# Piecewise smoothness for linear elliptic systems with piecewise smooth
coefficients

Youchan Kim  Department of Mathematics, University of Seoul, Seoul 02504,
Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

Li and Vogelius, and Li and Nirenberg obtained piecewise $\displaystyle
C^{1,\gamma}$-regularity for linear elliptic problems with piecewise
$\displaystyle C^{\gamma}$-coefficients which come from composite materials.
In this paper, we obtain piecewise $\displaystyle C^{m+1,\gamma}$-regularity
for linear elliptic systems with piecewise $\displaystyle
C^{m,\gamma}$-coefficients which also come from composite materials. This
answers the open problem suggested by Li and Vogelius, and Li and Nirenberg.

###### Key words and phrases:

Linear elliptic systems, Smoothness and regularity, composite materials

###### 2020 Mathematics Subject Classification:

35J47, 35B65, 74A40

##  1\. Introduction and main result

###  1.1. Introduction

We consider linear elliptic systems arising from composite materials such as
fiber reinforced materials. Assume that
$\displaystyle\Omega\subset\mathbb{R}^{2}$ is an open bounded $\displaystyle
C^{m,\gamma}$-domain, and $\displaystyle\\{\Omega^{k}\\}_{k=1}^{k_{0}}$ are
the mutually disjoint $\displaystyle C^{m,\gamma}$-subdomains in
$\displaystyle\Omega$ which are allowed to touch each other. We denote
$\displaystyle\Omega^{0}=\Omega\backslash\cup_{k=1}^{k_{0}}\Omega^{k}$. Let
$\displaystyle u$ be a weak solution of an linear elliptic equation

$D_{\alpha}\left[A^{\alpha\beta}_{ij}D_{\beta}u^{j}\right]=D_{\alpha}F_{\alpha}^{i}\quad\text{in}\quad\Omega,$

where $\displaystyle A^{\alpha\beta}_{ij}\in C^{m,\gamma}(\Omega^{k})$ and
$\displaystyle F_{\alpha}^{i}\in C^{m,\gamma}(\Omega^{k})$ for any
$\displaystyle 1\leq\alpha,\beta\leq n$, $\displaystyle 1\leq i,j\leq N$ and
$\displaystyle k\in[0,k_{0}]$. With our assumption, $\displaystyle
A^{\alpha\beta}_{ij}$ and $\displaystyle F_{\alpha}^{i}$ are allowed to be
discontinuous across the boundary of the subdomains
$\displaystyle\\{\partial\Omega^{k}\\}_{k=1}^{k_{0}}$. In this paper, we prove
that the weak solution $\displaystyle u$ is locally piecewise $\displaystyle
C^{m+1,\gamma}$, meaning that $\displaystyle u$ is $\displaystyle
C^{m+1,\gamma}(\tilde{\Omega}\cap\Omega^{k})$ for any compact subset
$\displaystyle\tilde{\Omega}\subset\subset\Omega$ and $\displaystyle
k\in[0,k_{0}]$.

An application of our result is the following. Suppose that the fibers and
resin of a fiber reinforced plastic(FRP) is assumed to be linearly elastic,
and described by a linear elliptic equation. Then due to the discontinuity of
the coefficients or elasticity, the gradient of the weak solution might be
discontinuous on the boundary of each materials. Nevertheless, our result
shows that the weak solution on each materials(fiber, resin) can be smooth,
even if the fibers are allowed to touch each other. To prove our result, we
generalize the difference quotient method to curves instead of traditional
fixed directions.

One of long standing problem for the regularity theory of composite material
is establishing a new regularity theory when the subdomains
$\displaystyle\\{\Omega^{k}\\}_{k=1}^{k_{0}}$ are allowed to touch each other.
Under this assumption, the flattening argument could not be applied at the
points where two subdomains touch each other, and the difficulty arises. With
this geometric assumption, local Lipschitz regularity can be derived as in [3,
8, 9]. But better regularity than Lipschitz regularity is not feasible,
because discontinuity of the gradient might occur on the boundary of
subdomains. However, from the observations made in the previous literatures,
one can consider so called piecewise regularity results. For example, if the
distance between the boundary of subdomains are bigger than some positive
constant, then one may use the flattening argument to prove local piecewise
$\displaystyle C^{m,\gamma}$-regularity. But this estimate depends on the
distance between the boundary of subdomains and could not be applied for
composite materials. When two subdomains are allowed to touch each other,
which might happen for composite materials, the best result known in the
previous literatures is piecewise $\displaystyle C^{1,\gamma}$-regularity by
Vogelius and Li [9], and Li and Nirenberg [8]. In that papers, they used
observation that in small scales, the boundaries of two touching subdomains
are almost flat near the touching point, and such geometric structures could
be viewed as linear laminates. Indeed, [9] and [8] proved that the solution is
$\displaystyle C^{1,\gamma}(\tilde{\Omega}\cap\Omega^{k})$ for any compact
subset $\displaystyle\tilde{\Omega}\subset\Omega$ and $\displaystyle
k\in[0,k_{0}]$ by using the regularity theory of linear laminates such as
Chipot, Kinderlehrer and Vergara-Caffarelli [4], and in fact their estimates
does not depend on the distance between the boundary of subdomains. However,
the method in [9] and [8] could not be extended to higher derivatives
$\displaystyle C^{m,\gamma}$ with $\displaystyle m\geq 2$, and in [8] they
suggest an open question that piecewise $\displaystyle
C^{m,\gamma}$-regularity even holds for $\displaystyle m\geq 2$. On the other-
hand, regularity results for higher derivatives have been obtained for special
geometric structures with dimension $\displaystyle 2$, when the domain
$\displaystyle\Omega$ is a disk, and two subdomains are also disks. In that
case, [9] proved $\displaystyle C^{m,\gamma}$ for $\displaystyle m\geq 2$ by
using conformal mapping theory, but unfortunately their method could not be
extended to higher dimensions. We also refer to [10] for a similar result by
using Green functions. With the observation in this paragraph, it has been
expected that the conjecture about piecewise $\displaystyle
C^{m,\gamma}$-regularity for $\displaystyle m\geq 2$ seems true, and in this
paper we prove that this conjecture indeed holds.

To prove our main result, we generalize the difference quotient method. As far
as we are concerned, the difference quotient method have been applied for a
fixed direction, but we generalize this method by applying to curves. In fact,
we use so called a ‘flow’ to obtain higher regularity, because the computation
of difference quotient method is complicated for the higher regularity and we
will use an approximation argument for obtaining higher regularity.

If a curve does not across the boundary of subdomains, then the coefficients
remain regular along that curve, even if the coefficients are discontinuous
between different subdomains. So we choose a suitable flow so that if we
restrict the domain of the flow to some subdomain $\displaystyle\Omega^{k}$,
then the range of that flow still remains in the same subdomain
$\displaystyle\Omega^{k}$. To explain our new method, let’s consider our model
case. Let $\displaystyle\Omega=[-2,2]\times[-4,4]\subset\mathbb{R}^{2}$,
$\displaystyle\varphi_{+}(x_{2})=\left(x^{2}\right)^{2}$ and
$\displaystyle\varphi_{-}(x^{2})=-\left(x^{2}\right)^{2}$ where $\displaystyle
x=\left(x^{1},x^{2}\right)$. Then $\displaystyle\varphi_{+}$ and
$\displaystyle\varphi_{-}$ touches each other at the origin
$\displaystyle(0,0)$. We choose a function
$\displaystyle\sigma:[-1,1]\times[-1,1]\times[-1,1]\rightarrow[-4,4]$ such
that

$\sigma(x,t)=\frac{[\varphi_{+}(x^{2}+t)-\varphi_{+}(x^{2})][x_{1}-\varphi_{-}(x^{2})]}{\varphi_{+}(x^{2})-\varphi_{-}(x^{2})}+\frac{[\varphi_{-}(x^{2}+t)-\varphi_{-}(x^{2})][\varphi_{+}(x^{2})-x_{1}]}{\varphi_{+}(x^{2})-\varphi_{-}(x^{2})}.$

Then $\displaystyle t\mapsto x+(\sigma(x,t),t)$ can be viewed as a flow in
$\displaystyle
Q_{1}^{0}:=\\{(x_{1},x^{2})\in[-1,1]\times[-1,1]:\varphi_{-}(x^{2})\leq
x_{1}\leq\varphi_{+}(x^{2})\\}$, and we find that
$\displaystyle\partial_{t}[f(x+(\sigma,t))]_{t=0}$ is the first difference
quotient or the first derivative along the curve $\displaystyle
t\mapsto\sigma(x,t)$.

However, this approach only works for piecewise $\displaystyle
C^{2,\gamma}$-regularity. The reason is that if two subdomains touches each
other, then the first derivative of two boundaries coincide at that touching
point, but the higher derivatives might not coincide at that touching point.
To explain, we calculate $\displaystyle 2$-nd derivative of $\displaystyle
h\in C^{\infty}(\mathbb{R}^{2})$ along the curves
$\displaystyle\\{\left(\varphi_{+}(x^{2}),x^{2}\right):x^{2}\in[-1,1]\\}$ and
$\displaystyle\\{\left(\varphi_{-}(x^{2}),x^{2}\right):x^{2}\in[-1,1]\\}$ at
the origin by using the following calculation:

$\displaystyle\partial_{t}^{2}[h(\varphi(t),t)]$
$\displaystyle=\varphi^{\prime}(x^{2}+t)^{2}D_{11}h(\varphi(t),t)+2\varphi^{\prime}(x^{2}+t)^{2}D_{12}h(\varphi(t),t)$
$\displaystyle\quad+D_{22}h(\varphi(t),t)+\varphi^{\prime\prime}(x^{2}+t)D_{1}h(\varphi(t),t).$

Since
$\displaystyle\varphi^{\prime\prime}_{+}(0)=2\not=\varphi^{\prime\prime}_{-}(0)=-2$
and $\displaystyle\varphi^{\prime}_{+}(0)=\varphi^{\prime}_{-}(0)=0$,
$\displaystyle D_{1}h(0,0)\not=0$ implies
$\displaystyle\partial_{t}^{2}[h(\varphi_{+}(t),t)]\big{|}_{t=0}\not=\partial_{t}^{2}[h(\varphi_{-}(t),t)]\big{|}_{t=0}$.
On the other-hand, it is possible to obtain $\displaystyle C^{2,\gamma}$ by
just using the approach in the previous paragraph because
$\displaystyle\partial_{t}[h(\varphi_{+}(t),t)]\big{|}_{t=0}=\partial_{t}[h(\varphi_{-}(t),t)]\big{|}_{t=0}$.

By our model case, we see that there are jumps of the higher order derivatives
on the boundary of subdomains. In fact, this jumps appears even if
$\displaystyle h$ is smooth which means that the jumps of the higher order
derivatives are mainly due to the geometry of the composite domains. Because
of this reason, we obtained geometric property of composite domains in [7, 2].
In [7], the author and Jang prove that the boundary of the subdomains are
almost parallel if they are sufficiently close, even if the subdomains are
fractal, say Reifenberg flat domains. Also [2] answers an question for the
typical examples of the composite domains. If $\displaystyle
C^{1,\gamma}$-domains satisfy certain inclusion conditions, they form a
composite domains.

To overcome the difficulty for handling the jumps of the higher order
derivatives on the boundary of subdomains, we extend functions defined in a
subdomain. We briefly explain this idea by an example. Let $\displaystyle u$
be a function which might have jumps on the boundary of subdomains. Let
$\displaystyle u^{k}$ be an extension of $\displaystyle u$ from
$\displaystyle\Omega^{k}$ to $\displaystyle\Omega$ such that $\displaystyle
u^{k}$ is regular in $\displaystyle\Omega\setminus\Omega^{k}$. Then we see
that $\displaystyle\sum_{k}u^{k}$ has no jumps on the boundary of the
subdomains. So by considering $\displaystyle\sum_{k}u^{k}$ instead of
$\displaystyle u$, jumps of the function is removed and the function became
easier for obtaining piecewise regularity. The value of $\displaystyle u$ and
$\displaystyle\sum_{k}u^{k}$ might not be the same in $\displaystyle\Omega$,
but $\displaystyle u-\sum_{k}u^{k}$ is piecewise regular because
$\displaystyle u^{k}$ is an extension of $\displaystyle u$ in
$\displaystyle\Omega^{k}$ and defined to be regular in
$\displaystyle\Omega\setminus\Omega^{k}$. If a suitable perturbations could be
done to $\displaystyle\sum_{k}u^{k}$ in each subdomains, then one can obtain
piecewise regularity of $\displaystyle u$. For partial differential equations,
the functions can be differentiated. But the ideas in the previous paragraphs
still work for our problem.

The results and the key idea of this paper was announced at ‘International
Conference on Partial Differential Equations Related to Material Science’, May
6-9, 2021.

###  1.2. Main result

We introduce the following notations in this paper.

  1. (1)

A $\displaystyle(n-1)$-dimensional point is denoted as $\displaystyle
x^{\prime}=(x^{2},\cdots,x^{n})$.

  2. (2)

$\displaystyle
Q_{r}^{\prime}(y^{\prime})=\left\\{x^{\prime}=(x^{2},\cdots,x^{n})\in\mathbb{R}^{n-1}:\max_{2\leq
i\leq n}|x^{i}-y^{i}|<r\right\\}$ is the open cube in
$\displaystyle\mathbb{R}^{n-1}$ with center $\displaystyle y^{\prime}$ and
size $\displaystyle r$. Also we denote $\displaystyle
Q_{r}^{\prime}=Q_{r}^{\prime}(0^{\prime})$.

  3. (3)

$\displaystyle Q_{r}(y)=\left\\{x\in\mathbb{R}^{n}:\max_{1\leq i\leq
n}|x^{\alpha}-y^{\alpha}|<r\right\\}=(y^{1}-r,y^{1}+r)\times
Q_{r}^{\prime}(y^{\prime})$ is the open cube in $\displaystyle\mathbb{R}^{n}$
with center $\displaystyle y$ and size $\displaystyle r$. Also we denote
$\displaystyle Q_{r}=Q_{r}(0)$.

  4. (4)

For a function $\displaystyle g(x)$ in $\displaystyle\mathbb{R}^{n}$,

$\displaystyle(g)_{U}=\,\mathchoice{{\vbox{\hbox{$\displaystyle\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\displaystyle\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\displaystyle\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\displaystyle\scriptscriptstyle-$}}\kern-2.29166pt}}\\!\int_{U}g(x)\,dx=\frac{1}{|U|}\int_{U}g(x)\,dx,$

where $\displaystyle U$ is an open subset in $\displaystyle\mathbb{R}^{n}$ and
$\displaystyle|U|$ is the $\displaystyle n$-dimensional Lebesgue measure of
$\displaystyle U$.

  5. (5)

$\displaystyle h\big{|}_{U}$ is the restriction of $\displaystyle h$ in
$\displaystyle U$.

  6. (6)

For an open set $\displaystyle U\subset\mathbb{R}^{n}$, $\displaystyle h\in
C^{l}\left(\overline{U}\right)$ means that $\displaystyle Du,\cdots,D^{l}u$
exist in $\displaystyle U$ and are uniformly continuous in $\displaystyle U$.

To handle the index in higher order derivatives, we use the following
notations.

  1. (1)

$\displaystyle p=(p_{1},\cdots,p_{n})\in\mathbb{R}^{n}$ denotes $\displaystyle
n$-dimensional indices satisfying that $\displaystyle p_{1},\cdots,p_{n}\geq
0$.

  2. (2)

$\displaystyle p^{\prime}=(p_{2},\cdots,p_{n})\in\mathbb{R}^{n-1}$ and
$\displaystyle q^{\prime}=(q_{2},\cdots,q_{n})\in\mathbb{R}^{n-1}$ denotes
$\displaystyle(n-1)$-dimensional indices satisfying that $\displaystyle
p_{2},\cdots,p_{n}\geq 0$ and $\displaystyle q_{2},\cdots,q_{n}\geq 0$.

  3. (3)

For any $\displaystyle p^{\prime}=(p_{2},\cdots,p_{n})$ and $\displaystyle
q^{\prime}=(q_{2},\cdots,q_{n})$, we write

$p^{\prime}\leq q^{\prime}\quad\text{ if }\quad p_{i}\leq q_{i}\quad\text{ for
all }i\in\\{2,\cdots,n\\},$

and

$p^{\prime}<q^{\prime}\quad\text{ if }\quad p^{\prime}\leq
q^{\prime}\quad\text{ and }\quad p^{\prime}\not=q^{\prime}.$

  4. (4)

For any $\displaystyle p^{\prime}=(p_{2},\cdots,p_{n})$ and $\displaystyle
q^{\prime}=(q_{2},\cdots,q_{n})$ with $\displaystyle p^{\prime}\geq
q^{\prime}\geq 0^{\prime}$, we define

$\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}=\Big{(}\begin{array}[]{c}p_{2}\\\
q_{2}\end{array}\Big{)}\cdots\Big{(}\begin{array}[]{c}p_{2}\\\
q_{2}\end{array}\Big{)}.$

  5. (5)

For any $\displaystyle p_{1}\geq 0$ and $\displaystyle
p^{\prime}=(p_{2},\cdots,p_{n})$ with $\displaystyle p^{\prime}\geq
0^{\prime}$, we define

$D^{p_{1},p^{\prime}}=D_{1}^{p_{1}}\cdots D_{n}^{p_{n}},\quad
D^{p^{\prime}}=D_{2}^{p_{2}}\cdots
D_{n}^{p_{n}}\quad\text{and}\quad|p^{\prime}|=p_{2}+\cdots+p_{n}.$

We introduce the notations for representing the subregions and the boundary of
the subregions. In composite materials, there exists a coordinate system such
that the boundaries of the subregions become almost flat graphs in
sufficiently small scale, see for instance [1]. So from now on, we assume that
the boundaries of the regions are represented by the graphs in $\displaystyle
Q_{7}$. We assume that the cube $\displaystyle Q_{7}$ is divided into the
components or the subregions by using $\displaystyle C^{1,\gamma}$-graph
functions. In $\displaystyle Q_{7}$, the component or the subregions will be
denoted as $\displaystyle Q_{7}^{k}(z)$ with the set $\displaystyle
K=\\{k_{-},\cdots,k_{+}\\}$. Also let $\displaystyle
K_{+}=\\{k_{-}+1,\cdots,k_{+}\\}$ be the $\displaystyle C^{1,\gamma}$-graph
functions. We remark that there can be only one element in $\displaystyle K$.
To coincide with the previous definition in [1], we define composite domain by
using $\displaystyle C^{1,\gamma}$-graph functions as follows.

######  Definition 1.1.

For the sets $\displaystyle K=\\{k_{-},k_{-}+1,\cdots,k_{+}\\}$,
$\displaystyle K_{+}=K\cup\left\\{k_{+}+1\right\\}$ and the graph functions
$\displaystyle\varphi_{k}\in
C^{1,\gamma}\left(Q_{r}^{\prime}(z^{\prime})\right)$ $\displaystyle(k\in
K_{+})$, $\displaystyle\left(Q_{r}(z),\left\\{\varphi_{k}:k\in
K_{+}\right\\}\right)$ is called a composite cube if

$\varphi_{k}(x^{\prime})\leq\varphi_{k+1}(x^{\prime})\qquad\left(x\in
Q_{r}^{\prime}(z),\,k\in K\right),$

and

$Q_{r}(z)=\mathop{\dot{\cup}}_{k\in K}Q_{r}^{k}(z),$

where

(1.1) $Q_{r}^{k}(y):=\big{\\{}(x^{1},x^{\prime})\in
Q_{\rho}(y):\varphi_{k}(x^{\prime})<x^{1}\leq\varphi_{k+1}(x^{\prime})\big{\\}}\qquad(k\in
K).$

Here, $\displaystyle\mathop{\dot{\cup}}_{k}U_{k}$ denotes disjoint union
meaning that $\displaystyle\mathop{\dot{\cup}}_{k}U_{k}$ is the union of the
sets $\displaystyle\\{U_{k}:k\in K\\}$ and that $\displaystyle\\{U_{k}:k\in
K\\}$ are mutually disjoint.

######  Remark 1.2.

For the graphs
$\displaystyle\left\\{(\varphi_{k}(x^{\prime}),x^{\prime}):x^{\prime}\in
Q_{r}^{\prime}(z^{\prime})\right\\}$ $\displaystyle(k\in K_{+})$ in Definition
1.1, the top
$\displaystyle\left\\{(\varphi_{k_{+}+1}(x^{\prime}),x^{\prime}):x^{\prime}\in
Q_{r}^{\prime}(z^{\prime})\right\\}$ and the bottom
$\displaystyle\left\\{(\varphi_{k_{-}}(x^{\prime}),x^{\prime}):x^{\prime}\in
Q_{r}^{\prime}(z^{\prime})\right\\}$ ones should be outside of $\displaystyle
Q_{r}(z)$, because other-wise we have that $\displaystyle
Q_{r}(z)\not=\mathop{\dot{\cup}}_{k\in K}Q_{r}^{k}(z)$.

For the composite cubes inside the cube, we use the following natural
definition.

######  Definition 1.3.

For the composite cube $\displaystyle\left(Q_{r}(z),\left\\{\varphi_{k}:k\in
K_{+}\right\\}\right)$, we denote

$Q_{\rho}^{k}(y):=\big{\\{}(x^{1},x^{\prime})\in
Q_{r}(y):\varphi_{k}(x^{\prime})<x^{1}\leq\varphi_{k+1}(x^{\prime})\big{\\}}\qquad\qquad(k\in
K),$

for any $\displaystyle Q_{\rho}(y)\subset Q_{r}(z)$.

In view of [1], there exists a coordinate system such that the graphs are
almost flat. So we also assume that

(1.2)
$\|D_{x^{\prime}}\varphi_{k}\|_{L^{\infty}(Q_{7}^{\prime})}\leq\frac{1}{20n}\qquad\big{(}k\in
K_{+}\big{)}.$

For a technical reason, we will focus on the set $\displaystyle Q_{5}$. Let
$\displaystyle K_{-}(\subset K)$ be the set of graph functions which
intersects $\displaystyle Q_{5}$. Then we see that

$K_{-}\subsetneq K\subsetneq K_{+}.$

From the definition of $\displaystyle K_{-}$, we discover that

(1.3) $k\in K_{-}\qquad\Longleftrightarrow\qquad
Q_{5}\cap\left\\{\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}:x^{\prime}\in
Q_{5}^{\prime}\right\\}\not=\emptyset,$

which implies that

(1.4) $k\in K_{-}\qquad\Longleftrightarrow\qquad
Q_{5}^{k}\not=\emptyset\quad\text{and}\quad Q_{5}^{k-1}\not=\emptyset.$

It follows from (1.2) and (1.3) that

(1.5) $\left\\{\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}:x^{\prime}\in
Q_{5}^{\prime}\right\\}\subset\partial Q_{6}^{k}\cap\partial
Q_{6}^{k-1}(\subset Q_{6})\qquad\big{(}k\in K_{-}\big{)}.$

######  Remark 1.4.

With the assumption that $\displaystyle a_{ij},u,F_{i}\in
C^{m,\gamma}\left(Q_{7}^{k}\right)$ $\displaystyle(k\in K)$, we will prove
that $\displaystyle u\in C^{m+1,\gamma}\left(Q_{3}^{k}\right)$
$\displaystyle(k\in K)$. The most of the calculations in this paper are
performed in $\displaystyle Q_{5}$ but $\displaystyle Q_{5}^{k}$
$\displaystyle(k\in K)$ might not be disconnected. So with (1.5), we consider
an extended domain $\displaystyle Q_{7}$ to overcome this difficulty.

For $\displaystyle A^{\alpha\beta,k}_{ij}:\mathbb{R}^{n}\to\mathbb{R}$
$\displaystyle(1\leq\alpha,\beta\leq n,\ 1\leq i,j\leq N,\ k\in K)$, assume
that

(1.6) $\lambda|\zeta|^{2}\leq\sum_{1\leq i,j\leq N}\sum_{1\leq\alpha,\beta\leq
n}A^{\alpha\beta,k}_{ij}(x)\zeta_{\alpha}^{i}\zeta_{\beta}^{j}\qquad\qquad(x\in\mathbb{R}^{n},~{}\zeta\in\mathbb{R}^{Nn},~{}k\in
K),$ (1.7)
$\left|A^{\alpha\beta,k}_{ij}(x)\right|\leq\Lambda\qquad\qquad\left(x\in\mathbb{R}^{n},~{}1\leq\alpha,\beta\leq
n,~{}1\leq i,j\leq N,~{}k\in K\right)$

and

(1.8) $A^{\alpha\beta,k}_{ij}\in
C^{m,\gamma}(\mathbb{R}^{n})\qquad\qquad\left(1\leq\alpha,\beta\leq n,~{}1\leq
i,j\leq N,~{}k\in K\right),$

for some $\displaystyle\Lambda\geq\lambda>0$ and
$\displaystyle\gamma\in(0,1/4]$. Set $\displaystyle
A^{\alpha\beta}_{ij}:Q_{7}\to\mathbb{R}$ as

(1.9) $A^{\alpha\beta}_{ij}=\sum_{k\in
K}A^{\alpha\beta,k}_{ij}\chi_{Q_{7}^{k}}$

for any $\displaystyle 1\leq\alpha,\beta\leq n$, $\displaystyle 1\leq i,j\leq
N$ and $\displaystyle k\in K$. Then by (1.6), (1.7) and (1.8),

(1.10) $\lambda|\zeta|^{2}\leq\sum_{1\leq i,j\leq
N}\sum_{1\leq\alpha,\beta\leq
n}A^{\alpha\beta}_{ij}(x)\zeta_{i}\zeta_{j}\qquad\qquad\left(x\in\mathbb{R}^{n},~{}\zeta\in\mathbb{R}^{Nn}\right),$
(1.11)
$\left|A^{\alpha\beta}_{ij}(x)\right|\leq\Lambda\qquad\qquad\left(x\in\mathbb{R}^{n},~{}1\leq\alpha,\beta\leq
n,\ 1\leq i,j\leq N\right)$

and

(1.12) $A^{\alpha\beta}_{ij}\in
C^{m,\gamma}\left(Q_{7}^{k}\right)\qquad\qquad\left(1\leq\alpha,\beta\leq
n,~{}1\leq i,j\leq N,~{}k\in K\right).$

Let $\displaystyle\left(Q_{7}(z),\left\\{\varphi_{k}:k\in
K_{+}\right\\}\right)$ be a compostie cube with the the assumption that

(1.13) $\varphi_{k}\in C^{m+1,\gamma}\left(Q_{7}^{\prime}\right)\qquad(k\in
K_{+}).$

Then let $\displaystyle u\in W^{1,2}(Q_{7})$ be a weak solution of

(1.14) $\sum_{1\leq\alpha\leq n}D_{\alpha}\left[\sum_{1\leq j\leq
N}\sum_{1\leq\beta\leq
n}A^{\alpha\beta}_{ij}D_{\beta}u^{j}\right]=\sum_{1\leq\alpha\leq
n}D_{\alpha}F_{\alpha}^{i}\quad\text{in}\quad Q_{7},$

with the assumption that

(1.15) $u,\,F\in C^{m,\gamma}\left(Q_{7}^{k}\right)\qquad(k\in K).$

In this paper, we might omit the summation by using Einstein notation.

To prove piece-wise Hölder continuity of $\displaystyle D^{m+1}u$, we will
first show in Theorem 1.6 that the following $\displaystyle
U^{p^{\prime}}:Q_{5}\to\mathbb{R}^{Nn}$ is Hölder continuous in $\displaystyle
Q_{5}$. For any $\displaystyle p^{\prime}\geq 0^{\prime}$ with
$\displaystyle|p^{\prime}|=m$, set $\displaystyle
U^{p^{\prime}}:Q_{5}\to\mathbb{R}^{Nn}$ as

$U_{1}^{i,p^{\prime}}=G_{1}^{i,p^{\prime}}+\sum_{1\leq\alpha\leq
n}\pi_{\alpha}\left[\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}\left[\sum_{1\leq
j\leq N}\sum_{1\leq\beta\leq
n}A^{\alpha\beta}_{ij}D_{\beta}u^{j}-F_{\alpha}^{i}\right]\right]$

with

$U_{\beta}^{i,p^{\prime}}=G_{\beta}^{i,p^{\prime}}+\sum_{0^{\prime}\leq
q^{\prime}\leq p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}\left[D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}D_{\beta}u^{i}+\pi_{\beta}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}D_{1}u^{i}\right]$

for any $\displaystyle 2\leq\beta\leq n$ and $\displaystyle 1\leq i\leq N$,
where $\displaystyle G^{p^{\prime}}:Q_{5}\to\mathbb{R}^{Nn}$ is defined as

$G_{1}^{i,p^{\prime}}=\sum_{k\in K_{-}}\sum_{1\leq\alpha\leq n}\
\mathbb{G}_{\alpha}^{p^{\prime},k}\left[U_{\alpha}^{i}\big{|}_{k-1}-U_{\alpha}^{i}\big{|}_{k}\right]H_{k}$

and

$G_{\beta}^{i,p^{\prime}}=\sum_{k\in
K_{-}}\mathbb{H}_{\beta}^{p^{\prime},k}\left[u\big{|}_{k-1}-u\big{|}_{k}\right]H_{k}$

for any $\displaystyle 2\leq\beta\leq n$ and $\displaystyle 1\leq i\leq N$.
Here, $\displaystyle H_{k}:Q_{7}\to\mathbb{R}$ $\displaystyle\big{(}k\in
K\big{)}$ is Heavyside function that

$H_{k}(x)=\left\\{\begin{array}[]{rl}1&\text{ if }x\in Q_{7}^{l}~{}\text{ with
}~{}l\geq k,\\\ 0&\text{ if }x\in Q_{7}^{l}~{}\text{ with
}~{}l<k.\end{array}\right.$

Also with (1.5), the linear operators$\displaystyle\
\mathbb{G}_{\alpha}^{p^{\prime},k}:C^{m-1,\gamma}\left(\partial
Q_{6}^{k}\cap\partial Q_{6}^{k-1}\right)\to
C^{\gamma}\left(Q_{5}^{\prime}\right)$ $\displaystyle(k\in K_{-})$ and
$\displaystyle\mathbb{H}_{\alpha}^{p^{\prime},k}:C^{m,\gamma}\left(\partial
Q_{6}^{k}\cap\partial Q_{6}^{k-1}\right)\to
C^{\gamma}\left(Q_{5}^{\prime}\right)$ $\displaystyle(k\in K_{-})$ are defined
as

$\displaystyle\ \mathbb{G}_{\alpha}^{p^{\prime},k}[h](x^{\prime})$
$\displaystyle=\sum_{|\xi|\leq|p^{\prime}|-1}\left[\pi_{\alpha}P_{\xi}^{p^{\prime}}D_{\xi}h\right]\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}$
$\displaystyle\quad+\sum_{0^{\prime}<q^{\prime}\leq
p^{\prime}}\left(\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\right)D^{q^{\prime}}D_{i}\varphi_{k}(x^{\prime})\,D^{p^{\prime}-q^{\prime}}\left[h\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\right],$

and

$\displaystyle\mathbb{H}_{\alpha}^{p^{\prime},k}[h](x^{\prime})$
$\displaystyle=\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}D_{\alpha}\left[\pi^{q^{\prime}}\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\right]D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}h\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}$
$\displaystyle\quad+\sum_{|\xi|\leq|p^{\prime}|-1}D_{\alpha}\left[P_{\xi}^{p^{\prime}}\big{[}\varphi_{k}(x^{\prime}),x^{\prime}\big{]}\,D_{\xi}h\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\right],$

for any $\displaystyle 1\leq\alpha\leq n$ and $\displaystyle k\in K$, where
the notation

$u\big{|}_{k}\in
C^{m,\gamma}\left(\overline{Q_{5}^{k}}\right)\qquad\text{and}\qquad
U_{\alpha}^{i}\big{|}_{k}\in C^{m-1,\gamma}\left(\overline{Q_{5}^{k}}\right)$

denote the extension of

$u\in C^{m,\gamma}\left(Q_{7}^{k}\right)\qquad\text{and}\qquad
U_{\alpha}^{i}=\sum_{1\leq j\leq N}\sum_{1\leq\beta\leq
n}A_{\alpha\beta}^{ij}D_{\beta}u^{j}-F_{\alpha}^{i}\in
C^{m-1,\gamma}\left(Q_{7}^{k}\right)$

for any $\displaystyle 1\leq\alpha\leq n$ and $\displaystyle 1\leq i\leq N$,
which exist by the assumption (1.15).

######  Remark 1.5.

By the definition,
$\displaystyle\mathbb{G}_{\alpha}^{p^{\prime},k}[h](x^{\prime})$ and
$\displaystyle\mathbb{H}_{\alpha}^{p^{\prime},k}[h](x^{\prime})$ depends only
on $\displaystyle x^{\prime}$-variables. So $\displaystyle G^{p^{\prime}}$
depends only on $\displaystyle x^{\prime}$-variables in $\displaystyle
Q_{5}^{k}$ and are Hölder continuous in each $\displaystyle Q_{5}^{k}$
$\displaystyle(k\in K)$ and might have big jumps on $\displaystyle\partial
Q_{5}^{k}\cap\partial Q_{5}^{k+1}$ $\displaystyle(k\in K_{-})$.

We will prove that $\displaystyle U^{p^{\prime}}$, $\displaystyle\pi_{\alpha}$
and $\displaystyle\pi^{q^{\prime}}$ are Hölder continuous in $\displaystyle
Q_{5}$. So one can only show that $\displaystyle D^{m+1}u$ is piece-wise
Hölder continuous due to vector-valued the function $\displaystyle
G^{p^{\prime}}$ which might have big jumps on $\displaystyle\partial
Q_{5}^{k}\cap\partial Q_{5}^{k+1}$ $\displaystyle(k\in K_{-})$.

In the previous [1], we discovered a term related to the gradient of the weak
solution which is Hölder continuous. In this paper, we obtain a corresponding
result related to higher order derivatives.

######  Theorem 1.6.

Let $\displaystyle u$ be a weak solution of (1.14). Then for any
$\displaystyle p^{\prime}\geq 0^{\prime}$ with $\displaystyle|p|=m$, we have
that $\displaystyle U^{p^{\prime}}\in C^{\gamma}\left(Q_{3}\right)$ with the
estimate

$\displaystyle\big{\|}U^{p^{\prime}}\big{\|}_{L^{\infty}(Q_{r}(z))}^{2}\leq
cr^{-2}E^{2},$ $\big{\|}U^{p^{\prime}}\big{\|}_{L^{\infty}(Q_{r}(z))}^{2}\leq
c\left[\,\mathchoice{{\vbox{\hbox{$\displaystyle\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\displaystyle\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\displaystyle\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\displaystyle\scriptscriptstyle-$}}\kern-2.29166pt}}\\!\int_{Q_{2r}(z)}\big{|}U^{p^{\prime}}\big{|}^{2}\,dx+E^{2}\right]$

and

$\big{[}U^{p^{\prime}}\big{]}_{C^{\gamma}(Q_{r}(z))}^{2}\leq
c\left[\frac{1}{r^{2\gamma}}\,\mathchoice{{\vbox{\hbox{$\displaystyle\textstyle-$}}\kern-4.86108pt}}{{\vbox{\hbox{$\displaystyle\scriptstyle-$}}\kern-3.25pt}}{{\vbox{\hbox{$\displaystyle\scriptscriptstyle-$}}\kern-2.29166pt}}{{\vbox{\hbox{$\displaystyle\scriptscriptstyle-$}}\kern-2.29166pt}}\\!\int_{Q_{2r}(z)}\left|U^{p^{\prime}}-\big{(}U^{p^{\prime}}\big{)}_{Q_{2r}(z)}\right|^{2}dx+\sum_{|q^{\prime}|=m}\big{\|}U^{q^{\prime}}\big{\|}_{L^{\infty}(Q_{2r}(z))}^{2}+E^{2}\right],$

for any $\displaystyle Q_{2r}(z)\subset Q_{5}$ where $\displaystyle
E=\sum_{k\in
K}\left[\|u\|_{C^{m,\gamma}(Q_{7}^{k})}+\|F\|_{C^{m,\gamma}(Q_{7}^{k})}\right]$.

$\displaystyle D^{m+1}u$ and $\displaystyle U^{p^{\prime}}$ will be compared
by using the following proposition.

######  Proposition 1.7.

For any $\displaystyle z\in Q_{5}$, we have that

$|D^{m+1}u(z)|\leq
c\left[E+\sum_{|p^{\prime}|=m}\big{|}U^{p^{\prime}}(z)\big{|}\right]\leq
c\left[E+|D^{m+1}u(z)|\right].$

In additional, for any $\displaystyle w,z\in Q_{5}^{k}$ $\displaystyle(k\in
K)$, we have that

$\displaystyle\big{|}D^{m+1}u(w)-D^{m+1}u(z)\big{|}$ $\displaystyle\leq
c\sum_{|p^{\prime}|=m}\left[\big{|}U^{p^{\prime}}(w)-U^{p^{\prime}}(z)\big{|}+|w-z|^{\gamma}\big{|}U^{p^{\prime}}(z)\big{|}\right]$
$\displaystyle\quad+c|w-z|^{\gamma}\sum_{k\in
K}\left[\|u\|_{C^{m,\gamma}(Q_{7}^{k})}+\|F\|_{C^{m,\gamma}(Q_{7}^{k})}\right].$

######  Theorem 1.8.

For a weak solution $\displaystyle u$ of (1.14), we have that $\displaystyle
u\in C^{m+1,\gamma}\left(Q_{3}^{k}\right)$ $\displaystyle(k\in K)$ and
$\displaystyle D^{m+1}u\in L^{\infty}(Q_{3}^{k})$ $\displaystyle(k\in K)$ with
the estimate

$\displaystyle\frac{1}{|Q_{r}|}\int_{Q_{r}^{k}(z)}\left|D^{m+1}u\right|^{2}\,dx\leq
cr^{-2}E^{2},$ $\left\|D^{m+1}u\right\|_{L^{\infty}(Q_{r}^{k}(z))}^{2}\leq
c\left[\sum_{l\in
K}\frac{1}{|Q_{2r}|}\int_{Q_{2r}^{l}(z)}\big{|}D^{m+1}u\big{|}^{2}\,dx+E^{2}\right]$

and

$\big{[}D^{m+1}u\big{]}_{C^{\gamma}(Q_{r}^{k}(z))}^{2}\leq\frac{c}{r^{2\gamma}}\left[\sum_{l\in
K}\frac{1}{|Q_{2r}|}\int_{Q_{2r}^{l}(z)}\big{|}D^{m+1}u\big{|}^{2}\,dx+E^{2}\right],$

for any $\displaystyle Q_{2r}(z)\subset Q_{5}$ and $\displaystyle k\in K$,
where $\displaystyle E=\sum_{k\in
K}\left[\|u\|_{C^{m,\gamma}(Q_{7}^{k})}+\|F\|_{C^{m,\gamma}(Q_{7}^{k})}\right]$.

######  Remark 1.9.

As in [1], one can in fact find a vector-valued function related to
$\displaystyle(m+1)$-th derivatives which $\displaystyle C^{\gamma}(Q_{2})$.
This function has big jumps on the boundary of the subregions.

##  2\. Geometric settings and the related properties

With the assumption (1.13) for some $\displaystyle m\geq 0$ and
$\displaystyle\gamma\in(0,1/4]$, let
$\displaystyle\left(Q_{7},\left\\{\varphi_{k}:k\in K_{+}\right\\}\right)$ be a
composite cube. For the approximation argument, we assume that

(2.1)
$\varphi_{k}(x^{\prime})+\delta\leq\varphi_{k+1}(x^{\prime})\qquad\big{(}x\in
Q_{7}^{\prime},\,k\in K\big{)},$

for some $\displaystyle\delta>0$. The desired estimate does not depend on
$\displaystyle\delta>0$ and one can use an approximation argument to handle
the case that $\displaystyle\delta=0$. Since
$\displaystyle\left(Q_{7},\left\\{\varphi_{k}:k\in K_{+}\right\\}\right)$ is a
composite cube, recall from Definition 1.1 that

(2.2) $Q_{7}=\mathop{\dot{\cup}}_{k\in K}Q_{7}^{k}.$

###  2.1. Time derivatives of the flow

To estimate the higher derivatives for linear elliptic systems in composite
cubes, we generalize the difference quotient from lines(or the fixed
directions) to general curves. To do it, we set the flow
$\displaystyle\psi(x,t^{\prime})$ as follows. For the graph functions,
$\displaystyle\left\\{\varphi_{k}:k\in K_{+}\right\\}$, set $\displaystyle
T_{k}:Q_{7}\to[0,1]$ ($\displaystyle k\in K$) as

(2.3)
$T_{k}(x^{1},x^{\prime})=\frac{x^{1}-\varphi_{k}(x^{\prime})}{\varphi_{k+1}(x^{\prime})-\varphi_{k}(x^{\prime})}\quad\text{in}\quad
Q_{7}^{k}.$

######  Remark 2.1.

For any $\displaystyle l>k$, $\displaystyle T_{k}=1$ in $\displaystyle
Q_{7}^{l}$ and for any $\displaystyle l<k$, $\displaystyle T_{k}=0$ in
$\displaystyle Q_{7}^{l}$. So with (2.1), $\displaystyle T_{k}$ linearly
increases from $\displaystyle\left(\varphi_{k}(x^{\prime}),x^{\prime}\right)$
to $\displaystyle\left(\varphi_{k+1}(x^{\prime}),x^{\prime}\right)$ in
$\displaystyle Q_{7}^{k}$.

For any $\displaystyle k\in K$, define $\displaystyle\psi:Q_{6}\times
Q_{1}^{\prime}\to\mathbb{R}$ so that

(2.4)
$\psi(x,t^{\prime})=\big{[}\varphi_{k+1}(x^{\prime}+t^{\prime})-\varphi_{k+1}(x^{\prime})\big{]}\cdot
T_{k}(x)+\big{[}\varphi_{k}(x^{\prime}+t^{\prime})-\varphi_{k}(x^{\prime})\big{]}\cdot\big{[}1-T_{k}(x)\big{]}$

in $\displaystyle Q_{6}^{k}\times Q_{1}^{\prime}$. If the flow starts at
$\displaystyle x\in Q_{6}^{k}$ then the flow $\displaystyle
x+\left(\psi(x,t^{\prime}),t^{\prime}\right)$ at time $\displaystyle
t^{\prime}\in Q_{1}^{\prime}$ still remains at the same region $\displaystyle
Q_{7}^{k}$ as in the following lemma.

######  Lemma 2.2.

For any $\displaystyle(x,t^{\prime})\in Q_{6}^{k}\times Q_{1}^{\prime}$, we
have that

$x+\left(\psi(x,t^{\prime}),t^{\prime}\right)=\left(x^{1}+\psi(x,t^{\prime}),x^{\prime}+t^{\prime}\right)\in
Q_{7}^{k}.$

###### Proof.

For any $\displaystyle(x,t^{\prime})\in Q_{6}\times Q_{1}^{\prime}$, we have
from (1.2) that

$\displaystyle\left|\left(x^{1}+\psi(x,t^{\prime})\right)\right|$
$\displaystyle\quad\leq|x^{1}|+\big{|}\varphi_{k+1}(x^{\prime}+t^{\prime})-\varphi_{k+1}(x^{\prime})\big{|}\cdot
T_{k}(x)+\big{|}\varphi_{k}(x^{\prime}+t^{\prime})-\varphi_{k}(x^{\prime})\big{|}\cdot\big{[}1-T_{k}(x)\big{]}$
$\displaystyle\quad\leq|x^{1}|+|t^{\prime}|<7,$

which implies that

(2.5) $\left\\{x+\left(\psi(x,t^{\prime}),t^{\prime}\right):(x,t^{\prime})\in
Q_{6}^{k}\times Q_{1}^{\prime}\right\\}\subset Q_{7}.$

Also from (2.3) and (2.4), one can check that for any
$\displaystyle(x,t^{\prime})\in Q_{6}^{k}\times Q_{1}^{\prime}$,

$\displaystyle x^{1}+\psi(x,t^{\prime})$
$\displaystyle\quad=\frac{[\varphi_{k+1}(x^{\prime}+t^{\prime})-\varphi_{k}(x^{\prime}+t^{\prime})]x^{1}+\varphi_{k}(x^{\prime}+t^{\prime})\varphi_{k+1}(x^{\prime})-\varphi_{k+1}(x^{\prime}+t^{\prime})\varphi_{k}(x^{\prime})}{\varphi_{k+1}(x^{\prime})-\varphi_{k}(x^{\prime})}.$

If $\displaystyle x\in Q_{6}^{k}$ then
$\displaystyle\varphi_{k}(x^{\prime})<x^{1}\leq\varphi_{k+1}(x^{\prime})$. So
for any $\displaystyle(x,t^{\prime})\in Q_{6}^{k}\times Q_{1}^{\prime}$, we
find that
$\displaystyle\varphi_{k}(x^{\prime}+t^{\prime})<x^{1}+\psi(x,t^{\prime})\leq\varphi_{k+1}(x^{\prime}+t^{\prime})$.
With (2.5), this proves the lemma. ∎

Fix $\displaystyle p^{\prime}=(p_{2},\cdots,p_{n})\geq 0^{\prime}$ with
$\displaystyle|p^{\prime}|=m$. Then we consider the following $\displaystyle
m$-th order time derivative of $\displaystyle h$ with respect to the flow
$\displaystyle\psi(x,t^{\prime})$ :

(2.6)
$\displaystyle\partial_{t^{\prime}}^{p^{\prime}}h\big{|}_{t^{\prime}=0^{\prime}}:=\partial_{t^{2}}^{p_{2}}\cdots\partial_{t^{n}}^{p_{n}}\big{[}h\big{(}x+(\psi(x,t^{\prime}),t^{\prime})\big{)}\big{]}\big{|}_{t^{\prime}=0^{\prime}}~{}\text{
in }~{}Q_{6}.$

We calculate (2.6) as follows.

For any $\displaystyle i\in\\{1,\cdots,n\\}$, to denote the first time
derivative of the flow, set

(2.7) $\pi_{\alpha}(x):=D_{i}\varphi_{k+1}(x^{\prime})\cdot
T_{k}(x)+D_{i}\varphi_{k}(x^{\prime})\cdot[1-T_{k}(x)]$

for any $\displaystyle x\in Q_{7}^{k}$ $\displaystyle(k\in K)$. Then we find
from (2.4) that

(2.8) $\pi_{\alpha}(x)=\partial_{t^{i}}\psi(x,t)\big{|}_{t=0}\qquad(x\in
Q_{6}).$

In addition, we set

(2.9)
$\pi^{\prime}=(\pi_{2},\cdots,\pi_{n})\quad\text{and}\quad\pi=(-1,\pi^{\prime})=(-1,\pi_{2},\cdots,\pi_{n})~{}\text{
in }~{}Q_{7}.$

Also we denote the power of the first time derivative of the flow as

(2.10) $\pi^{q^{\prime}}:=\pi_{2}^{q_{2}}\cdots\pi_{n}^{q_{n}}~{}\text{ in
}~{}Q_{7}.$

######  Lemma 2.3.

For any $\displaystyle p^{\prime}\geq 0^{\prime}$ with
$\displaystyle|p^{\prime}|=m$, there exists a polynomial
$\displaystyle\tilde{P}_{\xi}^{p^{\prime}}:\mathbb{R}^{\tiny\left(\begin{array}[]{c}n-1\\\
1\end{array}\right)\times\left(\begin{array}[]{c}n\\\
2\end{array}\right)\times\cdots\times\left(\begin{array}[]{c}n+m-2\\\
m\end{array}\right)}\to\mathbb{R}$ such that

(2.11)
$\displaystyle\partial_{t^{\prime}}^{p^{\prime}}h\big{|}_{t^{\prime}=0^{\prime}}=\sum_{0^{\prime}\leq
q^{\prime}\leq p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}h+\sum_{|\xi|\leq|p^{\prime}|-1}P_{\xi}^{p^{\prime}}\,D_{\xi}h~{}\text{
in }~{}Q_{6},$

where $\displaystyle P_{\xi}^{p^{\prime}}:Q_{7}\to\mathbb{R}$ is defined as

$P_{\xi}^{p^{\prime}}=\tilde{P}_{\xi}^{p^{\prime}}\left[T_{k}D_{x^{\prime}}\varphi_{k+1}+[1-T_{k}]\,D_{x^{\prime}}\varphi_{k},\cdots,T_{k}\,D_{x^{\prime}}^{m}\varphi_{k+1}+[1-T_{k}]\,D_{x^{\prime}}^{m}\varphi_{k}\right]~{}\text{
in }~{}Q_{7}^{k}$

for any $\displaystyle k\in K$.

###### Proof.

By the definition of
$\displaystyle\partial_{t^{\prime}}^{p^{\prime}}h\big{|}_{t^{\prime}=0^{\prime}}$
in (2.6), we obtain that

$\displaystyle\partial_{t^{\prime}}^{p^{\prime}}h\big{|}_{t^{\prime}=0^{\prime}}$
$\displaystyle\,=\partial_{t^{2}}^{p_{2}}\cdots\partial_{t^{n}}^{p_{n}}\big{[}h\big{(}x+(\psi(x,t^{\prime}),t^{\prime})\big{)}\big{]}\big{|}_{t^{\prime}=0^{\prime}}$
$\displaystyle\,=\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p_{2}\\\
q_{2}\end{array}\Big{)}\cdots\Big{(}\begin{array}[]{c}p_{n}\\\
q_{n}\end{array}\Big{)}\big{(}\partial_{t^{2}}\psi\big{|}_{t^{\prime}=0^{\prime}}\big{)}^{q_{2}}\cdots\big{(}\partial_{t^{n}}\psi\big{|}_{t^{\prime}=0^{\prime}}\big{)}^{q_{n}}D_{2}^{p_{2}-q_{2}}D_{1}^{q_{2}}\cdots
D_{n}^{p_{n}-q_{n}}D_{1}^{q_{n}}h$ $\displaystyle\,\quad+\sum_{|\xi|\leq
p_{2}+\cdots+p_{n}-1}\tilde{P}_{\xi}^{p_{2},\cdots,p_{n}}\big{(}\partial_{t^{2}}\psi\big{|}_{t^{\prime}=0^{\prime}},\cdots,\partial_{t^{n}}\psi\big{|}_{t^{\prime}=0^{\prime}},\cdots,\partial_{t^{2}}^{p_{2}}\cdots\partial_{t^{n}}^{p_{n}}\psi\big{|}_{t^{\prime}=0^{\prime}}\big{)}D_{\xi}h$

in $\displaystyle Q_{6}$, for some polynomial
$\displaystyle\tilde{P}_{\xi}^{p_{2},\cdots,p_{n}}:\mathbb{R}^{\tiny\left(\begin{array}[]{c}n-1\\\
1\end{array}\right)\times\left(\begin{array}[]{c}n\\\
2\end{array}\right)\times\cdots\times\left(\begin{array}[]{c}n+m-2\\\
m\end{array}\right)}\to\mathbb{R}$. The number of the choice for the first
derivative is $\displaystyle\left(\begin{array}[]{c}(n-1)+1-1\\\
1\end{array}\right)$, the second derivative is
$\displaystyle\left(\begin{array}[]{c}(n-1)+2-1\\\ 2\end{array}\right)$,
$\displaystyle\cdots$, the $\displaystyle m$-th derivative is
$\displaystyle\left(\begin{array}[]{c}(n-1)+m-1\\\ m\end{array}\right)$. We
simplify the above equality. Recall that

$D_{x^{\prime}}^{q^{\prime}}=D_{x^{2}}^{q_{2}}\cdots
D_{x^{n}}^{q_{n}},\qquad|p^{\prime}|=p_{2}+\cdots+p_{n}\qquad\text{and}\qquad\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}=\Big{(}\begin{array}[]{c}p_{2}\\\
q_{2}\end{array}\Big{)}\cdots\Big{(}\begin{array}[]{c}p_{2}\\\
q_{2}\end{array}\Big{)}.$

From (2.7) and (2.10), we obtain that for any $\displaystyle p^{\prime}\geq
q^{\prime}\geq 0^{\prime}$,

$\displaystyle\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p_{2}\\\
q_{2}\end{array}\Big{)}\cdots\Big{(}\begin{array}[]{c}p_{n}\\\
q_{n}\end{array}\Big{)}\big{(}\partial_{t^{2}}\psi\big{|}_{t^{\prime}=0^{\prime}}\big{)}^{q_{2}}\cdots\big{(}\partial_{t^{n}}\psi\big{|}_{t^{\prime}=0^{\prime}}\big{)}^{q_{n}}D_{2}^{p_{2}-q_{2}}D_{1}^{q_{2}}\cdots
D_{n}^{p_{n}-q_{n}}D_{1}^{q_{n}}h$ $\displaystyle\quad=\sum_{0^{\prime}\leq
q^{\prime}\leq p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}h.$

For any $\displaystyle q^{\prime}\geq 0$, we find from (2.4) that

$\partial_{t^{\prime}}^{q^{\prime}}\psi(x,t^{\prime})=D_{x^{\prime}}^{q^{\prime}}\varphi_{k+1}(x^{\prime}+t^{\prime})\,T_{k}(x)+D_{x^{\prime}}^{q^{\prime}}\varphi_{k}(x^{\prime}+t^{\prime})\,[1-T_{k}(x)]$

in $\displaystyle Q_{6}^{k}\times Q_{1}^{\prime}$, which implies that ,

$\partial_{t^{\prime}}^{q^{\prime}}\psi(x,t^{\prime})\big{|}_{t^{\prime}=0^{\prime}}=D_{x^{\prime}}^{q^{\prime}}\varphi_{k+1}(x^{\prime})\,T_{k}(x)+D_{x^{\prime}}^{q^{\prime}}\varphi_{k}(x^{\prime})\,[1-T_{k}(x)]$

for any $\displaystyle x\in Q_{6}^{k}$. So by letting
$\displaystyle\tilde{P}_{\xi}^{p^{\prime}}:=\tilde{P}_{\xi}^{p_{2},\cdots,p_{n}}$,
the lemma follows from (2.4). ∎

###  2.2. Decay estimate of the graph functions and
$\displaystyle\pi^{\prime}$ in (2.9)

To handle the two non-crossing graph functions $\displaystyle\varphi_{k}$
$\displaystyle(k\in K_{+})$, we use following result in [1], which naturally
holds from our geometric settings (also see [9, Section 5] or [8, Section 4]).

######  Lemma 2.4 ([1]).

Suppose that
$\displaystyle\varphi_{k},\varphi_{l}:C^{1,\gamma}(Q_{r+\rho}^{\prime})\to\mathbb{R}$
satisfy that

$[D_{x^{\prime}}\varphi_{k}]_{C^{\gamma}(Q_{r+\rho}^{\prime})},[D_{x^{\prime}}\varphi_{l}]_{C^{\gamma}(Q_{r+\rho}^{\prime})}\leq
c_{1},$
$\|\varphi_{k}\|_{L^{\infty}(Q_{r+\rho}^{\prime})},\|\varphi_{l}\|_{L^{\infty}(Q_{r+\rho}^{\prime})}\leq
c_{2},$

and

$\varphi_{k}\leq\varphi_{l}\text{ in }Q_{r+\rho}^{\prime}.$

Then we have that

(2.12)
$|D_{x^{\prime}}\varphi_{l}(x^{\prime})-D_{x^{\prime}}\varphi_{k}(x^{\prime})|\leq
3\rho^{-1}\left(\rho^{1+\gamma}c_{1}+2c_{2}\right)^{\frac{1}{\gamma+1}}[\varphi_{l}(x^{\prime})-\varphi_{k}(x^{\prime})]^{\frac{\gamma}{\gamma+1}}$

for any $\displaystyle x^{\prime}\in Q_{r}^{\prime}$.

Let $\displaystyle\left(Q_{7},\left\\{\varphi_{k}:k\in K_{+}\right\\}\right)$
be a composite cube. By applying Lemma 2.4 to $\displaystyle Q_{7}$, we obtain
from (1.13) and (2.1) that

(2.13)
$|D_{x^{\prime}}\varphi_{k}(x^{\prime})-D_{x^{\prime}}\varphi_{l}(x^{\prime})|\leq
c|\varphi_{k}(x^{\prime})-\varphi_{l}(x^{\prime})|^{\frac{1}{2}}\qquad\big{(}x^{\prime}\in
Q_{6}^{\prime},~{}k,l\in K_{+}\big{)}.$

Also we have from (1.13) that

(2.14)
$|D_{x^{\prime}}\varphi_{k}(x^{\prime})-D_{x^{\prime}}\varphi_{k}(y^{\prime})|\leq
c|x^{\prime}-y^{\prime}|^{\frac{1}{2}}\qquad\left(x^{\prime},y^{\prime}\in
Q_{6}^{\prime},~{}k\in K_{+}\right).$

Then we have the following lemma from [1].

######  Lemma 2.5 ([1]).

For a composite cube $\displaystyle\left(Q_{7},\left\\{\varphi_{k}:k\in
K_{+}\right\\}\right)$ and the corresponding $\displaystyle\pi^{\prime}$ in
(2.9), we have that

$\big{|}\pi^{\prime}(y)-\pi^{\prime}(z)\big{|}\leq
c|y-z|^{\frac{1}{2}}\qquad(y,z\in Q_{6}).$

###### Proof.

The lemma follows by applying (2.13) and (2.14) to [1, Lemma 2.3]. ∎

###  2.3. Estimate of the integrals related to $\displaystyle\pi$ in (2.9)

To use the weak formation, we will apply integration by parts formula. To do
it, we obtain the estimates related to $\displaystyle D\pi$ in Lemma 2.7 and
(2.17) where $\displaystyle\pi$ is defined in (2.9).

######  Lemma 2.6.

With the assumptions (2.1) and (1.2), we have that

$\big{|}D\pi(x)\big{|}\leq
c\Big{[}1+|\varphi_{k+1}(x^{\prime})-\varphi_{k}(x^{\prime})|^{-\frac{1}{2}}\Big{]}\qquad\qquad\left(x\in
Q_{6}^{k},~{}k\in K\right).$

###### Proof.

Since $\displaystyle\pi=(\pi_{1},\cdots,\pi_{n})$ and
$\displaystyle\pi_{1}=-1$, we estimate $\displaystyle D\pi_{\alpha}$
$\displaystyle(i=2,\cdots,n)$. Fix $\displaystyle i\in\\{2,\cdots,n\\}$ and
$\displaystyle k\in K$. By Definition 1.1,

(2.15) $\varphi_{k}(x^{\prime})\leq
x^{1}\leq\varphi_{k+1}(x^{\prime})\qquad\left(x\in Q_{6}^{k}\right).$

With (2.3) and (2.7), we find from (2.13) that

(2.16)
$|D_{1}\pi_{\alpha}|=\bigg{|}\frac{D_{i}\varphi_{k+1}-D_{i}\varphi_{k}}{\varphi_{k+1}-\varphi_{k}}\bigg{|}\leq\frac{c}{|\varphi_{k+1}-\varphi_{k}|^{\frac{1}{2}}}~{}\text{
in }~{}Q_{6}^{k}.$

It only remains to estimate $\displaystyle D_{j}\pi_{\alpha}$
$\displaystyle(j=2,\cdots,n)$ in $\displaystyle Q_{6}^{k}$. By (2.3) and (2.7)

$\displaystyle\pi_{\alpha}(x)$
$\displaystyle=\frac{D_{i}\varphi_{k+1}(x^{\prime})[x^{1}-\varphi_{k}(x^{\prime})]}{\varphi_{k+1}(x^{\prime})-\varphi_{k}(x^{\prime})}+\frac{D_{i}\varphi_{k}(x^{\prime})[\varphi_{k+1}(x^{\prime})-x^{1}]}{\varphi_{k+1}(x^{\prime})-\varphi_{k}(x^{\prime})},$

for any $\displaystyle x\in Q_{6}^{k}$, which implies that

$\displaystyle D_{j}\pi_{\alpha}(x)$
$\displaystyle=\frac{D_{ij}\varphi_{k+1}(x^{\prime})[x^{1}-\varphi_{k}(x^{\prime})]}{\varphi_{k+1}(x^{\prime})-\varphi_{k}(x^{\prime})}+\frac{D_{ij}\varphi_{k}(x^{\prime})[\varphi_{k+1}(x^{\prime})-x^{1}]}{\varphi_{k+1}(x^{\prime})-\varphi_{k}(x^{\prime})}$
$\displaystyle\quad-\frac{D_{i}\varphi_{k+1}(x^{\prime})D_{j}\varphi_{k}(x^{\prime})-D_{i}\varphi_{k}(x^{\prime})D_{j}\varphi_{k+1}(x^{\prime})}{\varphi_{k+1}(x^{\prime})-\varphi_{k}(x^{\prime})}$
$\displaystyle\quad-\frac{D_{i}\varphi_{k+1}(x^{\prime})[x^{1}-\varphi_{k}(x^{\prime})]}{\varphi_{k+1}(x^{\prime})-\varphi_{k}(x^{\prime})}\cdot\frac{D_{j}\varphi_{k+1}(x^{\prime})-D_{j}\varphi_{k}(x^{\prime})}{\varphi_{k+1}(x^{\prime})-\varphi_{k}(x^{\prime})}$
$\displaystyle\quad-\frac{D_{i}\varphi_{k}(x^{\prime})[\varphi_{k+1}(x^{\prime})-x^{1}]}{\varphi_{k+1}(x^{\prime})-\varphi_{k}(x^{\prime})}\cdot\frac{D_{j}\varphi_{k+1}(x^{\prime})-D_{j}\varphi_{k}(x^{\prime})}{\varphi_{k+1}(x^{\prime})-\varphi_{k}(x^{\prime})},$

for any $\displaystyle x\in Q_{6}^{k}$ and $\displaystyle j=2,\cdots,n$. Then
by the following calculation,

$\displaystyle\frac{D_{i}\varphi_{k+1}(x^{\prime})D_{j}\varphi_{k}(x^{\prime})-D_{i}\varphi_{k}(x^{\prime})D_{j}\varphi_{k+1}(x^{\prime})}{\varphi_{k+1}(x^{\prime})-\varphi_{k}(x^{\prime})}$
$\displaystyle\quad=\frac{D_{i}\varphi_{k+1}(x^{\prime})[D_{j}\varphi_{k+1}(x^{\prime})-D_{j}\varphi_{k}(x^{\prime})]-D_{j}\varphi_{k+1}(x^{\prime})[D_{i}\varphi_{k+1}(x^{\prime})-D_{i}\varphi_{k}(x^{\prime})]}{\varphi_{k+1}(x^{\prime})-\varphi_{k}(x^{\prime})},$

the lemma follows from (2.13) and (2.15), because $\displaystyle
i\in\\{2,\cdots,n\\}$ and $\displaystyle k\in K$ were arbitrary chosen. ∎

######  Lemma 2.7.

With the assumptions (2.1) and (1.2), we have that

$\int_{Q_{r}(z)}|D\pi|^{2}\eta^{2}\,dx\leq\epsilon
r^{\frac{1}{2}}\int_{Q_{r}(z)}|D\eta|^{2}\,dx+c\epsilon^{-2}r^{\frac{1}{2}}\int_{Q_{r}(z)}\eta^{2}\,dx\quad\left(\eta\in
C_{c}^{\infty}(Q_{r}(z))\right)$

for any $\displaystyle Q_{r}(z)\subset Q_{6}$ and
$\displaystyle\epsilon\in(0,1]$.

###### Proof.

Fix $\displaystyle\eta\in C_{c}^{\infty}(Q_{r}(z))$ and $\displaystyle k\in
K$. Then by Lemma 2.6 and Young’s inequality,

$\displaystyle\int_{Q_{r}^{k}(z)}|D\pi|^{2}\eta^{2}dx$ $\displaystyle\leq
c\int_{Q_{r}^{k}(z)}\eta^{2}+\frac{\eta^{2}\chi_{Q_{r}^{k}(z)}}{\varphi_{k+1}-\varphi_{k}}\,dx$
$\displaystyle=c_{1}\bigg{[}\int_{Q_{r}(z)}\eta^{2}dx+\int_{Q_{r}^{\prime}(z^{\prime})}\int_{\max\\{\varphi_{k}(x^{\prime}),z^{1}-r\\}}^{\min\\{\varphi_{k+1}(x^{\prime}),z^{1}+r\\}}\frac{\eta^{2}(x^{1},x^{\prime})}{\varphi_{k+1}-\varphi_{k}}dx^{1}dx^{\prime}\bigg{]}.$

Since $\displaystyle\eta\in C_{c}^{\infty}(Q_{r}(z))$, by the following
calculation

$\displaystyle\int_{\max\,\\{\varphi_{k}(x^{\prime}),z^{1}-r\\}}^{\min\,\\{\varphi_{k+1}(x^{\prime}),z^{1}+r\\}}\frac{\eta^{2}(x^{1},x^{\prime})}{\varphi_{k+1}-\varphi_{k}}\,dx^{1}$
$\displaystyle\quad=\int_{\max\,\\{\varphi_{k}(x^{\prime}),z^{1}-r\\}}^{\min\,\\{\varphi_{k+1}(x^{\prime}),z^{1}+r\\}}\int_{z^{1}-r}^{x^{1}}\frac{D_{y^{1}}[\eta^{2}(y^{1},x^{\prime})]}{\varphi_{k+1}-\varphi_{k}}\,dy^{1}dx^{1}$
$\displaystyle\quad\leq
2\int_{\max\,\\{\varphi_{k}(x^{\prime}),z^{1}-r\\}}^{\min\,\\{\varphi_{k+1}(x^{\prime}),z^{1}+r\\}}\int_{z^{1}-r}^{z^{1}+r}\frac{|D_{y^{1}}\eta(y^{1},x^{\prime})||\eta(y^{1},x^{\prime})|}{\varphi_{k+1}-\varphi_{k}}\,dy^{1}dx^{1}$
$\displaystyle\quad\leq
2\int_{z^{1}-r}^{z^{1}+r}|D_{y^{1}}\eta(y^{1},x^{\prime})||\eta(y^{1},x^{\prime})|\,dy^{1}.$

Since $\displaystyle Q_{r}(z)=(z^{1}-r,z^{1}+r)\times
Q_{r}^{\prime}(z^{\prime})$, we obtain from Young’s inequality that

$\displaystyle\int_{Q_{r}^{\prime}(z^{\prime})}\int_{\max\\{\varphi_{k}(x^{\prime}),z^{1}-r\\}}^{\min\\{\varphi_{k+1}(x^{\prime}),z^{1}+r\\}}\frac{\eta^{2}(x^{1},x^{\prime})}{\varphi_{k+1}-\varphi_{k}}dx^{1}dx^{\prime}$
$\displaystyle\quad\leq
2\int_{Q_{r}^{\prime}(z^{\prime})}\int_{z^{1}-r}^{z^{1}+r}\left|D_{y^{1}}\eta(y^{1},x^{\prime})\right|\left|\eta(y^{1},x^{\prime})\right|\,dy^{1}dx^{\prime}$
$\displaystyle\quad\leq\epsilon
r^{\frac{1}{2}}\int_{Q_{r}(z)}|D\eta|^{2}\,dx+\epsilon^{-1}r^{-\frac{1}{2}}\int_{Q_{r}(z)}\eta^{2}\,dx.$

Since $\displaystyle k\in K$ was arbitrary chosen, by combining the above
estimates,

$\sum_{k\in K}\int_{Q_{r}^{k}(z)}|D\pi|^{2}\eta^{2}\,dx\leq c\epsilon
r^{\frac{1}{2}}\int_{Q_{r}(z)}|D\eta|^{2}\,dx+c\left(1+\epsilon^{-1}r^{-\frac{1}{2}}\right)\int_{Q_{r}(z)}\eta^{2}\,dx.$

By Poincaré’s inequality,

$\displaystyle\int_{Q_{r}(z)}\eta^{2}\,dx$
$\displaystyle\leq\left(\int_{Q_{r}(z)}\eta^{2}\,dx\right)^{\frac{1}{2}}\left(\int_{Q_{r}(z)}\eta^{2}\,dx\right)^{\frac{1}{2}}$
$\displaystyle\leq
r\left(\int_{Q_{r}(z)}|D\eta|^{2}\,dx\right)^{\frac{1}{2}}\left(\int_{Q_{r}(z)}\eta^{2}\,dx\right)^{\frac{1}{2}}$
$\displaystyle\leq\epsilon^{2}r\int_{Q_{r}(z)}|D\eta|^{2}\,dx+\epsilon^{-2}r\int_{Q_{r}(z)}\eta^{2}\,dx.$

Since $\displaystyle\epsilon\in(0,1]$ and $\displaystyle r\in(0,6]$, by
combining the above two estimates, we find that

$\int_{Q_{r}(z)}|D\pi|^{2}\eta^{2}\,dx\leq c\epsilon
r^{\frac{1}{2}}\int_{Q_{r}(z)}|D\eta|^{2}\,dx+c\epsilon^{-2}r^{\frac{1}{2}}\int_{Q_{r}(z)}\eta^{2}\,dx.$

Since $\displaystyle\eta\in C_{c}^{\infty}(Q_{r}(z))$ and
$\displaystyle\epsilon\in(0,1]$ were arbitrary chosen, the lemma follows. ∎

With the similar reasoning for the proof of Lemma 2.7, by applying Lemma 2.6
to (2.10), one can also prove that

(2.17) $\int_{Q_{r}(z)}\left|D\left[\pi^{q^{\prime}}\right]\right|^{2}\,dx\leq
cr^{n-1}.$

###  2.4. Heavyside type functions

We define $\displaystyle H_{k}:Q_{5}\to\mathbb{R}$ $\displaystyle\big{(}k\in
K\big{)}$ so that

(2.18) $H_{k}(x)=\left\\{\begin{array}[]{rl}1&\text{ if }x\in Q_{5}^{l}\text{
with }l\geq k,\\\ 0&\text{ if }x\in Q_{5}^{l}\text{ with
}l<k.\end{array}\right.$

Then we have that

(2.19) $\chi_{Q_{5}^{k}}=H_{k}-H_{k+1}~{}\text{ in }~{}Q_{5}.$

##  3\. Perturbation

From now on, assume (1.13) for some $\displaystyle m\geq 1$. With the
assumption (1.8), (1.10), (1.13) and (1.2), let $\displaystyle u$ be a weak
solution of (1.14) which is

(3.1)
$D_{\alpha}\left[A_{\alpha\beta}^{ij}D_{\beta}u^{j}\right]=D_{i}F_{\alpha}^{i}\quad\text{in}\quad
Q_{7}.$

To estimate $\displaystyle D^{m+1}u$, assume that

(3.2) $u\in C^{m,\gamma}\left(Q_{7}^{k},\mathbb{R}^{N}\right)\qquad(k\in K).$

The higher order derivatives of $\displaystyle u$ in $\displaystyle Q_{5}^{k}$
will be perturbed with respect to the points on the graphs
$\displaystyle\left\\{(\varphi_{k}(x^{\prime}),x^{\prime}):x^{\prime}\in
Q_{5}^{\prime}\right\\}$ $\displaystyle(k\in K_{+})$. In view of (1.9),

(3.3) $A_{\alpha\beta}^{ij},F_{\alpha}^{i}\in
C^{m,\gamma}\left(Q_{6}^{k}\right)\text{ is a restriction of
}A_{\alpha\beta}^{ij,k},F_{\alpha}^{i,k}\in
C^{m,\gamma}\left(\mathbb{R}^{n}\right)$

for any $\displaystyle k\in K$, $\displaystyle 1\leq\alpha,\beta\leq n$ and
$\displaystyle 1\leq i,j\leq N$. With (2.1), one can extend $\displaystyle
u\in C^{m,\gamma}\left(Q_{6}^{k},\mathbb{R}^{N}\right)$ to $\displaystyle
u_{k}\in C^{m,\gamma}\left(\partial Q_{6}^{k}\cap\partial
Q_{6}^{k-1},\mathbb{R}^{N}\right)$ for any $\displaystyle k\in K$ so that

(3.4) $\|u_{k}\|_{C^{m,\gamma}\left(\partial Q_{6}^{k}\cap\partial
Q_{6}^{k-1},\mathbb{R}^{N}\right)}\leq
c\|u\|_{C^{m,\gamma}\left(Q_{7}^{k},\mathbb{R}^{N}\right)}.$

Here, we remark that the (higher order) derivatives of $\displaystyle u_{k}$
and $\displaystyle u_{k+1}$ might not coincide on $\displaystyle\partial
Q_{5}^{k}\cap\partial Q_{5}^{k+1}$ $\displaystyle\left(k\in K\right)$. To
obtain $\displaystyle m$-th time derivative, fix the index

(3.5) $p^{\prime}=(p_{2},\cdots,p_{n})\geq
0^{\prime}\quad\text{with}\quad|p^{\prime}|=p_{2}+\cdots+p_{m}=m.$

With an approximation argument by using (2.1), we may assume that

(3.6) $u\in C^{m+1,\gamma}\left(Q_{7}^{k},\mathbb{R}^{N}\right)\qquad(k\in
K).$

We remark that the term
$\displaystyle\left\|D^{m+1}u\right\|_{L^{\infty}(Q_{r}(z))}$ (for fixed
$\displaystyle Q_{r}(z)\subset Q_{5}$) might appear in the estimates. But
because the coefficient on
$\displaystyle\left\|D^{m+1}u\right\|_{L^{\infty}(Q_{r}(z))}$ can be
sufficiently small, this term will removed in Lemma 5.8.

###  3.1. Perturbation on the equation

In this subsection, we obtain a perturbation on the equation related to higher
order derivatives(which will be done in Lemma 3.8). Recall $\displaystyle
T_{k}(x)$ from (2.3). For for any $\displaystyle i\in\\{1,\cdots,n\\}$, set
$\displaystyle\psi_{i}:Q_{6}\times Q_{1}^{\prime}\to\mathbb{R}$ as

(3.7)
$\psi_{i}(x,t^{\prime})=T_{k}(x)\,D_{i}\varphi_{k+1}(x^{\prime}+t^{\prime})+[1-T_{k}(x)]\,D_{i}\varphi_{k}(x^{\prime}+t^{\prime})$

for any $\displaystyle(x,t^{\prime})\in Q_{6}^{k}\times Q_{1}^{\prime}$
$\displaystyle(k\in K)$. Then we find from (3.7) that

(3.8) $\psi_{1}(x,t^{\prime})=0$

for any $\displaystyle(x,t^{\prime})\in Q_{6}\times Q_{1}^{\prime}$. Also it
follows from (2.3) and (3.7) that

(3.9)
$\psi_{i}\big{(}(\varphi_{k}(x^{\prime}),x^{\prime}),t^{\prime}\big{)}=D_{i}\varphi_{k}(x^{\prime}+t^{\prime})$

for any $\displaystyle(x^{\prime},t^{\prime})\in Q_{6}^{\prime}\times
Q_{1}^{\prime}$ and $\displaystyle k\in K$.

To obtain piece-wise regularity, we use the weak formulation in the following
way. We differentiate the equation on the boundary $\displaystyle\partial
Q_{5}^{k}\cap\partial Q_{5}^{k+1}$ by using the flow (3.7). Then we apply
integration by parts formula with respect to each $\displaystyle Q_{5}^{k}$
$\displaystyle(k\in K)$.

In view of (3.6), for any $\displaystyle 1\leq\alpha\leq n$, $\displaystyle
1\leq i\leq N$ and $\displaystyle k\in K$, set

(3.10) $U_{\alpha}^{i,k}=\sum_{1\leq j\leq N}\sum_{1\leq\beta\leq
n}A_{\alpha\beta}^{ij,k}D_{\beta}u^{j,k}-F_{\alpha}^{i,k}\in
C^{m,\gamma}\left(\overline{Q_{6}^{k}}\right),$

and

(3.11)
$\partial_{t^{\prime}}^{p^{\prime}}U_{\alpha}^{i,k}\big{|}_{t^{\prime}=0^{\prime}}=\left.\partial_{t^{\prime}}^{p^{\prime}}\left[U_{\alpha}^{i,k}\big{(}x+(\psi(x,t^{\prime}),t^{\prime})\big{)}\right]\right|_{t^{\prime}=0^{\prime}}\quad\text{in}\quad
Q_{5}^{k}.$

Since $\displaystyle
D_{\alpha}\left[A^{\alpha\beta}_{ij}D_{\beta}u^{j,k}\right]=D_{\alpha}F_{\alpha}^{i,k}$
in $\displaystyle Q_{7}$, we find that for any $\displaystyle 1\leq i\leq N$,

(3.12) $\sum_{1\leq\alpha\leq
n}D_{\alpha}U_{\alpha}^{i,k}=\sum_{1\leq\alpha\leq
n}D_{\alpha}\left[\sum_{1\leq j\leq N}\sum_{1\leq\beta\leq
n}A_{\alpha\beta}^{ij,k}D_{\beta}u^{j,k}-F_{\alpha}^{i,k}\right]=0~{}\text{ in
}~{}Q_{6}^{k}.$

######  Lemma 3.1.

Let $\displaystyle\vec{n}_{k}=(n_{k}^{1},n_{k}^{2},\cdots,n_{k}^{n})$ be the
outward normal vector on $\displaystyle\partial Q_{6}^{k}$. Then for any
$\displaystyle\alpha=2,\cdots,n$ and $\displaystyle k\in K$, we have that

(3.13) $n_{k}^{\alpha}=-D_{\alpha}\varphi_{k}\,n_{k}^{1},\qquad
n_{k+1}^{\alpha}=-D_{\alpha}\varphi_{k}\,n_{k+1}^{1}\qquad\text{and}\qquad
n_{k}^{1}=-n_{k+1}^{1}$

on $\displaystyle\partial Q_{6}^{k}\cap\partial Q_{6}^{k+1}$. Also for any
$\displaystyle i=2,\cdots,n$ and $\displaystyle k\in K$, we have that

(3.14) $n_{k}^{\alpha}=-\psi(x,0^{\prime})\,n_{k}^{1}\quad\text{and}\quad
n_{k+1}^{\alpha}=-\psi(x,0^{\prime})\,n_{k+1}^{1}\quad\text{on}\quad\partial
Q_{6}^{k}\cap\partial Q_{6}^{k+1}.$

###### Proof.

(3.13) holds from Definition 1.1. (3.14) follows from (3.9) and (3.13). ∎

######  Lemma 3.2.

With the assumption (3.10), we have that

$\sum_{1\leq\alpha\leq
n}\left[U_{\alpha}^{i,k}n_{k}^{\alpha}+U_{\alpha}^{i,k+1}n_{k+1}^{\alpha}\right]=0\quad\text{in}\quad\partial
Q_{6}^{k}\cap\partial Q_{6}^{k+1}\qquad(k\in K).$

###### Proof.

Fix $\displaystyle\eta\in C_{c}^{\infty}(Q_{6},\mathbb{R}^{N})$. By (3.10) and
the weak formulation,

(3.15) $\displaystyle\sum_{k\in K}\int_{Q_{6}^{k}}\sum_{1\leq\alpha\leq
n}U_{\alpha}^{i,k}D_{\alpha}\eta^{i}\,dx$
$\displaystyle\quad=\int_{Q_{6}}\sum_{1\leq\alpha\leq n}\left[\sum_{1\leq
j\leq N}\sum_{1\leq\beta\leq
n}A_{\alpha\beta}^{ij,k}D_{\beta}u^{j,k}-F_{i}\right]D_{\alpha}\eta^{i}\,dx=0.$

for any $\displaystyle 1\leq i\leq N$. Apply the integration by parts. Then by
(3.12) and (LABEL:PET260),

$\sum_{k\in K}\int_{\partial Q_{6}^{k}}\sum_{1\leq\alpha\leq
n}\eta^{i}\,U_{\alpha}^{i,k}\,n_{k}^{\alpha}\,dS=\sum_{k\in
K}\int_{Q_{6}^{k}}\sum_{1\leq\alpha\leq
n}\left[D_{\alpha}U_{\alpha}^{i,k}\,\eta^{i}+U_{\alpha}^{i,k}D_{\alpha}\eta^{i}\right]\,dx=0,$

for any $\displaystyle 1\leq i\leq N$. Since $\displaystyle\eta\in
C_{c}^{\infty}\left(Q_{6},\mathbb{R}^{N}\right)$, we find that

(3.16) $0=\sum_{k\in K}\int_{Q_{6}^{\prime}}\sum_{1\leq\alpha\leq
n}\left[\eta^{i}\,U_{\alpha}^{i,k}\,n_{k}^{\alpha}+\eta^{i}\,U_{\alpha}^{i,k+1}\,n_{k+1}^{\alpha}\right]\left(\varphi_{k}(x^{\prime}),x^{\prime}\right)\,dx^{\prime},$

for any $\displaystyle 1\leq i\leq N$. In view of (2.1), fix $\displaystyle
z\in\partial Q_{6}^{k}\cap\partial Q_{6}^{k+1}$ satisfying that $\displaystyle
Q_{r}(z)\subset Q_{6}^{k}\cup Q_{6}^{k+1}$. Then we obtain from (1.13) and
(3.10) that

(3.17)
$\left|U_{\alpha}^{i,k}\,n_{k}^{\alpha}+U_{\alpha}^{i,k+1}\,n_{k+1}^{\alpha}\right|^{2}\left(\varphi_{k}(x^{\prime}),x^{\prime}\right)\text{
is bounded and continuous in }Q_{r}^{\prime}(z^{\prime})$

for any $\displaystyle 1\leq\alpha\leq n$, $\displaystyle 1\leq i\leq N$ and
$\displaystyle k\in K$.

With (3.17), for $\displaystyle m=1,2,\cdots$ and $\displaystyle 1\leq i\leq
N$, choose $\displaystyle\eta^{i,1}_{m}:\left(z^{1}-r,z^{1}+r\right)\to[0,1]$
with $\displaystyle\eta^{i,1}_{m}\in
C_{c}^{\infty}\left(z^{1}-r,z^{1}+r\right)$ and
$\displaystyle\eta^{i}_{m}:Q_{r}^{\prime}(z^{\prime})\to\mathbb{R}$ with
$\displaystyle\eta^{i}_{m}\in
C_{c}^{\infty}\left(Q_{r}^{\prime}(z^{\prime})\right)$ so that

$\eta^{i,1}_{m}=1\quad\text{ in
}\quad\left[z^{1}-(1-2^{-m})r,z^{1}+(1-2^{m})r\right]\qquad(m=1,2,\cdots),$

and

$\eta^{i}_{m}(x^{\prime})\xrightarrow[]{m\to\infty}\sum_{1\leq\alpha\leq
n}\left[U_{\alpha}^{i,k}\,n_{k}^{\alpha}+U_{\alpha}^{i,k+1}\,n_{k+1}^{i}\right]\left(\varphi_{k}(x^{\prime}),x^{\prime}\right)\quad\text{
in }\quad L^{2}\left(Q_{r}^{\prime}(z^{\prime})\right).$

Take $\displaystyle\eta^{i}_{m}=\eta^{i,1}_{m}\cdot\eta^{i}_{m}\in
C_{c}^{\infty}(Q_{r}(z))$ instead of $\displaystyle\eta^{i}$ in (3.16). Then
by letting $\displaystyle m\to\infty$,

$0=\int_{Q_{r}^{\prime}(z^{\prime})}\sum_{1\leq\alpha\leq
n}\left|\sum_{1\leq\alpha\leq
n}U_{\alpha}^{i,k}\,n_{k}^{\alpha}+U_{\alpha}^{i,k+1}\,n_{k+1}^{i}\right|^{2}\left(\varphi_{k}(x^{\prime}),x^{\prime}\right)\,dx^{\prime}.$

So we find from (3.17) that that

$\sum_{1\leq\alpha\leq
n}\left[U_{\alpha}^{i,k}(z)\,n_{k}^{\alpha}(z)+U_{\alpha}^{i,k+1}(z)\,n_{k+1}^{i}(z)\right]=0.$

Since $\displaystyle z\in\partial Q_{6}^{k}\cap\partial Q_{6}^{k+1}$ was
arbitrary chosen, the lemma follows. ∎

######  Lemma 3.3.

For any $\displaystyle 1\leq i\leq N$ and $\displaystyle\eta\in
C_{c}^{\infty}(Q_{5},\mathbb{R}^{N})$, we have that

$\displaystyle 0$ $\displaystyle=\sum_{k\in K}\int_{\partial
Q_{5}^{k}}\partial_{t^{\prime}}^{p^{\prime}}\Big{(}U_{1}^{i,k}\big{(}x+(\psi,t^{\prime})\big{)}\Big{)}\Big{|}_{t^{\prime}=0^{\prime}}\eta^{i}\,n_{k}^{1}\,dS$
$\displaystyle\quad-\sum_{k\in K}\sum_{2\leq\alpha\leq n}\int_{\partial
Q_{5}^{k}}\partial_{t^{\prime}}^{p^{\prime}}\Big{(}\psi_{\alpha}(x,t^{\prime})\,U_{\alpha}^{i,k}\big{(}x+(\psi,t^{\prime})\big{)}\Big{)}\Big{|}_{t^{\prime}=0^{\prime}}\eta^{i}\,n_{k}^{1}\,dS,$

where $\displaystyle\vec{n}_{k}=(n_{k}^{1},n_{k}^{2},\cdots,n_{k}^{n})$ is the
outward normal vector on $\displaystyle\partial Q_{6}^{k}$.

###### Proof.

It follows from Lemma 3.2 that for any $\displaystyle 1\leq i\leq N$ and
$\displaystyle k\in K$,

$\displaystyle U_{1}^{i,k}-\sum_{2\leq\alpha\leq
n}D_{\alpha}\varphi_{k}\,U_{\alpha}^{i,k}=U_{1}^{i,k+1}-\sum_{2\leq\alpha\leq
n}D_{\alpha}\varphi_{k}\,U_{\alpha}^{i,k+1}\quad\text{on}\quad\partial
Q_{6}^{k}\cap\partial Q_{6}^{k+1}.$

Fix $\displaystyle
x=\left(\varphi_{k}(x^{\prime}),x^{\prime}\right)\in\partial
Q_{5}^{k}\cap\partial Q_{5}^{k+1}$ $\displaystyle(k\in K)$. Then for any
$\displaystyle t^{\prime}\in Q_{1}^{\prime}$, we have that

$\displaystyle
U_{1}^{i,k}\left(\varphi_{k}(x^{\prime}+t^{\prime}),x^{\prime}+t^{\prime}\right)-\sum_{2\leq\alpha\leq
n}D_{\alpha}\varphi_{k}\left(x^{\prime}+t^{\prime}\right)\,U_{\alpha}^{i,k}\left(\varphi_{k}(x^{\prime}+t^{\prime}),x^{\prime}+t^{\prime}\right)$
$\displaystyle\
=U_{1}^{i,k+1}\left(\varphi_{k}(x^{\prime}+t^{\prime}),x^{\prime}+t^{\prime}\right)-\sum_{2\leq\alpha\leq
n}D_{\alpha}\varphi_{k}\left(x^{\prime}+t^{\prime}\right)\,U_{\alpha}^{i,k+1}\left(\varphi_{k}(x^{\prime}+t^{\prime}),x^{\prime}+t^{\prime}\right),$

for any $\displaystyle 1\leq i\leq N$. Since $\displaystyle
x=\left(\varphi_{k}(x^{\prime}),x^{\prime}\right)\in\partial
Q_{5}^{k}\cap\partial Q_{5}^{k+1}(\subset Q_{6})$, we find from (3.9) that

$\psi_{\alpha}(x,t)=\psi_{\alpha}\big{(}(\varphi_{k}(x^{\prime}),x^{\prime}),t^{\prime}\big{)}=D_{\alpha}\varphi_{k}\left(x^{\prime}+t^{\prime}\right)\qquad\left(t^{\prime}\in
Q_{1}^{\prime},~{}1\leq\alpha\leq n\right).$

Also from (2.4) and that $\displaystyle
x=\left(\varphi_{k}(x^{\prime}),x^{\prime}\right)\in\partial
Q_{5}^{k}\cap\partial Q_{5}^{k+1}(\subset Q_{6})$, we find that

$x+\left(\psi(x,t^{\prime}),t^{\prime}\right)=\left(\varphi_{k}(x^{\prime}+t^{\prime}),x^{\prime}+t^{\prime}\right)\qquad\left(t^{\prime}\in
Q_{1}^{\prime}\right).$

So by combining the above three equality,

$\displaystyle
U_{1}^{i,k}\left(x+(\psi,t^{\prime})\right)-\sum_{2\leq\alpha\leq
n}\psi_{\alpha}(x,t^{\prime})U_{\alpha}^{i,k}\left(x+(\psi,t^{\prime})\right)$
$\displaystyle\quad=U_{1}^{i,k+1}\left(x+(\psi,t^{\prime})\right)-\sum_{2\leq\alpha\leq
n}\psi_{\alpha}(x,t^{\prime})U_{\alpha}^{i,k+1}\left(x+(\psi,t^{\prime})\right)$

for any $\displaystyle t^{\prime}\in Q_{1}^{\prime}$ and $\displaystyle 1\leq
i\leq N$. Then by using that
$\displaystyle\partial_{t^{\prime}}^{p^{\prime}}=\partial_{t^{2}}^{p_{2}}\cdots\partial_{t^{n}}^{p_{n}}$,
we find that

$\displaystyle\left.\partial_{t^{\prime}}^{p^{\prime}}\left[U_{1}^{i,k}\left(x+(\psi,t^{\prime})\right)-\sum_{2\leq\alpha\leq
n}\psi_{\alpha}(x,t^{\prime})U_{\alpha}^{i,k}\left(x+(\psi,t^{\prime})\right)\right]\right|_{t^{\prime}=0^{\prime}}$
$\displaystyle\quad=\left.\partial_{t^{\prime}}^{p^{\prime}}\left[U_{1}^{i,k+1}\left(x+(\psi,t^{\prime})\right)-\sum_{2\leq\alpha\leq
n}\psi_{\alpha}(x,t^{\prime})U_{\alpha}^{i,k+1}\left(x+(\psi,t^{\prime})\right)\right]\right|_{t^{\prime}=0^{\prime}}.$

Since $\displaystyle
x=\left(\varphi_{k}(x^{\prime}),x^{\prime}\right)\in\partial
Q_{5}^{k}\cap\partial Q_{5}^{k+1}$ $\displaystyle(k\in K)$ was arbitrary
chosen and $\displaystyle\eta\in
C_{c}^{\infty}\left(Q_{5},\mathbb{R}^{N}\right)$, the lemma holds from that
$\displaystyle n_{k}^{1}=-n_{k+1}^{1}$ for any $\displaystyle k\in K$. ∎

Let $\displaystyle\vec{n}_{k}=\left(n_{k}^{1},\cdots,n_{k}^{n}\right)$ be the
outward normal vector on $\displaystyle\partial Q_{5}^{k}$. For any
$\displaystyle\eta\in C_{c}^{\infty}\left(Q_{5},\mathbb{R}^{N}\right)$, we
have from Lemma 3.3 that

(3.18) $I^{i}+II^{i}+III^{i}=0\qquad(1\leq i\leq N),$

where

$\displaystyle I^{i}$ $\displaystyle=\sum_{k\in K}\int_{\partial
Q_{5}^{k}}\partial_{t^{\prime}}^{p^{\prime}}\Big{(}U_{1}^{i,k}\big{(}x+(\psi,t^{\prime})\big{)}\Big{)}\Big{|}_{t^{\prime}=0^{\prime}}\eta^{i}\,n_{k}^{1}\,dS,$
$\displaystyle II^{i}$ $\displaystyle=-\sum_{k\in K}\sum_{2\leq\alpha\leq
n}\int_{\partial
Q_{5}^{k}}\partial_{t^{\prime}}^{p^{\prime}}\Big{(}\big{[}\psi_{\alpha}(x,t^{\prime})-\psi_{\alpha}(x,0^{\prime})\big{]}U_{\alpha}^{i,k}\big{(}x+(\psi,t^{\prime})\big{)}\Big{)}\Big{|}_{t^{\prime}=0^{\prime}}\eta^{i}\,n_{k}^{1}\,dS,$
$\displaystyle III^{i}$ $\displaystyle=-\sum_{k\in K}\sum_{2\leq\alpha\leq
n}\int_{\partial
Q_{5}^{k}}\partial_{t^{\prime}}^{p^{\prime}}\Big{(}\psi_{\alpha}(x,0^{\prime})\,U_{\alpha}^{i,k}\big{(}x+(\psi,t^{\prime})\big{)}\Big{)}\Big{|}_{t^{\prime}=0^{\prime}}\eta^{i}\,n_{k}^{1}\,dS,$

Then by Lemma 3.1, for any $\displaystyle\eta^{i}\in C_{c}^{\infty}(Q_{5})$.

$\displaystyle III^{i}$ $\displaystyle=-\sum_{k\in K}\sum_{2\leq\alpha\leq
n}\int_{\partial
Q_{5}^{k}}\partial_{t^{\prime}}^{p^{\prime}}\big{[}\psi_{\alpha}(x,0^{\prime})\,U_{\alpha}^{i,k}\big{(}x+(\psi,t^{\prime})\big{)}\big{]}\Big{|}_{t^{\prime}=0^{\prime}}\eta^{i}\,n_{k}^{1}\,dS$
$\displaystyle=\sum_{k\in K}\sum_{2\leq\alpha\leq n}\int_{\partial
Q_{5}^{k}}\partial_{t^{\prime}}^{p^{\prime}}\big{[}U_{\alpha}^{i,k}\big{(}x+(\psi,t^{\prime})\big{)}\big{]}\Big{|}_{t^{\prime}=0^{\prime}}\eta^{i}\,n_{k}^{\alpha}\,dS.$

Thus

(3.19) $I^{i}+III^{i}=\sum_{k\in K}\sum_{1\leq\alpha\leq n}\int_{\partial
Q_{5}^{k}}\partial_{t^{\prime}}^{p^{\prime}}\big{[}U_{\alpha}^{i,k}\big{(}x+(\psi,t^{\prime})\big{)}\big{]}\Big{|}_{t^{\prime}=0^{\prime}}\eta^{i}\,n_{k}^{\alpha}\,dS.$

We first estimate $\displaystyle I^{i}$ and $\displaystyle III^{i}$
$\displaystyle(1\leq i\leq N)$. Recall from (2.11) in Lemma 2.3 that

(3.20)
$\partial_{t^{\prime}}^{p^{\prime}}h\big{|}_{t^{\prime}=0^{\prime}}=\sum_{0^{\prime}\leq
q^{\prime}\leq p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}h+\sum_{|\xi|\leq|p^{\prime}|-1}P_{\xi}^{p^{\prime}}(x)D_{\xi}h~{}\text{
in }~{}Q_{5}.$

With (3.10), we find from (3.19) and (3.20), for any $\displaystyle\eta^{i}\in
C_{c}^{\infty}\left(Q_{5},\mathbb{R}^{N}\right)$,

(3.21) $\displaystyle I^{i}+III^{i}$ $\displaystyle=\sum_{1\leq\alpha\leq
n}\sum_{k\in K}\int_{\partial Q_{5}^{k}}\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}U_{\alpha}^{i,k}\,\eta^{i}\
n_{k}^{\alpha}\,dS$ $\displaystyle\quad+\sum_{1\leq\alpha\leq n}\sum_{k\in
K}\int_{\partial
Q_{5}^{k}}\sum_{|\xi|\leq|p^{\prime}|-1}P_{\xi}^{p^{\prime}}(x)D_{\xi}U_{\alpha}^{i,k}\,\eta^{i}\
n_{k}^{\alpha}\,dS$ $\displaystyle:=IV_{1}^{i}+IV_{2}^{i}.$

for any $\displaystyle 1\leq i\leq N$.

We estimate $\displaystyle IV_{1}^{i}$ by applying the integration by parts
with respect to $\displaystyle x^{1}$-variable. In view of (1.8), (1.15) and
(3.6), for the simplicity of the calculation, we set

(3.22) $E_{z,r}=\sum_{k\in
K}\left[\|D^{m+1}u\|_{L^{\infty}(Q_{r}^{k}(z))}+\|u\|_{C^{m,\gamma}(Q_{7}^{k})}+\|F\|_{C^{m,\gamma}(Q_{7}^{k})}\right],$

for any $\displaystyle Q_{r}(z)\subset Q_{5}$. To handle $\displaystyle
IV_{2}^{i}$, we obtain the following Lemma 3.4. In Lemma 3.4, $\displaystyle
f_{k}$ only need to be defined on the set

$\partial Q_{5}^{k}\cap\left(\bigcup_{l\in\left(\\{k,k+1\\}\cap
K_{-}\right)}\left\\{\left(\varphi_{l}(x^{\prime}),x^{\prime}\right):x\in
Q_{5}^{\prime}\right\\}\right)~{}\left(\subset\overline{Q_{6}^{k}}\right)$

(where the inclusion holds from (1.3)) but we define on the set
$\displaystyle\overline{Q_{6}^{k}}$ for the simplicity.

######  Lemma 3.4.

Under the assumption (1.3), suppose that $\displaystyle
g_{k}:\overline{Q_{6}^{k}}\to\mathbb{R}$ and $\displaystyle
h:\overline{Q_{6}}\to\mathbb{R}$ satisfies that $\displaystyle g_{k},h\in
C^{\gamma}\left(\overline{Q_{6}^{k}}\right)$ for any $\displaystyle k\in K$.
Then for any $\displaystyle\eta\in C_{c}^{\infty}(Q_{5})$,

$\sum_{k\in K}\int_{\partial Q_{5}^{k}}g_{k}h\eta\,n_{k}^{1}\,dS=\sum_{k\in
K_{-}}\int_{Q_{5}}\mathbb{G}_{k}\left[g_{k-1}-g_{k},h\right]H_{k}D_{1}\eta\,dx,$

where the linear operator
$\displaystyle\mathbb{G}_{k}:C^{\gamma}\left(\partial Q_{6}^{k}\cap\partial
Q_{6}^{k-1}\right)\times C^{\gamma}\left(\partial Q_{6}^{k}\cap\partial
Q_{6}^{k-1}\right)\to C^{\gamma}\left(Q_{5}^{\prime}\right)$ is defined as

$\mathbb{G}_{k}[g,h](x)=\left[gh\right]\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}.$

###### Proof.

By the definition of $\displaystyle K_{-}$ in (1.3),

$\displaystyle\sum_{k\in K}\int_{\partial Q_{5}^{k}}g_{k}h\eta\,n_{k}^{1}\,dS$
$\displaystyle=-\sum_{k+1\in
K_{-}}\int_{Q_{5}\cap\left\\{\left(\varphi_{k+1}(x^{\prime}),x^{\prime}\right):x^{\prime}\in
Q_{5}^{\prime}\right\\}}g_{k}h\eta\,n_{k}^{1}\,dS$
$\displaystyle\qquad-\sum_{k\in
K_{-}}\int_{Q_{5}\cap\left\\{\left(\varphi_{k}(x^{\prime}),x^{\prime}\right):x^{\prime}\in
Q_{5}^{\prime}\right\\}}g_{k}h\eta\,n_{k}^{1}\,dS.$

By using (2.18) and integration by parts with respect to $\displaystyle
x^{1}$-variable,

$\displaystyle\sum_{k+1\in
K_{-}}\int_{Q_{5}\cap\left\\{\left(\varphi_{k+1}(x^{\prime}),x^{\prime}\right):x^{\prime}\in
Q_{5}^{\prime}\right\\}}g_{k}h\eta\,n_{k}^{1}\,dS$
$\displaystyle\quad=\sum_{k+1\in
K_{-}}\int_{Q_{5}}\left[g_{k}h\right]\big{(}\varphi_{k+1}(x^{\prime}),x^{\prime}\big{)}D_{1}\eta\,H_{k+1}\,dx$
$\displaystyle\quad=\sum_{k\in
K_{-}}\int_{Q_{5}}\left[g_{k-1}h\right]\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}D_{1}\eta\,H_{k}\,dx,$

and

$\displaystyle\sum_{k\in
K_{-}}\int_{Q_{5}\cap\left\\{\left(\varphi_{k}(x^{\prime}),x^{\prime}\right):x^{\prime}\in
Q_{5}^{\prime}\right\\}}g_{k}h\eta\,n_{k}^{1}\,dS$
$\displaystyle\quad=\sum_{k\in
K_{-}}\int_{Q_{5}}\left[g_{k}h\right]\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}D_{1}\eta\,(1-H_{k})\,dx,$

where we used that $\displaystyle\eta\in C_{c}^{\infty}(Q_{5})$. So the lemma
holds by combining the above three equality and that $\displaystyle\sum_{k\in
K_{-}}\int_{Q_{5}}\left[g_{k}h\right]\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}D_{1}\eta\,dx=0$.
∎

We now estimate $\displaystyle I^{i}+III^{i}$ in Lemma 3.5 and $\displaystyle
II^{i}$ in Lemma 3.6. We remark that $\displaystyle G_{1}$ in Lemma 3.5 and
Lemma 3.6 are Hölder continuous in each subregion $\displaystyle Q_{5}^{k}$
$\displaystyle(k\in K)$. Moreover, $\displaystyle\mathbb{G}_{1,k}[h]$ only
depends on $\displaystyle x^{\prime}$-variables and $\displaystyle G_{1}$
might have big jumps on the graphs
$\displaystyle\left\\{\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}:x^{\prime}\in
Q_{5}^{\prime}\right\\}$ $\displaystyle(k\in K_{-})$.

######  Lemma 3.5.

For any $\displaystyle\eta\in
C_{c}^{\infty}\left(Q_{r}(z),\mathbb{R}^{N}\right)$ with $\displaystyle
Q_{r}(z)\subset Q_{5}$, we have that

$\displaystyle\left|I^{i}+III^{i}-\sum_{k\in
K}\int_{Q_{r}^{k}(z)}\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}D_{1}\left[D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}U_{1}^{i,k}\eta^{i}\right]\,+\,G_{1}^{i}D_{1}\eta^{i}\,dx\right|$
$\displaystyle\quad\leq
cr^{\frac{1}{4}}\int_{Q_{r}(z)}\epsilon|D\eta|^{2}+\epsilon^{-2}|\eta|^{2}+E_{z,r}^{2}\,dx,$

for any $\displaystyle 1\leq i\leq N$ and $\displaystyle\epsilon\in(0,1]$,
where

(3.23) $G_{1}^{i}=\sum_{k\in K_{-}}\sum_{1\leq\alpha\leq n}\
\mathbb{G}_{\alpha}^{k}\left[U_{\alpha}^{i,k-1}-U_{\alpha}^{i,k}\right]H_{k}$

with the linear operator $\displaystyle\
\mathbb{G}_{\alpha}^{k}:C^{m-1,\gamma}\left(\partial Q_{6}^{k}\cap\partial
Q_{6}^{k-1}\right)\to C^{\gamma}\left(Q_{5}^{\prime}\right)$
$\displaystyle(k\in K_{-})$ defined as

(3.24) $\
\mathbb{G}_{\alpha}^{k}[h](x^{\prime})=\sum_{|\xi|\leq|p^{\prime}|-1}\left[\pi_{\alpha}\,P_{\xi}^{p^{\prime}}D_{\xi}h\right]\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}.$

###### Proof.

With respect to $\displaystyle x^{1}$-variable, use integration by parts to
$\displaystyle IV_{1}$ in (3.21). Then

$\displaystyle IV_{1}^{i}$ $\displaystyle=\sum_{k\in K}\sum_{1\leq\alpha\leq
n}\int_{Q_{5}^{k}}\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}D_{\alpha}\left[\pi^{q^{\prime}}\right]D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}U_{\alpha}^{i,k}\,\eta^{i}\,dx$
$\displaystyle\quad+\sum_{k\in K}\sum_{1\leq\alpha\leq
n}\int_{Q_{5}^{k}}\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}D_{\alpha}\Big{[}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}U_{\alpha}^{i,k}\eta^{i}\Big{]}\,dx.$

By that $\displaystyle Q_{r}(z)\subset Q_{5}$,

(3.25) $\displaystyle IV_{1}^{i}$ $\displaystyle=\sum_{k\in
K}\sum_{1\leq\alpha\leq n}\int_{Q_{r}^{k}(z)}\sum_{0^{\prime}\leq
q^{\prime}\leq p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}D_{\alpha}\left[\pi^{q^{\prime}}\right]D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}U_{\alpha}^{i,k}\,\eta^{i}\,dx$
$\displaystyle\quad+\sum_{k\in K}\sum_{1\leq\alpha\leq
n}\int_{Q_{r}^{k}(z)}\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}D_{\alpha}\Big{[}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}U_{\alpha}^{i,k}\eta^{i}\Big{]}\,dx.$

From (1.2) and (2.7), we have that
$\displaystyle|\pi_{2}|,\cdots,|\pi_{n}|\leq c$. So by Young’s inequality,

$\displaystyle\bigg{|}\sum_{k\in K}\int_{Q_{r}^{k}(z)}\sum_{1\leq\alpha\leq
n}\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}D_{\alpha}\left[\pi^{q^{\prime}}\right]D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}U_{\alpha}^{i,k}\eta^{i}\,dx\bigg{|}$
$\displaystyle\quad\leq
c\int_{Q_{r}(z)}\big{|}D\pi\big{|}E_{z,r}\left|\eta\right|\,dx$
$\displaystyle\quad\leq
c\int_{Q_{r}(z)}r^{\frac{1}{4}}E_{z,r}^{2}+r^{-\frac{1}{4}}|D\pi|^{2}\left|\eta\right|^{2}\,dx,$

where the definition of $\displaystyle E_{z,r}$ in (3.22) is used. So by
(3.25) and Lemma 2.7,

$\displaystyle\left|IV_{1}^{i}-\sum_{k\in
K}\int_{Q_{r}^{k}(z)}\sum_{1\leq\alpha\leq n}\sum_{0^{\prime}\leq
q^{\prime}\leq p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}D_{1}\left[D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}U_{1}^{i,k}\eta^{i}\right]\,dx\right|$
$\displaystyle\quad\leq
cr^{\frac{1}{4}}\int_{Q_{r}(z)}\epsilon|D\eta|^{2}+\epsilon^{-2}|\eta|^{2}+E_{z,r}^{2}\,dx.$

We next estimate $\displaystyle IV_{2}$. By (2.9) and Lemma 3.1, we find that

$n_{k}^{\alpha}=-\pi_{\alpha}\,n_{k}^{1}\quad\text{ on }\quad\partial
Q_{5}^{k}\cap\partial Q_{5}^{k+1},$

for any $\displaystyle i=1,2,\cdots,n$ and $\displaystyle k\in K$. So by
(3.21),

$\displaystyle IV_{2}^{i}$ $\displaystyle=\sum_{k\in K}\int_{\partial
Q_{5}^{k}}\sum_{1\leq\alpha\leq
n}\sum_{|\xi|\leq|p^{\prime}|-1}P_{\xi}^{p^{\prime}}(x)D_{\xi}U_{\alpha}^{i,k}\,\eta^{i}\
n_{k}^{\alpha}\,dS$ $\displaystyle=-\sum_{k\in K}\int_{\partial
Q_{5}^{k}}\sum_{1\leq\alpha\leq
n}\sum_{|\xi|\leq|p^{\prime}|-1}P_{\xi}^{p^{\prime}}(x)D_{\xi}U_{\alpha}^{i,k}\,\eta^{i}\,\pi_{\alpha}\,n_{k}^{1}\,dS.$

With (2.1), we have from (3.10), (2.7) and the definition of $\displaystyle
P_{\xi}^{p^{\prime}}$ in Lemma 2.3 that

$D_{\xi}U_{\alpha}^{i,k},\pi_{\alpha}P_{\xi}^{p^{\prime}}\in
C^{\gamma}\left(\partial Q_{6}^{k}\cap\partial Q_{6}^{k-1}\right)\qquad(k\in
K,\,i\in\\{1,\cdots,n\\}),$

for any $\displaystyle|\xi|\leq|p^{\prime}|-1$. So take $\displaystyle
g_{k}=D_{\xi}U_{\alpha}^{i,k}$ and $\displaystyle
h=\pi_{\alpha}P_{\xi}^{p^{\prime}}$ in Lemma 3.4 to find that

$IV_{2}^{i}=-\sum_{k\in K}\int_{Q_{5}^{k}}\,G_{1}D_{1}\eta^{i}\,dx$

with

$G_{1}^{i}=\sum_{k\in K_{-}}\sum_{1\leq\alpha\leq
n}\sum_{|\xi|\leq|p^{\prime}|-1}\mathbb{G}_{k}\left[D_{\xi}U_{\alpha}^{i,k-1}-D_{\xi}U_{\alpha}^{i,k},\pi_{\alpha}P_{\xi}^{p^{\prime}}\right]H_{k}$

where $\displaystyle\mathbb{G}_{k}$ is defined in Lemma 3.4. So by comparing
the definition of $\displaystyle\mathbb{G}_{k}$ in Lemma 3.4 and
$\displaystyle\ \mathbb{G}_{\alpha}^{k}$ in (3.24), the lemma follows form
that $\displaystyle IV_{1}^{i}+IV_{2}^{i}=I^{i}+III^{i}$ in (3.21)
$\displaystyle(1\leq i\leq N)$ and that $\displaystyle Q_{r}(z)\subset Q_{5}$.
∎

######  Lemma 3.6.

For any $\displaystyle\eta\in
C_{c}^{\infty}\left(Q_{r}(z),\mathbb{R}^{N}\right)$ with $\displaystyle
Q_{r}(z)\subset Q_{5}$, we have that

$\displaystyle II^{i}=$
$\displaystyle-\int_{Q_{r}(z)}G_{1}^{i}D_{1}\eta^{i}\,dx,$

for any $\displaystyle 1\leq i\leq N$, where

$G_{1}^{i}=\sum_{k\in K_{-}}\sum_{1\leq\alpha\leq n}\
\mathbb{G}_{\alpha}^{k}\left[U_{\alpha}^{i,k-1}-U_{\alpha}^{i,k}\right]H_{k},$

with the linear operator $\displaystyle\
\mathbb{G}_{\alpha}^{k}:C^{m-1,\gamma}\left(\partial Q_{6}^{k}\cap\partial
Q_{6}^{k-1}\right)\to C^{\gamma}\left(Q_{5}^{\prime}\right)$
$\displaystyle(k\in K_{-})$ defined as

$\ \mathbb{G}_{\alpha}^{k}[h](x^{\prime})=\sum_{1\leq\alpha\leq
n}\sum_{0^{\prime}<q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}D^{q^{\prime}}D_{\alpha}\varphi_{k}(x^{\prime})\,D^{p^{\prime}-q^{\prime}}\left[h\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\right].$

###### Proof.

Recall from (3.18) that

$\displaystyle II^{i}=-\sum_{k\in K}\sum_{2\leq\alpha\leq n}\int_{\partial
Q_{5}^{k}}\partial_{t^{\prime}}^{p^{\prime}}\Big{(}\big{[}\psi_{\alpha}(x,t^{\prime})-\psi_{\alpha}(x,0^{\prime})\big{]}U_{\alpha}^{i,k}\big{(}x+(\psi,t^{\prime})\big{)}\Big{)}\Big{|}_{t^{\prime}=0^{\prime}}\eta^{i}\,n_{k}^{1}\,dS.$

By a direct calculation,

$\displaystyle II^{i}=-\sum_{k\in K}\int_{\partial
Q_{5}^{k}}\sum_{2\leq\alpha\leq n}\sum_{0^{\prime}<q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\partial_{t^{\prime}}^{q^{\prime}}\psi_{\alpha}(x,t^{\prime})\Big{|}_{t^{\prime}=0^{\prime}}\partial_{t^{\prime}}^{p^{\prime}-q^{\prime}}\big{[}U_{\alpha}^{i,k}\big{(}x+(\psi,t^{\prime})\big{)}\big{]}\Big{|}_{t^{\prime}=0^{\prime}}\eta^{i}\,n_{k}^{1}\,dS.$

We now compute the terms in the above integral. From (3.9), we have that

(3.26)
$\displaystyle\left.\partial_{t^{\prime}}^{q^{\prime}}\psi_{\alpha}\big{(}(\varphi_{k}(x^{\prime}),x^{\prime}),t^{\prime}\big{)}\right|_{t^{\prime}=0^{\prime}}=\left.\partial_{t^{\prime}}^{q^{\prime}}D_{\alpha}\varphi_{k}(x^{\prime}+t^{\prime})\right|_{t^{\prime}=0^{\prime}}=D^{q^{\prime}}D_{\alpha}\varphi_{k}(x^{\prime}),$

or any $\displaystyle i=2,\cdots,n$ and $\displaystyle k\in K_{-}$. By (2.4),
for any $\displaystyle k\in K_{-}$, we have that

$\displaystyle(\varphi_{k}(x^{\prime}),x^{\prime})+\big{(}\psi\big{(}(\varphi_{k}(x^{\prime}),x^{\prime}),t^{\prime}\big{)},t^{\prime}\big{)}$
$\displaystyle=\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}+\big{(}\varphi_{k}(x^{\prime}+t^{\prime})-\varphi_{k}(x^{\prime}),t^{\prime}\big{)}$
$\displaystyle=\big{(}\varphi_{k}(x^{\prime}+t^{\prime}),x^{\prime}+t^{\prime}\big{)},$

which implies that

(3.27)
$\displaystyle\left.\partial_{t^{\prime}}^{p^{\prime}-q^{\prime}}\left[h\Big{(}(\varphi_{k}(x^{\prime}),x^{\prime})+\big{(}\psi\big{(}(\varphi_{k}(x^{\prime}),x^{\prime}),t^{\prime}\big{)},t^{\prime}\big{)}\Big{)}\right]\right|_{t^{\prime}=0^{\prime}}$
$\displaystyle\quad=\left.\partial_{t^{\prime}}^{p^{\prime}-q^{\prime}}\left[h\big{(}\varphi_{k}(x^{\prime}+t^{\prime}),x^{\prime}+t^{\prime}\big{)}\right]\right|_{t^{\prime}=0^{\prime}}$
$\displaystyle\quad=D^{p^{\prime}-q^{\prime}}\big{[}h\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\big{]}.$

With (3.26) and (LABEL:PET780), for any $\displaystyle
0^{\prime}<q^{\prime}\leq p^{\prime}$, take $\displaystyle
g_{k}=\partial_{t^{\prime}}^{p^{\prime}-q^{\prime}}\big{[}U_{\alpha}^{i,k}\big{(}x+(\psi,t^{\prime})\big{)}\big{]}\Big{|}_{t^{\prime}=0^{\prime}}$
and $\displaystyle
h=\partial_{t^{\prime}}^{q^{\prime}}\psi_{\alpha}(x,t^{\prime})\Big{|}_{t^{\prime}=0^{\prime}}$
in Lemma 3.4 to find that

(3.28) $II^{i}=-\sum_{k\in K_{-}}\int_{Q_{5}^{k}}\sum_{2\leq\alpha\leq n}\
\mathbb{G}_{\alpha}^{k}\left[U_{\alpha}^{i,k-1}-U_{\alpha}^{i,k}\right]H_{k}D_{1}\eta^{i}\,dx,$

for the operator $\displaystyle\
\mathbb{G}_{\alpha}^{k}:C^{m-1,\gamma}\left(\overline{Q_{6}^{k}}\right)\to
C^{\gamma}\left(Q_{5}^{\prime}\right)$ defined as

$\ \mathbb{G}_{\alpha}^{k}[h](x)=\sum_{0^{\prime}<q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}D^{q^{\prime}}D_{\alpha}\varphi_{k}(x^{\prime})\,D^{p^{\prime}-q^{\prime}}\left[h\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\right].$

The lemma holds by that $\displaystyle\eta\in
C_{c}^{\infty}\left(Q_{r}(z),\mathbb{R}^{N}\right)$ and $\displaystyle
D_{1}\varphi_{k}(x^{\prime})=0$ $\displaystyle(k\in K_{-})$. ∎

By combining Lemma 3.5 and Lemma 3.6, we obtain the following lemma.

######  Lemma 3.7.

For any $\displaystyle\eta\in
C_{c}^{\infty}\left(Q_{r}(z),\mathbb{R}^{N}\right)$ with $\displaystyle
Q_{r}(z)\subset Q_{5}$, we have that

$\displaystyle\left|\sum_{k\in K}\int_{Q_{r}^{k}(z)}\sum_{0^{\prime}\leq
q^{\prime}\leq p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}U_{\alpha}^{i,k}D_{\alpha}\eta^{i}\,dx-\int_{Q_{r}(z)}G_{1}^{i}D_{1}\eta^{i}\,dx\right|$
$\displaystyle\quad\leq
cr^{\frac{1}{4}}\int_{Q_{r}(z)}\epsilon\left|D\eta\right|^{2}+\epsilon^{-2}\left|\eta\right|^{2}+E_{z,r}^{2}\,dx,$

for any $\displaystyle 1\leq i\leq N$ and $\displaystyle\epsilon\in(0,1]$,
where

(3.29) $G_{1}^{i}=\sum_{k\in K_{-}}\sum_{1\leq\alpha\leq n}\
\mathbb{G}_{\alpha}^{k}\left[U_{\alpha}^{i,k-1}-U_{\alpha}^{i,k}\right]H_{k}$

with the linear operator $\displaystyle\
\mathbb{G}_{\alpha}^{k}:C^{m-1,\gamma}\left(\partial Q_{6}^{k}\cap\partial
Q_{6}^{k-1}\right)\to C^{\gamma}\left(Q_{5}^{\prime}\right)$
$\displaystyle(k\in K_{-})$ defined as

$\displaystyle\ \mathbb{G}_{\alpha}^{k}[h](x^{\prime})$
$\displaystyle=\sum_{|\xi|\leq|p^{\prime}|-1}\left[\pi_{\alpha}P_{\xi}^{p^{\prime}}D_{\xi}h\right]\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}$
$\displaystyle\quad+\sum_{0^{\prime}<q^{\prime}\leq
p^{\prime}}\left(\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\right)D^{q^{\prime}}D_{\alpha}\varphi_{k}(x^{\prime})\,D^{p^{\prime}-q^{\prime}}\left[h\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\right].$

###### Proof.

Since $\displaystyle I+II+III=0$, by (3.21), Lemma 3.5 and Lemma 3.6,

$\displaystyle\bigg{|}\sum_{k\in K}\int_{Q_{r}^{k}(z)}\sum_{0^{\prime}\leq
q^{\prime}\leq p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}D_{\alpha}\Big{[}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}U_{\alpha}^{i,k}\eta^{i}\Big{]}\,dx-\int_{Q_{r}(z)}G_{1}^{i}D_{1}\eta^{i}\,dx\bigg{|}$
$\displaystyle\quad\leq
cr^{\frac{1}{4}}\int_{Q_{r}(z)}\epsilon\left|D\eta\right|^{2}+\epsilon^{-2}\left|\eta\right|^{2}+E_{z,r}^{2}\,dx.$

By a direct calculation,

$D_{\alpha}\Big{[}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}U_{\alpha}^{i,k}\eta^{i}\Big{]}=D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}D_{\alpha}U_{\alpha}^{i,k}\eta^{i}+D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}U_{\alpha}^{i,k}D_{\alpha}\eta^{i}.$

From (3.10) and (3.12), we have that $\displaystyle
D_{\alpha}U_{\alpha}^{i,k}=0$ in $\displaystyle Q_{r}^{k}(z)$. So the lemma
holds. ∎

So we obtain the desired estimate of this subsection.

######  Lemma 3.8.

For any $\displaystyle\eta\in
C_{c}^{\infty}\left(Q_{r}(z),\mathbb{R}^{N}\right)$ with $\displaystyle
Q_{r}(z)\subset Q_{5}$, we have that

$\displaystyle\left|\int_{Q_{r}(z)}\sum_{1\leq\alpha\leq n}\left[\sum_{1\leq
j\leq N}\sum_{1\leq\beta\leq n}A_{\alpha\beta}^{ij,k}\sum_{0^{\prime}\leq
q^{\prime}\leq p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}D_{\beta}u^{j,k}+\nu_{\alpha}^{i}\right]D_{\alpha}\eta^{i}-G_{1}^{i}D_{1}\eta^{i}\,dx\right|$
$\displaystyle\quad\leq
cr^{n+\frac{1}{4}}\left[\|D^{m+1}u\|_{L^{\infty}(Q_{r}(z))}^{2}+\sum_{k\in
K}\left[\|u\|_{C^{m,\gamma}(Q_{7}^{k})}^{2}+\|F\|_{C^{m,\gamma}(Q_{7}^{k})}^{2}\right]\right]$
$\displaystyle\qquad+cr^{\frac{1}{4}}\int_{Q_{r}(z)}\epsilon|D\eta|^{2}+\epsilon^{-2}|\eta|^{2}\,dx,$

for any $\displaystyle 1\leq i\leq N$, with $\displaystyle
G_{1}^{i}:Q_{5}\to\mathbb{R}$ in Lemma 3.7 and

$\nu_{\alpha}^{i}=\sum_{k\in K}\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}\Big{[}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}U_{\alpha}^{i,k}-\sum_{1\leq
j\leq N}\sum_{1\leq\beta\leq
n}A_{\alpha\beta}^{ij,k}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}D_{\beta}u^{j,k}\bigg{]}\chi_{Q_{7}^{k}}$

for any $\displaystyle 1\leq\alpha\leq n$ and $\displaystyle 1\leq i\leq N$.
Moreover, we have that

(3.30) $\left\|G_{1}^{i}\right\|_{C^{\gamma}(Q_{5}^{k})}+\sum_{k\in
K}\left\|\nu_{\alpha}^{i}\right\|_{C^{\gamma}(Q_{5}^{k})}\leq c\sum_{k\in
K}\left[\|u\|_{C^{m,\gamma}(Q_{7}^{k})}+\|F\|_{C^{m,\gamma}(Q_{7}^{k})}\right],$

for any $\displaystyle 1\leq\alpha\leq n$ and $\displaystyle 1\leq i\leq N$.

###### Proof.

With (3.22) and (2.19), the estimate holds from Lemma 3.7. With (3.10), we
find from (3.2) and (3.29) in Lemma 3.7 that (3.30) holds. ∎

###  3.2. Choosing test function

We will choose a suitable test function in $\displaystyle Q_{r}(z)$ by using
$\displaystyle\eta$ in (3.37) (which will be used in Lemma 5.1 and Lemma 5.2)
With (2.1), set $\displaystyle Z_{k}:Q_{7}\to\mathbb{R}$
$\displaystyle\big{(}k\in K\big{)}$ as

(3.31) $Z_{k}(x):=\left\\{\begin{array}[]{ccc}\varphi_{k+1}(x^{\prime})&\text{
if }&x^{1}>\varphi_{k+1}(x^{\prime}),\\\ x^{1}&\text{ if
}&\varphi_{k}(x^{\prime})<x^{1}\leq\varphi_{k+1}(x^{\prime}),\\\
\varphi_{k}(x^{\prime})&\text{ if
}&x^{1}\leq\varphi_{k}(x^{\prime}),\end{array}\right.$

which implies that

(3.32) $(x^{1},x^{\prime})\in Q_{7}^{l}\qquad\Longrightarrow\qquad
Z_{k}(x^{1},x^{\prime})=\left\\{\begin{array}[]{cl}\varphi_{k+1}(x^{\prime})&\text{
if }l>k,\\\ x^{1}&\text{ if }l=k,\\\ \varphi_{k}(x^{\prime})&\text{ if
}k>l.\end{array}\right.$

We use the perturbation on $\displaystyle Q_{5}^{k}$ with respect to the point
$\displaystyle z_{k}\left(\in\mathbb{R}^{n}\right)$ defined as follows. For
any $\displaystyle k\in K$ and fixed $\displaystyle z\in Q_{5}$, set

(3.33) $z_{k}=\big{(}Z_{k}(z),z^{\prime}\big{)}.$

To use the perturbation, with (1.2), we will check that $\displaystyle
z_{k}\in\overline{Q_{6}^{k}}$ in Corollary 3.10. However, $\displaystyle
z_{k}$ depends on $\displaystyle z$. So later in Lemma 3.14, we will handle
the points $\displaystyle z_{k}$ $\displaystyle(k\in K)$ by using the graph
functions $\displaystyle\varphi_{k}$ $\displaystyle(k\in K)$ which does not
depends on $\displaystyle z$. In fact, we use perturbation on $\displaystyle
Q_{5r}^{k}(z)$ with respect to the point $\displaystyle z_{k}$ in (3.33). So
with (1.2), we will check that $\displaystyle
z_{k}\in\overline{Q_{6r}^{k}(z)}$ in Lemma 3.9.

######  Lemma 3.9.

For any $\displaystyle Q_{6r}(z)\subset Q_{6}$ and the corresponding points
$\displaystyle z_{k}$ $\displaystyle(k\in K)$ in (3.32), we have that

(3.34) $Q_{5r}^{k}(z)\not=\emptyset\qquad\Longrightarrow\qquad
z_{k}\in\overline{Q_{6r}^{k}(z)}\qquad(k\in K).$

Moreover, for any $\displaystyle x\in Q_{5r}^{k}(z)$, there exists a path
belongs to $\displaystyle Q_{6r}^{k}(z)\cup\\{z_{k}\\}$ with length less than
$\displaystyle cr$ which connects between $\displaystyle x$ and $\displaystyle
z_{k}$.

###### Proof.

Fix $\displaystyle k\in K$. Assume that $\displaystyle Q_{6r}(z)\subset Q_{6}$
and $\displaystyle Q_{5r}^{k}(z)\not=\emptyset$. Then we have that
$\displaystyle Q_{5}^{k}\supset Q_{5r}^{k}(z)\not=\emptyset$, and it follows
from Corollary 3.10 that $\displaystyle z_{k}\in\overline{Q_{6}^{k}}$. So we
only need to prove that $\displaystyle z_{k}\in Q_{6r}(z)$.

Since $\displaystyle Q_{5r}^{k}(z)\not=\emptyset$, we have that
$\displaystyle\inf_{Q_{5r}^{\prime}(z^{\prime})}\varphi_{k}\leq z^{1}+5r$ and
$\displaystyle\sup_{Q_{5r}^{\prime}(z^{\prime})}\varphi_{k+1}\geq z^{1}-5r$,
other-wise we have that $\displaystyle Q_{5r}^{k}(z)=\emptyset$. So by (1.2),

(3.35)
$\varphi_{k}(x^{\prime})<z^{1}+\frac{11r}{2}\qquad\text{and}\qquad\varphi_{k+1}(x^{\prime})>z^{1}-\frac{11r}{2}\qquad\left(x^{\prime}\in
Q_{5r}^{\prime}(z^{\prime})\right).$

To prove that $\displaystyle z_{k}\in Q_{6r}(z)$, we consider three cases. (1)
Assume that $\displaystyle z^{1}>\varphi_{k+1}(z^{\prime})$. Then from (1.2)
and (3.35), we have that $\displaystyle
z^{1}-6r<\varphi_{k+1}(z^{\prime})<z^{1}$, which implies that $\displaystyle
z_{k}=(Z_{k}(z),z^{\prime})=\left(\varphi_{k+1}(z^{\prime}),z^{\prime}\right)\in\overline{Q_{6r}^{k}(z)}$
(2) Assume that $\displaystyle\varphi_{k}(z^{\prime})\leq
z^{1}\leq\varphi_{k+1}(z^{\prime})$. Then we have from (3.31) and (3.33) that
$\displaystyle z_{k}=z\in\overline{Q_{6r}^{k}(z)}$. (3) Assume that
$\displaystyle z_{1}<\varphi_{k}(z^{\prime})$. Then from (1.2) and (3.33), we
have that $\displaystyle z^{1}<\varphi_{k}(z^{\prime})<z^{1}+6r$, which
implies that $\displaystyle
z_{k}=(Z_{k}(z),z^{\prime})=\left(\varphi_{k}(z^{\prime}),z^{\prime}\right)\in\overline{Q_{6r}^{k}(z)}$.
Since $\displaystyle k\in K$ was arbitrary chosen, (3.34) holds.

We choose a path connecting $\displaystyle x\in Q_{5r}^{k}(z)$ and
$\displaystyle z_{k}\in\overline{Q_{6r}^{k}(z)}$ as follows. Set

${\it
l}(s)=\left(\min\left\\{z^{1}+\frac{11r}{2},\varphi_{k+1}(sz^{\prime}+(1-s)x^{\prime})\right\\},sz^{\prime}+(1-s)x^{\prime}\right)\qquad(s\in[0,1])$

The path is a connected path of (1) the line connecting $\displaystyle x$ and
$\displaystyle{\it l}(0)$, (2) the path $\displaystyle{\it l}(s)$
$\displaystyle(s\in[0,1])$ connecting $\displaystyle{\it l}(0)$ and
$\displaystyle{\it l}(1)$, (3) the line connecting $\displaystyle{\it l}(1)$
and $\displaystyle z_{k}$. Since $\displaystyle x\in Q_{5r}^{k}(z)$ and
$\displaystyle z_{k}\in\overline{Q_{6r}^{k}(z)}$, one can show that this new
path belongs to $\displaystyle Q_{6r}^{k}(z)\cup\\{z_{k}\\}$ and has length
less than $\displaystyle 50nr$. ∎

The following corollary is a special case of Lemma 3.9.

######  Corollary 3.10.

For any $\displaystyle z\in Q_{5}$ and the corresponding points $\displaystyle
z_{k}$ $\displaystyle(k\in K)$ in (3.32), we have that

$Q_{5}^{k}\not=\emptyset\qquad\Longrightarrow\qquad
z_{k}\in\overline{Q_{6}^{k}}\qquad(k\in K).$

###### Proof.

By taking $\displaystyle z=0$ and $\displaystyle r=1$ in Corollary 3.10, the
lemma follows. ∎

Recall from (2.11) that

(3.36)
$\partial_{t^{\prime}}^{p^{\prime}}h\big{|}_{t^{\prime}=0^{\prime}}=\sum_{0^{\prime}\leq
q^{\prime}\leq p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}h+\sum_{|\xi|\leq|p^{\prime}|-1}P_{\xi}^{p^{\prime}}(x)D_{\xi}h~{}\text{
in }~{}Q_{5}.$

For any $\displaystyle z\in Q_{5}$ and $\displaystyle p^{\prime}\geq
0^{\prime}$ with $\displaystyle|p^{\prime}|=m$, the test function will be
chosen by using the following $\displaystyle\eta:Q_{5}\to\mathbb{R}^{N}$
defined as

(3.37) $\displaystyle\eta$
$\displaystyle=\partial_{t}^{p^{\prime}}u\big{|}_{t=0}-\sum_{k\in
K,\,Q_{5}^{k}\not=\emptyset}\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}\big{(}Z_{k},x^{\prime}\big{)}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}u_{k}(z_{k})$
$\displaystyle\quad-\sum_{k\in
K,\,Q_{5}^{k}\not=\emptyset}\sum_{|\xi|\leq|p^{\prime}|-1}P_{\xi}^{p^{\prime}}(Z_{k},x^{\prime})D_{\xi}u_{k}\big{(}Z_{k},x^{\prime}\big{)}$
$\displaystyle\quad+\sum_{k\in
K_{-},\,Q_{5}^{k}\not=\emptyset}\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}u_{k}\big{(}\varphi_{k}(z^{\prime}),z^{\prime}\big{)}$
$\displaystyle\quad+\sum_{k\in
K_{-},\,Q_{5}^{k}\not=\emptyset}\sum_{|\xi|\leq|p^{\prime}|-1}P_{\xi}^{p^{\prime}}\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}D_{\xi}u_{k}\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}.$

It follows from (3.36) that

(3.38) $\tilde{\eta}=I+II+III\quad\text{in}\quad Q_{5},$

where

$I=\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\left[\pi^{q^{\prime}}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}u-\sum_{k\in
K,\,Q_{5}^{k}\not=\emptyset}\pi^{q^{\prime}}\big{(}Z_{k},x^{\prime}\big{)}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}u_{k}(z_{k})\chi_{Q_{5}^{k}}\right],$
$\displaystyle II$ $\displaystyle=-\sum_{k\in
K,\,Q_{5}^{k}\not=\emptyset}\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}\big{(}Z_{k},x^{\prime}\big{)}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}u_{k}(z_{k})\chi_{Q_{5}\setminus
Q_{5}^{k}}$ $\displaystyle\quad+\sum_{k\in
K_{-},\,Q_{5}^{k}\not=\emptyset}\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}u_{k}\big{(}\varphi_{k}(z^{\prime}),z^{\prime}\big{)}$

and

$\displaystyle III$ $\displaystyle=-\sum_{k\in
K,\,Q_{5}^{k}\not=\emptyset}\sum_{|\xi|\leq|p^{\prime}|-1}P_{\xi}^{p^{\prime}}\big{(}Z_{k},x^{\prime}\big{)}D_{\xi}u_{k}\big{(}Z_{k},x^{\prime}\big{)}\chi_{Q_{5}\setminus
Q_{5}^{k}}$ $\displaystyle\quad+\sum_{k\in
K_{-},\,Q_{5}^{k}\not=\emptyset}\sum_{|\xi|\leq|p^{\prime}|-1}P_{\xi}^{p^{\prime}}\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}D_{\xi}u_{k}\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}.$

Due to the approximation argument, we have that $\displaystyle\delta>0$ in
(2.1) and (3.6) hold. So we find that $\displaystyle\eta$ in (3.37) is weakly
differentiable in $\displaystyle Q_{5}$ and that $\displaystyle D\eta\in
L^{\infty}(Q_{5})$.

######  Lemma 3.11.

$\displaystyle\eta$ in (3.37) is weakly differentiable in $\displaystyle
Q_{5}$ and that $\displaystyle D\eta\in L^{\infty}(Q_{5})$.

###### Proof.

We only need to check that
$\displaystyle\partial_{t}^{p^{\prime}}u\big{|}_{t=0}$ is weakly
differentiable in $\displaystyle Q_{5}$ and the derivative is bounded in
$\displaystyle Q_{5}$. Fix $\displaystyle\alpha\in\\{1,\cdots,n\\}$.

In view of (3.6), we have that
$\displaystyle\partial_{t}^{p^{\prime}}u_{k}\big{|}_{t=0}\in
C^{1}\left(\overline{Q_{5}^{k}},\mathbb{R}^{N}\right)$. Thus for any
$\displaystyle\varphi\in C_{c}^{\infty}(Q_{5},\mathbb{R}^{N})$,

$\displaystyle\int_{Q_{5}}\partial_{t}^{p^{\prime}}u\big{|}_{t=0}D_{i}\varphi\,dx$
$\displaystyle=\sum_{k\in
K}\int_{Q_{5}^{k}}\partial_{t}^{p^{\prime}}u\big{|}_{t=0}D_{i}\varphi\,dx$
$\displaystyle=\sum_{k\in
K}\int_{Q_{5}^{k}}\partial_{t}^{p^{\prime}}u_{k}\big{|}_{t=0}\,dx$
$\displaystyle=-\sum_{k\in
K}\int_{Q_{5}^{k}}D_{i}\left(\partial_{t}^{p^{\prime}}u_{k}\right)\big{|}_{t=0}\varphi\,dx+\sum_{k\in
K}\int_{\partial
Q_{5}^{k}}\partial_{t}^{p^{\prime}}u_{k}\big{|}_{t=0}\varphi\,n_{k}^{\alpha}\,dS.$

where $\displaystyle\vec{n}_{k}=(n_{k}^{1},n_{k}^{2},\cdots,n_{k}^{n})$ is the
outward normal vector on $\displaystyle\partial Q_{5}^{k}$.

From that $\displaystyle u_{k}=u_{k-1}$ on $\displaystyle\partial
Q_{5}^{k}\cap\partial Q_{5}^{k-1}$, we obtain that
$\displaystyle\partial_{t}^{p^{\prime}}u_{k}\big{|}_{t=0}=\partial_{t}^{p^{\prime}}u_{k-1}\big{|}_{t=0}$
on $\displaystyle\partial Q_{5}^{k}\cap\partial Q_{5}^{k-1}$. So from that
$\displaystyle n_{k}^{\alpha}=-n_{k-1}^{i}$ $\displaystyle(k\in K)$, we obtain
that

$\sum_{k\in K}\int_{\partial
Q_{5}^{k}}\partial_{t}^{p^{\prime}}u_{k}\big{|}_{t=0}\varphi\,n_{k}^{\alpha}\,dS=0.$

Thus

$\displaystyle\int_{Q_{5}}\partial_{t}^{p^{\prime}}u\big{|}_{t=0}D_{i}\varphi\,dx$
$\displaystyle=-\sum_{k\in
K}\int_{Q_{5}^{k}}D_{\alpha}\left(\partial_{t}^{p^{\prime}}u_{k}\right)\big{|}_{t=0}\varphi\,dx$
$\displaystyle=-\int_{Q_{5}}\left[\sum_{k\in
K}D_{\alpha}\left(\partial_{t}^{p^{\prime}}u_{k}\right)\big{|}_{t=0}\chi_{Q_{5}^{k}}\right]\varphi\,dx.$

With (3.6), we discover that
$\displaystyle\partial_{t}^{p^{\prime}}u\big{|}_{t=0}$ is weakly
differentiable in $\displaystyle Q_{5}$ and that $\displaystyle
D_{\alpha}\left(\partial_{t}^{p^{\prime}}u\big{|}_{t=0}\right)=\sum_{k\in
K}D_{\alpha}\left(\partial_{t}^{p^{\prime}}u_{k}\right)\big{|}_{t=0}\chi_{Q_{5}^{k}}\in
L^{\infty}(Q_{5})$. Since $\displaystyle i\in\\{1,\cdots,n\\}$ was arbitrary
chosen, the lemma holds. ∎

Recall from (2.18) that

(3.39) $x\in Q_{5}^{l}\qquad\Longrightarrow\qquad
H_{k}(x)=\left\\{\begin{array}[]{cl}1&~{}\text{ if }~{}l\geq k,\\\ 0&~{}\text{
if }~{}l<k.\end{array}\right.$

To handle $\displaystyle I$, $\displaystyle II$ and $\displaystyle III$, we
obtain three following lemmas related to (3.39).

######  Lemma 3.12.

For any $\displaystyle h:Q_{5}\to\mathbb{R}$ and $\displaystyle k\in K$, we
have that

$h\big{(}Z_{k},x^{\prime}\big{)}\chi_{Q_{5}\setminus
Q_{5}^{k}}=(1-H_{k})\,h\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}+H_{k+1}\,h\big{(}\varphi_{k+1}(x^{\prime}),x^{\prime}\big{)}.$

###### Proof.

Fix $\displaystyle k\in K$. From (3.39), we find that

$\sum_{l<k,\,l\in
K}h\big{(}Z_{k},x^{\prime}\big{)}\chi_{Q_{5}^{l}}=\sum_{l<k,\,l\in
K}h\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\chi_{Q_{5}^{l}}=(1-H_{k})\,h\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}~{}\text{
in }~{}Q_{5}.$

Also from (3.39), we find that

$\sum_{l>k,\,l\in
K}h\big{(}Z_{k},x^{\prime}\big{)}\chi_{Q_{5}^{l}}=\sum_{l>k,\,l\in
K}h\big{(}\varphi_{k+1}(x^{\prime}),x^{\prime}\big{)}\chi_{Q_{5}^{l}}=H_{k+1}\,h\big{(}\varphi_{k+1}(x^{\prime}),x^{\prime}\big{)}~{}\text{
in }~{}Q_{5}.$

Since $\displaystyle k\in K$ was arbitrary chosen, the lemma follows from the
above equalities. ∎

With Lemma 3.13, we only need to handle the indices in $\displaystyle K_{-}$
in (1.3) not all the indices in $\displaystyle K$ . Here, $\displaystyle
K_{-}$ represents the index of graph functions intersecting $\displaystyle
Q_{5}$.

######  Lemma 3.13.

For any $\displaystyle k\in K$ with $\displaystyle Q_{5}^{k}\not=\emptyset$,
we have that

(3.40) $k+1\not\in K_{-}\qquad\Longrightarrow\qquad H_{k+1}=0~{}\text{ in
}~{}Q_{5},$

and

(3.41) $k\not\in K_{-}\qquad\Longrightarrow\qquad H_{k}=1~{}\text{ in
}~{}Q_{5}.$

###### Proof.

Let $\displaystyle K_{-}=\left\\{l_{-},l_{-}+1,\cdots,l_{+}\right\\}$. From
(1.3) and that $\displaystyle Q_{5}^{k}\not=\emptyset$, we find that

(3.42) $k\in\left\\{l_{-}-1,l_{-},l_{-}+1,\cdots,l_{+}\right\\}.$

To prove (3.40), assume that $\displaystyle k+1\not\in K_{-}$. Then from
(3.42) and that $\displaystyle k+1\not\in K_{-}$, we obtain that
$\displaystyle k+1=l_{+}+1$. Since $\displaystyle
K_{-}=\left\\{l_{-},l_{-}+1,\cdots,l_{+}\right\\}$, we have from Definition
1.1 that $\displaystyle H_{k+1}=0$ in $\displaystyle Q_{5}$.

To prove (3.41), assume that $\displaystyle k\not\in K_{-}$. Then by (3.42)
and that $\displaystyle k\not\in K_{-}$, we find that $\displaystyle
k=l_{-}-1$. Since $\displaystyle
K_{-}=\left\\{l_{-},l_{-}+1,\cdots,l_{+}\right\\}$, we have that
$\displaystyle H_{k}=1$ in $\displaystyle Q_{5}$. ∎

We use perturbations with respect to the points $\displaystyle z_{k}$
$\displaystyle(k\in K)$. In view of Lemma 3.12 and Lemma 3.14, $\displaystyle
z_{k}$ can be handled by the graph functions $\displaystyle\varphi_{k}$ for
any $\displaystyle k\in K$.

######  Lemma 3.14.

For any $\displaystyle k\in K$, we have that

(3.43) $H_{k}(x)\not=1\text{ for some }x\in
Q_{r}(z)\qquad\Longrightarrow\qquad\left|z_{k}-\left(\varphi_{k}(z^{\prime}),z^{\prime}\right)\right|<2r,$

and

(3.44) $H_{k+1}(x)\not=0\text{ for some }x\in
Q_{r}(z)\qquad\Longrightarrow\qquad\left|z_{k}-\left(\varphi_{k+1}(z^{\prime}),z^{\prime}\right)\right|<2r.$

###### Proof.

We first prove (3.43). Suppose that $\displaystyle H_{k}(x)\not=1$ for some
$\displaystyle x\in Q_{r}(z)$. Then we have from (1.1) and (3.39) that
$\displaystyle x^{1}\leq\varphi_{k}(x^{\prime})$.

(1) Assume that $\displaystyle\varphi_{k}(z^{\prime})\geq z^{1}$. Then from
(3.32) and (3.33), we have that $\displaystyle
z_{k}=\left(\varphi_{k}(z^{\prime}),z^{\prime}\right)$, and so (3.43) holds.

(2) Assume that
$\displaystyle\varphi_{k}(z^{\prime})<z^{1}\leq\varphi_{k+1}(z^{\prime})$.
Then we have from (3.32) and (3.33) that $\displaystyle z_{k}=z$. Since
$\displaystyle x^{1}\leq\varphi_{k}(x^{\prime})$, choose a point
$\displaystyle y=(y^{1},y^{\prime})\in Q_{r}(z)$ on the line connecting
$\displaystyle z$ and $\displaystyle x$ such that $\displaystyle
y^{1}=\varphi_{k}(y^{\prime})$. Then by (1.2) and that $\displaystyle
z_{k}=z$,

$\left|\varphi_{k}(z^{\prime})-z_{k}^{1}\right|\leq\left|\varphi_{k}(z^{\prime})-\varphi_{k}(y^{\prime})\right|+\left|\varphi_{k}(y^{\prime})-z^{1}\right|\leq\frac{\left|z^{\prime}-y^{\prime}\right|}{10n}+\left|y^{1}-z^{1}\right|.$

Since $\displaystyle y=(y^{1},y^{\prime})\in Q_{r}(z)$ is on the line
connecting $\displaystyle z$ and $\displaystyle x$, we have that
$\displaystyle|z^{\prime}-y^{\prime}|\leq|z^{\prime}-x^{\prime}|<2nr$ and
$\displaystyle|z^{1}-y^{1}|\leq|z^{1}-x^{1}|<r$, and so (3.43) holds.

(3) Assume that $\displaystyle
z^{1}>\varphi_{k+1}(z^{\prime})\left(>\varphi_{k}(z^{\prime})\right)$. Then we
have that $\displaystyle
z_{k}=\left(\varphi_{k+1}(z^{\prime}),z^{\prime}\right)$. Since $\displaystyle
x^{1}\leq\varphi_{k}(x^{\prime})\left(<\varphi_{k+1}(x^{\prime})\right)$,
choose $\displaystyle y,t\in Q_{r}(z)$ on the line connecting $\displaystyle
z$ and $\displaystyle x$ such that $\displaystyle
y^{1}=\varphi_{k}(y^{\prime})$ and $\displaystyle
t^{1}=\varphi_{k+1}(t^{\prime})$. Then by (1.2) and that $\displaystyle
z_{k}=\left(\varphi_{k+1}(z^{\prime}),z^{\prime}\right)$,

$\displaystyle\left|\varphi_{k}(z^{\prime})-z_{k}^{1}\right|$
$\displaystyle\leq\left|\varphi_{k}(z^{\prime})-\varphi_{k}(y^{\prime})\right|+\left|\varphi_{k}(y^{\prime})-\varphi_{k+1}(t^{\prime})\right|+\left|\varphi_{k+1}(t^{\prime})-\varphi_{k+1}(z^{\prime})\right|$
$\displaystyle\leq\frac{\left|z^{\prime}-y^{\prime}\right|}{10n}+|y^{1}-t^{1}|+\frac{\left|t^{\prime}-z^{\prime}\right|}{10n}.$

Since $\displaystyle y,t\in Q_{r}(z)$ are on the line connecting
$\displaystyle z$ and $\displaystyle x$, we have that
$\displaystyle|z^{\prime}-y^{\prime}|,|t^{\prime}-z^{\prime}|\leq|z^{\prime}-x^{\prime}|<2nr$
and $\displaystyle|y^{1}-t^{1}|\leq|z^{1}-x^{1}|<r$, and so (3.43) holds.

We now prove (3.44). Suppose that $\displaystyle H_{k+1}(x)\not=0$ for some
$\displaystyle x\in Q_{r}(z)$. Then we have from (1.1) and (3.39) that
$\displaystyle x^{1}>\varphi_{k+1}(x^{\prime})$.

(1) Assume that $\displaystyle\varphi_{k+1}(z^{\prime})<z^{1}$. Then from
(3.32) and (3.33), we have that $\displaystyle
z_{k}=\left(\varphi_{k+1}(z^{\prime}),z^{\prime}\right)$ and so (3.44) holds.

(2) Assume that
$\displaystyle\varphi_{k}(z^{\prime})<z^{1}\leq\varphi_{k+1}(z^{\prime})$.
Then we have from (3.32) and (3.33) that $\displaystyle z_{k}=z$. Since
$\displaystyle x^{1}>\varphi_{k+1}(x^{\prime})$, choose a point $\displaystyle
y=(y^{1},y^{\prime})\in Q_{r}(z)$ on the line connecting $\displaystyle z$ and
$\displaystyle x$ such that $\displaystyle y^{1}=\varphi_{k+1}(y^{\prime})$.
Then by (1.2) and that $\displaystyle z_{k}=z$,

$\left|\varphi_{k+1}(z^{\prime})-z_{k}^{1}\right|\leq\left|\varphi_{k+1}(z^{\prime})-\varphi_{k+1}(y^{\prime})\right|+\left|\varphi_{k+1}(y^{\prime})-z^{1}\right|\leq\frac{\left|z^{\prime}-y^{\prime}\right|}{10n}+\left|y^{1}-z^{1}\right|.$

Since $\displaystyle y=(y^{1},y^{\prime})\in Q_{r}(z)$ is on the line
connecting $\displaystyle z$ and $\displaystyle x$, we have that
$\displaystyle|z^{\prime}-y^{\prime}|\leq|z^{\prime}-x^{\prime}|<2nr$ and
$\displaystyle|z^{1}-y^{1}|\leq|z^{1}-x^{1}|<r$, and so (3.44) holds.

(3) Assume that $\displaystyle
z^{1}\leq\varphi_{k}(z^{\prime})\left(<\varphi_{k+1}(z^{\prime})\right)$. Then
we have that $\displaystyle
z_{k}=\left(\varphi_{k}(z^{\prime}),z^{\prime}\right)$. Since $\displaystyle
x^{1}>\varphi_{k+1}(x^{\prime})\left(>\varphi_{k}(x^{\prime})\right)$, choose
$\displaystyle y,t\in Q_{r}(z)$ on the line connecting $\displaystyle z$ and
$\displaystyle x$ such that $\displaystyle y^{1}=\varphi_{k+1}(y^{\prime})$
and $\displaystyle t^{1}=\varphi_{k}(t^{\prime})$. Then by (1.2) and that
$\displaystyle z_{k}=\left(\varphi_{k}(z^{\prime}),z^{\prime}\right)$,

$\displaystyle\left|\varphi_{k+1}(z^{\prime})-z_{k}^{1}\right|$
$\displaystyle\leq\left|\varphi_{k+1}(z^{\prime})-\varphi_{k+1}(y^{\prime})\right|+\left|\varphi_{k+1}(y^{\prime})-\varphi_{k}(t^{\prime})\right|+\left|\varphi_{k}(t^{\prime})-\varphi_{k}(z^{\prime})\right|$
$\displaystyle\leq\frac{\left|z^{\prime}-y^{\prime}\right|}{10n}+|y^{1}-t^{1}|+\frac{\left|t^{\prime}-z^{\prime}\right|}{10n}.$

Since $\displaystyle y,t\in Q_{r}(z)$ are on the line connecting
$\displaystyle z$ and $\displaystyle x$, we have that
$\displaystyle|z^{\prime}-y^{\prime}|,|t^{\prime}-z^{\prime}|\leq|z^{\prime}-x^{\prime}|<2nr$
and $\displaystyle|y^{1}-t^{1}|\leq|z^{1}-x^{1}|<r$, and so (3.44) holds. ∎

Set $\displaystyle S=\cup_{k\in
K_{+}}\big{\\{}\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\in
x^{\prime}\in Q_{5}^{\prime}\big{\\}}$. Then $\displaystyle|S|=0$. By using
Corollary 3.10, Lemma 3.12 and Lemma 3.13 and Lemma 3.14, we estimate
$\displaystyle DI$, $\displaystyle D(II)$ and $\displaystyle D(III)$ in the
set $\displaystyle Q_{r}(z)\setminus S$ as in the following three lemmas.

######  Lemma 3.15.

We have the following estimate for $\displaystyle I$ in (3.38) :

$\int_{Q_{r}(z)\setminus S}\bigg{|}DI-\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}Du\bigg{|}^{2}\,dx\leq
cr^{n+1}\|D^{m+1}u\|_{L^{\infty}(Q_{2r}(z))}^{2}.$

###### Proof.

From (1.1) and (3.31), we have that
$\displaystyle\big{(}Z_{k}(x),x^{\prime}\big{)}=x$ for any $\displaystyle x\in
Q_{5}^{k}$ $\displaystyle(k\in K)$. It follows that
$\displaystyle\pi^{q^{\prime}}\big{(}Z_{k}(x),x^{\prime}\big{)}=\pi^{q^{\prime}}(x)$
for any $\displaystyle x\in Q_{5}^{k}$ $\displaystyle(k\in K)$. So by (3.38),

$I=\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\bigg{[}\pi^{q^{\prime}}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}u-\pi^{q^{\prime}}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}u_{k}(z_{k})\bigg{]}~{}\text{
in }~{}Q_{r}^{k}(z)\quad(k\in K).$

It follows that

$\displaystyle\sum_{k\in K}\int_{Q_{r}^{k}(z)\setminus
S}\left|DI-\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}Du\right|^{2}\,dx$
$\displaystyle\quad=\sum_{k\in K}\int_{Q_{r}^{k}(z)\setminus
S}\left|\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}D\left[\pi^{q^{\prime}}\right]\left[D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}u-D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}u_{k}(z_{k})\right]\right|^{2}\,dx.$

Since $\displaystyle Q_{2r}(z)\subset Q_{5}$ and $\displaystyle D^{m}u_{k}\in
C^{\gamma}\left(\overline{Q_{2r}^{k}(z)}\right)$, we have from Lemma 3.9 that

$|D^{m}u-D^{m}u_{k}(z_{k})|\leq
cr\|D^{m+1}u\|_{L^{\infty}(Q_{2r}^{k}(z))}~{}\text{ in
}~{}Q_{r}^{k}(z)\qquad(k\in K),$

by separating the cases that $\displaystyle Q_{r}^{k}(z)=\emptyset$ and
$\displaystyle Q_{r}^{k}(z)\not=\emptyset$. It follows that

$\displaystyle\int_{Q_{r}(z)\setminus S}\left|DI-\sum_{0^{\prime}\leq
q^{\prime}\leq p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}Du\right|^{2}\,dx$
$\displaystyle\quad\leq
cr^{2}\|D^{m+1}u\|_{L^{\infty}(Q_{2r}(z))}^{2}\int_{Q_{r}(z)}\left|D\left[\pi^{q^{\prime}}\right]\right|^{2}\,dx,$

and so the lemma follows from (2.17). ∎

######  Lemma 3.16.

For any $\displaystyle\alpha\in\\{1,\cdots,n\\}$, the next estimate holds for
$\displaystyle II$ in (3.38):

$\left|D_{\alpha}(II)+\sum_{k\in
K_{-}}\mathbb{H}_{\alpha}^{k}\left[u_{k-1}-u_{k}\right]H_{k}\right|\leq
cr^{\gamma}\sum_{k\in
K_{-}}\|u_{k}\|_{C^{m,\gamma}\left(\overline{Q_{6}^{k}}\right)},$

in $\displaystyle Q_{r}(z)\setminus S$ where the linear operator
$\displaystyle\mathbb{H}_{\alpha}^{k}:C^{m,\gamma}\left(\partial
Q_{6}^{k}\cap\partial Q_{6}^{k-1}\right)\to
C^{\gamma}\left(Q_{5}^{\prime}\right)$ $\displaystyle(k\in K_{-})$ defined as

$\mathbb{H}_{\alpha}^{k}[h](x^{\prime})=\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}D_{j}\left[\pi^{q^{\prime}}\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\right]D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}h\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}.$

###### Proof.

In Lemma 3.12, we take $\displaystyle h=\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}u_{k}(z_{k})$.
Then

$\displaystyle II$ $\displaystyle=-\sum_{k\in
K,\,Q_{5}^{k}\not=\emptyset}\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}(1-H_{k})\pi^{q^{\prime}}\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}u_{k}(z_{k})$
$\displaystyle\quad-\sum_{k\in
K,\,Q_{5}^{k}\not=\emptyset}\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}H_{k+1}\pi^{q^{\prime}}\big{(}\varphi_{k+1}(x^{\prime}),x^{\prime}\big{)}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}u_{k}(z_{k})$
$\displaystyle\quad+\sum_{k\in
K_{-},\,Q_{5}^{k}\not=\emptyset}\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}u_{k}\big{(}\varphi_{k}(z^{\prime}),z^{\prime}\big{)},$

in $\displaystyle Q_{r}(z)$. Fix $\displaystyle\alpha\in\\{1,\cdots,n\\}$. We
find from Lemma 3.13 that

$D_{\alpha}(II)=II_{1}+II_{2}+II_{3}\quad\text{in}\quad Q_{r}(z)\setminus S,$

where

$\displaystyle II_{1}$ $\displaystyle=-\sum_{k\in
K_{-},\,Q_{5}^{k}\not=\emptyset}\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}(1-H_{k})D_{\alpha}\left[\pi^{q^{\prime}}\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\right]D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}u_{k}(z_{k})$
$\displaystyle\quad+\sum_{k\in
K_{-},\,Q_{5}^{k}\not=\emptyset}\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}(1-H_{k})D_{\alpha}\left[\pi^{q^{\prime}}\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\right]D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}u_{k}\big{(}\varphi_{k}(z^{\prime}),z^{\prime}\big{)},$
$\displaystyle II_{2}$ $\displaystyle=\sum_{k\in
K_{-},\,Q_{5}^{k}\not=\emptyset}\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}H_{k}D_{\alpha}\left[\pi^{q^{\prime}}\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\right]D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}u_{k}\big{(}\varphi_{k}(z^{\prime}),z^{\prime}\big{)},$

and

$\displaystyle II_{3}$ $\displaystyle=-\sum_{k+1\in
K_{-},\,Q_{5}^{k}\not=\emptyset}\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}H_{k+1}D_{\alpha}\left[\pi^{q^{\prime}}\big{(}\varphi_{k+1}(x^{\prime}),x^{\prime}\big{)}\right]D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}u_{k}(z_{k})$
$\displaystyle=-\sum_{k\in
K_{-},\,Q_{5}^{k-1}\not=\emptyset}\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}H_{k}D_{\alpha}\left[\pi^{q^{\prime}}\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\right]D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}u_{k-1}(z_{k-1}),$

in $\displaystyle Q_{r}(z)\setminus S$. In view of (2.7), we discover that

(3.45)
$\left|D\left[\pi^{q^{\prime}}\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\right]\right|\leq
c\quad\text{in}\quad Q_{5}^{\prime}.$

By Corollary 3.10, if $\displaystyle Q_{5}^{k}\not=\emptyset$ for some
$\displaystyle k\in K$ then $\displaystyle z_{k}\in\overline{Q_{6}^{k}}$. Also
we have from (1.5) that
$\displaystyle\big{(}\varphi_{k}(z^{\prime}),z^{\prime}\big{)}\in\overline{Q_{6}^{k}}$
for any $\displaystyle k\in K_{-}$. So by Lemma 3.14 and (3.45),

$\displaystyle|II_{1}|$ $\displaystyle\leq c\sum_{k\in
K_{-},\,Q_{5}^{k}\not=\emptyset}\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}(1-H_{k})\left|D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}u_{k}(z_{k})-D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}u_{k}\big{(}\varphi_{k}(z^{\prime}),z^{\prime}\big{)}\right|$
$\displaystyle\leq cr^{\gamma}\sum_{k\in
K_{-},\,Q_{5}^{k}\not=\emptyset}\|u_{k}\|_{C^{m,\gamma}\left(\overline{Q_{6}^{k}}\right)}$

in $\displaystyle Q_{r}(z)\setminus S$.

From (1.5), we have that
$\displaystyle\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\in\overline{Q_{6}^{k}}$
for any $\displaystyle x^{\prime}\in Q_{r}^{\prime}(z^{\prime})$ and
$\displaystyle k\in K_{-}$. So it follows from (3.45) that

$\displaystyle\left|II_{2}-\sum_{k\in
K_{-},\,Q_{5}^{k}\not=\emptyset}\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}H_{k}D_{\alpha}\left[\pi^{q^{\prime}}\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\right]D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}u_{k}\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\right|$
$\displaystyle\quad\leq cr^{\gamma}\sum_{k\in
K_{-},\,Q_{5}^{k}\not=\emptyset}\|u_{k}\|_{C^{m,\gamma}\left(\overline{Q_{6}^{k}}\right)},$

in $\displaystyle Q_{r}(z)\setminus S$.

By Corollary 3.10, if $\displaystyle Q_{5}^{k-1}\not=\emptyset$ for some
$\displaystyle k-1\in K$ then $\displaystyle
z_{k-1}\in\overline{Q_{6}^{k-1}}$. Also we have from (1.5) that
$\displaystyle\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\in\overline{Q_{6}^{k-1}}$
for any $\displaystyle x^{\prime}\in Q_{r}^{\prime}(z^{\prime})$ and
$\displaystyle k\in K_{-}$. So by Lemma 3.14 and (3.45) that

$\displaystyle\left|II_{3}+\sum_{k\in
K_{-},\,Q_{5}^{k-1}\not=\emptyset}\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}H_{k}D_{\alpha}\left[\pi^{q^{\prime}}\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\right]D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}u_{k-1}\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\right|$
$\displaystyle\quad\leq cr^{\gamma}\sum_{k\in
K_{-},\,Q_{5}^{k-1}\not=\emptyset}\|u_{k}\|_{C^{m,\gamma}\left(\overline{Q_{6}^{k}}\right)},$

in $\displaystyle Q_{r}(z)\setminus S$.

Since $\displaystyle\alpha\in\\{1,\cdots,n\\}$ was arbitrary, the lemma holds
from (1.4). ∎

######  Lemma 3.17.

For any $\displaystyle\alpha\in\\{1,\cdots,n\\}$, the following holds for
$\displaystyle III$ in (3.38):

$D_{\alpha}(III)=-\sum_{k\in
K_{-}}\mathbb{H}_{\alpha}^{k}\left[u_{k-1}-u_{k}\right]H_{k}$

in $\displaystyle Q_{r}(z)\setminus S$ with the linear operator
$\displaystyle\mathbb{H}_{\alpha}^{k}:C^{m,\gamma}\left(\partial
Q_{6}^{k}\cap\partial Q_{6}^{k-1}\right)\to
C^{\gamma}\left(Q_{5}^{\prime}\right)$ $\displaystyle(k\in K_{-})$ defined as

$\mathbb{H}_{\alpha}^{k}[h](x)=\sum_{|\xi|\leq|p^{\prime}|-1}D_{\alpha}\left[P_{\xi}^{p^{\prime}}\big{[}\varphi_{k}(x^{\prime}),x^{\prime}\big{]}\,D_{\xi}h\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\right]\,H_{k}.$

###### Proof.

In Lemma 3.12, we take $\displaystyle
h=\sum_{|\xi|\leq|p^{\prime}|-1}P_{\xi}^{p^{\prime}}\big{(}Z_{k},x^{\prime}\big{)}D_{\xi}u_{k}\big{(}Z_{k},x^{\prime}\big{)}$.
Then

$\displaystyle III$ $\displaystyle=-\sum_{k\in
K,\,Q_{5}^{k}\not=\emptyset}\sum_{|\xi|\leq|p^{\prime}|-1}(1-H_{k})P_{\xi}^{p^{\prime}}\big{[}\varphi_{k}(x^{\prime}),x^{\prime}\big{]}\,D_{\xi}u_{k}\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}$
$\displaystyle\quad-\sum_{k\in
K,\,Q_{5}^{k}\not=\emptyset}\sum_{|\xi|\leq|p^{\prime}|-1}H_{k+1}P_{\xi}^{p^{\prime}}\big{[}\varphi_{k+1}(x^{\prime}),x^{\prime}\big{]}\,D_{\xi}u_{k}\big{(}\varphi_{k+1}(x^{\prime}),x^{\prime}\big{)}$
$\displaystyle\quad+\sum_{k\in
K_{-},\,Q_{5}^{k}\not=\emptyset}\sum_{|\xi|\leq|p^{\prime}|-1}P_{\xi}^{p^{\prime}}\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}D_{\xi}u_{k}\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}$

in $\displaystyle Q_{r}(z)\setminus S$. Fix
$\displaystyle\alpha\in\\{1,\cdots,n\\}$. Then from Lemma 3.13, we find that

$D_{\alpha}(III)=III_{1}+III_{2}\quad\text{in}\quad Q_{5}\setminus S,$

where

$\displaystyle III_{1}$ $\displaystyle=\sum_{k\in
K_{-},\,Q_{5}^{k}\not=\emptyset}\sum_{|\xi|\leq|p^{\prime}|-1}H_{k}D_{\alpha}\left[P_{\xi}^{p^{\prime}}\big{[}\varphi_{k}(x^{\prime}),x^{\prime}\big{]}\,D_{\xi}u_{k}\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\right],$

and

$\displaystyle III_{2}$ $\displaystyle=-\sum_{k+1\in
K_{-},\,Q_{5}^{k}\not=\emptyset}\sum_{|\xi|\leq|p^{\prime}|-1}H_{k+1}D_{\alpha}\left[P_{\xi}^{p^{\prime}}\big{[}\varphi_{k+1}(x^{\prime}),x^{\prime}\big{]}\,D_{\xi}u_{k}\big{(}\varphi_{k+1}(x^{\prime}),x^{\prime}\big{)}\right]$
$\displaystyle=-\sum_{k\in
K_{-},\,Q_{5}^{k-1}\not=\emptyset}\sum_{|\xi|\leq|p^{\prime}|-1}H_{k}D_{\alpha}\left[P_{\xi}^{p^{\prime}}\big{[}\varphi_{k}(x^{\prime}),x^{\prime}\big{]}\,D_{\xi}u_{k-1}\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\right],$

in $\displaystyle x\in Q_{r}(z)\setminus S$. Since
$\displaystyle\alpha\in\\{1,\cdots,n\\}$ was arbitrary chosen, the lemma
follows from (1.4). ∎

In the following lemma, $\displaystyle G_{\alpha}$
$\displaystyle(\alpha\in\\{1,\cdots,N\\})$ does not depend on $\displaystyle
Q_{r}(z)$.

######  Lemma 3.18.

For any $\displaystyle Q_{2r}(z)\subset Q_{5}$ and
$\displaystyle\alpha\in\\{1,\cdots,n\\}$, the following

$\displaystyle\int_{Q_{r}(z)}\bigg{|}D_{\alpha}\eta-\sum_{0^{\prime}\leq
q^{\prime}\leq p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}D_{\alpha}u+G_{j}\bigg{|}^{2}\,dx$
$\displaystyle\quad\leq
cr^{n+2\gamma}\left[\|D^{m+1}u\|_{L^{\infty}(Q_{2r}(z))}^{2}+\sum_{k\in
K_{-}}\|u\|_{C^{m,\gamma}(Q_{7}^{k})}\right],$

holds for $\displaystyle\eta$ in (3.37) where

$G_{\alpha}=\sum_{k\in
K_{-}}\mathbb{H}_{\alpha}^{k}\left[u_{k-1}-u_{k}\right]H_{k}$

with the linear operator
$\displaystyle\mathbb{H}_{\alpha}^{k}:C^{m,\gamma}\left(\partial
Q_{6}^{k}\cap\partial Q_{6}^{k-1}\right)\to
C^{\gamma}\left(Q_{5}^{\prime}\right)$ $\displaystyle(k\in K_{-})$ defined as

$\displaystyle\mathbb{H}_{\alpha}^{k}[h](x)$
$\displaystyle=\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}D_{\alpha}\left[\pi^{q^{\prime}}\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\right]D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}h\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}$
$\displaystyle\quad+\sum_{|\xi|\leq|p^{\prime}|-1}D_{\alpha}\left[P_{\xi}^{p^{\prime}}\big{[}\varphi_{k}(x^{\prime}),x^{\prime}\big{]}\,D_{\xi}h\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\right].$

###### Proof.

We first estimate $\displaystyle\eta$. From (3.11), $\displaystyle\eta$ is
weakly differentiable in $\displaystyle Q_{r}(z)$ with the estimate
$\displaystyle D\eta\in L^{\infty}(Q_{r}(z))$, which implies that

$\int_{S}\bigg{|}D_{\alpha}\eta-\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}D_{\alpha}u+\sum_{k\in
K_{-}}\mathbb{H}_{\alpha}^{k}\left[u_{k-1}-u_{k}\right]H_{k}\bigg{|}^{2}dx=0.$

With (3.38), we obtain from Lemma 3.15, Lemma 3.16 and Lemma 3.17 that

$\displaystyle\int_{Q_{r}(z)\setminus
S}\bigg{|}D_{\alpha}\eta-\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}D_{\alpha}u+\sum_{k\in
K_{-}}\mathbb{H}_{\alpha}^{k}\left[u_{k-1}-u_{k}\right]H_{k}\bigg{|}^{2}\,dx$
$\displaystyle\quad\leq
cr^{n+2\gamma}\left[\|D^{m+1}u\|_{L^{\infty}(Q_{2r}(z))}^{2}+\sum_{k\in
K_{-}}\|u\|_{C^{m,\gamma}(\overline{Q_{6}^{k}})}\right].$

So the lemma follows from (3.4). ∎

To obtain the piece-wise regularity, we will use the following lemma.

######  Lemma 3.19.

For $\displaystyle\mathbb{H}_{\alpha}^{k}$ and $\displaystyle G_{\alpha}$ in
Lemma 3.18, we have that

$\displaystyle\sum_{k\in
K_{-}}\left\|\mathbb{H}_{\alpha}^{k}\left[u_{k-1}-u_{k}\right]\right\|_{C^{\gamma}(Q_{5}^{\prime})}+\sum_{k\in
K}\|G_{\alpha}\|_{C^{\gamma}(Q_{5}^{k})}\leq c\sum_{k\in
K}\|u\|_{C^{m,\gamma}(Q_{7}^{k})},$

for any $\displaystyle\alpha\in\\{1,\cdots,n\\}$.

###### Proof.

By that
$\displaystyle\pi^{q^{\prime}}\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}=[D_{2}\varphi_{k}(x^{\prime})]^{q_{2}}\cdots[D_{n}\varphi_{k}(x^{\prime})]^{q_{n}}$
$\displaystyle\big{(}x^{\prime}\in Q_{5}^{\prime},\,k\in K\big{)}$, the lemma
holds from (1.3), (3.3), (3.4) and the definition of $\displaystyle
P_{\xi}^{p^{\prime}}$ in Lemma 2.3. ∎

##  4\. Higher order derivatives

In the main equation, we assume that the coefficients are piece-wise
continuous, and so the gradient of the weak solution might have big jumps on
each $\displaystyle\partial Q_{5}^{k}\cap\partial Q_{5}^{k-1}$
$\displaystyle(k\in K)$. To show that $\displaystyle D^{m+1}u$ is piece-wise
Hölder continuous, we first show that $\displaystyle U^{p^{\prime}}$
$\displaystyle(|p^{\prime}|=m)$ defined in (4.1) are Hölder continuous. Then
we will prove that $\displaystyle D^{m+1}u$ is piece-wise Hölder continuous by
comparing $\displaystyle D^{m+1}u$ and $\displaystyle U^{p^{\prime}}$. Here,
$\displaystyle U^{p^{\prime}}$ is defined by using the perturbation results in
Lemma 3.8 and Lemma 3.18.

For the simplicity of the calculation, we abuse the notation $\displaystyle
D_{\xi}u=D_{\xi}u_{k}$ in $\displaystyle Q_{7}^{k}$. Recall from (2.9) that

$\pi_{1}=-1,\quad\pi^{\prime}=(\pi_{2},\cdots,\pi_{n}),\quad\pi=(\pi_{1},\pi^{\prime})=(-1,\pi_{2},\cdots,\pi_{n})\quad\text{
in }\quad Q_{7}.$

With the assumption (3.6), for any $\displaystyle p^{\prime}\geq 0^{\prime}$
with $\displaystyle|p^{\prime}|=m$, set $\displaystyle G_{1}^{p^{\prime}}$ as
in Lemma 3.8 and $\displaystyle G_{2}^{p^{\prime}},\cdots,G_{n}^{p^{\prime}}$
as in Lemma 3.18. Then define $\displaystyle
U^{p^{\prime}}:Q_{5}\to\mathbb{R}^{Nn}$ and $\displaystyle
G^{p^{\prime}}:Q_{5}\to\mathbb{R}^{Nn}$ as follows :

(4.1)
$U^{p^{\prime}}=\left(\begin{array}[]{ccc}U_{1}^{1,p^{\prime}}&\cdots&U_{n}^{1,p^{\prime}}\\\
\vdots&\ddots&\vdots\\\
U_{1}^{N,p^{\prime}}&\cdots&U_{n}^{N,p^{\prime}}\end{array}\right)\qquad\text{and}\qquad
G^{p^{\prime}}=\left(\begin{array}[]{ccc}G_{1}^{1,p^{\prime}}&\cdots&G_{n}^{1,p^{\prime}}\\\
\vdots&\ddots&\vdots\\\
G_{1}^{N,p^{\prime}}&\cdots&G_{n}^{N,p^{\prime}}\end{array}\right),$

where

$U_{1}^{i,p^{\prime}}=G_{1}^{i,p^{\prime}}+\sum_{1\leq\alpha\leq
n}\pi_{\alpha}\left[\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}\left[\sum_{1\leq
j\leq N}\sum_{1\leq\beta\leq
n}A^{\alpha\beta}_{ij}D_{\beta}u^{j}-F_{\alpha}^{i}\right]\right]$

in $\displaystyle Q_{5}$, and

(4.2) $U_{\beta}^{i,p^{\prime}}=G_{\beta}^{i,p^{\prime}}+\sum_{0^{\prime}\leq
q^{\prime}\leq p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}\left[D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}D_{\beta}u^{i}+\pi_{\beta}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}D_{1}u^{i}\right]$

in $\displaystyle Q_{5}$ for any $\displaystyle\beta\in\\{2,\cdots,n\\}$ and
$\displaystyle i\in\\{1,\cdots,N\\}$. Recall from Lemma 3.7 and (3.10) that

(4.3) $G_{1}^{i,p^{\prime}}=\sum_{k\in K_{-}}\sum_{1\leq\alpha\leq n}\
\mathbb{G}_{\alpha}^{p^{\prime},k}\left[U_{\alpha}^{i,k-1}-U_{\alpha}^{i,k}\right]H_{k}\quad\text{in}\quad
Q_{5}$

and

$U_{\alpha}^{i,k}=\sum_{1\leq j\leq N}\sum_{1\leq\beta\leq
n}A_{\alpha\beta}^{ij,k}D_{\beta}u^{j,k}-F_{\alpha}^{i,k}\quad\text{in}\quad
Q_{5}$

for any $\displaystyle 1\leq\alpha\leq n,\ 1\leq i\leq N$ and $\displaystyle
k\in K$ with the linear operator $\displaystyle\
\mathbb{G}_{\alpha}^{p^{\prime},k}:C^{m-1,\gamma}\left(\partial
Q_{6}^{k}\cap\partial Q_{6}^{k-1}\right)\to
C^{\gamma}\left(Q_{5}^{\prime}\right)$ $\displaystyle(k\in K_{-})$ defined as

$\displaystyle\ \mathbb{G}_{\alpha}^{p^{\prime},k}[h](x^{\prime})$
$\displaystyle=\sum_{|\xi|\leq|p^{\prime}|-1}\left[\pi_{\alpha}P_{\xi}^{p^{\prime}}D_{\xi}h\right]\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}$
$\displaystyle\quad+\sum_{0^{\prime}<q^{\prime}\leq
p^{\prime}}\left(\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\right)D^{q^{\prime}}D_{i}\varphi_{k}(x^{\prime})\,D^{p^{\prime}-q^{\prime}}\left[h\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\right],$

for any $\displaystyle x^{\prime}\in Q_{5}^{\prime}$, $\displaystyle
1\leq\alpha\leq n$ and $\displaystyle k\in K$. Also recall from Lemma 3.18
that

(4.4) $G_{\beta}^{i,p^{\prime}}=\sum_{k\in
K_{-}}\mathbb{H}_{\beta}^{p^{\prime},k}\left[u_{k-1}-u_{k}\right]H_{k}\quad\text{in}\quad
Q_{5}\qquad(\beta=2,3,\cdots,n)$

with the linear operator
$\displaystyle\mathbb{H}_{\beta}^{p^{\prime},k}:C^{m,\gamma}\left(\partial
Q_{6}^{k}\cap\partial Q_{6}^{k-1}\right)\to
C^{\gamma}\left(Q_{5}^{\prime}\right)$ $\displaystyle(k\in K_{-})$ defined as

$\displaystyle\mathbb{H}_{\beta}^{p^{\prime},k}[h](x^{\prime})$
$\displaystyle=\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}D_{\beta}\left[\pi^{q^{\prime}}\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\right]D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}h\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}$
$\displaystyle\quad+\sum_{|\xi|\leq|p^{\prime}|-1}D_{\beta}\left[P_{\xi}^{p^{\prime}}\big{[}\varphi_{k}(x^{\prime}),x^{\prime}\big{]}\,D_{\xi}h\big{(}\varphi_{k}(x^{\prime}),x^{\prime}\big{)}\right],$

for any $\displaystyle x^{\prime}\in Q_{5}^{\prime}$, $\displaystyle
2\leq\beta\leq n$ and $\displaystyle k\in K$. Then by Lemma 3.8 and Lemma
3.19,

(4.5) $\left\|G^{p^{\prime}}\right\|_{C^{\gamma}(Q_{5}^{k})}\leq c\sum_{k\in
K}\Big{[}\|u\|_{C^{m,\gamma}(Q_{7}^{k})}+\|F\|_{C^{m,\gamma}(Q_{7}^{k})}\Big{]}\quad\text{
in }\quad Q_{5}.$

To handle $\displaystyle U_{1}^{p^{\prime}}$, we show Lemma 4.1. In view of
Lemma 3.8, set $\displaystyle\nu^{p^{\prime}}:Q_{5}\to\mathbb{R}^{Nn}$ as

(4.6)
$\nu^{p^{\prime}}=\left(\begin{array}[]{ccc}\nu_{1}^{1,p^{\prime}}&\cdots&\nu_{n}^{1,p^{\prime}}\\\
\vdots&\ddots&\vdots\\\
\nu_{1}^{N,p^{\prime}}&\cdots&\nu_{n}^{N,p^{\prime}}\end{array}\right)\quad\text{
in }\quad Q_{5},$

where

$\displaystyle\nu_{\alpha}^{i,p^{\prime}}$ $\displaystyle=\sum_{0^{\prime}\leq
q^{\prime}\leq p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}\left(\sum_{1\leq
j\leq N}\sum_{1\leq\beta\leq
n}A^{\alpha\beta}_{ij}D_{\beta}u^{j}-F_{\alpha}^{i}\right)$
$\displaystyle\quad-\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}\sum_{1\leq j\leq
N}\sum_{1\leq\beta\leq
n}A^{\alpha\beta}_{ij}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}D_{\beta}u^{j},$

for any $\displaystyle 1\leq\alpha\leq n$ and $\displaystyle 1\leq i\leq N$.
For the simplicity, set
$\displaystyle\tilde{u}^{p^{\prime}}:Q_{5}\to\mathbb{R}^{Nn}$ as

(4.7) $\tilde{u}_{\alpha}^{j,p^{\prime}}=\sum_{0^{\prime}\leq q^{\prime}\leq
p^{\prime}}\Big{(}\begin{array}[]{c}p^{\prime}\\\
q^{\prime}\end{array}\Big{)}\pi^{q^{\prime}}D^{(|q^{\prime}|,p^{\prime}-q^{\prime})}D_{\beta}u^{j}\quad\text{in}\quad
Q_{5},$

for any $\displaystyle 1\leq\alpha\leq n$ and $\displaystyle 1\leq j\leq N$.
By recalling (4.6) and that $\displaystyle U_{1}^{i,p}$ in (4.1),

(4.8) $U_{1}^{i,p^{\prime}}=G_{1}^{i,p^{\prime}}+\sum_{1\leq\alpha\leq
n}\pi_{\alpha}\left[\nu_{\alpha}^{i,p^{\prime}}+\sum_{1\leq j\leq
N}\sum_{1\leq\beta\leq
n}A^{\alpha\beta}_{ij}\tilde{u}_{\beta}^{j,p^{\prime}}\right]\quad\text{in}\quad
Q_{5},$

and

(4.9)
$U_{\beta}^{j,p^{\prime}}=G_{\beta}^{j,p^{\prime}}+\tilde{u}_{\beta}^{j,p^{\prime}}+\pi_{\beta}\tilde{u}_{1}^{j,p^{\prime}}\quad\text{in}\quad
Q_{5},$

for any $\displaystyle 2\leq\beta\leq n$ and $\displaystyle 1\leq j\leq N$.