Uniform boundary estimates in homogenization of higher-order elliptic systems

Abstract

This paper focuses on uniform boundary estimates in homogenization of a family of higher-order elliptic operators $\mathcal {L}_\varepsilon$, with rapidly oscillating periodic coefficients. We derive uniform boundary $C^{m-1,\lambda } (0\!<\!\lambda \!<\!1)$ and $W^{m,p}$ estimates in $C^1$ domains, as well as uniform boundary $C^{m-1,1}$ estimate in $C^{1,\theta } (0\!<\!\theta \!<\!1)$ domains without the symmetry assumption on the operator. The proof, motivated by the works “Armstrong and Smart in Ann Sci Éc Norm Supér (4) 49(2):423–481 (2016) and Shen in Anal PDE 8(7):1565–1601 (2015),” is based on a suboptimal convergence rate in $H^{m-1}(\Omega )$. Compared to “Kenig et al. in Arch Ration Mech Anal 203(3):1009–1036 (2012) and Shen (2015),” the convergence rate obtained here does not require the symmetry assumption on the operator, nor additional assumptions on the regularity of $u_0$ (the solution to the homogenized problem), and thus might be of some independent interests even for second-order elliptic systems.

1 Introduction

This paper is aimed to investigate the uniform boundary estimates in homogenization of the following 2m-order elliptic system,

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {L}_\varepsilon u_\varepsilon =f &{} \quad \text { in } \Omega , \\ \hbox {Tr} (D^\gamma u_\varepsilon )=g_\gamma &{} \quad \text { on } \partial \Omega \quad \text {for } 0\le |\gamma |\le m-1, \end{array}\right. } \end{aligned}$$

(1.1)

where $\Omega \subset \mathbb {R}^d, d\ge 1,$ is a bounded Lipschitz domain,

$$\begin{aligned} (\mathcal {L}_\varepsilon u_\varepsilon )_i = (-1)^{m}\sum _{|\alpha |=|\beta |=m} D^\alpha (A_{ij}^{\alpha \beta }(x/\varepsilon )D^\beta u_{\varepsilon j}), ~~ 1\le i, j\le n, \end{aligned}$$

$u_{\varepsilon j}$ denotes the jth component of the $\mathbb {R}^n$-valued vector function $u_\varepsilon$, $\alpha , \beta ,\gamma$ are multi-indexes with components $\alpha _k, \beta _k,\gamma _k, k=1,2,\ldots ,d$, and

$$\begin{aligned} |\alpha |=\sum _{k=1}^d \alpha _k, ~~D^\alpha =D_{x_1}^{\alpha _1} D_{x_2}^{\alpha _2}\cdot \cdot \cdot D_{x_d}^{\alpha _d}. \end{aligned}$$

The coefficients matrix $A(y)=(A_{ij}^{\alpha \beta }(y)), 1 \le i, j \le n,$ is real, bounded measurable, satisfying the strong ellipticity condition

$$\begin{aligned} \mu |\xi |^2\le \sum _{|\alpha |=|\beta |=m} A_{ij}^{\alpha \beta }(y)\xi ^i_\alpha \xi ^j_\beta \le \frac{1}{\mu } |\xi |^2 \quad \text {for }\, \hbox {a.e.}\, y\in \mathbb {R}^{d}, \end{aligned}$$

(1.2)

where $\mu >0, \xi =(\xi _\alpha )_{|\alpha |=m}, \xi _\alpha =(\xi ^1_\alpha ,\ldots ,\xi _\alpha ^n)\in \mathbb {R}^n$, as well as the periodicity condition

$$\begin{aligned} A(y+z)=A(y), \quad \text {for any } z\in \mathbb {Z}^{d} \text { and }\, \hbox {a.e.}\, y\in \mathbb {R}^{d}. \end{aligned}$$

(1.3)

The regularity estimate uniform in $\varepsilon >0$ is one of the main concerns in quantitative homogenization. For second-order elliptic operators, this issue has been studied extensively. In the celebrated work of Avellaneda and Lin [5,6,7], by using a compactness method, the interior and boundary Hölder estimate, $W^{1,p}$ estimate and Lipschitz estimate were obtained for second-order elliptic systems with Hölder continuous coefficients and Dirichlet conditions in bounded $C^{1,\theta }$ domains. The uniform boundary Lipschitz estimate for the Neumann problem has been a longstanding open problem and was recently settled by Kenig et al. [23]. Interested readers may refer to [20, 24, 34, 36] and references therein for more applications of compactness method in quantitative homogenization. More recently, another fabulous scheme, which is based on convergence rates, was formulated in [4] to investigate uniform (interior) estimates in stochastic homogenization. The approach was further developed in [3, 35], where the large-scale interior or boundary Lipschitz estimates for second-order elliptic operators with periodic and almost periodic coefficients were studied systematically. We also refer readers to [2, 16, 17, 46] for more related results.

Relatively speaking, few quantitative results were known in the homogenization of higher-order elliptic equations previously, although results on qualitative homogenization have been obtained for many years [9]. Very recently, the optimal $O(\varepsilon )$ convergence rate in $L^2(\mathbb {R}^d)$ for higher-order elliptic equations was obtained in [25, 29, 30]. In [39, 40], some interesting two-parameter resolvent estimates were established in homogenization of general higher-order elliptic systems with periodic coefficients in bounded $C^{2m}$ domains. Meanwhile, in [28, 45] we investigated the sharp $O(\varepsilon )$ convergence rate in periodic and almost periodic homogenization of higher-order elliptic systems in Lipschitz domains. Particularly, under the assumptions that A is symmetric and $u_0\!\in \!H^{m+1}(\Omega )$, the optimal $O(\varepsilon )$ convergence rate was obtained in $W^{m-1, q_0}(\Omega ), q_0=2d/(d-1)$ in [28]. Moreover, the uniform interior $W^{m,p}$ and $C^{m-1,1}$ estimates were also established therein.

As a continuation of [28], in this paper we investigate the uniform boundary estimates in homogenization of higher-order elliptic system (1.1). Let $\psi : \mathbb {R}^{d-1}\rightarrow \mathbb {R}$ be a $C^1$ function with

$$\begin{aligned} \begin{aligned}&\psi (0)=0, \quad |\nabla \psi |\le M,\\&\sup \Big \{ |\nabla \psi (x')-\nabla \psi (y') |: x',y'\in \mathbb {R}^{d-1} \text { and } |x'-y'|\le t \Big \}\le \tau (t), \end{aligned} \end{aligned}$$

(1.4)

where $\tau (t)\longrightarrow 0$ as $t\longrightarrow 0^+.$ Set

$$\begin{aligned} \begin{aligned}&D_r=D(r,\psi )=\Big \{ (x',x_d)\in \mathbb {R}^d: |x'|<r \text { and } \psi (x')<x_d<\psi (x')+r\Big \},\\&\Delta _r=\Delta (r,\psi )=\Big \{ (x',\psi (x'))\in \mathbb {R}^d: |x'|<r \Big \}. \end{aligned} \end{aligned}$$

(1.5)

The main results of this paper are stated as follows.

Theorem 1.1

Suppose that the coefficient matrix $A=A(y)$ satisfies the conditions (1.2)–(1.3) and $u_\varepsilon \in H^{m}(D_1; \mathbb {R}^n)$ is a weak solution to

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {L}_\varepsilon u_\varepsilon =F &{}\text { in } D_1, \\ \hbox {Tr} (D^\gamma u_\varepsilon )=D^\gamma G &{}\text { on } \Delta _1 \quad \text {for } 0\le |\gamma |\le m-1, \end{array}\right. } \end{aligned}$$

where $G\in C^{m-1,1}(D_1; \mathbb {R}^n), F\in L^p(D_1; \mathbb {R}^n)$ with $p>\max \big \{d/(m+1), 2d/(d+2m-2), 1\big \}$. Then, for any $0<\lambda <\min \{m+1-d/p, 1\}$ and any $\varepsilon \le r < 1,$

$$\begin{aligned} \Big (\fint _{D_r}|\nabla ^m u_\varepsilon |^2\Big )^{1/2}\le C r^{\lambda -1}\bigg \{&\Big (\fint _{D_1}|u_\varepsilon |^2 \Big )^{1/2}+\Big (\fint _{D_1} |F|^p\Big )^{1/p} + \Vert G\Vert _{C^{m-1,1}(D_1)} \bigg \}, \end{aligned}$$

(1.6)

where C depends only on $d,n,m, \lambda ,\mu ,p$ and $\tau (t)$ in (1.4).

Estimate (1.6) can be viewed as the $C^{m-1,\lambda }$ estimate uniform down to the scale $\varepsilon$ in $C^1$ domains for higher-order elliptic operators $\mathcal {L}_\varepsilon$. In addition to the assumptions in Theorem 1.1, if $A\in V\!M\!O (\mathbb {R}^d),$ i.e.,

$$\begin{aligned} \sup _{ x\in \mathbb {R}^d,\,0<r<t } \fint _{B(x,r)}|A(y)-\fint _{B(x,r)}A|\hbox {d}y\le \varrho (t), ~~~ 0<t\le 1, \end{aligned}$$

(1.7)

for some nondecreasing continuous function $\varrho (t)$ on [0, 1] with $\varrho (0)= 0$, then a standard blow-up argument gives the following full-scale boundary $C^{m-1,\lambda }$ estimate

$$\begin{aligned} \Vert u_\varepsilon \Vert _{C^{m-1,\lambda }(D_{1/4})}\le C \bigg \{\Big (\fint _{D_1}|u_\varepsilon |^2 \Big )^{1/2}&+\Big (\fint _{D_1} |F|^p\Big )^{1/p} + \Vert G\Vert _{C^{m-1,1}(D_1)}\bigg \}. \end{aligned}$$

(1.8)

We also mention that the restriction $p>\max \{ d/(m+1), 1\}$ is made to ensure $C^{m-1,\lambda }$ estimate of the solution $u_0$ to the homogenized system, which plays an essential role in the proof of the theorem. The restriction $p>2d/(d+2m-2)$ is used to ensure that $F\in H^{-m+1}(\Omega )$, since our proof is based on the convergence result in Theorem 1.4 (see Lemma 4.1 for details). Although the assumption on the regularity of F in Theorem 1.1 is not sharp, see Corollary 5.1 for the full-scale uniform $C^{m-1,\lambda }$ estimate of $u_\varepsilon$, it is enough for us to derive the following uniform $W^{m,p}$ estimate on $u_\varepsilon$.

Theorem 1.2

Let $\Omega$ be a bounded $C^1$ domain in $\mathbb {R}^d$. Suppose that the coefficient matrix $A\in V\!M\!O(\mathbb {R}^d)$ satisfies (1.2)–(1.3) and $u_\varepsilon \in H^m(\Omega ; \mathbb {R}^n)$ is a weak solution to

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {L}_\varepsilon u_\varepsilon =\sum _{|\alpha |\le m}D^\alpha f^\alpha &{}\text { in } \Omega ,\\ \hbox {Tr}(D^\gamma u_\varepsilon )=g_\gamma &{}\text { on } \partial \Omega \quad \text {for}~ 0\le |\gamma |\le m-1, \end{array}\right. } \end{aligned}$$

where $\dot{g} =\{g_\gamma \}_{|\gamma |\le m-1}\in \dot{B}^{m-1/p}_p(\partial \Omega ; \mathbb {R}^n)$ and $f^\alpha \in L^p(\Omega ;\mathbb {R}^n)$ for $|\alpha |\le m, 2\le p< \infty$. Then,

$$\begin{aligned} \Vert u_\varepsilon \Vert _{W^{m,p}(\Omega )} \le C_p \,\bigg \{ \sum _{|\alpha |\le m}\Vert f^\alpha \Vert _{L^p(\Omega )} + \Vert \dot{g}\Vert _{\dot{B}^{m-1/p}_p(\partial \Omega ) }\bigg \}, \end{aligned}$$

(1.9)

where the constant $C_p$ depends only on $p,d,n,m,\mu ,\Omega$ and $\varrho (t)$ in (1.7).

We refer readers to Sect. 2 for the definition of the Whitney–Besov space $\dot{B}^{s}_p(\partial \Omega ; \mathbb {R}^n)$. Note that although the result presented in Theorem 1.2 focuses on the case $p\ge 2$, by a standard duality argument, it still holds for $1\!<\!p\!<\!2$. We also mention that the uniform $W^{1,p}$ estimates in the homogenization of second-order elliptic systems have been studied largely, see e.g., [14, 15, 33, 43, 44]. Theorem 1.2 generalizes the uniform $W^{1,p}$ estimates for second-order elliptic systems to higher-order elliptic systems. It also extends, in some sense, the $W^{m,p}$ estimate for higher-order elliptic equations (or systems) with non-oscillating coefficients, see e.g., [10, 12, 13].

Our third result gives the uniform boundary $C^{m-1,1}$ estimate of $u_\varepsilon$ in $C^{1,\theta }\, (0<\theta <1)$ domains. Let $D_r, \Delta _r$ be defined as in (1.5), and let the defining function $\psi \in C^{1,\theta } (\mathbb {R}^{d-1})$ with

$$\begin{aligned} \psi (0)=0, \quad \Vert \nabla \psi \Vert _{C^\theta (\mathbb {R}^{d-1})}\le M_1. \end{aligned}$$

(1.10)

Theorem 1.3

Assume that A satisfies (1.2)–(1.3). Let $u_\varepsilon \in H^m(D_{1}; \mathbb {R}^n )$ be a weak solution to

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {L}_\varepsilon u_\varepsilon = \sum _{|\alpha |\le m-1}D^\alpha f^\alpha &{}\text { in } D_1, \\ \hbox {Tr} (D^\gamma u_\varepsilon )=D^\gamma G &{}\text { on } \Delta _1 \quad \text {for } 0\le |\gamma |\le m-1, \end{array}\right. } \end{aligned}$$

where $f^\alpha \in L^{q}(D_1; \mathbb {R}^n)$ with $q>d, q\ge 2$, and $G\in C^{m,\sigma }(D_1; \mathbb {R}^n)$ for some $0<\sigma \le \theta$. Then, for any $\varepsilon \le r<1$, we have

$$\begin{aligned} \Big (\fint _{D_r}|\nabla ^m u_\varepsilon |^2\Big )^{1/2}\le C \bigg \{\Big (\fint _{D_1}|u_\varepsilon |^2 \Big )^{1/2} + \sum _{|\alpha |\le m-1}\Big (\fint _{D_1}|f^\alpha |^q\Big )^{1/q} + \Vert G \Vert _{C^{m,\sigma }(D_1)}\bigg \}, \end{aligned}$$

(1.11)

where C depends only on $d,n,m,\mu , q, \sigma , \theta$ and $M_1$.

Similar to (1.6), estimate (1.11) is the $C^{m-1,1}$ estimate uniform down to the scale $\varepsilon$ for the operator $\mathcal {L}_\varepsilon$, which separates the large-scale estimates due to the homogenization process from the small-scale estimates related to the smoothness of the coefficients. If in addition, A is Hölder continuous, i.e., there exist $\Lambda _0>0, \tau _0\in (0,1)$ such that

$$\begin{aligned} |A(x)-A(y)|\le \Lambda _0 |x-y|^{\tau _0} \quad \text { for any } x, y\in \mathbb {R}^d, \end{aligned}$$

(1.12)

we can derive the full-scale boundary $C^{m-1,1}$ estimate

$$\begin{aligned} \Vert \nabla ^mu_\varepsilon \Vert _{L^{\infty }(D_{1/4})}\le C \bigg \{\Big (\fint _{D_1}|u_\varepsilon |^2 \Big )^{1/2} +\sum _{|\alpha |\le m-1}\Big (\fint _{D_1}|f^\alpha |^q\Big )^{1/q} + \Vert G \Vert _{C^{m,\sigma }(D_1)}\bigg \}. \end{aligned}$$

(1.13)

This generalizes the boundary Lipschitz estimates in [5, 35] for second-order elliptic systems to higher-order elliptic systems.

Note that Theorem 1.3 does not require the symmetry assumption on the coefficient matrix A. Therefore, it may be of some independent interests even for second-order elliptic systems [21, p. 485]. Recall that the symmetry assumption on the coefficient matrix A is made in [23] to establish the uniform boundary Lipschitz estimate for second-order elliptic systems with Neumann boundary conditions. Such an assumption was removed in [3], where the boundary Lipschitz estimate was obtained for second-order elliptic systems with almost periodic coefficients. However, our investigations do not rely on the nontangential maximum function estimates, which had played an essential role in [3, p. 1896]. This may allow one to treat more general elliptic systems which arise in the homogenization theory [9], see also [21] for some discussions on this topic.

Finally, we mention that the requirements on smoothness of coefficients and the domain for uniform estimates in Theorems 1.1–1.3 are the same as those for second-order elliptic systems [35]. Therefore, results in theorems above, combined with the interior estimates in our previous paper [28], present a unified description on the uniform regularity estimates in homogenization of 2m-order elliptic systems in the divergence form. The counterpart for higher-order elliptic operators of non-divergence form will be studied in future.

The proofs of theorems above rely on the following convergence result.

Theorem 1.4

Suppose that $\Omega$ is a bounded Lipschitz domain in $\mathbb {R}^d, d\ge 1$, and the coefficient matrix A satisfies (1.2)–(1.3). Let $u_\varepsilon , u_0$ be the weak solutions to the Dirichlet problem (1.1) and the homogenized problem (2.2), respectively. Then, for $0<\varepsilon <1$ and any $0<\nu <1$, we have

$$\begin{aligned} \Vert u_\varepsilon -u_0\Vert _{H^{m-1}_0(\Omega )}\le C_\nu \varepsilon ^{1-\nu } \left\{ \Vert \dot{g}\Vert _{W\!A^{m,2}(\partial \Omega )}+\Vert f\Vert _{H^{-m+1}(\Omega )}\right\} , \end{aligned}$$

(1.14)

where $C_\nu$ depends only on $d, n, m,\nu ,\mu$ and $\Omega .$ If in addition A is symmetric, i.e., $A=A^*$, then

$$\begin{aligned} \Vert u_\varepsilon -u_0\Vert _{H^{m-1}_0(\Omega )} \le C \varepsilon \ln (1/\varepsilon ) \left\{ \Vert \dot{g}\Vert _{W\!A^{m,2}(\partial \Omega )}+\Vert f\Vert _{H^{-m+1}(\Omega )}\right\} , \end{aligned}$$

(1.15)

where C depends only on $d, n, m,\mu$ and $\Omega .$

The error estimates above can be viewed as a counterpart in general Lipschitz domains for the convergence rates obtained in [25, 29, 30, 39]. Estimate (1.14) is new even for second-order elliptic systems. Recall that sharp convergence rate is also one of the central issues in quantitative homogenization theory. The estimate

$$\begin{aligned} \Vert u_\varepsilon -u_0\Vert _{L^{2} (\Omega )} \le C \varepsilon \Vert u_0\Vert _{H^{2}(\Omega )} \end{aligned}$$

has been obtained for second-order elliptic equations in divergence form in $C^{1,1}$ domains [18, 37, 38], as well as in Lipschitz domains with additional assumptions $u_0\in H^2(\Omega )$ and $A=A^*$ [28, 35]. In [22, 42], the $O[\varepsilon \ln (1/\varepsilon )]$ convergence rate like (1.15) was obtained for second-order elliptic systems under the assumption that $A=A^*$. Compared with the reference aforementioned, our estimate (1.14), although suboptimal, holds in general Lipschitz domains and needs neither the symmetry of A, nor additional regularity assumptions of $u_0$. Moreover, the assumptions on the regularity of $A, \dot{g}$ and f are also rather general. To the best of the authors’ knowledge, optimal or suboptimal convergence rate under such weak conditions seems to be unknown previously even for second-order elliptic systems.

The proof of Theorem 1.4 follows the line of [22, 37]. The first step is to derive an estimate like

$$\begin{aligned} \Vert u_\varepsilon -u_0\Vert _{H^{m-1} (\Omega )} \le C \varepsilon ^{(1/2)^-} \left\{ \Vert \dot{g}\Vert _{W\!A^{m,2}(\partial \Omega )}+\Vert f\Vert _{H^{-m+1}(\Omega )}\right\} . \end{aligned}$$

(1.16)

When A is symmetric, this was done with the help of the nontangential maximum function estimate, which gives proper controls on $u_0$ near the boundary $\partial \Omega$, see [22, 35, 42] for the details. Unfortunately, if A is not symmetric and the domain is just Lipschitz (or even $C^1$) the nontangential maximum function estimate is not in hand. Instead, we will take advantage of some weighted estimate of $u_0$ (see Lemma 3.2) to achieve the goal. With the estimate (1.16) at our disposal, we then modify the duality argument in [37] with proper weight to derive the desired convergence rate. This idea is also partially motivated by [22, 42], see Remark 3.1.

Armed with Theorem 1.4, our proof of Theorems 1.1 and 1.3 follows the scheme in [3, 4, 35], which roughly speaking is a three-step argument:

• (i) Establish the convergence rate in $L^2(\Omega )$ in terms of boundary data g and the forcing term f, i.e., the error estimate like

  $$\begin{aligned} \Vert u_\varepsilon -u_0\Vert _{L^2(\Omega )}\le C \varepsilon ^{\sigma _0}\big \{ \text { norms of data } g\text { and } f \big \} \quad \text { for some } 0<\sigma _0\le 1; \end{aligned}$$

• (ii) Prove that $u_\varepsilon$ satisfies the so-called flatness property, i.e., how well it could be approximated by affine functions as $u_0$ does;

• (iii) Iterate step (ii) down to the scale $\varepsilon$, with the help of the error estimate in the first step.

Note that (1.14) gives (i), and we can thus pass to Step (ii). We shall adapt some ideas in [3, 4, 35] to verify that $u_\varepsilon$ satisfies the “flatness property.” However, instead of estimating how well $u_\varepsilon$ is approximated by affine functions as in [3, 4, 35], we estimate how well $u_\varepsilon$ is approximated by polynomials of degree $m-1$ and m, respectively. By a proper iteration argument, we then derive the desired large-scale $C^{m-1,\lambda } (0\!<\!\lambda \!<1)$ and $C^{m-1,1}$ estimates. The corresponding full-scale estimates (4.14) and (6.16) follow from a standard blow-up argument.

Finally, the proof of Theorem 1.2 relies on the boundary Hölder estimate (1.6) and a real-variable argument originated from [11] and further developed in [31, 32]. The key idea is to reduce the $W^{m,p}$ estimate (1.9) to a reverse Hölder inequality of the corresponding homogeneous problem, see Lemma 5.1 for the details.

2 Preliminaries

2.1 Function spaces

To begin with, let us give the definitions of some function spaces involved next. Let $\Omega$ be a bounded Lipschitz domain in $\mathbb {R}^d.$ Let $H^m(\Omega ; \mathbb {R}^n )$ and $H_0^m(\Omega ; \mathbb {R}^n )$ with dual $H^{-m}(\Omega ; \mathbb {R}^n )$ be the conventional Sobolev spaces of $\mathbb {R}^n$-valued functions. For $0<s<1, 1<p<\infty$ and any nonnegative integer k, let $B^{k+s}_p(\Omega )$ be the Besov space with norm (see e.g., [19, p. 17])

$$\begin{aligned} \Vert u\Vert _{B^{k+s}_p(\Omega )}= \sum _{0\le \ell \le k}\Vert \nabla ^\ell u\Vert _{L^{p}(\Omega )}+\sum _{|\zeta |=k}\bigg \{\int _{\Omega } \int _{\Omega } \frac{|D^\zeta f(x)-D^\zeta f(y)|^p}{|x-y|^{d +sp}}\,\hbox {d}x\,\hbox {d}y\bigg \}^{1/p}. \end{aligned}$$

Since $\Omega$ is a bounded Lipschitz domain, $B^{k+s}_p(\Omega )$ consists of the restrictions to $\Omega$ of functions in $B^{k+s}_p(\mathbb {R}^d)$ [19, p. 25].

Also, define the Whitney–Besov space $\dot{B}^{m-1+s}_p(\partial \Omega ; \mathbb {R}^n)$ as the closure of the set of arrays

$$\begin{aligned} \left\{ \{ D^\alpha \mathcal {U} \}_{|\alpha |\le m-1}: \mathcal {U}\in C_c^\infty (\mathbb {R}^d) \right\} , \end{aligned}$$

under the norm

$$\begin{aligned} \Vert \dot{u} \Vert _{\dot{B}^{m\!-1\!+s}_p(\partial \Omega )} =\sum _{|\alpha |\le m-1}\Big \{\Vert u_\alpha \Vert _{L^p(\partial \Omega )}+\Big (\int _{\partial \Omega } \int _{\partial \Omega } \frac{|u_\alpha (x)-u_\alpha (y)|^p}{|x-y|^{d-1+sp}}\hbox {d}S_x \hbox {d}S_y\Big )^{1/p}\Big \}, \end{aligned}$$

where $\dot{u}= \{u_\alpha \}_{|\alpha |\le m-1},$ see e.g., [1].

Define the Whitney–Sobolev space $W\!A^{m,p}(\partial \Omega , \mathbb {R}^n)$ as the completion of the set of arrays of $\mathbb {R}^n$-valued functions

$$\begin{aligned} \left\{ \{ D^\alpha \mathcal {U}\mid _{\partial \Omega }\}_{|\alpha |\le m-1}: \mathcal {U}\in C_c^\infty (\mathbb {R}^d; \mathbb {R}^n ) \right\} , \end{aligned}$$

under the norm [26]

$$\begin{aligned} \Vert \dot{g} \Vert _{W\!A^{m,p}(\partial \Omega )} =\sum _{|\alpha |\le m-1} \Vert g_\alpha \Vert _{L^p(\partial \Omega )} + \sum _{|\alpha |=m-1} \Vert \nabla _{tan} g_\alpha \Vert _{L^p(\partial \Omega )}. \end{aligned}$$

2.2 Qualitative homogenization

Under the ellipticity condition (1.2), for any $\dot{g}\in W\!A^{m,2}(\partial \Omega , \mathbb {R}^n)$ and $f\in H^{-m}(\Omega ;\mathbb {R}^n)$, Dirichlet problem (1.1) admits a unique weak solution $u_\varepsilon$ in $H^m(\Omega ;\mathbb {R}^n)$ such that

$$\begin{aligned} \Vert u_\varepsilon \Vert _{H^m(\Omega )} \le C \left\{ \Vert f\Vert _{H^{-m}(\Omega )} +\Vert \dot{g}\Vert _{W\!A^{m,2}(\partial \Omega )}\right\} , \end{aligned}$$

where C depends only on d, m, n, $\mu$ and $\Omega$. It is known that (see e.g., [9, 29]) under the additional periodicity condition (1.3), the operator $\mathcal {L}_\varepsilon$ is G-convergent to $\mathcal {L}_0$, where

$$\begin{aligned} (\mathcal {L}_0 u )_i = \sum _{|\alpha |=|\beta |=m} (-1)^m D^\alpha (\bar{A}_{ij}^{\alpha \beta }D^\beta u_j) \end{aligned}$$

is an elliptic operator of order 2m with constant coefficients,

$$\begin{aligned} \bar{A}_{ij}^{\alpha \beta }=\sum _{|\gamma |=m}\frac{1}{|Q|}\int _Q \Big [A_{ij}^{\alpha \beta }(y) + A_{i\ell }^{\alpha \gamma }(y)D^\gamma \chi _{\ell j}^\beta (y)\Big ]\hbox {d}y. \end{aligned}$$

Here, $Q=[0,1)^d$, $\chi =(\chi _j^\gamma )= (\chi ^\gamma _{ij})$ is the matrix of correctors for the operator $\mathcal {L}_\varepsilon$ given by

$$\begin{aligned} {\left\{ \begin{array}{ll} \sum _{|\alpha |=|\beta |=m} D^\alpha \big \{A_{ik}^{\alpha \beta }(y)D^\beta \chi _{kj}^\gamma (y)\big \}=- \sum _{|\alpha | =m} D^\alpha A_{ij}^{\alpha \gamma }(y) ~~ \text { in } \mathbb {R}^d,\\ \chi _j^\gamma (y) \quad \text { is 1-periodic } \quad \text { and }\quad \int _Q \chi _j^\gamma (y)=0. \end{array}\right. } \end{aligned}$$

(2.1)

The matrix $(\bar{A}_{ij}^{\alpha \beta })$ is bounded and satisfies the coercivity condition (1.2). Thus, the following homogenized problem of (1.1),

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {L}_0 u_0 =f &{}\text { in } \Omega , \\ \hbox {Tr} (D^\gamma u_0)=g_\gamma &{} \text { on } \partial \Omega \quad \text {for } 0\le |\gamma |\le m-1, \end{array}\right. } \end{aligned}$$

(2.2)

admits a unique weak solution $u_0\in H^m(\Omega ; \mathbb {R}^n )$, satisfying

$$\begin{aligned} \Vert u_0\Vert _{H^m(\Omega )} \le C \left\{ \Vert f\Vert _{H^{-m}(\Omega )} +\Vert \dot{g}\Vert _{W\!A^{m,2}(\partial \Omega )}\right\} . \end{aligned}$$

For $1\le i,j\le n$ and multi-indexes $\alpha ,\beta$ with $|\alpha |=|\beta |=m,$ set

$$\begin{aligned} B_{ij}^{\alpha \beta }(y)=A_{ij}^{\alpha \beta }(y)+ \sum _{|\gamma |=m}A_{ik}^{\alpha \gamma }(y) D^\gamma \chi _{kj}^\beta (y)-\bar{A}_{ij}^{\alpha \beta }. \end{aligned}$$

(2.3)

By the definitions of $\chi ^\gamma (y)$ and $\bar{A},$ for any $1\le i,j\le n$ and any multi-indexes $\alpha ,\beta$ with $|\alpha |=|\beta |=m,$$B_{ij}^{\alpha \beta }(y)\in L^2(Q)$ is 1-periodic with zero mean, and $\sum _{|\alpha |=m}D^\alpha B_{ij}^{\alpha \beta }(y)=0.$ Therefore, there exists a function $\mathfrak {B}_{ij}^{\gamma \alpha \beta }$ such that

$$\begin{aligned} \mathfrak {B}_{ij}^{\gamma \alpha \beta } =- \mathfrak {B}_{ij}^{\alpha \gamma \beta } , ~~\sum _{|\gamma |=m}D^\gamma \mathfrak {B}_{ij}^{\gamma \alpha \beta } = B_{ij}^{\alpha \beta } \quad \text {and} \quad \Vert \mathfrak {B}_{ij}^{\gamma \alpha \beta } \Vert _{H^m(Q)}\le C \Vert B_{ij}^{\alpha \beta }\Vert _{L^2(Q)}, \end{aligned}$$

where C depends only on d, n, m, see [28, Lemma 2.1].

Let $\mathcal {L}^*_\varepsilon$ be the adjoint operators of $\mathcal {L}_\varepsilon$, i.e.,

$$\begin{aligned} \mathcal {L}^*_\varepsilon = (-1)^{m}\sum _{|\alpha |=|\beta |=m} D^\alpha \left( A^{*\alpha \beta } (x/\varepsilon )D^\beta \right) , \quad ~A^*=(A_{ij}^{*\alpha \beta })= ( A_{ji}^{\beta \alpha }). \end{aligned}$$

(2.4)

Parallel to (2.1), we introduce the matrix of correctors $\chi ^*= (\chi ^{*\gamma }_{j})= (\chi ^{*\gamma }_{ij})$ for $\mathcal {L}^*_\varepsilon$,

$$\begin{aligned} {\left\{ \begin{array}{ll} \sum _{|\alpha |=|\beta |=m} D^\alpha \big \{A_{ik}^{*\alpha \beta }(y)D^\beta \chi _{kj}^{*\gamma }(y)\big \}=- \sum _{|\alpha | =m} D^\alpha A_{ij}^{*\alpha \gamma }(y) ~~ \text { in } \mathbb {R}^d,\\ \chi _j^{*\gamma } (y) \quad \text { is 1-periodic } \quad \text { and }\quad \int _Q \chi _j^{*\gamma } (y)=0. \end{array}\right. } \end{aligned}$$

(2.5)

We also introduce the dual correctors $\mathfrak {B}^{*\gamma \alpha \beta }(y)$ of $\mathcal {L}^*_\varepsilon$. It is not difficult to see that $\chi ^{*\gamma }$ and $\mathfrak {B}^{*\gamma \alpha \beta }$ satisfy the same properties as $\chi ^\gamma$ and $\mathfrak {B}^{\gamma \alpha \beta }$, since $A^*$ satisfies the same conditions as A.

2.3 Smoothing operators and auxiliary estimates

For any fixed $\varphi \in C_c^\infty (B(0,\frac{1}{2}))$ such that $\varphi >0$ and $\int _{\mathbb {R}^d} \varphi (x)\hbox {d}x=1$, set $\varphi _\varepsilon =\frac{1}{\varepsilon ^d} \varphi (\frac{x}{\varepsilon })$ and define

$$\begin{aligned} S_\varepsilon (f)(x)=\int _{\mathbb {R}^d} \varphi _\varepsilon (x-y)f(y)\, \hbox {d}y, \quad \text {and}\quad S^2_\varepsilon = S_\varepsilon \circ S_\varepsilon . \end{aligned}$$

Denote $\delta (x)=dist(x, \partial \Omega )$, $\Omega ^{ \varepsilon }=\{x\in \Omega : \delta (x)> \varepsilon \}$ and $\Omega _{ \varepsilon }=\{x\in \Omega : \delta (x)< \varepsilon \}.$

Lemma 2.1

Assume that $f\in L^p(\mathbb {R}^d)$ for some $1\le p<\infty$ and $g\in L_{loc}^p(\mathbb {R}^d)$ is 1-periodic. Let $h\in L^\infty (\mathbb {R}^d)$ with compact support $\Omega ^{3\varepsilon }$. Then,

$$\begin{aligned}&\Vert g(x/\varepsilon )S_\varepsilon (f)(x) h(x) \Vert _{L^p(\Omega ^{3\varepsilon };\, \delta )}\le C \Vert h\Vert _{L^\infty } \Vert g\Vert _{L^p(Q)} \Vert f\Vert _{L^p(\Omega ^{2\varepsilon };\, \delta )}, \end{aligned}$$

(2.6)

$$\begin{aligned}&\Vert g(x/\varepsilon )S_\varepsilon (f)(x) h(x) \Vert _{L^p(\Omega ^{3\varepsilon };\, \delta ^{-1})}\le C \Vert h\Vert _{L^\infty } \Vert g\Vert _{L^p(Q)} \Vert f\Vert _{L^p(\Omega ^{2\varepsilon };\, \delta ^{-1} )}, \end{aligned}$$

(2.7)

where $\Vert u\Vert _{L^p(\Omega ;\, \delta )}$ (similar for $\Vert u\Vert _{L^p(\Omega ;\, \delta ^{-1} )}$ ) denotes the weighted norm

$$\begin{aligned} \Vert u\Vert _{L^p(\Omega ;\, \delta )}=\Big (\int _\Omega |u(x)|^p \delta (x) \, \hbox {d}x\Big )^{1/p}. \end{aligned}$$

Proof

Observe that

$$\begin{aligned}&\int _{\mathbb {R}^d} \big |g(x/\varepsilon )h(x) \int _{\mathbb {R}^d} \varphi _\varepsilon (x-y)f(y)\, \hbox {d}y \big |^p\delta (x)\, \hbox {d}x \nonumber \\&\quad \le C\int _{\Omega ^{3\varepsilon }} |g(x/\varepsilon ) |^p \int _{\Omega ^{2\varepsilon }} \varphi _\varepsilon (x-y)|f(y)|^p \delta (y)\, \nonumber \\&\qquad \times \, \hbox {d}y \Big \{\int _{\Omega ^{2\varepsilon }}\varphi _\varepsilon (x-y) \delta (y)^{-q/p} \hbox {d}y \Big \}^{p/q}\delta (x)\, \hbox {d}x \nonumber \\&\quad \le C \int _{\Omega ^{2\varepsilon }} \int _{\Omega ^{3\varepsilon }} |g(x/\varepsilon )|^p \varphi _\varepsilon (x-y)\hbox {d}x\,|f(y)|^p \delta (y)\, \hbox {d}y \nonumber \\&\quad \le C \int _{Q} |g(z)|^pdz \int _{\Omega ^{2\varepsilon }} |f(y)|^p \delta (y)\, \hbox {d}y, \end{aligned}$$

(2.8)

where we have used Fubini’s theorem and the observation

$$\begin{aligned} \int _{\Omega ^{2\varepsilon }}\varphi _\varepsilon (x-y) [\delta (y)]^{-q/p}\, \hbox {d}y \le C [\delta (x)]^{-q/p} \end{aligned}$$

for the second inequality. This gives (2.6). The proof of (2.7) is the same. $\square$

Lemma 2.2

Let $\widetilde{\Omega }_\varepsilon = \{x\in \mathbb {R}^d: \delta (x)<\varepsilon \}, f\in H^\ell (\mathbb {R}^d), \ell \ge 0$. Then, for any multi-index $\alpha , |\alpha |=\ell$,

$$\begin{aligned}&\Vert S_\varepsilon (D^\alpha f)\Vert _{L^p(\Omega _{\varepsilon })}\le C \varepsilon ^{-\ell }\Vert f\Vert _{L^p( \widetilde{\Omega }_{2\varepsilon })}, \end{aligned}$$

(2.9)

$$\begin{aligned}&\Vert S_\varepsilon (D^\alpha f)\Vert _{L^p(\Omega ^{3\varepsilon };\, \delta )}\le C \varepsilon ^{-\ell } \Vert f\Vert _{L^p(\Omega ^{\varepsilon };\,\delta )}. \end{aligned}$$

(2.10)

Proof

Inequality (2.9) was proved in [28, Lemma 2.3], and the proof of (2.10) is quite similar. We provide it just for completeness.

$$\begin{aligned}&\Vert S_\varepsilon (D^\alpha f )\Vert ^p_{L^p(\Omega ^{3\varepsilon };\, \delta )}=\int _{\Omega ^{3\varepsilon }} \Big |\int _{\mathbb {R}^d} D^\alpha \varphi _\varepsilon (x-y) f(y) \, \hbox {d}y \Big |^p \delta (x) \, \hbox {d}x\\&\quad \le \int _{\Omega ^{3\varepsilon }} \int _{\Omega ^{2\varepsilon }} |D^\alpha \varphi _\varepsilon (x-y)|\, |f(y)|^p \delta (y)\\&\qquad \times \, \hbox {d}y \Big \{\int _{\Omega ^{2\varepsilon }} |D^\alpha \varphi _\varepsilon (x-y)|\, [\delta (y)]^{-q/p}\, \hbox {d}y \Big \}^{p/q} \, \delta (x)\, \hbox {d}x\\&\quad \le \frac{C}{\varepsilon ^{p\ell }} \int _{\Omega ^{2\varepsilon }} |f(y) |^p \delta (y)\, \hbox {d}y, \end{aligned}$$

where, for the last step, we have used Fubini’s theorem and the observation

$$\begin{aligned} \int _{\Omega ^{2\varepsilon }} |D^\alpha \varphi _\varepsilon (x-y)|\, [\delta (y)]^{-q/p} \, \hbox {d}y&\le C\int _{\Omega ^{2\varepsilon }} |D^\alpha \varphi _\varepsilon (x-y)|\, [\delta (x)]^{-q/p} \, \hbox {d}y \\&\le C \varepsilon ^{-\ell } [\delta (x)]^{-q/p} \end{aligned}$$

for $x\in \Omega ^{3\varepsilon }.$$\square$

Lemma 2.3

Suppose that $f\in W^{1,q}(\mathbb {R}^d)$ for some $1< q<\infty$. Let $\nabla ^s f=(D^\alpha f)_{|\alpha |=s}.$ Then,

$$\begin{aligned} \Vert S_\varepsilon (f)-f\Vert _{L^q(\Omega ^{2\varepsilon };\, \delta )}\le C \varepsilon \Vert \nabla f\Vert _{L^q(\Omega ^\varepsilon ;\, \delta )}. \end{aligned}$$

(2.11)

Proof

See [42, Lemma 3.3] and also [35, Lemma 2.2] for the case $q=2$. $\square$

Lemma 2.4

Assume that A satisfies (1.2)–(1.3), and $u_\varepsilon \in H^m(B(x_0,R)\cap \Omega ; \mathbb {R}^n )$ is a solution to $\mathcal {L}_\varepsilon u= \sum _{|\alpha |\le m}D^\alpha f^\alpha$ in $B(x_0,R)\cap \Omega$ with $\hbox {Tr} (D^\gamma u_\varepsilon )=D^\gamma G$ on $B(x_0,R)\cap \partial \Omega$ for some $G\in H^{m}(B(x_0,R)\cap \Omega ; \mathbb {R}^n )$ where $x_0\in \partial \Omega$. Let $f^\alpha \in L^2(B(x_0,R)\cap \Omega ; \mathbb {R}^n )$ for $|\alpha |\le m$. Then, for $0\le j\le m$ and $0<r<R$, we have

$$\begin{aligned}&\int _{B(x_0,r)\cap \Omega } |\nabla ^j (u_\varepsilon -G)|^2\nonumber \\&\quad \le \frac{C}{(R-r)^{2j}} \int _{B(x_0,R)\cap \Omega }(|u_\varepsilon |^2+|G|^2)+ C R^{2m-2j} \int _{B(x_0,R)\cap \Omega }|\nabla ^m G|^2 \nonumber \\&\qquad +C\sum _{|\alpha |\le m} R^{4m-2j-2|\alpha |} \int _{B(x_0,R)\cap \Omega } |f^\alpha |^2, \end{aligned}$$

(2.12)

where C depends only on $d,n, m,\mu$ and $\Omega$.

Proof

It is obvious that $v_\varepsilon =u_\varepsilon -G$ is a solution to

$$\begin{aligned} \begin{aligned}&\mathcal {L}_\varepsilon v_\varepsilon = \sum _{|\alpha |\le m}D^\alpha f^\alpha + \sum _{|\alpha |=|\beta |=m}D^\alpha \{A^{\alpha \beta }(x/\varepsilon ) D^\beta G\} \quad \text { in } B(x_0,R)\cap \Omega ,\\&\hbox {Tr} (D^\gamma v_\varepsilon )=0 \quad \text { on } B(x_0,R)\cap \partial \Omega \quad \text {for } 0\le |\gamma |\le m-1 . \end{aligned} \end{aligned}$$

Let $\phi \in C_c^\infty (B(x_0,R))$ with $\phi =1$ in $B(x_0,r)$ and $|\nabla ^k \phi |\le C (R-r)^{-k}$. Multiplying $v_\varepsilon \phi ^2$ and using integration by parts, we obtain that

$$\begin{aligned} \int _{B(x_0,R)\cap \Omega } |\nabla ^mv_\varepsilon |^2\phi ^2 \le&\,\sum _{|\alpha |\le m} \Big \{C(\epsilon _0)R^{2m-2|\alpha |} \int _{B(x_0,R)\cap \Omega } |f^\alpha |^2 \nonumber \\&+ \frac{\epsilon _0}{R^{2m-2|\alpha |}} \int _{B(x_0,R)\cap \Omega } |D^\alpha (v_\varepsilon \phi ^2)|^2 \Big \} \nonumber \\&+C(\epsilon _0) \sum _{|\alpha |=m} \int _{B(x_0,R)\cap \Omega } |D^\alpha G|^2 + \epsilon _0\int _{B(x_0,R)\cap \Omega } |\nabla ^m v_\varepsilon |^2\phi ^2 \nonumber \\&+ \sum _{j=0}^{m-1}\frac{C\epsilon _0+C}{(R-r)^{2m-2j}}\int _{(B(x_0,R)\setminus B(x_0,r))\cap \Omega } |\nabla ^jv_\varepsilon |^2. \end{aligned}$$

(2.13)

Note that $v_\varepsilon \phi ^2 \in H^m_0( B(x_0,R)\cap \Omega )$. Using Poincaré’s inequality and setting $\epsilon _0$ small enough, we may obtain from (2.13) that

$$\begin{aligned} \int _{B(x_0,r)\cap \Omega } |\nabla ^m(u_\varepsilon -G)|^2\le&\, \sum _{j=0}^{m-1}\frac{C}{(R-r)^{2m-2j}} \int _{(B(x_0,R)\setminus B(x_0,r))\cap \Omega }|\nabla ^j (u_\varepsilon -G)|^2 \nonumber \\&+ C\Big \{\sum _{|\alpha |\le m} R^{2m-2|\alpha |}\int _{B(x_0,R)\cap \Omega } |f^\alpha |^2\nonumber \\&+ \sum _{|\alpha |= m} \int _{B(x_0,R)\cap \Omega }|D^\alpha G|^2 \Big \}, \end{aligned}$$

(2.14)

where C depends only on d, n, m and $\mu$, but never on $\varepsilon , R.$ The estimate (2.12) follows from (2.14) in the same way as Corollary 23 in [8] through an induction argument. $\square$

Remark 2.1

It is possible to replace the $L^2$ norm of $f^\alpha$ in (2.12) by the $L^p$ norm for some $1<p<2$ when $|\alpha |<m$. For example, assuming that $f^\alpha =0$ for $1\le |\alpha |\le m$, we may prove that

$$\begin{aligned} \int _{B(x_0,r)\cap \Omega } |\nabla ^j (u_\varepsilon -G)|^2\le&\,\frac{C}{(R-r)^{2j}} \int _{B(x_0,R)\cap \Omega }\big (|u_\varepsilon |^2+|G|^2\big ) \\&+ C R^{2m-2j} \int _{B(x_0,R)\cap \Omega }|\nabla ^m G|^2 \\&+C R^{4m-2j+d-\frac{2d}{p}}\Big (\int _{B(x_0,R)\cap \Omega } |f^0|^p\Big )^{\frac{2}{p}},\\&\quad \text {for } p>\max \{1, 2d/(d+2m)\}. \end{aligned}$$

3 Convergence rates in Lipschitz domains

Let $0\le \rho _\varepsilon \le 1$ be a function in $C_c^\infty (\Omega )$ with $supp(\rho _\varepsilon )\subset \Omega ^{3\varepsilon }, \rho _\varepsilon =1$ on $\Omega ^{4\varepsilon }$ and $|\nabla ^m \rho _\varepsilon |\le C \varepsilon ^{-m}.$

Lemma 3.1

Suppose that $\Omega$ is a bounded Lipschitz domain in $\mathbb {R}^d$, and A satisfies (1.2)–(1.3). Let $u_\varepsilon , u_0$ be the weak solutions to Dirichlet problems (1.1) and (2.2), respectively. Define

$$\begin{aligned} w_\varepsilon = u_\varepsilon -u_0-\varepsilon ^m\sum _{|\gamma |=m}\chi ^\gamma (x/\varepsilon ) S^2_\varepsilon (D^\gamma u_0 )\rho _\varepsilon . \end{aligned}$$

(3.1)

Then, for any $\phi \in H^m_0(\Omega ;\mathbb {R}^n),$ we have

$$\begin{aligned}&\Big |\sum _{|\alpha |=|\beta |=m} \int _\Omega D^\alpha \phi _i A_{ij}^{\alpha \beta }(x/\varepsilon ) D^\beta w_{\varepsilon j}\Big | \nonumber \\&\quad \le C \Vert \nabla ^m \phi \Vert _{L^2(\Omega _{4\varepsilon })}\Vert \nabla ^m u_0\Vert _{L^{2}(\Omega _{4\varepsilon })} \nonumber \\&\qquad + C \Vert \nabla ^m \phi \Vert _{L^2(\Omega _{4\varepsilon })} \sum _{ 0\le k \le m-1, }\varepsilon ^{k } \Vert S_\varepsilon (\nabla ^{m+k} u_0 )\Vert _{L^2(\Omega _{5\varepsilon }\setminus \Omega _{2\varepsilon })}\nonumber \\&\qquad + C\Vert \nabla ^m \phi \Vert _{L^2(\Omega ^{2\varepsilon };\, \vartheta ^{-1})} \Vert \nabla ^m u_0 -S_\varepsilon (\nabla ^m u_0) \Vert _{L^2(\Omega ^{2\varepsilon };\, \vartheta )}\nonumber \\&\qquad + C \Vert \nabla ^m \phi \Vert _{L^2(\Omega ^{2\varepsilon };\, \vartheta ^{-1})} \sum _{0\le k\le m-1}\varepsilon ^{m-k } \Vert S_\varepsilon (\nabla ^{2m-k} u_0 )\Vert _{L^2(\Omega ^{2\varepsilon };\, \vartheta )}, \end{aligned}$$

(3.2)

where $\vartheta (x)=\delta (x)$ or 1, C depends only on $d, n, m, \mu$ and $\Omega .$

Proof

See [28, Lemma 3.1] for $\vartheta \equiv 1$. The proof for the case $\vartheta (x)=\delta (x)$ is almost the same with the help of Lemmas 2.1,  2.2 and  2.3. $\square$

Lemma 3.2

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb {R}^d$. Let A satisfy (1.2)–(1.3) and let $u_0$ be the weak solution to homogenized problem (2.2) with $\dot{g} \in W\!A^{m,2}(\partial \Omega ;\mathbb {R}^n) , f\in H^{-m+1}(\Omega ;\mathbb {R}^n)$. Then, for any $0<\nu < 1/2,$

$$\begin{aligned}&\Vert \nabla ^m u_0\Vert _{L^2(\Omega _{2\varepsilon })}\le C_\nu \varepsilon ^{1/2-\nu } \left\{ \Vert \dot{g}\Vert _{W\!A^{m,2}(\partial \Omega )}+\Vert f\Vert _{H^{-m+1} (\Omega )}\right\} , \end{aligned}$$

(3.3)

$$\begin{aligned}&\Vert \nabla ^{m+1} u_0\Vert _{L^2(\Omega ^{2\varepsilon })}\le C_\nu \varepsilon ^{-1/2-\nu } \left\{ \Vert \dot{g}\Vert _{W\!A^{m,2}(\partial \Omega )}+\Vert f\Vert _{H^{-m+1} (\Omega )}\right\} , \end{aligned}$$

(3.4)

$$\begin{aligned}&\Vert \nabla ^m u_0\Vert _{L^2(\Omega ^{2\varepsilon };\, \delta ^{-1} )}\le C_\nu \varepsilon ^{-\nu } \left\{ \Vert \dot{g}\Vert _{W\!A^{m,2}(\partial \Omega )}+\Vert f\Vert _{H^{-m+1} (\Omega )}\right\} , \end{aligned}$$

(3.5)

$$\begin{aligned}&\Vert \nabla ^{m+1}u_0\Vert _{L^{2}(\Omega ^{2\varepsilon };\, \delta )}\le C_\nu \varepsilon ^{-\nu } \left\{ \Vert \dot{g}\Vert _{W\!A^{m,2}(\partial \Omega )}+\Vert f\Vert _{H^{-m+1} (\Omega )}\right\} , \end{aligned}$$

(3.6)

where $C_\nu$ depends only on $d,n,m,\nu ,\mu$ and $\Omega$.

Proof

Recall that $f \in H^{-m+1}(\Omega )$ can be written as

$$\begin{aligned} f=\sum _{|\zeta |\le m-1} D^\zeta f^\zeta ~~\text { with } ~~ \Vert f\Vert _{H^{-m+1}(\Omega )} \approx \sum _{|\zeta |\le m-1} \Vert f^\zeta \Vert _{L^{2}(\Omega )}. \end{aligned}$$

Let $\widetilde{f}^\zeta$ be the extension of $f^\zeta$, being zero in $\mathbb {R}^d\setminus \Omega .$ Let $\widehat{\Omega }$ be a smooth bounded domain such that $\Omega \subset \widehat{\Omega }$. Let $v_0$ be the solution to

$$\begin{aligned} \mathcal {L}_0 v_0=\sum _{|\zeta |\le m-1}D^\zeta \widetilde{f}^\zeta \quad \text {in } \widehat{\Omega } ,\quad \quad \hbox {Tr}(D^\gamma v_0)=0 \quad \text {on } \partial \widehat{\Omega } \quad \text {for } 0\le |\gamma |\le m-1. \end{aligned}$$

(3.7)

Standard regularity estimates for higher-order elliptic systems imply that

$$\begin{aligned} \sum _{1\le \ell \le m+1}\Vert \nabla ^{\ell } v_0\Vert _{L^{2}(\widehat{\Omega })}\le C \Vert f\Vert _{H^{-m+1}(\Omega )}. \end{aligned}$$

(3.8)

Denote $\Sigma _t= \{x\in \Omega : dist(x,\Omega )=t\}, 0\le t\le c_0.$ Similar to [28, Theorem 3.1] (see (3.23) and (3.24) therein), by trace theorem, we may prove that

$$\begin{aligned}&\sum _{1\le \ell \le m} \Vert \nabla ^\ell v_0\Vert _{L^2(\Sigma _t)}\le C \Vert f\Vert _{H^{-m+1}(\Omega )}, \end{aligned}$$

(3.9)

$$\begin{aligned}&\sum _{1\le \ell \le m} \Vert \nabla ^\ell v_0\Vert _{L^2(\Omega _\varepsilon )}\le C\varepsilon ^{1/2} \Vert f\Vert _{H^{-m+1}(\Omega )}. \end{aligned}$$

(3.10)

On the other hand, setting $u_0(x)=v_0(x)+v(x),$ we have

$$\begin{aligned} \mathcal {L}_0 v=0 \quad \text {in } \Omega , \quad \hbox {Tr}(D^\gamma v)=g_\gamma -D^\gamma v_0 \quad \text {on } \partial \Omega \, \quad \text {for } 0\le |\gamma |\le m-1. \end{aligned}$$

(3.11)

Thanks to Theorem $3'$ and Theorem $5'$ in [1], we have $v \in B_2^{m-1/2+s}(\Omega )$ for any $1/2 \!<\!s\!<\!1$, and,

$$\begin{aligned} \Vert v\Vert _{B^{m-1/2+s}_2(\Omega )}&\le C_s \big \{\Vert \dot{g}\Vert _{W\!A^{m,2}(\partial \Omega )}+\Vert \dot{v}_0\Vert _{W\!A^{m,2}(\partial \Omega )} \big \} \nonumber \\&\le C_s\big \{\Vert \dot{g}\Vert _{W\!A^{m,2}(\partial \Omega )}+ \Vert f\Vert _{H^{-m+1}(\Omega )}\big \}, \end{aligned}$$

(3.12)

where $\dot{v}_0=\big \{D^\gamma v_0|_{\partial \Omega }\big \}_{|\gamma |\le m-1}$, and (3.9) has been used for the last step. Therefore, we have $D^\alpha v \in B_2^{s-1/2}(\Omega )$ for $|\alpha |=m$. Thanks to Theorems 1.4.2.4 and 1.4.4.4 in [19],

$$\begin{aligned} \int _\Omega |\nabla ^m v(x)|^2 \delta (x)^{1-2s}\hbox {d}x\le C_s \Vert v\Vert ^2_{B^{m-1/2+s}_2(\Omega )}\le C_s\big \{\Vert \dot{g}\Vert ^2_{W\!A^{m,2}(\partial \Omega )}+ \Vert f\Vert ^2_{H^{-m+1}(\Omega )}\big \}. \end{aligned}$$

(3.13)

This implies that

$$\begin{aligned} \int _{\Omega _{2\varepsilon }} |\nabla ^m v(x)|^2 \, \hbox {d}x&= \int _{\Omega _{2\varepsilon }} |\nabla ^m v(x)|^2 \delta (x)^{1-2s} \delta (x)^{2s-1}\, \hbox {d}x \nonumber \\&\le C_s \varepsilon ^{2s-1}\big \{ \Vert \dot{g}\Vert ^2_{W\!A^{m,2}(\partial \Omega )}+ \Vert f\Vert ^2_{H^{-m+1}(\Omega )} \big \} \end{aligned}$$

(3.14)

for any $1/2 \!<\!s\!<\!1$. By combining (3.10) with (3.14), we derive (3.3) by setting $\nu =1-s$.

In view of (3.11) and interior estimates for higher-order elliptic systems with constant coefficients, we have

$$\begin{aligned} |\nabla ^{m+1} v(x)|\le \frac{C}{\delta (x)} \Big (\fint _{B(x, \frac{\delta (x)}{8})} |\nabla ^{m} v |^2\Big )^{1/2}. \end{aligned}$$

Thus, by using (3.13) we deduce that

$$\begin{aligned} \Vert \nabla ^{m+1} v \Vert ^2_{L^2(\Omega ^{2\varepsilon })}&\le C \int _{\Omega ^{2\varepsilon }}\frac{1}{\delta (x)^{3-2s}} \fint _{B(x, \frac{\delta (x)}{8})} |\nabla ^{m} v(y)|^2 \delta (y)^{1-2s}\hbox {d}y\, \hbox {d}x \nonumber \\&\le C\varepsilon ^{2s-3} \Vert \nabla ^m v\Vert ^2_{L^2(\Omega ;\, \delta ^{1-2s})} \nonumber \\&\le C_s\varepsilon ^{ 2s-3} \left\{ \Vert \dot{g}\Vert ^2_{W\!A^{m,2}(\partial \Omega )}+ \Vert f\Vert ^2_{H^{-m+1}(\Omega )}\right\} , \end{aligned}$$

(3.15)

which, together with (3.8), gives (3.4).

For (3.5), it is easy to conclude from (3.12) and (3.13) that

$$\begin{aligned} \int _{\Omega ^{2\varepsilon }} |\nabla ^m v(x)|^2 \delta (x)^{-1}\, \hbox {d}x&= \int _{\Omega ^{2\varepsilon }} |\nabla ^m v(x)|^2 \delta (x)^{1-2s}\delta (x)^{2s-2}\, \hbox {d}x\nonumber \\&\le C_s \varepsilon ^{2s-2}\big \{ \Vert \dot{g}\Vert ^2_{W\!A^{m,2}(\partial \Omega )}+ \Vert f\Vert ^2_{H^{-m+1}(\Omega )} \big \}. \end{aligned}$$

(3.16)

On the other hand, by the co-area formula and (3.9) we deduce that

$$\begin{aligned} \int _{\Omega ^{2\varepsilon }} |\nabla ^m v_0(x)|^2 \delta (x)^{-1}\, \hbox {d}x&= \int _{\Omega ^{2\varepsilon }\setminus \Omega ^{c_0} } |\nabla ^m v_0(x)|^2 \delta (x)^{-1}\, \hbox {d}x +\int _{\Omega ^{c_0}} |\nabla ^m v_0(x)|^2 \delta (x)^{-1}\, \hbox {d}x\nonumber \\&\le C \int _{2\varepsilon }^{c_0}\int _{\Sigma _t} |\nabla ^m v_0(x)|^2 \frac{1}{t} \hbox {d}S \hbox {d}t+ C \int _{\Omega ^{c_0}} |\nabla ^m v_0(x)|^2 \, \hbox {d}x\nonumber \\&\le C \ln (1/\varepsilon )\, \big \{ \Vert \dot{g}\Vert ^2_{W\!A^{m,2}(\partial \Omega )}+ \Vert f\Vert ^2_{H^{-m+1}(\Omega )} \big \} \end{aligned}$$

(3.17)

for $0<\varepsilon <1/2$. This, combined with (3.16), gives (3.5). The proof for (3.6) is almost the same as (3.4), and thus we omit the details. $\square$

Lemma 3.3

Suppose that the assumptions of Lemma 3.2 are satisfied, and A is symmetric, i.e., $A=A^*$. Then, we have

$$\begin{aligned}&\displaystyle \Vert \nabla ^m u_0\Vert _{L^2(\Omega _{2\varepsilon } )}\le C \varepsilon ^{1/2} \big \{\Vert \dot{g}\Vert _{W\!A^{m,2}(\partial \Omega )}+\Vert f\Vert _{H^{-m+1} (\Omega )}\big \}, \end{aligned}$$

(3.18)

$$\begin{aligned}&\displaystyle \Vert \nabla ^m u_0\Vert _{L^2(\Omega ^{2\varepsilon };\, \delta ^{-1} )}\le C\, [\ln (1/\varepsilon )]^{1/2} \big \{\Vert \dot{g}\Vert _{W\!A^{m,2}(\partial \Omega )}+\Vert f\Vert _{H^{-m+1} (\Omega )}\big \}, \end{aligned}$$

(3.19)

$$\begin{aligned}&\displaystyle \Vert \nabla ^{m+1}u_0\Vert _{L^{2}(\Omega ^{2\varepsilon };\, \delta )}\le C\, [\ln (1/\varepsilon )]^{1/2} \big \{\Vert \dot{g}\Vert _{W\!A^{m,2}(\partial \Omega )}+\Vert f\Vert _{H^{-m+1} (\Omega )}\big \}, \end{aligned}$$

(3.20)

where C depends only on $d,n,m,\mu$ and $\Omega$.

Proof

The proof is the same as that of Lemma 3.2 except for three places. Firstly, since A is symmetric, in place of (3.13) we have the nontangential maximum function estimate, see e.g., [41, Theorem 6.1],

$$\begin{aligned} \Vert \mathcal {M}(\nabla ^m v)\Vert _{L^2(\partial \Omega )} \le C\Vert \dot{v}\Vert _{W\!A^{m,2}(\partial \Omega )} \le C\big \{\Vert \dot{g}\Vert _{W\!A^{m,2}(\partial \Omega )}+ \Vert f\Vert _{H^{-m+1}(\Omega )}\big \}, \end{aligned}$$

(3.21)

where $\mathcal {M}(\nabla ^m v)$ denotes the nontangential maximal function of $\nabla ^m v$. Therefore, instead of (3.14) we have

$$\begin{aligned} \int _{\Omega _{2\varepsilon }} |\nabla ^m v(x)|^2 \, \hbox {d}x \le C \varepsilon \left\{ \Vert \dot{g}\Vert ^2_{W\!A^{m,2}(\partial \Omega )}+ \Vert f\Vert ^2_{H^{-m+1}(\Omega )} \right\} , \end{aligned}$$

which, combined with (3.10), implies (3.18).

Secondly, in substitution for (3.16) we use the nontangential estimate (3.21) and the co-area formula to deduce that

$$\begin{aligned} \int _{\Omega ^{2\varepsilon }} |\nabla ^m v(x)|^2 \delta (x)^{-1}\, \hbox {d}x&\le C \int _{2\varepsilon }^{c_0}\int _{\Sigma _t} \frac{1}{t}|\nabla ^m v(x)|^2 dSdt+C\int _{\Omega ^{c_0}} |\nabla ^m v(x)|^2 \, \hbox {d}x \\&\le C\ln (1/\varepsilon )\left\{ \Vert \dot{g}\Vert ^2_{W\!A^{m,2}(\partial \Omega )}+ \Vert f\Vert ^2_{H^{-m+1}(\Omega )} \right\} , \end{aligned}$$

which, combined with (3.17), gives (3.19).

Finally, instead of (3.15), we have

$$\begin{aligned} \Vert \nabla ^{m+1} v \Vert ^2_{L^2(\Omega ^{\varepsilon }; \, \delta )}&\le C \int _{\Omega ^{\varepsilon }\setminus \Omega ^{c_0} }\frac{1}{\delta (x) } \fint _{B(x, \frac{\delta (x)}{8})} |\nabla ^{m} v(y)|^2 \hbox {d}y\, \hbox {d}x \\&\quad +C \int _{\Omega ^{c_0}} \fint _{B(x, \frac{\delta (x)}{8})} |\nabla ^{m} v(y)|^2 \hbox {d}y\, \hbox {d}x\\&\le C \int ^{c_0}_\varepsilon \int _{\Sigma _t} \frac{1}{t} |\mathcal {M}(\nabla ^{m} v) |^2 \hbox {d}S\, \hbox {d}t + C\left\{ \Vert \dot{g}\Vert ^2_{W\!A^{m,2}(\partial \Omega )}+ \Vert f\Vert ^2_{H^{-m+1}(\Omega )}\right\} \\&\le C \ln (1/\varepsilon ) \left\{ \Vert \dot{g}\Vert ^2_{W\!A^{m,2}(\partial \Omega )}+ \Vert f\Vert ^2_{H^{-m+1}(\Omega )}\right\} . \end{aligned}$$

This, together with (3.8), gives (3.20). The proof is thus completed. $\square$

Lemma 3.4

Assume that $\Omega$ is a bounded Lipschitz domain in $\mathbb {R}^d$ and A satisfies (1.2)–(1.3). Let $u_\varepsilon , u_0$ be the weak solutions to Dirichlet problems (1.1) and (2.2), respectively, with $\dot{g} \in W\!A^{m,2}(\partial \Omega ;\mathbb {R}^n) , f\in H^{-m+1}(\Omega ;\mathbb {R}^n)$. Let $w_\varepsilon$ be defined as in (3.1). Then, for any $0<\nu < 1/2$, we have

$$\begin{aligned} \Vert w_\varepsilon \Vert _{H^m_0(\Omega )}\le C_\nu \varepsilon ^{1/2-\nu } \left\{ \Vert \dot{g}\Vert _{W\!A^{m,2}(\partial \Omega )}+\Vert f\Vert _{H^{-m+1}(\Omega )}\right\} , \end{aligned}$$

(3.22)

where $C_\nu$ depends only on $d, n, m,\nu ,\mu$ and $\Omega .$ If in addition A is symmetric, i.e., $A=A^*$, then

$$\begin{aligned} \Vert w_\varepsilon \Vert _{H^m_0(\Omega )} \le C \varepsilon ^{1/2} \left\{ \Vert \dot{g}\Vert _{W\!A^{m,2}(\partial \Omega )}+\Vert f\Vert _{H^{-m+1}(\Omega )}\right\} , \end{aligned}$$

(3.23)

where C depends only on $d, n, m,\mu$ and $\Omega .$

Proof

The estimate (3.23) has been proved in [28, Theorem 3.1], we only need to consider (3.22) here. Using Lemmas  2.2 and 2.3, we deduce from (3.2) that

$$\begin{aligned}&\Big |\sum _{|\alpha |=|\beta |=m} \int _\Omega D^\alpha \phi _i A_{ij}^{\alpha \beta }(x/\varepsilon ) D^\beta w_{\varepsilon j}\Big |\nonumber \\&\le \, C\Big \{ \Vert \nabla ^m \phi \Vert _{L^2(\Omega _{4\varepsilon })} \Vert \nabla ^m u_0\Vert _{L^2(\Omega _{5\varepsilon })}+\varepsilon \Vert \nabla ^{m+1} u_0\Vert _{L^2(\Omega ^{2\varepsilon };\, \vartheta )}\Vert \nabla ^m \phi \Vert _{L^2(\Omega ^{2\varepsilon };\, \vartheta ^{-1})} \Big \}. \end{aligned}$$

(3.24)

Taking $\vartheta =1, \phi =w_\varepsilon$ and using the ellipticity condition (1.2), we obtain that

$$\begin{aligned} \Vert \nabla ^m w_{\varepsilon }\Vert _{H_0^m(\Omega )} \le \, C\Big \{ \Vert \nabla ^m u_0\Vert _{L^2(\Omega _{5\varepsilon })}+\varepsilon \Vert \nabla ^{m+1} u_0\Vert _{L^2(\Omega ^{2\varepsilon })} \Big \}, \end{aligned}$$

from which and (3.3), (3.4), we obtain (3.22) immediately. $\square$

We are now prepared to prove Theorem 1.4.

Proof of Theorem 1.4

We only provide the details for (1.14), as the proof of (1.15) is similar. By scaling, we may assume that

$$\begin{aligned} \Vert \dot{g}\Vert _{W\!A^{m,2}(\partial \Omega )}+\Vert f\Vert _{H^{-m+1}(\Omega )}=1. \end{aligned}$$

For any fixed $F\in H^{-m+1}(\Omega ;\mathbb {R}^n)$, let $\psi _\varepsilon \in H^m_0(\Omega ;\mathbb {R}^n)$ be the weak solution to the Dirichlet problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {L}^*_\varepsilon \psi _\varepsilon =F &{}\text { in } \Omega , \\ \hbox {Tr} (D^\gamma \psi _\varepsilon )=0 &{} \text { on } \partial \Omega \ \ \text { for } 0\le |\gamma |\le m-1, \end{array}\right. } \end{aligned}$$

and let $\psi _0\in H^m_0(\Omega ;\mathbb {R}^n)$ be the weak solution to the homogenized problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {L}_0^* \psi _0 =F &{}\text { in } \Omega , \\ \hbox {Tr} (D^\gamma \psi _0)=0 &{} \text { on } \partial \Omega \ \ \text { for } 0\le |\gamma |\le m-1. \end{array}\right. } \end{aligned}$$

Here, $\mathcal {L}^*_\varepsilon$ and $\mathcal {L}^*_0$ are the adjoint operators of $\mathcal {L}_\varepsilon$ and $\mathcal {L}_0$, respectively. Let $0\le \widetilde{\rho }_\varepsilon \le 1$ be a function in $C_c^\infty (\Omega )$ with $supp(\widetilde{\rho }_\varepsilon )\subset \Omega ^{6\varepsilon }, \widetilde{\rho }_\varepsilon =1$ on $\Omega ^{8\varepsilon }$ and $|\nabla ^m \widetilde{\rho }_\varepsilon |\le C \varepsilon ^{-m}.$ Set

$$\begin{aligned} \Psi _\varepsilon =\psi _\varepsilon -\psi _0-\varepsilon ^m\sum _{|\gamma |=m}\chi ^{*\gamma }(x/\varepsilon ) S^2_\varepsilon (D^\gamma \psi _0 )\widetilde{\rho }_\varepsilon . \end{aligned}$$

It satisfies the same properties as $w_\varepsilon ,$ since $A^*$ satisfies the same properties as A. Note that $w_\varepsilon \in H^m_0(\Omega ;\mathbb {R}^n)$, and we deduce that

$$\begin{aligned} \langle F, w_\varepsilon \rangle _{ H^{-m+1}(\Omega )\times H_0^{m-1}(\Omega ) }&= \sum _{|\alpha |=|\beta |=m} \int _{\Omega } A^{\beta \alpha }(x/\varepsilon ) D^\alpha w_\varepsilon D^\beta \Psi _\varepsilon \nonumber \\&\qquad + \sum _{|\alpha |=|\beta |=m}\int _{\Omega } A^{\beta \alpha }(x/\varepsilon )D^\alpha w_\varepsilon D^\beta \psi _0 \nonumber \\&\qquad + \sum _{|\alpha |=|\beta |=m}\int _{\Omega } A^{\beta \alpha }(x/\varepsilon ) D^\alpha w_\varepsilon D^\beta \nonumber \\&\qquad \times \Big \{\sum _{|\gamma |=m}\varepsilon ^m\chi ^{*\gamma }(x/\varepsilon ) S^2_\varepsilon (D^\gamma \psi _0 )\widetilde{\rho }_\varepsilon \Big \} \nonumber \\&\doteq J_1+J_2+J_3. \end{aligned}$$

(3.25)

By (3.22), we obtain that

$$\begin{aligned} |J_1|\le C\Vert w_\varepsilon \Vert _{H^m_0(\Omega )}\Vert \Psi _\varepsilon \Vert _{H^m_0(\Omega )}\le C_\nu \varepsilon ^{1-2\nu } \Vert F\Vert _{H^{-m+1}(\Omega )}. \end{aligned}$$

Using (3.24) and taking $\vartheta (x)=\delta (x)$, we have

$$\begin{aligned} |J_2| \le&\, C \varepsilon \Vert \nabla ^m \psi _0 \Vert _{L^2(\Omega ^{2\varepsilon };\, \delta ^{-1})} \Vert \nabla ^{m+1}u_0\Vert _{L^2(\Omega ^{2\varepsilon };\, \delta )}\nonumber \\&+ C\Vert \nabla ^mu_0\Vert _{L^{2}(\Omega _{5\varepsilon })} \Vert \nabla ^m \psi _0\Vert _{L^2(\Omega _{4\varepsilon })} . \end{aligned}$$

(3.26)

By (3.3) (note that $\psi _0$ also satisfies (3.3)), we get

$$\begin{aligned} \Vert \nabla ^{m}u_0 \Vert _{L^2(\Omega _{5\varepsilon })} \Vert \nabla ^{m}\psi _0 \Vert _{L^2(\Omega _{4\varepsilon })}\le C_\nu \varepsilon ^{1-2\nu }\Vert F\Vert _{H^{-m+1}(\Omega )}. \end{aligned}$$

Furthermore, taking (3.5) and (3.6) into consideration, we conclude from (3.26) that

$$\begin{aligned} |J_2| \le C_\nu \varepsilon ^{1-2\nu }\Vert F\Vert _{H^{-m+1}(\Omega )}. \end{aligned}$$

We now turn to $J_3$. By (3.24), we obtain that

$$\begin{aligned} |J_3 |\le&\, C \varepsilon \sum _{|\gamma |=m} \Vert \nabla ^{m+1}u_0\Vert _{L^2 (\Omega ^{2\varepsilon };\, \delta )} \varepsilon ^{m}\Vert \nabla ^m\big \{ \chi ^{*\gamma }(x/\varepsilon ) S^2_\varepsilon (D^\gamma \psi _0 )\widetilde{\rho }_\varepsilon \big \}\Vert _{L^2(\Omega ^{2\varepsilon };\, \delta ^{-1})} \nonumber \\&+ C\sum _{|\gamma |=m} \varepsilon ^m\Vert \nabla ^m u_0\Vert _{L^{2}(\Omega _{5\varepsilon })}\Vert \nabla ^m \big \{ \chi ^{*\gamma }(x/\varepsilon ) S^2_\varepsilon (D^\gamma \psi _0 )\widetilde{\rho }_\varepsilon \big \}\Vert _{L^2(\Omega _{4\varepsilon } )}, \end{aligned}$$

(3.27)

where the last term is zero by the definition of $\widetilde{\rho }_\varepsilon .$ To estimate the first term, we note that

$$\begin{aligned}&\,\varepsilon ^m\Vert D^\beta \big \{ \chi ^{*\gamma }( x/\varepsilon ) S^2_\varepsilon (D^\gamma \psi _0 )\widetilde{\rho }_\varepsilon \big \}\Vert _{L^2(\Omega ^{2\varepsilon };\, \delta ^{-1})}\\&\quad \le C \Vert (D^\beta \chi ^{*\gamma })(x/\varepsilon ) S^2_\varepsilon (D^\gamma \psi _0 )\widetilde{\rho }_\varepsilon \Vert _{L^2(\Omega ^{2\varepsilon };\, \delta ^{-1})} \\&\qquad +C \varepsilon ^m \Vert \chi ^{*\gamma }(x/\varepsilon ) S^2_\varepsilon (D^{\beta +\gamma } \psi _0 )\widetilde{\rho }_\varepsilon \Vert _{L^2(\Omega ^{2\varepsilon };\, \delta ^{-1})} \\&\qquad + C\varepsilon ^m \Vert \chi ^{*\gamma }(x/\varepsilon ) S^2_\varepsilon (D^\gamma \psi _0 )D^\beta \widetilde{\rho }_\varepsilon \Vert _{L^2(\Omega ^{2\varepsilon };\, \delta ^{-1})}\\&\qquad +C\sum _{\begin{array}{c} |\zeta +\eta +\xi |=m\\ 1\le |\zeta |, |\eta |, |\xi | \end{array}} \varepsilon ^{|\eta |+|\xi |} \Vert (D^\zeta \chi ^{*\gamma })(x/\varepsilon ) S^2_\varepsilon (D^{\gamma +\eta } \psi _0 )D^\xi \widetilde{\rho }_\varepsilon \Vert _{L^2(\Omega ^{2\varepsilon };\, \delta ^{-1})}\\&\quad \doteq J_{31}+J_{32}+J_{33} +J_{34}, \end{aligned}$$

for all multi-indexes $\beta ,\gamma$ with length m. By Lemmas 2.1 and 3.2, we obtain that

$$\begin{aligned}&J_{31}\le C \Vert S_\varepsilon (\nabla ^m \psi _0 ) \Vert _{L^2(\Omega ^{6\varepsilon };\, \delta ^{-1})} \le C\Vert \nabla ^m {\psi }_0 \Vert _{L^2(\Omega ^{5\varepsilon };\, \delta ^{-1})} \le C_\nu \varepsilon ^{-\nu } \Vert F\Vert _{H^{-m+1}(\Omega )},\\&J_{33}\le C \Vert S_\varepsilon (\nabla ^m \psi _0 )\Vert _{L^2(\Omega _{9\varepsilon }\setminus \Omega _{4\varepsilon };\, \delta ^{-1})} \le C_{\nu } \varepsilon ^{-\nu } \Vert F\Vert _{H^{-m+1}(\Omega )}. \end{aligned}$$

Furthermore, by Lemmas  2.1, 2.2 and 3.2, we see that

$$\begin{aligned} J_{32}&\le C \varepsilon ^m \Vert S_\varepsilon (\nabla ^{2m} \psi _0 ) \Vert _{L^2(\Omega ^{4\varepsilon };\, \delta ^{-1})} \le C_\nu \varepsilon ^{-\nu } \Vert F\Vert _{H^{-m+1}(\Omega )},\\ J_{34}&= C\sum _{\begin{array}{c} k_1+k_2+k_3=m\\ 1\le k_i ,i=1,2,3 \end{array}}\varepsilon ^{k_2+k_3} \Vert (\nabla ^{k_1}\chi ^{*})(x/\varepsilon ) S^2_\varepsilon (\nabla ^{k_2+m} \psi _0 ) \nabla ^{k_3}\widetilde{\rho }_\varepsilon \Vert _{L^2(\Omega _{8\varepsilon }\setminus \Omega _{6\varepsilon };\, \delta ^{-1})}\\&\le C\sum _{1\le k_2\le m-2} \varepsilon ^{k_2 }\Vert S_\varepsilon (\nabla ^{k_2+m} \psi _0 )\Vert _{L^2(\Omega _{9\varepsilon }\setminus \Omega _{5\varepsilon };\, \delta ^{-1})}\\&\le C_\nu \varepsilon ^{-\nu } \Vert F\Vert _{H^{-m+1}(\Omega )}. \end{aligned}$$

Taking the estimates on $J_{31}, J_{32}, J_{33}, J_{34}$ into (3.27), and using (3.6), we obtain that

$$\begin{aligned} |J_3|\le C_\nu \varepsilon ^{1-2\nu } \Vert F\Vert _{H^{-m+1}(\Omega )}. \end{aligned}$$

In view of the estimates on $J_1, J_2, J_3$ and (3.25), we have proved that

$$\begin{aligned} \Big | \langle F, w_\varepsilon \rangle _{H^{-m+1}(\Omega )\times H_0^{m-1}(\Omega ) } \big | \le C_\nu \varepsilon ^{1-2\nu } \Vert F\Vert _{H^{-m+1}(\Omega )}, \end{aligned}$$

which, combined with the following estimate

$$\begin{aligned} \Vert \varepsilon ^m\sum _{|\gamma |=m}\chi ^\gamma (x/\varepsilon ) S^2_\varepsilon (D^\gamma u_0 )\rho _\varepsilon \Vert _{H_0^{m-1}(\Omega )}\le C\varepsilon , \end{aligned}$$

gives (1.14). The proof is complete. $\square$

Remark 3.1

Part of our motivation for the proof of (1.14) is the finding that $u_0$ satisfying certain weighted estimates such as (3.3)–(3.6), which give a proper control on the solution $u_0$ in $\Omega ^\varepsilon$ and $\Omega _\varepsilon$. This also inspires us to modify the duality method with weight $\delta (x)$. We mention that weight functions have been used previously in [22, 42] to derive the suboptimal $O(\varepsilon \ln (1/\varepsilon ))$ convergence rate for second-order elliptic systems with symmetric coefficients. Our consideration on the suboptimal convergence rate is also in debt to these works.

4 Uniform $C^{m-1, \lambda }$ estimates

In this section, we consider the uniform boundary $C^{m-1,\lambda }, 0<\lambda <1,$ estimate of $u_\varepsilon$ in $C^1$ domains. Throughout the section, we always assume that A satisfies (1.2) and (1.3). Recall that locally the boundary of a $C^1$ domain is the graph of a $C^1$ function; we thus restrict our considerations to equations on $(D_r, \Delta _r)$ defined in (1.5) with the defining function satisfying (1.4). Let

$$\begin{aligned} \mathfrak {P}_k=\Big \{(P^1_k, P^2_k,\ldots , P^n_k)\mid P^i_k, 1\le i\le n, \text { are polynomials of degree } k \Big \}. \end{aligned}$$

Let $u_\varepsilon \in H^m(D_{2r}; \mathbb {R}^n )$ be a weak solution to

$$\begin{aligned} \mathcal {L}_\varepsilon u_\varepsilon = F \quad \text {in } D_{2r}, \quad \quad \hbox {Tr} (D^\gamma u_\varepsilon )=D^\gamma G \quad \text {on } \Delta _{2r} \quad \text {for } 0\le |\gamma |\le m-1, \end{aligned}$$

(4.1)

where $G\in C^{m-1,1}(D_{2r};\mathbb {R}^n), F\in L^p(D_{2r}; \mathbb {R}^n)$ with $p> \max \{1, 2d/(d+2m-2)\}$. Define

$$\begin{aligned} \Phi _\lambda (r,u_\varepsilon )=&\, \frac{1}{r^{m-1+\lambda }}\inf _{P_{m-1}\in \mathfrak {P}_{m-1}} \bigg \{\Big (\fint _{D_{r}}|u_\varepsilon -P_{m-1}|^2 \Big )^{1/2}+ r^{2m} \Big (\fint _{D_{r}} |F|^{p}\Big )^{1/p} \nonumber \\&+ \sum _{j=0}^{m} r^j \Vert \nabla ^j (G-P_{m-1})\Vert _{L^\infty (D_r)} \bigg \},\quad \quad 0<\lambda <1 . \end{aligned}$$

(4.2)

Lemma 4.1

Let $0<\varepsilon \le r\le 1$ and $\Phi _\lambda (r,u_\varepsilon )$ be defined as above. There exists $u_0\in H^m(D_r; \mathbb {R}^n )$ such that $\mathcal {L}_0 u_0=F$ in $D_r$, $\hbox {Tr} (D^\gamma u_0)=D^\gamma G$ on $\Delta _r$ for $0\le |\gamma |\le m-1$, and

$$\begin{aligned} \Big (\fint _{D_r}|u_\varepsilon -u_0|^2 \Big )^{1/2}\le Cr^{m-1+\lambda } \Big (\frac{\varepsilon }{r}\Big )^{1/4} \Phi _\lambda (2r,u_\varepsilon ), \end{aligned}$$

(4.3)

where C depends only on $d, n,m, p, \mu$ and M in (1.4).

Proof

Let us first assume that $r=1$. By Caccioppoli’s inequality (see Remark 2.1), we have

$$\begin{aligned} \Vert u_\varepsilon \Vert _{H^m( D_{3/2})} \le C \Big \{\Vert u_\varepsilon \Vert _{L^2(D_2)}+ \Vert F\Vert _{L^{p}(D_2)}+ \sum _{j=0}^{m} \Vert \nabla ^j G\Vert _{L^2(D_2)} \Big \}, \end{aligned}$$

(4.4)

for $p>\max \{1, 2d/(d+2m)\}$. By the co-area formula, there exists $t\in [5/4, 3/2]$ such that

$$\begin{aligned} \Vert u_\varepsilon \Vert _{H^m(\partial D_t\setminus \Delta _2) } \le C \Big \{\Vert u_\varepsilon \Vert _{L^2(D_2)}+ \Vert F\Vert _{L^{p}(D_2)} +\sum _{j=0}^{m} \Vert \nabla ^j G\Vert _{L^2(D_2)} \Big \}, \end{aligned}$$

(4.5)

where C depends only on $d,n, m,\mu$. Now let $u_0$ be the weak solution to

$$\begin{aligned} \mathcal {L}_0u_0=F ~\text { in } D_t, ~~~~~~ \hbox {Tr} (D^\gamma u_0)= \hbox {Tr}(D^\gamma u_\varepsilon ) \text { on } \partial D_t. \end{aligned}$$

Note that $F\in L^p\hookrightarrow H^{-m+1}$ when $p>\max \{ 2d/(d+2m-2), 1\}$. As a consequence of (1.14) in Theorem 1.4, we have

$$\begin{aligned} \Vert u_\varepsilon -u_0\Vert _{L^2(D_t)}&\le C \varepsilon ^{1/4} \Big \{ \Vert u_\varepsilon \Vert _{H^m(\partial D_t)}+ \Vert F\Vert _{L^{p}(D_2)} \Big \}, \end{aligned}$$

where C depends only on $d,n,m,\mu$ and M in (1.4). This, together with (4.5), yields

$$\begin{aligned} \Vert u_\varepsilon -u_0\Vert _{L^2(D_1)} \le \Vert u_\varepsilon -u_0\Vert _{L^2(D_t)} \le C \varepsilon ^{1/4} \Big \{\Vert u_\varepsilon \Vert _{L^2(D_2)}+ \Vert F\Vert _{L^{p}(D_2)}+ \Vert G\Vert _{C^{m-1,1}(D_2)} \Big \}, \end{aligned}$$

(4.6)

for $p>\max \{1, 2d/(d+2m-2) \}.$

We now perform scaling for general $\varepsilon \le r<1$. Set

$$\begin{aligned} v_\varepsilon (x)=u_\varepsilon (r x),\quad \widetilde{G}(x)=G(rx), \quad \widetilde{F}(x) =r^{2m}F(rx). \end{aligned}$$

By (4.1), we know that $\mathcal {L}_\frac{\varepsilon }{r} v_\varepsilon =\widetilde{F}(x)$ in $\widetilde{D}_2,$ and $\hbox {Tr} (D^\gamma v_\varepsilon )=D^\gamma \widetilde{G}(x)$ on $\widetilde{\Delta }_2$ for $0\le |\gamma |\le m-1,$ where

$$\begin{aligned} \widetilde{D}_2&=\big \{ (x',x_d) \in \mathbb {R}^d: |x'|< 2 \text { and } \psi _r(x')< x_d<\psi _r(x')+2 \big \},\\ \widetilde{\Delta }_2&=\big \{ (x',\psi _r(x')) \in \mathbb {R}^d: |x'| < 2 \big \}, \text { with } \psi _r(x')=r^{-1}\psi (rx'). \end{aligned}$$

Thanks to (4.6), there exists $v_0$ with $\mathcal {L}_0 v_0=\widetilde{F}(x)$ in $\widetilde{D}_1$, $\hbox {Tr} (D^\gamma v_0)=\hbox {Tr} (D^\gamma v_\varepsilon )$ on $\partial \widetilde{D}_1$ for $0\le |\gamma |\le m-1,$ such that

$$\begin{aligned} \Vert v_\varepsilon -v_0\Vert _{L^2(\widetilde{D}_1)}&\le C \Big (\frac{\varepsilon }{r}\Big )^{1/4} \Big \{\Vert v_\varepsilon \Vert _{L^2(\widetilde{D}_2)}+ \Vert \widetilde{F}(x)\Vert _{L^{p}(\widetilde{D}_2)} +\Vert \widetilde{G}\Vert _{C^{m-1,1}(\widetilde{D}_2)} \Big \}. \end{aligned}$$

Setting $u_0(x)=v_0(x/r)$, we then obtain by the change of variables,

$$\begin{aligned} \Big (\fint _{D_r}|u_\varepsilon -u_0|^2 \Big )^{1/2} \le&\, C \Big (\frac{\varepsilon }{r}\Big )^{1/4} \bigg \{\Big (\fint _{D_{2r}}|u_\varepsilon |^2 \Big )^{1/2} + r^{2m} \Big (\fint _{D_{2r}} |F|^{p}\Big )^{1/p} \\&+ \sum _{j=0}^{m} r^j \Vert \nabla ^j G\Vert _{L^\infty (D_{2r})} \bigg \}. \end{aligned}$$

Note that the above inequality still hold if we subtract a polynomial $P_{m-1}\in \mathfrak {P}_{m-1}$ from $u_\varepsilon , u_0$ and G simultaneously. This gives (4.3) by taking the infimum with respect to $P_{m-1}$. $\square$

Lemma 4.2

For $0<\varepsilon \le r\le 1$, let $u_0\in H^m(D_2; \mathbb {R}^n)$ be a weak solution to

$$\begin{aligned} \mathcal {L}_0 u_0=F \quad \text {in } D_2, \quad \quad \hbox {Tr} (D^\gamma u_0)=D^\gamma G \quad \text {on } \Delta _2 \quad \text {for } 0\le |\gamma |\le m-1, \end{aligned}$$

where $G\in C^{m-1,1}( D _2; \mathbb {R}^n), F\in L^p(D_2; \mathbb {R}^n)$ with $p>\max \{ d/(m+1), 2d/(d+2m-2), 1\}$. Then, for any $0<\lambda < \min \{m+1-d/p, 1\}$, there exist $\lambda _0<\min \{m+1-d/p, 1\}$ and a constant C depending only on $d,n,m, p, \mu , M$ and $\tau (t)$ in (1.4), such that

$$\begin{aligned} \Phi _\lambda (\delta r;u_0)\le C \delta ^{\lambda _0-\lambda } \Phi _\lambda (r;u_0) \quad \text {for }\, 0<\delta <1/4. \end{aligned}$$

(4.7)

Proof

By rescaling, we assume that $r=1$. For $0< \lambda < \min \{m+1-d/p, 1\}$, fix $\lambda _0$ such that $\lambda<\lambda _0<\min \{m+1-d/p, 1\}.$ Set $P_{m-1}$ in (4.2) as

$$\begin{aligned} P_{m-1}(x)= \sum _{|\alpha |=0}^{m-1} \frac{1}{\alpha !} D^\alpha u_0(0) x^{\alpha }=\sum _{|\alpha |=0}^{m-1} \frac{1}{\alpha !} D^\alpha G(0) x^{\alpha }. \end{aligned}$$

It is not difficult to find that

$$\begin{aligned} \Phi _\lambda (\delta ,u_0)\le&\, C \delta ^{\lambda _0-\lambda } \Vert u_0\Vert _{C^{m-1,\lambda _0}(D_\delta )} \nonumber \\&+ \delta ^{m+1-\lambda -d/p} \Big ( \fint _{D_{1}} |F|^{p}\Big )^{1/p} + \delta ^{1-\lambda } \sum _{j=0}^m \Vert \nabla ^jG\Vert _{L^\infty (D_1)} \end{aligned}$$

(4.8)

for any $0<\delta <1/4$. Observe that when $p>\max \{d/(m+1),1\}$, $L^p(D_2;\mathbb {R}^n)\hookrightarrow W^{-m,q}(D_2;\mathbb {R}^n)$ for some $q>d.$ Combining the $C^{m-1,\lambda _0}$ estimate for higher-order elliptic systems with constant coefficients in $C^1$ domains (see e.g., [10, 12]) and a localization argument (see e.g., the proof of Corollary 5.1), we have

$$\begin{aligned} \Vert u_0\Vert _{C^{m-1,\lambda _0}(D_\delta )}&\le C\Vert u_0\Vert _{C^{m-1,\lambda _0}(D_{1/4})}\nonumber \\&\le C\bigg \{ \Big (\fint _{D_1} |u_0|^2\Big )^{1/2} +\Big (\fint _{D_1} |F|^p\Big )^{1/p} + \sum _{j=0}^m \Vert \nabla ^jG\Vert _{L^\infty (D_1)} \bigg \}. \end{aligned}$$

(4.9)

Taking (4.9) into (4.8), we derive that

$$\begin{aligned} \Phi _\lambda (\delta ,u_0)\le C \delta ^{\lambda _0-\lambda } \bigg \{ \Big (\fint _{D_1} |u_0|^2\Big )^{1/2} +\Big (\fint _{D_1} |F|^p\Big )^{1/p} +\sum _{j=0}^m \Vert \nabla ^jG\Vert _{L^\infty (D_1)} \bigg \}. \end{aligned}$$

Substituting $u_0, G$ by $u_0-P_{m-1}$ and $G-P_{m-1}$, respectively, and taking the infimum with respect to $P_{m-1}\in \mathfrak {P}_{m-1}$, we obtain (4.7) immediately. $\square$

Lemma 4.3

For $0<\varepsilon \le r\le 1/2$, let $\Phi _\lambda (r,u_\varepsilon )$ be defined as in (4.2). Then, there exists $\delta \in (0, 1/4)$ depending only on $d,n,m, p,\lambda ,\mu , M$ and $\tau (t)$ in (1.4), such that

$$\begin{aligned} \Phi _\lambda (\delta r;u_\varepsilon )\le \frac{1}{2}\Phi _\lambda (2r;u_\varepsilon )+C\Big (\frac{\varepsilon }{r}\Big )^{1/4}\Phi _\lambda (2r;u_\varepsilon ), \end{aligned}$$

(4.10)

where C depends only on $d, n, m, p, \lambda , \mu , M$ and $\tau (t)$ in (1.4).

Proof

By the definition, it is easy to find that

$$\begin{aligned} \Phi _\lambda (\delta r;u_\varepsilon )&\le \Phi _\lambda (\delta r;u_0)+\frac{1}{(\delta r)^{m-1+\lambda }}\Big (\fint _{D_{\delta r}}|u_\varepsilon -u_0|^2 \Big )^{1/2}\\&\le C \delta ^{\lambda _0-\lambda }\Phi _\lambda (r;u_0)+\frac{1}{(\delta r)^{m-1+\lambda }}\Big (\fint _{D_{\delta r}}|u_\varepsilon -u_0|^2 \Big )^{1/2}\\&\le C \delta ^{\lambda _0-\lambda }\Phi _\lambda (r;u_\varepsilon ) +\frac{C\delta ^{\lambda _0-\lambda }}{r^{m-1+\lambda }}\Big (\fint _{D_{r}}|u_\varepsilon -u_0|^2 \Big )^{1/2} \\&\quad +\frac{1}{(\delta r)^{m-1+\lambda }}\Big (\fint _{D_{\delta r}}|u_\varepsilon -u_0|^2 \Big )^{1/2}\\&\le C \delta ^{\lambda _0-\lambda }\Phi _\lambda (2r;u_\varepsilon ) +\frac{C_\delta }{r^{m-1+\lambda }}\Big (\fint _{D_{r}}|u_\varepsilon -u_0|^2 \Big )^{1/2}. \end{aligned}$$

Taking $\delta$ small enough such that $C \delta ^{\lambda _0-\lambda }<1/2$, and then using Lemma 4.1, we obtain (4.10) directly. $\square$

Proof of Theorem 1.1

We only need to consider the case $\varepsilon \le r<1/4$, since the estimate (1.6) is trivial when $1/4\le r\le 1$, following directly from Caccioppoli’s inequality. Thanks to Lemma 4.3, we can take $N_0$ large enough such that

$$\begin{aligned} \Phi _\lambda (\delta r;u_\varepsilon )\le \frac{1}{2}\Phi _\lambda (2r;u_\varepsilon )+C\Big (\frac{1}{N_0}\Big )^{1/4}\Phi _\lambda (2r;u_\varepsilon )\le \Phi _\lambda (2r;u_\varepsilon ), \end{aligned}$$

(4.11)

for $r \ge N_0\varepsilon ,$ where $\delta$ given by Lemma 4.3 is fixed. Hence, by iteration we have

$$\begin{aligned} \Phi _\lambda (r;u_\varepsilon )\le C \Phi _\lambda (1;u_\varepsilon ) \quad \text { for } r\in [N_0\varepsilon , 1/2). \end{aligned}$$

(4.12)

On the other hand, for $\varepsilon \le r<N_0\varepsilon ,$ it is obvious that

$$\begin{aligned} \Phi _\lambda (r;u_\varepsilon )\le C \Phi _\lambda (N_0\varepsilon ;u_\varepsilon )\le C \Phi _\lambda (1;u_\varepsilon ), \end{aligned}$$

where C depends on $N_0$. This, together with (4.12), gives

$$\begin{aligned} \Phi _\lambda (r;u_\varepsilon )\le C\Phi _\lambda (1;u_\varepsilon ) \quad \text { for } \varepsilon \le r\le 1/2. \end{aligned}$$

(4.13)

By Caccioppoli’s inequality, we deduce that

$$\begin{aligned} \Big ( \fint _{D_r}|\nabla ^m (u_\varepsilon -P_{m-1})|^2\Big )^{1/2}&\le Cr^{-m}\inf _{P_{m-1}\in \mathfrak {P}_{m-1}}\bigg \{\Big (\fint _{D_{2r}}|u_\varepsilon -P_{m-1} |^2 \Big )^{1/2} \\&\quad + r^{2m} \Big (\fint _{D_{2r}} |F|^{p}\Big )^{1/p} \\&\quad + \sum _{j=0}^{m} r^j \Big ( \fint _{D_{2r}} |\nabla ^j (G-P_{m-1})|^{2}\Big )^{1/2} \bigg \}.\\&=Cr^{\lambda -1} \Phi _\lambda (2r, u_\varepsilon ) \le C r^{\lambda -1} \Phi _\lambda (1, u_\varepsilon )\\&\le C r^{\lambda -1}\Big \{ \Big (\fint _{D_1}|u_\varepsilon |^2 \Big )^{1/2}\\&\quad +\Big (\fint _{D_1} |F|^p\Big )^{1/p} +\sum _{j=0}^{m} \Vert \nabla ^jG\Vert _{L^\infty (D_1)} \Big \}, \end{aligned}$$

for all $\varepsilon \le r<1/2$ and any $P_{m-1}\in \mathfrak {P}_{m-1},$ which is exactly (1.6). $\square$

Corollary 4.1

In addition to the assumptions of Theorem 1.1, if $A\in V\!M\!O (\mathbb {R}^d),$ then for any $0<\lambda <\min \{m+1-d/p, 1\}$,

$$\begin{aligned} \Vert u_\varepsilon \Vert _{C^{m-1,\lambda }(D_{1/4})}\le C \Big \{\Big (\fint _{D_1}|u_\varepsilon |^2 \Big )^{1/2}&+\Big (\fint _{D_1} |F|^p\Big )^{1/p} + \Vert G\Vert _{C^{m-1,1}(D_1)}\Big \}, \end{aligned}$$

(4.14)

where C depends only on $d,n,m, p, \mu$ as well as $M, \tau (t)$ in (1.4) and $\varrho (t)$ in (1.7).

Proof

It is enough to assume $0<\varepsilon < 1/2$, as the other case is trivial. Setting

$$\begin{aligned} v_\varepsilon (x)=u_\varepsilon (\varepsilon x), ~~~ \widetilde{F}(x) =\varepsilon ^{2m} F(\varepsilon x), ~~~\widetilde{G}(x)=G(\varepsilon x), \end{aligned}$$

then $v_\varepsilon$ satisfies

$$\begin{aligned} \mathcal {L}_1 v_\varepsilon =\widetilde{F}(x) \quad \text { in } D_1, \quad \quad \hbox {Tr} (D^\gamma v_\varepsilon )=D^\gamma \widetilde{G}(x) \quad \text { on } \Delta _1\quad \text {for } 0\le |\gamma |\le m-1. \end{aligned}$$

(4.15)

By $C^{m-1,\lambda }$ estimates of operator $\mathcal {L}_1$ in $C^1$ domains [10, 12] and a localization argument, we have for any $0<\lambda <\min \{m+1-d/p, 1\}$ and $0<s< 1/2 ,$

$$\begin{aligned} \Big (\fint _{D_s}|\nabla ^m v_\varepsilon |^2\Big )^{1/2}\le C s^{\lambda -1} \Big \{&\Big (\fint _{D_1} |v_\varepsilon |^2\Big )^{1/2} +\Big (\fint _{D_1} |\widetilde{F}|^p\Big )^{1/p} + \sum _{j=0}^{m} \Vert \nabla ^j\widetilde{G}\Vert _{L^\infty (D_1)} \Big \}. \end{aligned}$$

(4.16)

By the change of variables, we obtain for $0<r< \varepsilon /2 ,$

$$\begin{aligned} \Big (\fint _{D_r}|\nabla ^m u_\varepsilon |^2\Big )^{1/2}\le&\, C \Big ( \frac{r}{\varepsilon }\Big )^{\lambda -1} \frac{1}{\varepsilon ^m} \Big \{\Big (\fint _{D_\varepsilon }| u_\varepsilon |^2\Big )^{1/2}\nonumber \\&+\varepsilon ^{2m}\Big ( \fint _{D_\varepsilon }|F|^p\Big )^{1/p} + \sum _{j=0}^{m} \varepsilon ^j \Vert \nabla ^jG\Vert _{L^\infty (D_\varepsilon )}\Big \} . \end{aligned}$$

(4.17)

Subtracting $P_{m-1}$ from $u_\varepsilon$ and G simultaneously, and taking (4.13) in consideration, we obtain that

$$\begin{aligned} \Big (\fint _{D_r}|\nabla ^m u_\varepsilon |^2\Big )^{1/2}\le C r^{\lambda -1} \Big \{&\Big (\fint _{D_1}|u_\varepsilon |^2 \Big )^{1/2} +\Big (\fint _{D_1} |F|^p\Big )^{1/p} +\sum _{j=0}^{m} \Vert \nabla ^jG\Vert _{L^\infty (D_1)}\Big \} \end{aligned}$$

(4.18)

for any $0<r\le \varepsilon .$ In view of (1.6), we know that (4.18) holds for $0\le r< 1/2$. Combining (4.18) with similar interior $C^{m-1,\lambda }$ estimate in [28, Corollary 5.1], we obtain that

$$\begin{aligned} \Big (\fint _{B(x,r)\cap D_{1/4}}|\nabla ^m u_\varepsilon |^2\Big )^{1/2}\le&\, C r^{\lambda -1} \Big \{\Big (\fint _{D_1}|u_\varepsilon |^2 \Big )^{1/2}\\&+\Big (\fint _{D_1} |F|^p\Big )^{1/p} +\sum _{j=0}^{m} \Vert \nabla ^jG\Vert _{L^\infty (D_1)}\Big \} \end{aligned}$$

for any $0<r<r_0$ ($r_0$ is small) and $x\in D_{1/4}$. This gives (4.14) by the Campanato characterization of Hölder spaces. $\square$

Remark 4.1

Under the assumptions of Corollary 4.1, if $F,\, G\equiv 0$ we may use Poincaré’s inequality to deduce from (4.17) that

$$\begin{aligned} \Big (\fint _{D_r}|\nabla ^m u_\varepsilon |^2\Big )^{1/2}\le C \Big ( \frac{r}{\varepsilon }\Big )^{\lambda -1} \Big (\fint _{D_\varepsilon }| \nabla ^m u_\varepsilon |^2\Big )^{1/2} \end{aligned}$$

(4.19)

for any $0<r\le \varepsilon .$ This will be used to establish the uniform $W^{m,p}$ estimate in the next section.

5 Uniform $W^{m,p}$ estimates

This section is devoted to the uniform $W^{m,p}$ estimate for $u_\varepsilon$ in $C^1$ domains under the assumption $A\in V\!M\!O(\mathbb {R}^d)$.

Lemma 5.1

Assume that $\Omega$ is a bounded $C^1$ domain in $\mathbb {R}^d$ and the coefficient matrix $A\in V\!M\!O(\mathbb {R}^d)$ satisfies (1.2)–(1.3). Let $B(x_0,r), r<r_0,$ be a ball centered at $x_0\in \partial \Omega$ with radius r, and $u_\varepsilon \in H^m(B(x_0,2r)\cap \Omega ; \mathbb {R}^n)$ be a weak solution to

$$\begin{aligned} \mathcal {L}_\varepsilon u_\varepsilon =0 \quad \text {in } B(x_0,2r)\cap \Omega , \quad \quad \hbox {Tr}(D^\gamma u_\varepsilon )=0 \quad \text {on } B(x_0, 2r) \cap \partial \Omega \quad \text {for }\, 0 \le |\gamma |\le m-1. \end{aligned}$$

Then, for any $2\le p<\infty$,

$$\begin{aligned} \Big (\fint _{B(x_0,r)\cap \Omega } |\nabla ^m u_\varepsilon |^p\Big )^{1/p} \le C \Big (\fint _{B(x_0,2r)\cap \Omega } |\nabla ^m u_\varepsilon |^2\Big )^{1/2}, \end{aligned}$$

(5.1)

where C depends only on $d,n,m,p,\mu$ as well as $M, \tau (t)$ in (1.4) and $\varrho (t)$ in (1.7).

Proof

We only need to consider the case $\varepsilon <\frac{1}{4}$. Since if else $A(x/\varepsilon )$ satisfies (1.7) uniformly, and (5.1) follows from the existing $W^{m,p}$ estimates for higher-order elliptic systems with $V\!M\!O$ coefficients, see e.g., [10, 12]. Also, note that the function $\psi _r(x')=r^{-1}\psi (rx')$ satisfies condition (1.4) uniformly. We can then fix our considerations on the case $r=1$ by rescaling. By the uniform interior $W^{m,p}$ estimates derived by the authors in [28, Theorem 1.3], we have

$$\begin{aligned} \Big (\fint _{B(x, t)} |\nabla ^m u_\varepsilon |^p \Big )^{1/p}\le C \Big (\fint _{B(x, 2t)} |\nabla ^m u_\varepsilon |^2\hbox {d}x\Big )^{1/2}, \end{aligned}$$

whenever $u_\varepsilon$ is a weak solution to $\mathcal {L}_\varepsilon u_\varepsilon =0$ in B(x, 2t). Therefore, in view of (1.6) and (4.19), we have for any $0<\lambda <1$ and $y\in B(x_0,1)\cap \Omega$,

$$\begin{aligned} \begin{aligned} \Big (\fint _{B(y,\delta (y)/8)} |\nabla ^m u_\varepsilon |^p \Big )^{1/p}&\le C \Big (\fint _{B(y,\delta (y)/4)} |\nabla ^m u_\varepsilon |^2\Big )^{1/2}\\&\le C [\delta (y)]^{\lambda -1}\Big (\int _{B(x_0,2)\cap \Omega }|\nabla ^m u_\varepsilon |^2\Big )^{1/2}, \end{aligned} \end{aligned}$$

(5.2)

where $\delta (y)$ denotes the distance of y to $\partial (B(x_0,2)\cap \Omega )$. Fix $\lambda \in (1-1/p, 1)$ and integrate (5.2) with respect to y in $B(x_0,1)\cap \Omega$. We obtain that

$$\begin{aligned} \int _{B(x_0,1)\cap \Omega }\fint _{B(y,\delta (y)/8)} |\nabla ^m u_\varepsilon |^p\hbox {d}x\, \hbox {d}y\le C \Vert \nabla ^m u_\varepsilon \Vert ^p_{L^2(B(x_0,2)\cap \Omega )}. \end{aligned}$$

(5.3)

We then deduce from Fubini’s theorem that

$$\begin{aligned} \int _{B(x_0,1)\cap \Omega }|\nabla ^m u_\varepsilon (x)|^p \int _{\left\{ y\in B(x_0,1)\cap \Omega :\, |x-y|<\delta (y)/8\right\} } \frac{1}{\delta (y)^d} \hbox {d}y\, \hbox {d}x\le C \Vert \nabla ^m u_\varepsilon \Vert ^p_{L^2(B(x_0,2)\cap \Omega )}. \end{aligned}$$

(5.4)

Observe that when $|x-y|<\delta (y)/8$, it holds that

$$\begin{aligned} \frac{1}{2}\delta (y)\le \delta (x) \le 2\delta (y). \end{aligned}$$

(5.5)

We thus conclude that

$$\begin{aligned} B(x_0,1)\cap \Omega \cap B(x, \delta (x)/16)\subset \big \{y\in B(x_0,1)\cap \Omega : |x-y|<\delta (y)/8\big \} \end{aligned}$$

for any $x\in B(x_0,1)\cap \Omega .$ This, together with (5.5), implies that

$$\begin{aligned} \int _{\{y\in B(x_0,1)\cap \Omega : |x-y|<\delta (y)/8\} } \frac{1}{\delta (y)^d}\hbox {d}y \ge C_0>0. \end{aligned}$$

Taking this into (5.4), we obtain (5.1) immediately. $\square$

With Lemma 5.1 at our disposal, we are ready to prove Theorem 1.2. The proof is based on a real-variable argument in the following theorem, which is formulated in [31, 32].

Theorem 5.1

Let $q>2$ and $\Omega$ be a bounded Lipschitz domain. Let $F\in L^2(\Omega )$ and $f\in L^p(\Omega )$ for some $2<p<q<\infty$. Suppose that for each ball $B\subset \mathbb {R}^d$ with the property that $|B|<c_0|\Omega |$, and either $4B\subset \Omega$ or B is centered on $\partial \Omega$, there exists two measurable functions $F_B$ and $R_B$ on $2B\cap \Omega$ such that

$$\begin{aligned}&\displaystyle |F|\le |F_B|+|R_B| \text { on } 2B\cap \Omega , \end{aligned}$$

(5.6)

$$\begin{aligned}&\displaystyle \Big (\fint _{2B\cap \Omega } |R_B|^q\Big )^{1/q}\le C_1 \Big \{ \Big (\fint _{4B\cap \Omega } |F|^2\Big )^{1/2} +\sup _{B\subset B' \subset 4B_0} \Big (\fint _{B'\cap \Omega } |f|^2\Big )^{1/2}\Big \}, \end{aligned}$$

(5.7)

$$\begin{aligned}&\displaystyle \Big (\fint _{2B\cap \Omega } |F_B|^2\Big )^{1/2} \le C_2 \sup _{B\subset B' \subset 4B_0} \Big (\fint _{B'\cap \Omega } |f|^2\Big )^{1/2}+\delta \Big (\int _{4B\cap \Omega } |F|^2\Big )^{1/2}, \end{aligned}$$

(5.8)

where $C_1,C_2>0, 0<c_0<1.$ Then, there exists $\delta _0>0$, depending only on $C_1,C_2,c_0, p,q$ and the Lipschitz character of $\Omega ,$ such that, for any $0<\delta <\delta _0$, $F\in L^p(\Omega )$ and

$$\begin{aligned} \Big (\fint _{\Omega } |F|^p\Big )^{1/p} \le C \Big (\fint _{\Omega } |F|^2\Big )^{1/2}+ \Big (\fint _{\Omega } |f|^p\Big )^{1/p}, \end{aligned}$$

(5.9)

where C depends only on $d, C_1,C_2, c_0, p,q$ and the Lipschitz character of $\Omega$.

Proof of Theorem 1.2

Since the desired estimate is trivial when $p=2$, it suffices to consider the case $p>2.$ Thanks to the extension theorem in [26, p. 223], for any $\dot{g}=\{g_\gamma \}_{|\gamma |\le m-1}\in \dot{B}^{m-1/p}_p(\partial \Omega )$ there exist a $G\in W^{m,p}(\Omega )$ such that

$$\begin{aligned} \hbox {Tr}(D^\gamma G)=g_\gamma \quad \text {for } 0\le |\gamma |\le m-1, \quad \quad \Vert G\Vert _{ W^{m,p}(\Omega )}\le C \Vert \dot{g}\Vert _{\dot{B}^{m-1/p}_p(\partial \Omega )}. \end{aligned}$$

Therefore, we can restrict our investigations to the problem with homogeneous boundary conditions.

$$\begin{aligned} \mathcal {L}_\varepsilon \overline{u}_\varepsilon =\sum _{|\alpha |\le m}D^\alpha \overline{f}^\alpha \quad \text { in } \Omega , \quad \quad \hbox {Tr}(D^\gamma \overline{u}_\varepsilon )=0 \quad \text { on } \partial \Omega \quad \text {for}~0\le |\gamma |\le m-1, \end{aligned}$$

where $\overline{u}_\varepsilon =u_\varepsilon -G$ and

$$\begin{aligned} \overline{f}^\alpha =f^\alpha +(-1)^{m+1}\sum _{|\beta |=m}A^{\alpha \beta }(x/\varepsilon ) D^\beta G \quad \text { for } |\alpha |=m , \quad \text {and}\quad \overline{f}^\alpha =f^\alpha \quad \text { for } |\alpha |< m . \end{aligned}$$

Let $F=|\nabla ^m\overline{u}_\varepsilon |$ and $f(x)=\sum _{|\alpha |\le m}|\overline{f}^\alpha |$. We only need to construct the functions $F_B$, $R_B$ and then verify the conditions (5.6), (5.7) and (5.8) to hold for balls $B(x_0,r)$ with the property $|B|<c_0|\Omega |$ and either $4B\subset \Omega$ or B is centered on $\partial \Omega$. The case of $4B\subset \Omega$ has been investigated for interior $W^{m,p}$ estimates in [28]. So here we only consider the situation that B is centered on $\partial \Omega$.

Let $B=B(x_0,r)$ for some $x_0\in \partial \Omega$ and $0<r<r_0/16.$ Let $v_\varepsilon \in H_0^{m}(4B\cap \Omega ; \mathbb {R}^n)$ be the solution to $\mathcal {L}_\varepsilon v_\varepsilon =\sum _{|\alpha |\le m}D^\alpha \overline{f}^\alpha$ in $4B\cap \Omega$ and set

$$\begin{aligned} F_B=|\nabla ^m v_\varepsilon |,\quad R_B=|\nabla ^m w_\varepsilon |,\quad w_\varepsilon =\overline{u}_\varepsilon -v_\varepsilon . \end{aligned}$$

Then, it is obvious that

$$\begin{aligned}&|F|\le |F_B|+|R_B| \text { on } 2B\cap \Omega ,\\&\Big (\fint _{2B\cap \Omega } |F_B|^2\Big )^{1/2}\le C\Big (\fint _{4B\cap \Omega } |\nabla ^m v_\varepsilon |^2\Big )^{1/2} \le C \Big (\fint _{4B\cap \Omega } |f|^2\Big )^{1/2}, \end{aligned}$$

which imply the conditions (5.6) and (5.8). Furthermore, note that

$$\begin{aligned} \mathcal {L}_\varepsilon w_\varepsilon =0 \text { in } 4B\cap \Omega ,~~~~~~~\hbox {Tr}(D^\gamma w_\varepsilon )=0 \quad \text {on } 4B \cap \partial \Omega \quad \text {for } 0 \le |\gamma |\le m-1. \end{aligned}$$

By Lemma 5.1, we know that for any $2<p<\infty$,

$$\begin{aligned} \Big (\fint _{2B\cap \Omega } |\nabla ^m w_\varepsilon |^p\Big )^{1/p}&\le C \Big (\fint _{4B\cap \Omega } |\nabla ^m w_\varepsilon |^2\Big )^{1/2}\\&\le C \Big (\fint _{4B\cap \Omega } |\nabla ^m \overline{u}_\varepsilon |^2\Big )^{1/2} +C \Big (\fint _{4B\cap \Omega } |\nabla ^m v_\varepsilon |^2\Big )^{1/2}\\&\le C \Big (\fint _{4B\cap \Omega } |\nabla ^m \overline{u}_\varepsilon |^2\Big )^{1/2} +C \Big (\fint _{4B\cap \Omega } |f|^2\Big )^{1/2}, \end{aligned}$$

which implies (5.7). Noticing that all the conditions in Theorem 5.1 are verified, (1.9) follows from (5.9) immediately. $\square$

Note that if $u_\varepsilon \in W_0^{m,p}(\Omega , \mathbb {R}^n)$ is a weak solution to $\mathcal {L}_\varepsilon u_\varepsilon =f$ in $\Omega$, and $u_\varepsilon ^*\in W_0^{m,p'}(\Omega , \mathbb {R}^n)$ is a weak solution to $\mathcal {L}^*_\varepsilon u^*_\varepsilon = f^{*}$ in $\Omega$, where $p'=p/(p-1)$. Then, we have

$$\begin{aligned} \langle f , u^*_\varepsilon \rangle _{W^{-m,p}\times W_0^{m,p'}}=\sum _{|\alpha |=|\beta |=m}\int _\Omega A^{\alpha \beta }(x/\varepsilon )D^\beta u_\varepsilon D^\alpha u^*_\varepsilon =\langle f^{*}, u_\varepsilon \rangle _{W^{-m,p'}\times W_0^{m,p}}. \end{aligned}$$

Therefore, Theorem 1.2 also holds for $1\!<\!p\!<\!2$ by a standard duality argument.

As a consequence of Theorem 1.2, one can obtain $C^{m-1,\lambda }$ estimate on $u_\varepsilon$ in $\Omega$ immediately. However, we choose to provide a local version using the localization argument mentioned in (4.9) and (4.16) (where we did not provide any details). The result will also provide a direct comparison to Theorem 1.1 as well as Corollary 4.1.

Corollary 5.1

Suppose that $\Omega$ is a bounded $C^1$ domain in $\mathbb {R}^d$, $A\in \!V\!M\!O(\mathbb {R}^d)$ satisfies (1.2)–(1.3). For any $x_0\in \partial \Omega$ and $0<r \le c_0$, let $u_\varepsilon \in H^m(\Omega \cap B(x_0,4r); \mathbb {R}^n)$ be a weak solution to

$$\begin{aligned} \mathcal {L}_\varepsilon u_\varepsilon= & {} \sum _{|\zeta |\le m} D^\zeta f^\zeta \quad \text {in } \Omega \cap B(x_0,4r), \quad \quad \hbox {Tr}(D^\gamma u_\varepsilon )=D^\gamma G \quad \text {on } \partial \Omega \cap B(x_0,4r),\, 0 \\\le & {} \,|\gamma |\le m-1, \end{aligned}$$

where $f^\zeta \in L^p(\Omega \cap B(x_0,4r); \mathbb {R}^n)$ for all $|\zeta |\le m$, and $G \in W^{m,p}(\Omega \cap B(x_0,4r); \mathbb {R}^n)$ with $p> d$ and $p\ge 2$. Then, for any $x,y\in \Omega \cap B(x_0,r)$,

$$\begin{aligned}&|\nabla ^{m-1} u_\varepsilon (x)-\nabla ^{m-1} u_\varepsilon (y)| \le \, C\frac{|x-y|^\lambda }{r^{m-1+\lambda }} \Big \{\Big (\fint _{\Omega \cap B(x_0,4r)}|u_\varepsilon |^2\Big )^{1/2} \nonumber \\&\quad + r^{m-d/p} \Vert G\Vert _{W^{m,p}(\Omega \cap B(x_0,4r))}\nonumber \\&\quad +\sum _{|\zeta |\le m} r^{2m-|\zeta |}\Big (\fint _{\Omega \cap B(x_0,4r)}|f^\zeta |^p\Big )^{1/p} \Big \} , \end{aligned}$$

(5.10)

$$\begin{aligned}&\Vert \nabla ^k u_\varepsilon \Vert _{L^\infty (\Omega \cap B(x_0,r))}\le C \, r^{-k} \Big \{\Big (\fint _{\Omega \cap B(x_0,4r)}|u_\varepsilon |^2\Big )^{1/2}+ r^{m-d/p} \Vert G\Vert _{W^{m,p}(\Omega \cap B(x_0,4r))} \nonumber \\&\quad + \sum _{|\zeta |\le m} r^{2m-|\zeta |}\Big (\fint _{\Omega \cap B(x_0,4r)}|f^\zeta |^p\Big )^{1/p} \Big \} , \end{aligned}$$

(5.11)

where $0\le k \le m-1, \lambda =1-d/p,$ and C depends only on $d, n,m, p, \mu , \Omega$ as well as $\varrho (t)$ in (1.7).

Proof

By rescaling and translation, we may assume that $r=1, x_0=0.$ Denote B(0, r) as $B_r$ and let $\widetilde{D}$ be a $C^1$ domain such that $B_{3/2}\cap \Omega \subset \widetilde{D}\subset B_2\cap \Omega$. Set $v_\varepsilon =u_\varepsilon -G$. It is obvious that $v_\varepsilon$ satisfies

$$\begin{aligned} \begin{aligned}&\mathcal {L}_\varepsilon v_\varepsilon =\sum _{|\alpha |=m} D^\alpha \{A^{\alpha \beta }(x/\varepsilon ) D^\beta G\} + \sum _{|\zeta |\le m} D^\zeta f^\zeta \, \quad \text { in } \Omega \cap B_4, \\&\hbox {Tr}(D^\gamma v_\varepsilon )=0 \,\quad \text { on } \partial \Omega \cap B_4 \quad \text {for } 0<|\gamma |\le m-1. \end{aligned} \end{aligned}$$

(5.12)

Let $\phi \in C_c^\infty (B_{3/2})$ with $\phi =1$ in $B_1$ and $|\nabla ^k\phi |\le C2^k$. We have

$$\begin{aligned}&\mathcal {L}_\varepsilon (v_\varepsilon \phi )=\sum _{|\alpha |=|\beta |=m}\Big \{ D^\alpha \big \{A^{\alpha \beta }(x/\varepsilon ) D^\beta G\big \}\phi \\&\quad +\sum _{\alpha '+\alpha ''=\alpha , |\alpha ''|\ge 1} C(\alpha ')D^{\alpha '}\big \{A^{\alpha \beta }(x/\varepsilon ) D^\beta v_\varepsilon \big \} D^{\alpha ''} \phi \\&\quad + \sum _{\beta '+\beta ''=\beta , |\beta ''|\ge 1} C(\beta ') D^\alpha \big \{A^{\alpha \beta }(x/\varepsilon ) D^{\beta '} v_\varepsilon D^{\beta ''}\phi \big \} \Big \}+ \sum _{|\zeta |\le m} D^\zeta f^\zeta \phi \quad \text { in } \widetilde{D},\\&\hbox {Tr}(D^\gamma (v_\varepsilon \phi ))=0, \quad \text { on } \partial \widetilde{D} \quad \text {for}\quad 0<|\gamma |\le m-1. \end{aligned}$$

Observe that for $0\le \ell =|\alpha '|\le m-1$,

$$\begin{aligned}&D^{\alpha '} \big \{A^{\alpha \beta }(x/\varepsilon ) D^\beta v_\varepsilon \big \} D^{\alpha ''} \phi \in W^{-m,p}(\widetilde{D}) \quad \text { if }\, \nabla ^m v_\varepsilon \in L^{q_\ell }(\widetilde{D}) \text { with } q_\ell \\&\quad = \frac{dp}{(m-\ell )p+d}~(<p). \end{aligned}$$

Thus, we may deduce from Theorem 1.2 that

$$\begin{aligned} \Vert v_\varepsilon \Vert _{W^{m,p}(B_1\cap \Omega )}&\le C \Big (\int _{\widetilde{D}} |\nabla ^m G|^p \Big )^{1/p} + C\sum _{0\le k\le m-1} \Big (\int _{\widetilde{D}} |\nabla ^k v_\varepsilon |^p \Big )^{1/p} \nonumber \\&\quad + C\sum _{0\le \ell \le m-1} \Big (\int _{\widetilde{D}} |\nabla ^m v_\varepsilon |^{q_\ell } \Big )^{1/{q_\ell }}+ C\sum _{|\zeta |\le m}\Big ( \int _{\widetilde{D}} |f^\zeta |^p \Big )^{1/p}\nonumber \\&\le C \Big (\int _{B_2\cap \Omega } |\nabla ^m G|^p \Big )^{1/p} + C\sum _{0\le k\le m-1} \Big (\int _{B_2\cap \Omega } |\nabla ^k v_\varepsilon |^p \Big )^{1/p} \nonumber \\&\quad + C \sum _{0\le \ell \le m-1} \Big (\int _{B_2\cap \Omega } |\nabla ^m v_\varepsilon |^{q_\ell } \Big )^{1/{q_\ell }} + C\sum _{|\zeta |\le m} \Big (\int _{B_2\cap \Omega } |f^\zeta |^p \Big )^{1/p}. \end{aligned}$$

(5.13)

Let $p_1= dp/(d+p).$ Thanks to the Poincaré inequality and Sobolev imbedding, we have

$$\begin{aligned}&\Vert \nabla ^k v_\varepsilon \Vert _{L^p(B_2\cap \Omega )}\le C \Vert \nabla ^m v_\varepsilon \Vert _{L^{p_1}(B_2\cap \Omega )} \quad \text {for } 0\le k\le m-1,\\&\Vert \nabla ^m v_\varepsilon \Vert _{L^{q_\ell }(B_2\cap \Omega )}\le C \Vert \nabla ^m v_\varepsilon \Vert _{L^{p_1}(B_2\cap \Omega )} \quad \text {for } q_\ell =\frac{dp}{(m-\ell )p+d},\, 0\le \ell \le m-1, \end{aligned}$$

which, combined with (5.13), implies that

$$\begin{aligned} \Vert v_\varepsilon \Vert _{W^{m,p}(B_1\cap \Omega )} \le C \Vert \nabla ^m G\Vert _{L^p(B_2\cap \Omega )} + C \Vert \nabla ^m v_\varepsilon \Vert _{L^{p_1}(B_2\cap \Omega )} +C\sum _{|\zeta |\le m} \Vert f^\zeta \Vert _{L^p(B_2\cap \Omega )}. \end{aligned}$$

(5.14)

If $p_1>2$, we can perform a bootstrap argument for finite times to obtain that

$$\begin{aligned} \Vert v_\varepsilon \Vert _{W^{m,p}(B_1\cap \Omega )} \le C \Vert \nabla ^m G\Vert _{L^p(B_3\cap \Omega )} + C \Vert \nabla ^m v_\varepsilon \Vert _{L^{2}(B_3\cap \Omega )}+C\sum _{|\zeta |\le m} \Vert f^\zeta \Vert _{L^p(B_3\cap \Omega )}. \end{aligned}$$

By Caccioppoli’s inequality, this implies that

$$\begin{aligned} \Vert u_\varepsilon \Vert _{W^{m,p}(B_1\cap \Omega )}&\le C \Big \{ \Vert \nabla ^m v_\varepsilon \Vert _{L^2(B_3\cap \Omega )} + \Vert G\Vert _{W^{m,p}( B_4\cap \Omega )}+ \sum _{|\zeta |\le m} \Vert f^\zeta \Vert _{L^p(B_4\cap \Omega )}\Big \}\\&\le C \Big \{ \Vert u_\varepsilon \Vert _{L^2(B_4\cap \Omega )}+ \Vert G\Vert _{W^{m,p}( B_4\cap \Omega )}+\sum _{|\zeta |\le m} \Vert f^\zeta \Vert _{L^p(B_4\cap \Omega )} \Big \}, \end{aligned}$$

which gives (5.10) and (5.11) by Sobolev imbedding. $\square$

6 Uniform $C^{m-1,1}$ estimates

In this section, we consider uniform boundary $C^{m-1, 1}$ estimates for $u_\varepsilon$ in $C^{1,\theta } (0<\theta <1)$ domains. Throughout the section, we always assume that A satisfies (1.2) and (1.3). Similar to Sect. 4, we only need to consider equations in $(D_r, \Delta _r)$ defined as in (1.5) with the defining function $\psi \in C^{1,\theta }(\mathbb {R}^{d-1})$ satisfying $\psi (0)=0, \Vert \nabla \psi \Vert _{C^\theta (\mathbb {R}^{d-1})}\le M_1$.

Let $u_\varepsilon \in H^m(D_{2}; \mathbb {R}^n )$ be a weak solution to

$$\begin{aligned} \mathcal {L}_\varepsilon u_\varepsilon = \sum _{|\alpha |\le m-1}D^\alpha f^\alpha \quad \text { in } D_1, \quad \quad \hbox {Tr} (D^\gamma u_\varepsilon )=D^\gamma G \quad \text { on } \Delta _1 \quad \text {for } 0\le |\gamma |\le m-1, \end{aligned}$$

where $f^\alpha \in L^{q}(D_1; \mathbb {R}^n)$ with $q>d, q\ge 2$, and $G\in C^{m,\sigma }(D_1; \mathbb {R}^n)$ for some $0<\sigma \le \theta$. For $0<r\le 1$, define the following auxiliary quantities,

$$\begin{aligned} \Phi (r,u_\varepsilon )=&\, \frac{1}{r^m}\inf _{P_{m-1}\in \mathfrak {P}_{m-1}}\Big \{\Big (\fint _{D_{r}}|u_\varepsilon -P_{m-1} |^2 \Big )^{1/2} + \sum _{|\alpha |\le m-1}r^{2m-|\alpha |} \Big (\fint _{D_r}|f^\alpha |^q\Big )^{1/q}\nonumber \\&+ \sum _{j=0}^{m} r^j \Vert \nabla ^j (G-P_{m-1})\Vert _{L^\infty (D_r)} \Big \}, \end{aligned}$$

(6.1)

$$\begin{aligned} H(r; u_\varepsilon )=&\, \frac{1}{r^{m}}\inf _{P_{m}\in \mathfrak {P}_{m}} \Big \{\Big (\fint _{D_r}|u_\varepsilon -P_{m}|^2 \Big )^{1/2}+ \sum _{|\alpha |\le m-1}r^{2m-|\alpha |} \Big (\fint _{D_r}|f^\alpha |^q\Big )^{1/q} \nonumber \\&+ \sum _{j=0}^{m} r^j \Vert \nabla ^j (G-P_{m})\Vert _{L^\infty (D_r)} +r^{m+\sigma }\Vert \nabla ^m (G -P_{m})\Vert _{C^{0,\sigma }(D_r)} \Big \} . \end{aligned}$$

(6.2)

Lemma 6.1

For $0<\varepsilon \le r\le 1$, let $\Phi (r; u_\varepsilon )$ be defined as in (6.1). Then, there exists $u_0\in H^m(D_r; \mathbb {R}^n )$ such that $\mathcal {L}_0 u_0= \sum _{|\alpha |\le m-1}D^\alpha f^\alpha$ in $D_r$, $\hbox {Tr} (D^\gamma u_0)=D^\gamma G$ on $\Delta _r$ for $0\le |\gamma |\le m-1$, and

$$\begin{aligned} \frac{1}{r^m}\Big (\fint _{D_r}|u_\varepsilon -u_0|^2 \Big )^{1/2}\le C \Big (\frac{\varepsilon }{r}\Big )^{1/4} \Phi (2r; u_\varepsilon ), \end{aligned}$$

(6.3)

where C depends only on $d, n,m, q, \sigma , \mu$ and M in (1.4).

Proof

The proof is the same as Lemma 4.1, and we therefore omit the details. $\square$

Lemma 6.2

Let $u_0\in H^m(D_{r}; \mathbb {R}^n )$ be a weak solution to $\mathcal {L}_0 u_0= \sum _{|\alpha |\le m-1} D^\alpha f^\alpha$ in $D_{r}$ with $\hbox {Tr} (D^\gamma u_0)=D^\gamma G$ on $\Delta _{r}$ for $0\le |\gamma |\le m-1$. Then, there exists a $\delta \in (0, 1/4)$, depending only on $d,n,m, q, \sigma ,\mu ,\theta$ and $M_1$ in (1.10), such that

$$\begin{aligned} H(\delta r;u_0)\le \frac{1}{2} H (r;u_0). \end{aligned}$$

(6.4)

Proof

The proof, parallel to that of Lemma 4.2, is mainly based on $C^{m,\sigma }$ estimates for higher-order elliptic systems with constant coefficients in $C^{1,\theta } (0<\sigma \le \theta )$ domains. By rescaling, we assume that $r=1$. Taking

$$\begin{aligned} P_{m}(x)= \sum _{|\alpha |\le m} \frac{1}{\alpha !} D^\alpha u_0(0) x^{\alpha }=\sum _{|\alpha |\le m} \frac{1}{\alpha !} D^\alpha G(0) x^{\alpha }, \end{aligned}$$

it is not difficult to find that for any $0< \delta <1/4$ and any $0<\sigma '< \min \{1-d/q, \sigma \}$,

$$\begin{aligned} H(\delta ,u_0)\le&\, C \delta ^{\sigma '} \Vert u_0\Vert _{C^{m,\sigma '}(D_\delta )}+ C \delta ^{m-|\alpha |-d/q} \sum _{|\alpha |\le m-1} \Big ( \fint _{D_{1}} |f^\alpha |^{q}\Big )^{1/q} \nonumber \\&+ C \delta ^{\sigma } \Vert G\Vert _{C^{m,\sigma }(D_1)}. \end{aligned}$$

(6.5)

By the localization argument and the $C^{m,\sigma }$ estimate for higher-order elliptic systems with constant coefficients (see e.g., [27, Corollary 2.4]), we have

$$\begin{aligned} \Vert u_0\Vert _{C^{m,\sigma '}(D_\delta )}&\le C\Vert u_0\Vert _{C^{m,\sigma '}(D_{1/4})}\nonumber \\&\le C\Big \{ \Big (\fint _{D_1} |u_0|^2\Big )^{1/2} + \sum _{|\alpha |\le m-1}\Big (\fint _{D_1}|f^\alpha |^q\Big )^{1/q}+ \Vert G \Vert _{C^{m,\sigma }(D_1)} \Big \} \end{aligned}$$

(6.6)

for $0<\sigma '< \min \{ 1-d/q, \sigma \}.$ Taking (6.6) into (6.5) and setting $\delta$ small enough, we get

$$\begin{aligned} H(\delta ,u_0)\le \frac{1}{2} \Big \{ \Big (\fint _{D_1} |u_0|^2\Big )^{1/2} + \sum _{|\alpha |\le m-1}\Big (\fint _{D_1}|f^\alpha |^q\Big )^{1/q}+\Vert G \Vert _{C^{m,\sigma }(D_1)}\Big \}. \end{aligned}$$

For any $P_{m}\in \mathfrak {P}_{m}$, substituting $u_0, G$ by $u_0-P_{m}$ and $G-P_{m}$, respectively, and taking the infimum, we obtain (6.4) immediately. $\square$

Lemma 6.3

Let $0<\varepsilon <1/2$ and $\Phi (r; u_\varepsilon ), H(r; u_\varepsilon )$ be defined as in (6.1) and (6.2). Let $\delta$ be given by Lemma 6.2. Then, for any $r\in [\varepsilon ,1/2],$

$$\begin{aligned} H(\delta r; u_\varepsilon )\le \frac{1}{2} H(r; u_\varepsilon )+ C\Big ( \frac{\varepsilon }{r}\Big )^{1/4} \Phi (2r; u_\varepsilon ), \end{aligned}$$

(6.7)

where C depends only on $d,n, m, q,\mu ,\sigma ,\theta$ and $M_1$ in (1.10).

Proof

Similar to Lemma 4.3, the result follows from Lemmas 6.1 and 6.2. We thus omit the details. $\square$

Lemma 6.4

Let H(r) and h(r) be two nonnegative continuous functions on the interval (0, 1],  and let $\varepsilon \in (0,1/4).$ Assume that

$$\begin{aligned} \max _{r\le t\le 2r} H(t)\le C_0H(2r), ~~~~~\max _{r\le t,s\le 2r} | h(t)-h(s)|\le C_0H(2r), \end{aligned}$$

(6.8)

for any $r\in [\varepsilon , 1/2],$ and also

$$\begin{aligned} H(\delta r) \le \frac{1}{2} H(r) + C_0 \omega (\varepsilon /r)\big \{ H(2r)+h(2r)\big \} \quad \text { for any } r\in [\varepsilon , 1/2], \end{aligned}$$

(6.9)

where $\delta \in (0,1/4)$ and $\omega$ is a nonnegative increasing function on [0, 1] such that $\omega (0)=0$ and $\int _0^1 \omega (\varsigma )/\varsigma \, d\varsigma < \infty .$ Then, there exists a constant C depending only on $C_0,\delta$ and $\omega$, such that

$$\begin{aligned} \max _{\varepsilon \le r\le 1} \big \{H(r) +h(r)\big \}\le C \big \{H(1) +h(1)\big \}. \end{aligned}$$

(6.10)

Proof

See Lemma 8.5 in [35]. $\square$

Armed with lemmas above, we are ready to prove Theorem 1.3.

Proof of Theorem 1.3

We assume that $0<\varepsilon \le r< 1/4$, since if else (1.11) is just a consequence of Caccioppoli’s inequality. Let $u_\varepsilon \in H^m(D_{1}; \mathbb {R}^n )$ be a weak solution to

$$\begin{aligned} \mathcal {L}_\varepsilon u_\varepsilon = \sum _{|\alpha |\le m-1}D^\alpha f^\alpha \quad \text {in } D_1, \quad \quad \hbox {Tr} (D^\gamma u_\varepsilon )=D^\gamma G \quad \text {on } \Delta _1 \quad \text {for } 0\le |\gamma |\le m-1, \end{aligned}$$

where $f^\alpha \in L^{q}(D_1; \mathbb {R}^n)$ with $q>d, q\ge 2$, and $G\in C^{m,\sigma }(D_1; \mathbb {R}^n)$ for some $0<\sigma \le \theta$. For $r\in (0,1)$, let $H(r)= H(r, u_\varepsilon ), \Phi (r)=\Phi (r,u_\varepsilon )$ and $\omega (y)=y^{1/4}$. Define

$$\begin{aligned} h(r)= \sum _{|\alpha |=m}\frac{1}{\alpha !} |D^\alpha P_{mr}(x)|, \end{aligned}$$

where $P_{mr}\in \mathfrak {P}_{m}$ such that

$$\begin{aligned} H(r)=&\,\frac{1}{r^{m}} \Big \{\Big (\fint _{D_r}|u_\varepsilon -P_{mr}|^2 \Big )^{1/2} + \sum _{|\alpha |\le m-1}r^{2m-|\alpha |} \Big (\fint _{D_r}|f^\alpha |^q\Big )^{1/q} \nonumber \\&\quad + \sum _{j=0}^{m} r^j \Vert \nabla ^j (G-P_{mr})\Vert _{L^\infty (D_r)} +r^{m+\sigma }\Vert \nabla ^m (G -P_{mr})\Vert _{C^{0,\sigma }(D_r)} \Big \}. \end{aligned}$$

(6.11)

Next, let us check that H(r), h(r) satisfy conditions (6.8) and (6.9). From the definition, it is obvious that

$$\begin{aligned} H(t)\le C H(2r)\quad \text {for any } t\in [r,2r]. \end{aligned}$$

(6.12)

On the other hand, by the definition of h(r), 

$$\begin{aligned} |h(t)-h(s)|&\le \sum _{|\alpha |=m} \frac{1}{\alpha !} |D^\alpha (P_{mt}-P_{ms})| = \sum _{|\alpha |=m} \frac{1}{\alpha !} \Big (\fint _{D_r} |D^\alpha (P_{mt}-P_{ms}) |^2 \Big )^{1/2}\nonumber \\&\le C\Big (\fint _{D_{t}} | \nabla ^m( G-P_{mt} ) |^2 \Big )^{1/2}+ C\Big (\fint _{D_{s}} | \nabla ^m(G-P_{ms} ) |^2 \Big )^{1/2}\nonumber \\&\le C \big \{ H(t) +H(s) \big \} \le C H(2r), \end{aligned}$$

(6.13)

where we have used the fact $r\le t,s\le 2r$, the definition of $P_{mr}$ and (6.12), respectively, for the last three inequalities. Combining (6.12) with (6.13), we know that condition (6.8) is satisfied. Finally, from the definitions of $\Phi (r), H(r)$ and h(r), we obtain that

$$\begin{aligned} \Phi (r) \le&\, \frac{1}{r^m}\Big \{ \Big (\fint _{D_{r}}|u_\varepsilon -P_{mr} |^2 \Big )^{1/2} + \sum _{|\alpha |\le m-1}r^{2m-|\alpha |} \Big (\fint _{D_r}|f^\alpha |^q\Big )^{1/q}\\&+ \sum _{j=0}^{m} r^j \Vert \nabla ^j (G-P_{mr})\Vert _{L^\infty (D_r)} \Big \}\\&+ \inf _{P_{m-1}\in \mathfrak {P}_{m-1}}\frac{1}{r^m}\Big \{ \Big (\fint _{D_{r}}|P_{mr}-P_{m-1} |^2 \Big )^{1/2} + \sum _{j=0}^{m} r^j \Vert \nabla ^j (P_{mr}-P_{m-1})\Vert _{L^\infty (D_r)}\Big \}\\&\le H(r)+ C h(r) , \end{aligned}$$

which, together with (6.7), implies (6.9). Note that all conditions of Lemma 6.4 are verified. Therefore, for all $\varepsilon \le r\le 1,$

$$\begin{aligned} \frac{1}{r^m} \inf _{P_{m-1}\in \mathfrak {P}_{m-1}} \Big (\fint _{D_r} |u_\varepsilon -P_{m-1}|^2\Big )^{1/2} \le \Phi (r) \le C \big \{ H(r)+h(r) \big \} \le C \big \{ H(1)+h(1) \big \}, \end{aligned}$$

(6.14)

where C depends only on $d,n, m, q,\mu ,\sigma ,\theta$ and $M_1$ in (1.10). From the definition of H(1), we have

$$\begin{aligned} h(1)&\le \sum _{|\alpha |=m} \Big (\fint _{D_1} |D^\alpha (G- P_{m1}) |^2\Big )^{1/2} + C \Vert \nabla ^m G\Vert _{L^\infty (D_1)} \nonumber \\&\le C \big \{H(1) + \Vert \nabla ^m G\Vert _{L^\infty (D_1)} \big \} . \end{aligned}$$

(6.15)

It then follows that

$$\begin{aligned} \frac{1}{r^m}\!\inf _{P_{m-1}\in \mathfrak {P}_{m-1}}\!\Big (\fint _{D_r} |u_\varepsilon \!-\!P_{m-1} |^2\Big )^{1/2}\! \le&\, C \Big \{\Big (\fint _{D_1}|u_\varepsilon |^2 \Big )^{1/2} +\!\sum _{|\alpha |\le m-1}\Big (\fint _{D_1}|f^\alpha |^q\Big )^{1/q} \\&+ \Vert G \Vert _{C^{m,\sigma }(D_1)}\Big \}, \end{aligned}$$

which gives (1.11) through Caccioppoli’s inequality. $\square$

Corollary 6.1

In addition to the assumptions of Theorem 1.3, if A satisfies (1.12), then

$$\begin{aligned} \Vert \nabla ^mu_\varepsilon \Vert _{L^{\infty }(D_{1/4})}\le C \Big \{\Big (\fint _{D_1}|u_\varepsilon |^2 \Big )^{1/2} +\sum _{|\alpha |\le m-1}\Big (\fint _{D_1}|f^\alpha |^q\Big )^{1/q} + \Vert G \Vert _{C^{m,\sigma }(D_1)}\Big \}, \end{aligned}$$

(6.16)

where C depends only on $d,n,m,q, \sigma ,\mu$ as well as $\Lambda _0, \tau _0$ in (1.12) and $\theta , M_1$ in (1.10).

Proof

It is enough to consider the case $0<\varepsilon <1/2$, since otherwise the coefficient is uniformly Hölder continuous and the result (6.16) is known, see e.g., [27, Corollary 2.4]. Setting

$$\begin{aligned} v_\varepsilon (x)=u_\varepsilon (\varepsilon x)-\widetilde{G}(x), \quad \,\widetilde{G}(x)= G(\varepsilon x), \quad \, \widetilde{f}^\alpha (x)=\varepsilon ^{2m-|\alpha |} f(\varepsilon x), \end{aligned}$$

we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {L}_1 v_\varepsilon = \sum _{|\alpha |\le m-1}D^\alpha \widetilde{f}^\alpha (x)+ \sum _{|\alpha |=|\beta |=m}D^\alpha \left\{ A^{\alpha \beta }D^\beta \widetilde{G}(x)\right\} \quad \text { in } D_1, \\ \hbox {Tr} (D^\gamma v_\varepsilon )=0, \quad \quad \quad \text {on } \Delta _1\quad \text {for } 0\le |\gamma |\le m-1. \end{array}\right. } \end{aligned}$$

(6.17)

Let $\phi \in C_c^\infty (B_1)$ with $\phi =1$ in $B_{1/4}$ and $|\nabla ^k\phi |\le C2^k$, and let $\widetilde{D}$ be a $C^{1,\theta }$ domain such that $D_{1/4}\subseteq \widetilde{D}\subseteq D_{1/2}$. We have

$$\begin{aligned} \mathcal {L}_1 (v_\varepsilon \phi )&= E(x) \phi + \sum _{\begin{array}{c} |\alpha |=|\beta |=m\\ \zeta +\eta =\beta \\ |\eta |\ge 1 \end{array}}C(\zeta ) D^\alpha \big \{A^{\alpha \beta } D^\zeta v_\varepsilon D^\eta \phi \big \}\\&\quad +\sum _{\begin{array}{c} |\alpha |=|\beta |=m\\ \zeta '+\eta '=\alpha \\ |\eta '|\ge 1 \end{array}} C(\zeta ') D^{\zeta '}\big \{A^{\alpha \beta } D^\beta v_\varepsilon \big \} D^{\eta '} \phi ~~\text { in } \widetilde{D}, \\ \hbox {Tr} (D^\gamma (v_\varepsilon \phi ))&=0 \quad \quad \text { on } \partial \widetilde{D} \quad \text {for } 0\le |\gamma |\le m-1, \end{aligned}$$

where

$$\begin{aligned} E(x)= \sum _{|\alpha |\le m-1}D^\alpha \widetilde{f}^\alpha (x)+ \sum _{|\alpha |=|\beta |=m}D^\alpha \left\{ A^{\alpha \beta }D^\beta \widetilde{G}(x)\right\} . \end{aligned}$$

Thanks to the boundary $C^{m,\lambda }$ estimate for operator $\mathcal {L}_1$ in $C^{1,\theta }$ domains [27], we know that for any $q, p>d,$

$$\begin{aligned} \Vert \nabla ^m v_\varepsilon \Vert _{L^\infty (D_{1/2})} \le&\, C \Big \{ \Big (\fint _{D_1} |v_\varepsilon |^2\Big )^{1/2} + \Vert \widetilde{G}\Vert _{C^{m,\sigma }(D_1)} \nonumber \\&+ \sum _{|\alpha |\le m-1}\Big (\fint _{D_1}|\widetilde{f}^\alpha |^q\Big )^{1/q} + \Vert v_\varepsilon \Vert _{W^{m,p}(\widetilde{D})} \Big \}. \end{aligned}$$

(6.18)

Thanks to the $W^{m,p}$ estimate for (6.17), there exists some $p>d$ such that

$$\begin{aligned} \Vert v_\varepsilon \Vert _{W^{m,p}(\widetilde{D})} \le C \Big \{ \Big (\fint _{D_1} |v_\varepsilon |^2\Big )^{1/2} + \sum _{|\alpha |\le m-1}\Big (\fint _{D_1}|\widetilde{f}^\alpha |^q\Big )^{1/q} + \Vert \widetilde{G}\Vert _{C^{m,\sigma }(D_1)} \Big \}, \end{aligned}$$

which, combined with (6.18), implies that

$$\begin{aligned} \Vert \nabla ^m v_\varepsilon \Vert _{L^\infty (D_{1/2})} \le C \Big \{ \Big (\fint _{D_1} |v_\varepsilon |^2\Big )^{1/2} + \sum _{|\alpha |\le m-1}\Big (\fint _{D_1}|\widetilde{f}^\alpha |^q\Big )^{1/q} + \Vert \widetilde{G}\Vert _{C^{m,\sigma }(D_1)} \Big \}. \end{aligned}$$

It then follows from the change of variables that

$$\begin{aligned} \Vert \nabla ^m u_\varepsilon \Vert _{L^\infty (D_{\varepsilon /2})}&\le C \frac{1}{\varepsilon ^m} \Big \{\Big (\fint _{D_\varepsilon }| u_\varepsilon |^2\Big )^{1/2} + \varepsilon ^{2m-|\alpha |}\sum _{|\alpha |\le m-1}\Big (\fint _{D_\varepsilon }|f^\alpha |^q\Big )^{1/q} \nonumber \\&\quad + \sum _{j=0}^{m} \varepsilon ^j \Vert \nabla ^j G \Vert _{L^\infty (D_\varepsilon )} +\varepsilon ^{m+\sigma }\Vert \nabla ^m G\Vert _{C^{0,\sigma }(D_\varepsilon )}\Big \} . \end{aligned}$$

(6.19)

Using (6.11), (6.14) and (6.15), we may conclude from (6.19) that

$$\begin{aligned} \Vert \nabla ^m u_\varepsilon \Vert _{L^\infty (D_{\varepsilon /2})}&\le C \big \{H(\varepsilon )+h(\varepsilon )\big \}\le C \big \{H(1)+h(1)\big \}\\&\le C \Big \{\Big (\fint _{D_1}|u_\varepsilon |^2 \Big )^{1/2} + \sum _{|\alpha |\le m-1}\Big (\fint _{D_1}|f^\alpha |^q\Big )^{1/q} + \Vert G \Vert _{C^{m,\sigma }(D_1)}\Big \}. \end{aligned}$$

This, together with the interior uniform $C^{m-1,1}$ estimate for $u_\varepsilon$ derived in [28, Theorem 1.2], gives (6.16). $\square$