcal{M}}\xi^{i}\xi^{i^{\prime}}\partial_{i}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})d^{l}_{121}d\mu_{y}\\\
=&d_{21}+d_{22}+d_{23}+d_{24}+d_{25}=d_{21}+2d_{22}+d_{24}+d_{25},\end{split}$
(4.53)

Still, ${n}_{\Omega}(\bm{\alpha})$ is the unit outward normal vector of
$\Omega_{J({x})}$ at $\bm{\alpha}\in\partial\Omega_{J({y})}$, $\sqrt{det\
G(\bm{\alpha})}$ is the Jacobian matrix defined in (1.4) such that
$d\mu_{{y}}=\sqrt{det\ G(\bm{\alpha})}d\bm{\alpha}$. We are then ready to
control the error terms $d_{2i},i=1,2,3,4,5$. For $d_{21}$, we have

$\begin{split}&\left\lVert d_{21}\right\lVert^{2}_{L^{2}(\mathcal{M})}\\\
=&4\delta^{4}\int_{\mathcal{M}}\Big{(}\int_{\Omega_{J({x})}}\xi^{i}\xi^{i^{\prime}}\partial_{l}\big{(}\partial_{i}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})\sqrt{det\
G(\bm{\alpha})}\big{)}\\\
&\bar{R}_{\delta}({x},{y})d\bm{\alpha}\Big{)}^{2}d\mu_{{x}}\\\
=&4\delta^{4}\sum\limits_{p=1}^{N}\int_{\mathcal{O}_{p}}\Big{(}\int_{\Omega_{p}}\xi^{i}\xi^{i^{\prime}}\partial_{l}\big{(}\partial_{i}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})\sqrt{det\
G(\bm{\alpha})}\big{)}\\\
&\bar{R}_{\delta}({x},{y})d\bm{\alpha}\Big{)}^{2}d\mu_{{x}}\\\
\leq&4\delta^{8}\sum\limits_{p=1}^{N}\int_{\mathcal{O}_{p}}\Big{(}\int_{\Omega_{p}}\max\limits_{i,i^{\prime}}\big{|}\partial_{l}\big{(}\partial_{i}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})\sqrt{det\
G(\bm{\alpha})}\big{)}\big{|}\\\
&\bar{R}_{\delta}({x},{y})d\bm{\alpha}\Big{)}^{2}d\mu_{{x}}\\\
\leq&4\delta^{8}\sum\limits_{p=1}^{N}\int_{\mathcal{O}_{p}}\Big{(}\int_{\Omega_{p}}\max\limits_{i,i^{\prime}}\big{|}\partial_{l}\big{(}\partial_{i}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})\sqrt{det\
G(\bm{\alpha})}\big{)}\big{|}^{2}\\\
&\bar{R}_{\delta}({x},{y})d\bm{\alpha}\Big{)}\Big{(}\int_{\Omega_{J({x})}}\bar{R}_{\delta}({x},{y})d\bm{\alpha}\Big{)}d\mu_{{x}}\\\
\leq&4\delta^{8}\sum\limits_{p=1}^{N}\int_{\mathcal{O}_{p}}\Big{(}\int_{\Omega_{p}}\max\limits_{i,i^{\prime}}\big{|}\partial_{l}\big{(}\partial_{i}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})\sqrt{det\
G(\bm{\alpha})}\big{)}\big{|}^{2}\\\
&\bar{R}_{\delta}({x},{y})d\bm{\alpha}\Big{)}d\mu_{{x}}\\\
\leq&C\delta^{8}\sum\limits_{p=1}^{N}\int_{\Omega_{p}}\big{(}\int_{\mathcal{O}_{p}}\bar{R}_{\delta}({x},{y})d\mu_{{x}}\big{)}\max\limits_{i,i^{\prime}}\big{|}\partial_{l}\big{(}\partial_{i}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})\\\
&\sqrt{det\ G(\bm{\alpha})}\big{)}\big{|}^{2}d\bm{\alpha}\\\
\leq&C\delta^{8}\sum\limits_{p=1}^{N}\int_{\Omega_{p}}\max\limits_{i,i^{\prime}}\big{|}\partial_{l}\big{(}\partial_{i}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})\sqrt{det\
G(\bm{\alpha})}\big{)}\big{|}^{2}d\bm{\alpha}\\\
\leq&C\delta^{8}\sum\limits_{p=1}^{N}\left\lVert
u\right\rVert_{H^{2}(\Omega_{p})}^{2}\leq C\delta^{8}\left\lVert
u\right\rVert_{H^{3}(\mathcal{M})}^{2},\end{split}$ (4.54)

by the similar argument, we discover

$\begin{split}&\left\lVert\nabla
d_{21}\right\lVert^{2}_{L^{2}(\mathcal{M})}\\\
\leq&C\delta^{4}\int_{\mathcal{M}}\Big{(}\int_{\Omega_{J({x})}}(\nabla_{\mathcal{M}}^{{x}}\xi^{i}\xi^{i^{\prime}})\partial_{l}\big{(}\partial_{i}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})\sqrt{det\
G(\bm{\alpha})}\big{)}\\\
&\bar{R}_{\delta}({x},{y})d\bm{\alpha}\Big{)}^{2}d\mu_{{x}}+\int_{\mathcal{M}}\int_{\Omega_{J({x})}}\Big{(}\xi^{i}\xi^{i^{\prime}}\partial_{l}\big{(}\partial_{i}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})\\\
&\sqrt{det\
G(\bm{\alpha})}\big{)}(\nabla_{\mathcal{M}}^{{x}}\bar{R}_{\delta}({x},{y}))d\bm{\alpha}\Big{)}^{2}d\mu_{{x}}\\\
\leq&C\delta^{4}\int_{\mathcal{M}}\Big{(}\int_{\Omega_{J({x})}}\delta\max\limits_{i,i^{\prime}}\big{|}\partial_{l}\big{(}\partial_{i}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})\sqrt{det\
G(\bm{\alpha})}\big{)}\big{|}\\\
&\\!\\!\\!\\!\bar{R}_{\delta}({x},{y})d\bm{\alpha}\Big{)}^{2}d\mu_{{x}}+C\delta^{4}\int_{\mathcal{M}}\Big{(}\int_{\Omega_{J({x})}}\delta^{2}\max\limits_{i,i^{\prime}}\big{|}\partial_{l}\big{(}\partial_{i}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})\\\
&\sqrt{det\ G(\bm{\alpha})}\big{)}\big{|}\
\frac{1}{2\delta}{R}_{\delta}({x},{y})\ d\bm{\alpha}\Big{)}^{2}d\mu_{{x}}\\\
\leq&C\delta^{6}\sum\limits_{p=1}^{N}\int_{\mathcal{O}_{p}}\Big{(}\int_{\Omega_{p}}\max\limits_{i,i^{\prime}}\big{|}\partial_{l}\big{(}\partial_{i}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})\sqrt{det\
G(\bm{\alpha})}\big{)}\big{|}\\\
&(\bar{R}_{\delta}({x},{y})+{R}_{\delta}({x},{y}))d\bm{\alpha}\Big{)}^{2}d\mu_{{x}}\\\
\leq&C\delta^{6}\left\lVert
u\right\rVert_{H^{3}(\mathcal{M})}^{2}.\end{split}$ (4.55)

For $d_{22}$ we apply integration by parts again:

$\begin{split}&d_{22}({x})\\\
=&2\delta^{2}\int_{\Omega_{J({x})}}\xi^{i^{\prime}}\partial_{l}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})\bar{R}_{\delta}({x},{y})\sqrt{det\
G(\bm{\alpha})}d\bm{\alpha}\\\
=&-4\delta^{4}\int_{\partial\Omega_{J({x})}}\partial_{l}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})g^{i^{\prime}l^{\prime}}(\bm{\alpha})n_{\Omega}^{l^{\prime}}(\bm{\alpha})\overset{=}{R}_{\delta}({x},{y})\sqrt{det\
G(\bm{\alpha})}d\bm{\alpha}\\\
&+4\delta^{4}\int_{\Omega_{J({x})}}\partial_{l^{\prime}}\Big{(}\partial_{l}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})g^{i^{\prime}l^{\prime}}(\bm{\alpha})\sqrt{det\
G(\bm{\alpha})}\Big{)}\overset{=}{R}_{\delta}({x},{y})d\bm{\alpha}\\\
&+4\delta^{4}\int_{\Omega_{J({x})}}\partial_{l}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})g^{i^{\prime}l^{\prime}}d_{122}^{l^{\prime}}(\bm{\alpha})\sqrt{det\
G(\bm{\alpha})}d\bm{\alpha}\\\
=&-4\delta^{4}\int_{\partial\Omega_{J({x})}}\partial_{l}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})g^{i^{\prime}l^{\prime}}(\bm{\alpha})n_{\Omega}^{l^{\prime}}(\bm{\alpha})\overset{=}{R}_{\delta}({x},{y})\sqrt{det\
G(\bm{\alpha})}d\bm{\alpha}\\\
&+4\delta^{4}\int_{\mathcal{M}}\frac{1}{\sqrt{det\
G(\bm{\alpha})}}\partial_{l^{\prime}}\Big{(}\partial_{l}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})g^{i^{\prime}l^{\prime}}(\bm{\alpha})\sqrt{det\
G(\bm{\alpha})}\Big{)}\\\
&\overset{=}{R}_{\delta}({x},{y})d\mu_{{y}}+4\delta^{4}\int_{\mathcal{M}}\partial_{l}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})g^{i^{\prime}l^{\prime}}(\bm{\alpha})d_{122}^{l^{\prime}}(\bm{\alpha})d\mu_{{y}}\\\
=&\ d_{221}+d_{222}+d_{223}.\end{split}$ (4.56)

Here $d_{221}$ is a term supported in the layer of $\partial\mathcal{M}$ with
width $2\delta$ due to the support of $\bar{R}$, and

$\begin{split}&\left\lVert d_{221}\right\rVert^{2}_{L^{2}(\mathcal{M})}\\\
=&16\delta^{8}\int_{\mathcal{M}}\Big{(}\int_{\partial\Omega_{J({x})}}\partial_{l}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})g^{i^{\prime}l^{\prime}}(\bm{\alpha})n_{\Omega}^{l^{\prime}}(\bm{\alpha})\overset{=}{R}_{\delta}({x},{y})\\\
&\sqrt{det\ G(\bm{\alpha})}d\bm{\alpha}\Big{)}^{2}d\mu_{{x}}\\\
\leq&C\delta^{8}\int_{\mathcal{M}}\int_{\partial\Omega_{J({x})}}(\partial_{l}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})g^{i^{\prime}l^{\prime}}(\bm{\alpha})n_{\Omega}^{l^{\prime}}(\bm{\alpha}))^{2}\
\overset{=}{R}_{\delta}({x},{y})\\\ &det\ G(\bm{\alpha})d\bm{\alpha}\
(\int_{\partial\Omega_{J({x})}}\overset{=}{R}_{\delta}({x},{y})d\alpha)d\mu_{{x}}\\\
\leq&C\delta^{7}\sum\limits_{p=1}^{N}\int_{\mathcal{O}_{p}}\int_{\partial\Omega_{p}}(\partial_{l}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})g^{i^{\prime}l^{\prime}}(\bm{\alpha})n_{\Omega}^{l^{\prime}}(\bm{\alpha}))^{2}\overset{=}{R}_{\delta}({x},{y})\\\
&det\ G(\bm{\alpha})d\bm{\alpha}d\mu_{{x}},\\\
\leq&C\delta^{7}\sum\limits_{p=1}^{N}\int_{\partial\Omega_{p}}(\int_{\mathcal{O}_{p}}\overset{=}{R}_{\delta}({x},{y})d\mu_{{x}})(\partial_{l}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})g^{i^{\prime}l^{\prime}}(\bm{\alpha}))^{2}\\\
&(n_{\Omega}^{l^{\prime}}(\bm{\alpha}))^{2}det\ G(\bm{\alpha})d\bm{\alpha}\\\
\leq&C\delta^{7}\sum\limits_{p=1}^{N}\int_{\partial\Omega_{p}}(\partial_{l}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})g^{i^{\prime}l^{\prime}}(\bm{\alpha})n_{\Omega}^{l^{\prime}}(\bm{\alpha}))^{2}det\
G(\bm{\alpha})d\bm{\alpha}\\\ \leq&C\delta^{7}\sum\limits_{p=1}^{N}\left\lVert
u\right\rVert^{2}_{H^{3}(\Omega_{p})}\leq C\delta^{7}\left\lVert
u\right\rVert^{2}_{H^{3}(\mathcal{M})},\end{split}$ (4.57)

Similarly, we have

$\begin{split}&\left\lVert\nabla
d_{221}\right\rVert^{2}_{L^{2}(\mathcal{M})}\\\
=&16\delta^{8}\sum\limits_{p=1}^{N}\int_{\mathcal{O}_{p}}\int_{\partial\Omega_{p}}\partial_{l}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})g^{i^{\prime}l^{\prime}}(\bm{\alpha})n_{\Omega}^{l^{\prime}}(\bm{\alpha})(\nabla_{\mathcal{M}}^{{x}}\overset{=}{R}_{\delta}({x},{y}))\\\
&\sqrt{det\ G(\bm{\alpha})}d\bm{\alpha}\Big{)}^{2}d\mu_{{x}}\\\
\leq&C\delta^{5}\left\lVert
u\right\rVert^{2}_{H^{3}(\mathcal{M})}.\end{split}$ (4.58)

In addition, let us verify the property (3.6) of $d_{221}$. In fact, for any
$f_{1}\in H^{1}(\mathcal{M})$, we have

$\begin{split}&\int_{\mathcal{M}}d_{221}({x})\ f_{1}({x})d\mu_{x}\\\
=&-4\delta^{4}\int_{\mathcal{M}}f_{1}({x})\int_{\partial\Omega_{J({x})}}\partial_{l}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})g^{i^{\prime}l^{\prime}}(\bm{\alpha})n_{\Omega}^{l^{\prime}}(\bm{\alpha})\overset{=}{R}_{\delta}({x},{y})\\\
&\sqrt{det\ G(\bm{\alpha})}d\bm{\alpha}d\mu_{{x}}\\\
=&-4\delta^{4}\sum\limits_{p=1}^{N}\int_{\mathcal{O}_{p}}\int_{\partial\Omega_{p}}\partial_{l}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})g^{i^{\prime}l^{\prime}}(\bm{\alpha})n_{\Omega}^{l^{\prime}}(\bm{\alpha})\sqrt{det\
G(\bm{\alpha})}\\\
&(f_{1}({x})\overset{=}{R}_{\delta}({x},{y}))d\bm{\alpha}d\mu_{{x}}\\\
=&-4\delta^{4}\sum\limits_{p=1}^{N}\int_{\partial\Omega_{p}}(\int_{\mathcal{O}_{p}}f_{1}({x})\overset{=}{R}_{\delta}({x},{y})d\mu_{{x}})\partial_{l}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})g^{i^{\prime}l^{\prime}}(\bm{\alpha})\\\
&n_{\Omega}^{l^{\prime}}(\bm{\alpha})\sqrt{det\ G(\bm{\alpha})}d\bm{\alpha}\\\
=&-4\delta^{4}\sum\limits_{p=1}^{N}\int_{\partial\Omega_{p}}\overset{=}{f}_{1}({y})\partial_{l}\phi^{j}(\bm{\alpha})\partial_{i^{\prime\prime}}\partial_{i^{\prime}}\phi^{j^{\prime}}(\bm{\alpha})\nabla^{j^{\prime}}\nabla^{j}u(\bm{\alpha})g^{i^{\prime\prime}l}(\bm{\alpha})g^{i^{\prime}l^{\prime}}(\bm{\alpha})n_{\Omega}^{l^{\prime}}(\bm{\alpha})\sqrt{det\
G(\bm{\alpha})}d\bm{\alpha}\\\
\leq&C\delta^{4}\int_{\partial\mathcal{M}}\overset{=}{f}_{1}({y})|\nabla^{2}u({y})|d\mu_{{y}}\leq
C\delta^{4}\left\lVert\overset{=}{f}_{1}\right\rVert_{L^{2}(\partial\mathcal{M})}\left\lVert
u\right\rVert_{H^{2}(\partial\mathcal{M})}\leq
C\delta^{4}\left\lVert\overset{=}{f}_{1}\right\rVert_{H^{1}(\mathcal{M})}\left\lVert
u\right\rVert_{H^{3}(\mathcal{M})}.\end{split}$ (4.59)

For the next terms, $d_{222}$ and $d_{223}$ are two lower order terms defined
on the interior, where the control is by the similar argument of $d_{21}$.

The control for $d_{24}$ is similar as $d_{221}$, where two more integration
by parts than $d_{221}$ are needed, thus heavy calculations are need in the
control of $d_{24}$. However, as a consequence, $d_{24}$ satisfy the same
property of $d_{221}$.

$d_{25}$ is obviously a lower order term due to the calculation of $d_{121}$
in (4.50).

Hence we have finished the control on the term $d_{1}$ and $d_{2}$. For the
term $d_{3}$, we denote

$r_{11}=\frac{1}{\delta^{2}}\int_{\mathcal{M}}d_{3}({x},{y})R_{\delta}({x},{y})d{y}=\frac{1}{6\delta^{2}}\int_{\mathcal{M}}\eta^{i}\eta^{j}\eta^{k}(\nabla^{i}\nabla^{j}\nabla^{k}u({y}))R_{\delta}({x},{y})d\mu_{y}$
(4.60)

Similar as $r_{1}$, we can decompose $r_{11}$ into
$r_{11}=\sum\limits_{i=2}^{7}r_{1i}$, where

$\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!r_{12}=\frac{1}{6\delta^{2}}(\int_{\mathcal{M}}\eta^{i}\eta^{j}\eta^{k}(\nabla^{i}\nabla^{j}\nabla^{k}u({y}))R_{\delta}({x},{y})d\mu_{y}-\int_{\mathcal{M}}3\eta^{i}\eta^{j}(\nabla^{i}\nabla^{j}\nabla^{k}u({y}))\nabla^{k}\bar{R}_{\delta}({x},{y})d\mu_{y}),$
(4.61)

$\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!r_{13}({x})=\frac{1}{2\delta^{2}}(\int_{\mathcal{M}}\eta^{i}\eta^{j}(\nabla^{i}\nabla^{j}\nabla^{k}u({y}))\nabla^{k}\bar{R}_{\delta}({x},{y})d\mu_{y}+\int_{\mathcal{M}}div\
(\eta^{i}\eta^{j}(\nabla^{i}\nabla^{j}\nabla
u({y})))\bar{R}_{\delta}({x},{y})d\mu_{y}),$ (4.62)

$\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!r_{14}({x})=\frac{1}{2\delta^{2}}(2\int_{\mathcal{M}}\eta^{i}(\nabla^{i}\nabla^{j}\nabla^{j}u({y}))\bar{R}_{\delta}({x},{y})d\mu_{y}-\int_{\mathcal{M}}div\
(\eta^{i}\eta^{j}(\nabla^{i}\nabla^{j}\nabla
u({y})))\bar{R}_{\delta}({x},{y})d\mu_{y}),$ (4.63)

$r_{15}({x})=-\frac{1}{\delta^{2}}(\int_{\mathcal{M}}\eta^{i}(\nabla^{i}\nabla^{j}\nabla^{j}u({y}))\bar{R}_{\delta}({x},{y})d\mu_{y}-\int_{\mathcal{M}}(\nabla^{i}\nabla^{j}\nabla^{j}u({y}))\nabla^{i}\overset{=}{R}_{\delta}({x},{y})d\mu_{y}),$
(4.64)

$r_{16}({x})=\frac{1}{\delta^{2}}(\int_{\mathcal{M}}(\nabla^{i}\nabla^{j}\nabla^{j}u({y}))\nabla^{i}\overset{=}{R}_{\delta}({x},{y})d\mu_{y}+\int_{\mathcal{M}}(\nabla^{i}\nabla^{i}\nabla^{j}\nabla^{j}u({y}))\overset{=}{R}_{\delta}({x},{y})d\mu_{y}),$
(4.65)

$r_{17}({x})=-\frac{1}{\delta^{2}}\int_{\mathcal{M}}(\nabla^{i}\nabla^{i}\nabla^{j}\nabla^{j}u({y}))\overset{=}{R}_{\delta}({x},{y})d\mu_{y}.$
(4.66)

The control for $r_{12}$ and $r_{15}$ is the same as $r_{2}$ in [37]. For
$r_{13}$, we can calculate

$\begin{split}r_{13}=\int_{\partial\mathcal{M}}(\eta^{i}\eta^{j}n^{k}(\nabla^{i}\nabla^{j}\nabla^{k}u({y})))\bar{R}_{\delta}({x},{y})d\tau_{y}\end{split}$
(4.67)

Since $\eta^{i}=\xi^{i^{\prime}}\partial_{i^{\prime}}\phi^{i}$, we can refer
to the control of the term (4.23) in the proof of lemma 3.4 to deduce

$\int_{\mathcal{M}}r_{13}({x})f_{1}({x})d\mu_{x}\leq C\delta^{2}(\left\lVert
f_{1}\right\rVert_{H^{1}(\mathcal{M})}+\left\lVert\bar{f}_{1}\right\rVert_{H^{1}(\mathcal{M})})\left\lVert
u\right\rVert_{H^{4}(\mathcal{M})},$ (4.68)

and the same argument hold for $r_{16}$. For $r_{14}$, since
$\partial_{i^{\prime}}{\xi^{l}}=-\delta_{i^{\prime}}^{l}$, we have

$\begin{split}div&\ (\eta^{i}\eta^{j}(\nabla^{i}\nabla^{j}\nabla u({y}))\\\
=&\frac{1}{\sqrt{det\ G}}\partial_{i^{\prime}}(\sqrt{det\
G}g^{i^{\prime}j^{\prime}}(\partial_{j^{\prime}}\phi^{k})\eta^{i}\eta^{j}\nabla^{i}\nabla^{j}\nabla^{k}u({y}))\\\
=&\frac{\xi^{l}}{\sqrt{det\ G}}\partial_{i^{\prime}}(\sqrt{det\
G}g^{i^{\prime}j^{\prime}}(\partial_{j^{\prime}}\phi^{k})\eta^{i}(\partial_{l}\phi^{j})\nabla^{i}\nabla^{j}\nabla^{k}u({y}))\\\
&-g^{i^{\prime}j^{\prime}}(\partial_{j^{\prime}}\phi^{k})\eta^{i}(\partial_{i^{\prime}}\phi^{j})(\nabla^{i}\nabla^{j}\nabla^{k}u({y}))\\\
=&\frac{\xi^{l}\xi^{k^{\prime}}}{\sqrt{det\
G}}\partial_{i^{\prime}}(\sqrt{det\
G}g^{i^{\prime}j^{\prime}}(\partial_{j^{\prime}}\phi^{k})(\partial_{k^{\prime}}\phi^{i})(\partial_{l}\phi^{j})\nabla^{i}\nabla^{j}\nabla^{k}u({y}))\\\
&-2g^{i^{\prime}j^{\prime}}(\partial_{j^{\prime}}\phi^{k})\eta^{i}(\partial_{i^{\prime}}\phi^{j})(\nabla^{i}\nabla^{j}\nabla^{k}u({y}))\\\
=&\frac{\xi^{l}\xi^{k^{\prime}}}{\sqrt{det\
G}}\partial_{i^{\prime}}(\sqrt{det\
G}g^{i^{\prime}j^{\prime}}(\partial_{j^{\prime}}\phi^{k})(\partial_{k^{\prime}}\phi^{i})(\partial_{l}\phi^{j})\nabla^{i}\nabla^{j}\nabla^{k}u({y}))\\\
&-2\eta^{i}(\nabla^{i}\nabla^{j}\nabla^{j}u({y})),\end{split}$ (4.69)

where the last inequality results from the equation $(31)$ of [37]. This
implies

$\begin{split}r_{14}({x})=&\int_{\mathcal{M}}div\
(\eta^{i}\eta^{j}(\nabla^{i}\nabla^{j}\nabla
u({y}))\bar{R}_{\delta}({x},{y})d\mu_{y}+2\int_{\mathcal{M}}\eta^{i}(\nabla^{i}\nabla^{j}\nabla^{j}u({y}))\bar{R}_{\delta}({x},{y})d\mu_{y}\\\
=&\int_{\mathcal{M}}\frac{\xi^{l}\xi^{k^{\prime}}}{\sqrt{det\
G}}\partial_{i^{\prime}}(\sqrt{det\
G}g^{i^{\prime}j^{\prime}}(\partial_{j^{\prime}}\phi^{k})(\partial_{k^{\prime}}\phi^{i})(\partial_{l}\phi^{j})\nabla^{i}\nabla^{j}\nabla^{k}u({y}))\bar{R}_{\delta}({x},{y})d\mu_{y}.\end{split}$
(4.70)

By the same calculation as the term $d_{21}$ we can conclude

$\left\lVert r_{14}\right\rVert_{L^{2}(\mathcal{M})}\leq
C\delta^{2}\left\lVert u\right\rVert_{H^{4}(\mathcal{M})},$ (4.71)
$\left\lVert\nabla r_{14}\right\rVert_{L^{2}(\mathcal{M})}\leq
C\delta\left\lVert u\right\rVert_{H^{4}(\mathcal{M})}.$ (4.72)

And similarly, we have the same bound for $r_{17}$ as $r_{14}$.
Thereafter, we have completed the control on all the terms that composes
$r_{1}$.

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