Uniform bound and convergence for elliptic homogenization problems

Abstract

Uniform bound and convergence for the solutions of elliptic homogenization problems are concerned. The problem domain has a periodic microstructure; it consists of a connected subregion with high permeability and a disconnected matrix block subset with low permeability. Let $\epsilon \in (0,1)$ denote the size ratio of the period to the whole domain, and let $\omega ^2\in (0,1)$ denote the permeability ratio of the disconnected matrix block subset to the connected subregion. For elliptic equations with diffusion depending on the permeability, the elliptic solutions are smooth in the connected subregion but change rapidly in the disconnected matrix block subset. More precisely, the solutions in the connected subregion can be bounded uniformly in $\omega ,\epsilon$ in Hölder norm, but not in the matrix block subset. It is known that the elliptic solutions converge to a solution of some homogenized elliptic equation as $\omega ,\epsilon$ converge to 0. In this work, the $L^p$ convergence rate for $p\in (2,\infty ]$ is derived. Depending on strongly coupled or weakly coupled case, the convergence rate is related to the factors $\omega ,\epsilon ,\frac{\omega }{\epsilon }$ for the former and related to the factors $\omega ,\epsilon$ for the latter.

1 Introduction

Uniform bound and convergence for the solutions of elliptic homogenization problems are presented. The problems have applications in contaminant transport in the subsurface, heat transfer in two-phase media, the stress in composite materials, and so on (see [3, 10, 17, 18]). The problem domain $\Omega \subset {\mathbb {R}}^n\ (n= 2,3)$ contains two subsets, a periodic connected subregion with high permeability and a periodic disconnected matrix block subset with low permeability. Let $\epsilon \in (0,1)$ be a parameter, $Y\equiv (0,1)^n$ consist of a subdomain $Y_m$ completely surrounded by another connected subdomain $Y_f(\equiv Y{\setminus }\overline{Y_m})$, $\Omega (\epsilon )\equiv \{x\in \Omega | dist( x,\partial \Omega )>\epsilon \}$, $\Omega _m^\epsilon \equiv \{x| x\in \epsilon (Y_m+j)\subset \Omega (\epsilon ) \text{ for } \text{ some } j\in {\mathbb {Z}}^n\}$ be the disconnected matrix block subset of $\Omega$, $\Omega _f^\epsilon \equiv \Omega \setminus \overline{\Omega _m^\epsilon }$ be the connected subregion of $\Omega$, and $\partial \Omega$ (resp. $\partial \Omega ^\epsilon _m$) be the boundary of $\Omega$ (resp. $\Omega ^\epsilon _m$). The problem that we consider is

$$\begin{aligned} {\left\{ \begin{array}{ll} -\nabla \cdot (\mathbf{E}_{\omega ^2,\epsilon }{\mathbf{K}}_{\epsilon }\nabla \Psi _{\omega ,\epsilon }+V_{\omega ,\epsilon })+{\mathbf{T}}_{\omega ,\epsilon }\Psi _{\omega ,\epsilon } =G_{\omega ,\epsilon } &{} \text{ in } \Omega ,\\ \Psi _{\omega ,\epsilon }=0 &{} \text{ on } \partial \Omega ,\\ \end{array}\right. } \end{aligned}$$

(1.1)

where $\omega ,\epsilon \in (0,1)$, $\mathbf{E}_{\nu ,\epsilon }\equiv {\left\{ \begin{array}{ll} 1 &{} \text{ in } \Omega ^\epsilon _f\\ \nu &{} \text{ in } \Omega ^\epsilon _m\\ \end{array}\right. }$ for any $\nu >0$, ${\mathbf{K}}_\epsilon (x)\equiv {\mathbf{K}}(\frac{x}{\epsilon })$, ${\mathbf{K}}$ is a positive periodic function in ${\mathbb {R}}^n$ with period Y, ${\mathbf{T}}_{\omega ,\epsilon }$ is a nonnegative function, and $V_{\omega ,\epsilon }, G_{\omega ,\epsilon }$ are given functions. It is known that if ${\mathbf{K}}_\epsilon , {\mathbf{T}}_{\omega ,\epsilon }, V_{\omega ,\epsilon }, G_{\omega ,\epsilon }$ are smooth in $\Omega ^\epsilon _f\cup \Omega ^\epsilon _m$, a piecewise smooth solution of (1.1) exists uniquely [19]. The $H^1$ norm of the solution in the high-permeability subregion $\Omega _f^\epsilon$ is bounded uniformly in $\omega ,\epsilon$ when $V_{\omega ,\epsilon }, G_{\omega ,\epsilon }$ are small in $\Omega _m^\epsilon$. However, that may not be the case for the solution in the low-permeability subset $\Omega _m^\epsilon$ (see Remark 2.2). Also the second-order derivatives of the solution of (1.1) may not be bounded uniformly in $\omega ,\epsilon$ in the high-permeability subregion $\Omega _f^\epsilon$ even when $V_{\omega ,\epsilon }, G_{\omega ,\epsilon }$ are bounded uniformly in $\omega ,\epsilon$ and are small in $\Omega _m^\epsilon$ (see Remark 3.1). By homogenization theory (see [7, 17, 23]), if $\omega ,\epsilon$ become small, the solution of (1.1) approaches to a solution of some homogenized elliptic differential equation. So it seems that, if both $\omega ,\epsilon$ are small, the solution of the homogenized elliptic differential equation is a good approximation of the solution of (1.1). We shall see in Sect. 2 that the solution of (1.1) can be approximated by the solution of the homogenized elliptic differential equation plus some functions, which are the solutions of mutually independent local problems.

Lipschitz estimate and $W^{2,p}$ estimate for the solutions of the uniform elliptic equations with discontinuous coefficients had been considered in [20, 21]. For the uniform elliptic case of (1.1) (that is, $\omega =1$), uniform bound and convergence results were also studied. For example, uniform Hölder, $W^{1,p}$, and Lipschitz estimates in $\epsilon$ for uniform elliptic case of (1.1) with Hölder coefficients were proved in [4, 5]. Uniform $W^{1,p}$ estimate in $\epsilon$ for uniform elliptic case of (1.1) with continuous coefficients was shown in [13], and the same problem with VMO coefficients could be found in [25]. Uniform Lipschitz estimate in $\epsilon$ for the Laplace equation in periodic perforated domains was studied in [24]. By [7, 17, 23], the solution of uniform elliptic case of (1.1) with Dirichlet boundary condition converges to a solution of some homogenized elliptic equation with convergence rate $\epsilon$ in $L^2$ norm and with convergence rate $\epsilon ^{1/2}$ in $H^1$ norm as $\epsilon$ closes to 0.

In this work, we consider the non-uniform elliptic case of (1.1) with discontinuous coefficients. We derive uniform Hölder estimates in $\omega , \epsilon$ for the solution of (1.1) as well as derive $L^p$ convergence estimates for $p\in (2,\infty ]$ for the approximation of the solution of (1.1).

One interesting related problem is the study of the equations with contrasting diffusivity in a fibered medium, that is, a conductivity medium reinforced by an $\epsilon$-periodic lattice of highly or lowly conducting thin rods (see [6, 8, 9, 11, 12, 26] and references therein). In [9], homogenization problem of degenerate Poisson equations in a fiber-reinforced structure was considered. In [12], the article tried to find a suitable conductivity medium which corresponding the prescribed Dirichlet problem with the non-local term. In [26], the authors analyzed nonlinear monotone conduction problems in a fibered medium. A two-scale convergence result to a non-local homogenized equation was shown. In [8], weak convergence of the solution of a p-Laplacian-type equation in a fiber-reinforced structure was shown. In [6], a spectral problem of a Poisson equations in a bounded domain with a high contrast in both stiffness and density was studied. In [11], the article considered the uniform regularity of the elliptic solutions in a fibered medium with $\omega =\omega (\epsilon )\gg 1$ and $\epsilon \ll 1$. Uniform $W^{1,6}$ bound in $\epsilon$ and uniform $C^{1,\nu }$ convergence estimate in $\epsilon$ of the solutions were derived in an interior region with a distance $\epsilon ^{\tau }$ away from the highly conducting thin rods for some $\nu ,\tau >0$ (the distance constraint is required). Different from [11], we derive uniform Hölder estimates in $\omega , \epsilon$ for the solution of (1.1) in the high-permeability region $\Omega _f^\epsilon$ (the distance constraint is not required). Moreover, we obtain the uniform convergence estimates in $\omega , \epsilon$ for the solution of (1.1) in the whole domain $\Omega$.

The rest of the work is organized as follows: Notation and main results are stated in Sect. 2. In Sect. 3, we derive a priori uniform estimates for interface problems. Uniform Hölder estimates for the non-uniform elliptic solutions in heterogeneous media are considered in Sect. 4. $L^p$ convergence estimates for elliptic homogenization problems are presented in Sect. 5.

2 Notation and main result

If $D\subset {\mathbb {R}}^n$ is a set, $\overline{D}$ denotes the closure of the set D, ${\mathcal {X}}_D$ is the characteristic function on D, |D| is the volume of D, $\partial D$ is the boundary of D, and $D/r=\{x| rx\in D\}$ for $r>0$. $B_r(x)$ denotes a ball centered at x with radius $r>0$. If ${\mathbf{B}}_1,{\mathbf{B}}_2$ are Banach spaces, $\Vert \varphi _1,\cdots ,\varphi _m\Vert _{{\mathbf{B}}_1} \equiv \Vert \varphi _1\Vert _{{\mathbf{B}}_1}+\cdots +\Vert \varphi _m\Vert _{{\mathbf{B}}_1}$ and $\Vert \varphi \Vert _{{\mathbf{B}}_1\cap {\mathbf{B}}_2}\equiv \Vert \varphi \Vert _{{\mathbf{B}}_1}+\Vert \varphi \Vert _{{\mathbf{B}}_2}$. $C^{k,\alpha }$ denotes the Hölder space with norm $\Vert \cdot \Vert _{C^{k,\alpha }}$; $W^{s,p}$ denotes the Sobolev space with norm $\Vert \cdot \Vert _{W^{s,p}}$; $[\varphi ]_{C^{0,\alpha }}$ is the Hölder semi-norm of $\varphi$; $L^p(D)\equiv W^{0,p}(D)$; $H^s(D)\equiv W^{s,2}(D)$ for $k\ge 0$, $\alpha \in (0,1], s\ge -1, p\in [1,\infty ]$ (see [2, 16]). $C^\infty _0(D)$ is the space of infinitely differentiable functions with support in D; $C^\infty _{per}({\mathbb {R}}^n)$ is the space of infinitely differentiable Y-periodic functions in ${\mathbb {R}}^n$; $W^{s,p}_{per}(D)$ is the closure of $C^\infty _{per}({\mathbb {R}}^n)$ under the $W^{s,p}$ norm for $s\ge 0, p\in [1,\infty ]$; $H^1_{per}(D)\equiv W^{1,2}_{per}(D)$; $L^\infty _{per}(D)\equiv W^{0,\infty }_{per}(D)$. For $p\ge 2$, $W^{1,p}_0(D)\equiv \{\varphi \in W^{1,p}(D) |\ \varphi =0\ \text{ on } \partial D\}$ and $H^1_0(D)=W^{1,2}_0(D)$. For any $\varphi \in L^1(D)$ and $r>0$,

If $\vec {\mathbf{n}}$ is an outward normal vector on $\partial Y_m$, we define, for any function $\varphi$ and $x\in \partial Y_m$,

$$\begin{aligned} \varphi _{,\pm }(x)\equiv \lim \limits _{t\rightarrow 0^+}\varphi (x\pm t\vec {\mathbf{n}}), \qquad \lfloor \varphi \rfloor (x)=\varphi _{,+}(x)-\varphi _{,-}(x). \end{aligned}$$

(2.1)

Similarly, if $\vec {\mathbf{n}}^\epsilon$ is an outward normal vector on $\partial \Omega ^\epsilon _m$, we define, for any $x\in \partial \Omega ^\epsilon _m$,

$$\begin{aligned} \varphi _{,\pm }(x)\equiv \lim \limits _{t\rightarrow 0^+}\varphi (x\pm t\vec {\mathbf{n}}^\epsilon ), \qquad \lfloor \varphi \rfloor (x)=\varphi _{,+}(x)-\varphi _{,-}(x). \end{aligned}$$

(2.2)

Next we recall an extension result in [1].

Remark 2.1

For any $\epsilon \in (0,1)$ and $p\in [1,\infty )$, there are a constant $\ell _1(Y_f,p)$ and a linear continuous extension operator $\Pi _\epsilon : W^{1,p}(\Omega _f^\epsilon ) \rightarrow W^{1,p}(\Omega )$ such that

• (1) if $\varphi \in W^{1,p}(\Omega _f^\epsilon )$, then

  $$\begin{aligned} {\left\{ \begin{array}{ll} \Pi _\epsilon \varphi = \varphi \quad \text{ in } \Omega _f^\epsilon \hbox { almost everywhere},\\ \Vert \Pi _\epsilon \varphi \Vert _{L^{p}(\Omega )}\le \ell _1\Vert \varphi \Vert _{L^{p}(\Omega ^\epsilon _f)},\\ \Vert \nabla \Pi _\epsilon \varphi \Vert _{L^{p}(\Omega )}\le \ell _1\Vert \nabla \varphi \Vert _{L^{p}(\Omega ^\epsilon _f)},\\ \ell _2\le \Pi _\epsilon \varphi \le \ell _3 \qquad \text{ if } \varphi \in L^\infty (\Omega ^\epsilon _f) \hbox { and } \ell _2\le \varphi \le \ell _3,\\ \Pi _\epsilon \varphi =\zeta \quad \text{ in } \Omega \hbox { if } \varphi =\zeta |_{\Omega _f^\epsilon } \hbox { for some linear function } \zeta \hbox { in } \Omega ,\\ \end{array}\right. } \end{aligned}$$

• (2) if $r>0$, $\epsilon /r<1$, and $\zeta (x)\equiv \varphi (r x)$, then $\Pi _{\epsilon /r}\zeta (x)=(\Pi _\epsilon \varphi )(r x)$.

If $\varphi \in W^{1,p}(\Omega )$ for any $p\ge 1$, then $\Pi _\epsilon \varphi |_{\Omega _f^\epsilon }\in W^{1,p}(\Omega )$ denotes the extension function of $\varphi |_{\Omega _f^\epsilon }\in W^{1,p}(\Omega _f^\epsilon )$ in $\Omega$.

We briefly state our main results. Theorems 2.1–2.2 are uniform bound estimates for the solution of (1.1). Theorem 2.1 is for the strongly coupled case (i.e., $0<{\mathbf{M}}_0\le {\mathbf{T}}_{\omega ,\epsilon }\le {\mathbf{M}}$ in $\Omega$), and Theorem 2.2 is for the weakly coupled case (i.e., $0\le \mathbf{E}_{1/\omega ,\epsilon }{\mathbf{T}}_{\omega ,\epsilon }\le {\mathbf{M}}$ in $\Omega$). Proofs of Theorems 2.1–2.2 are given in Sect. 4. Theorems 2.3–2.5 are convergence estimates for the solution of (1.1). Theorem 2.3 is for the strongly coupled case, and Theorems 2.4–2.5 are for the weakly coupled case. Proofs of Theorems 2.3–2.5 are given in Sect. 5.

Theorem 2.1

Suppose

• A1. $\Omega \subset {\mathbb {R}}^n$ is a $C^{2,1}$ domain for $n\in \{2,3\}$, $Y_m$ is a smooth simply connected subdomain of Y, $\overline{Y_m}\subset Y$,

• A2. ${\mathbf{K}}\in H^1_{per}({\mathbb {R}}^n)$ is a positive function, $\Vert \nabla {\mathbf{K}}\Vert _{L^\infty (Y)}$ is small compared with $\min _{x\in Y}{\mathbf{K}}(x)$, and ${\mathbf{K}}\in C^{1,\alpha }(\overline{Y_f})\cap C^{1,\alpha }(\overline{Y_m})$ for some $\alpha \in (0,1)$,

• A3. $\omega ,\epsilon \in (0,1)$, $\delta \in (0,3)$, $V_{\omega ,\epsilon }, G_{\omega ,\epsilon }\in L^{n+\delta }(\Omega )$,

• A4. ${\mathbf{M}}_0,{\mathbf{M}}>0$, ${\mathbf{T}}_{\omega ,\epsilon }(x)\in [{\mathbf{M}}_0,{\mathbf{M}}]$ for all $x\in \Omega$,

then a $H^{1}(\Omega )$ solution of (1.1) exists uniquely and there is a constant $\omega _0\in (0,1)$ (depending on $\delta ,{\mathbf{K}},{\mathbf{M}},Y_f,\Omega$) such that, for $\omega <\omega _0$ and $\frac{\omega }{\epsilon }>\ell _4>0$,

$$\begin{aligned}&[\Psi _{\omega ,\epsilon }]_{C^{0,\mu }(\overline{\Omega ^\epsilon _f})} +\mathop {\mathop {\sup }_{j\in {\mathbb {Z}}^n}} \limits _{\epsilon (Y_m+j)\subset \Omega _m^\epsilon } \omega [\Psi _{\omega ,\epsilon }]_{C^{0,\mu }(\epsilon (\overline{Y_m}+j))} \le c\Vert \mathbf{E}_{1/\omega ,\epsilon }V_{\omega ,\epsilon }, G_{\omega ,\epsilon }\Vert _{L^{n+\delta }(\Omega )}, \end{aligned}$$

(2.3)

where $\ell _4$ is any number, $\mu \equiv \frac{\delta }{n+\delta }$, and c is a positive constant independent of $\omega ,\epsilon$.

Theorem 2.2

Besides A1–A3, if

• A4’. $\mathbf{E}_{1/\omega ,\epsilon }{\mathbf{T}}_{\omega ,\epsilon }(x)\in [0,{\mathbf{M}}]$ for all $x\in \Omega$,

then a $H^{1}(\Omega )$ solution of (1.1) exists uniquely and there is a constant $\omega _0\in (0,1)$ (depending on $\delta ,{\mathbf{K}},{\mathbf{M}},Y_f,\Omega$) such that, for any $\omega <\omega _0$,

$$\begin{aligned}&[\Psi _{\omega ,\epsilon }]_{C^{0,\mu }(\overline{\Omega ^\epsilon _f})} + \mathop {\mathop {\sup }_{j\in {\mathbb {Z}}^n}}\limits _{\epsilon (Y_m+j)\subset \Omega _m^\epsilon } \omega ^{\lambda }[\Psi _{\omega ,\epsilon }]_{C^{0,\mu }(\epsilon (\overline{Y_m}+j))}\nonumber \\&\qquad \le c(\Vert \mathbf{E}_{1/\omega ,\epsilon }V_{\omega ,\epsilon },\ \mathbf{E}_{\max \{1,\epsilon /\omega \}} G_{\omega ,\epsilon }\Vert _{L^{n+\delta }(\Omega )}, \end{aligned}$$

(2.4)

where $\mu \equiv \frac{\delta }{n+\delta }$ and c is independent of $\omega ,\epsilon$. In (2.4), $\lambda ={\left\{ \begin{array}{ll} \frac{3}{2} &{} \text{ if } \ {\mathbf{T}}_{\omega ,\epsilon }\not =0\\ 1 &{} \text{ if } \ {\mathbf{T}}_{\omega ,\epsilon }=0\\ \end{array}\right. }$.

From Theorem 2.1 and Theorem 2.2, we know that if the right-hand side of (2.3) or (2.4) is bounded, the Hölder norm of the solution of (1.1) in the connected high-permeability region $\Omega _f^\epsilon$ is bounded uniformly in $\omega ,\epsilon$, but the solution in $\Omega _m^\epsilon$ may change rapidly when $\omega ,\epsilon$ are small. This is different from uniform elliptic equation case, where the solution is bounded uniformly in the whole domain. To obtain the uniform Hölder estimate for the solution of (1.1), the condition $\frac{\omega }{\epsilon }\ge \ell _4>0$ is needed in strongly coupled case but not in weakly coupled case. Below is one example to show that the Hölder norm and the $H^1$ norm of the solution of (1.1) in $\Omega _m^\epsilon$ in general are not be bounded uniformly in $\omega ,\epsilon$.

Remark 2.2

Suppose $\varphi \in C_{per}^\infty ({\mathbb {R}}^n)$ and $\varphi$ in the cell $Y\equiv (0,1)^n$ has support in $Y_m$, we define, for any $\epsilon \in (0,1)$,

$$\begin{aligned} \Psi _{\omega ,\epsilon }(x)\equiv {\left\{ \begin{array}{ll} 0 &{} \hbox { if } x\in \Omega _f^\epsilon ,\\ \varphi (\frac{x}{\epsilon }) &{} \,\,\text{ if } x\in \Omega _m^\epsilon .\\ \end{array}\right. } \end{aligned}$$

Then $\Psi _{\omega ,\epsilon }$ satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} -\nabla \cdot (\mathbf{E}_{\omega ^2,\epsilon }\nabla \Psi _{\omega ,\epsilon })=G_{\omega ,\epsilon } &{} \text{ in } \Omega ,\\ \Psi _{\omega ,\epsilon }=0 &{} \text{ on } \partial \Omega ,\\ \end{array}\right. } \end{aligned}$$

where $G_{\omega ,\epsilon }(x)=-\omega ^2\epsilon ^{-2}\Delta \varphi (\frac{x}{\epsilon }){\mathcal {X}}_{\Omega _m^\epsilon }$. Note $[\Psi _{\omega ,\epsilon }]_{C^{0,\mu }(\epsilon (\overline{Y_m}+j))} =\epsilon ^{-\mu }[\varphi ]_{C^{0,\mu }(\overline{Y_m}+j)}$, $\Vert \nabla \Psi _{\omega ,\epsilon }\Vert _{L^2(\Omega _m^\epsilon )}\approx \epsilon ^{-1}\Vert \nabla \varphi \Vert _{L^2(Y_m)}$, and $\Vert G_{\omega ,\epsilon }\Vert _{L^{n+\delta }(\Omega _m^\epsilon )}\approx \omega ^{2}\epsilon ^{-2}\Vert \Delta \varphi \Vert _{L^{n+\delta }(Y_m)}$ where $\delta >0, \mu \equiv \frac{\delta }{n+\delta }$. Here $A\approx B$ means that A is almost like B times a constant when $\epsilon$ is small. If $\omega \le \epsilon <1$, then the right-hand side of (2.4) is finite and (2.4) holds for $\Psi _{\omega ,\epsilon }$. But the $C^{0,\mu }$ norm and the $H^1$ norm of $\Psi _{\omega ,\epsilon }$ in $\Omega _m^\epsilon$ are not bounded uniformly in $\epsilon \in (0,1)$.

Next we state convergence results. Set ${\mathbb {A}}_m\equiv \{x\in {\mathbb {R}}^n| x\in \cup _{j\in {\mathbb {Z}}^n}(Y_m+j)\}$, ${\mathbb {A}}_f\equiv {\mathbb {R}}^n{\setminus }\overline{{\mathbb {A}}_m}$, and ${\mathbb {E}}_{\nu }\equiv {\left\{ \begin{array}{ll} 1 &{} \text{ in } Y_f\\ \nu &{} \text{ in } Y_m\\ \end{array}\right. }$ for any $\nu >0$. We find ${{\mathbb {X}}}^{(i)}_\nu \in H^1_{per}({\mathbb {R}}^n)$ for $\nu \in (0,1]$ and $i\in \{1,2,\cdots ,n\}$ satisfying

$$\begin{aligned} {\left\{ \begin{array}{ll} -\nabla \cdot ({\mathbb {E}}_{\nu ^2}{\mathbf{K}}(\nabla {{\mathbb {X}}}^{(i)}_\nu + \vec {e}_i))=0 &{} \text{ in } Y,\\ \displaystyle \int _{Y_f} {{\mathbb {X}}}^{(i)}_\nu (y) {\hbox {d}}y=0, \end{array}\right. } \end{aligned}$$

(2.5)

find ${\mathbb {X}}^{(i)}_0\in H_{per}^1({\mathbb {A}}_f)\cap H_{per}^1({\mathbb {A}}_m)$ for $i\in \{1,2,\cdots ,n\}$ satisfying

$$\begin{aligned} {\left\{ \begin{array}{ll} -\nabla \cdot ({\mathbf{K}}(\nabla {{\mathbb {X}}}^{(i)}_0+ \vec {e}_i))=0 &{} \text{ in } Y_f,\\ {{\mathbb {X}}}^{(i)}_0=0 &{} \text{ in } Y_m,\\ {\mathbf{K}}(\nabla {{\mathbb {X}}}^{(i)}_0+ \vec {e}_i)_{,+}\cdot \vec {{\mathbf{n}}}=0 &{} \text{ on } \partial Y_m,\\ \displaystyle \int _{Y_f}{\mathbb {X}}^{(i)}_0(y) {\hbox {d}}y=0, \end{array}\right. } \end{aligned}$$

(2.6)

and find ${\mathbb {W}}_\beta \in H_{per}^1({\mathbb {A}}_f)\cap H_{per}^1({\mathbb {A}}_m)$ for $\beta >0$, ${\mathbf{T}}\in L^\infty _{per}({\mathbb {R}}^n)$, and ${\mathbf{T}}\ge 0$ satisfying

$$\begin{aligned}&{\left\{ \begin{array}{ll} \beta ^2\nabla \cdot ({\mathbf{K}}\nabla {\mathbb {W}}_\beta )- {\mathbf{T}}{\mathbb {W}}_\beta =0 &{} \text{ in } Y_m,\\ {\mathbb {W}}_\beta =1 &{} \text{ on } \partial Y_m,\\ \end{array}\right. } \end{aligned}$$

(2.7)

$$\begin{aligned}&{\left\{ \begin{array}{ll} \displaystyle \nabla \cdot ({\mathbf{K}}\nabla {\mathbb {W}}_\beta )- {\mathbf{T}}=\frac{-1}{|Y_f|}\bigg (\int _{Y_f}{\mathbf{T}}{\hbox {d}}y+\int _{Y_m}{\mathbf{T}}{\mathbb {W}}_\beta {\hbox {d}}y\bigg ) &{} \text{ in } Y_f,\\ \lfloor {\mathbb {E}}_{\beta ^2}{\mathbf{K}}\nabla {\mathbb {W}}_\beta \rfloor \cdot \vec {{\mathbf{n}}}=0 &{} \text{ on } \partial Y_m,\\ \displaystyle \int _{Y_f}{\mathbb {W}}_\beta (y) {\hbox {d}}y=0, \end{array}\right. } \end{aligned}$$

(2.8)

where $\vec {e}_i$ is a unit vector in the ith coordinate direction, $|Y_f|$ is the volume of $Y_f$, and $\vec {{\mathbf{n}}}$ is an outward normal vector on $\partial Y_m$. See (2.1) for (2.6)$_3$ and (2.8)$_2$. Let ${\mathbb {X}}^{(i)}_{\nu ,s}(x)\equiv s{\mathbb {X}}^{(i)}_{\nu }(\frac{x}{s})$ ${\mathbb {X}}_{\nu ,s}\equiv ({\mathbb {X}}^{(1)}_{\nu ,s},\cdots ,{\mathbb {X}}^{(n)}_{\nu ,s})$, and ${\mathbb {W}}_{\beta ,s,i}(x)\equiv s^{i}{\mathbb {W}}_\beta (\frac{x}{s})$ for any $\nu \in [0,1]$, $s\in (0,1)$, $\beta >0$, $i\in {\mathbb {Z}}$. By Lax–Milgram Theorem [16], (2.5)–(2.8) are uniquely solvable. Denote by $\Xi _\nu$ for $\nu \in [0,1]$ a $n\times n$ matrix function whose (i, j)-component is $\partial _{i}{{\mathbb {X}}}_\nu ^{(j)}$. By remark in pages 17–19, 94–95 [17],

$$\begin{aligned} {\mathcal {K}}_\nu \equiv \int _{Y_f\cup Y_m}{\mathbb {E}}_{\nu ^2}{\mathbf{K}}(I+\Xi _\nu (y)){\hbox {d}}y\quad \text{ for } \quad \nu \in [0,1] \end{aligned}$$

(2.9)

is a constant symmetric positive definite matrix. Here I is the identity matrix.

If, in addition to A1–A4,

• A5. ${\mathbf{T}}_{\omega ,\epsilon }(x)={\mathbf{T}}(\frac{x}{\epsilon })>0$ and ${\mathbf{T}}\in C_{per}^{0,\alpha }({\mathbb {R}}^n)$ for some $\alpha >0$,

• A6. $\Vert \mathbf{E}_{1/\omega ,\epsilon } G_{\omega ,\epsilon }\Vert _{L^{n+\delta }(\Omega )} +\Vert G_{\omega ,\epsilon }\Vert _{W^{1,n+\delta }(\Omega ^\epsilon _f)}$ is bounded independent of $\omega ,\epsilon$,

the solution of (1.1) with $V_{\omega ,\epsilon }=0$ satisfies $\Vert \mathbf{E}_{\omega ,\epsilon }\nabla \Psi _{\omega ,\epsilon },\Psi _{\omega ,\epsilon }\Vert _{L^2(\Omega )} \le c$ (independent of $\omega$, $\epsilon$). Suppose $\omega ,\epsilon \rightarrow 0$ and $\frac{\omega }{\epsilon }\rightarrow \sigma \in [0, \infty ]$, by tracing the proof of Theorem 2.3 [3], we can extract a subsequence (same notation for subsequence) such that

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathbf{E}_{\omega ^2,\epsilon }{\mathbf{K}}_\epsilon \nabla \Psi _{\omega ,\epsilon } \rightarrow {\mathcal {K}}_0\nabla \Psi \\ {\mathbf{T}}_{\omega ,\epsilon } \Psi _{\omega ,\epsilon } \rightarrow |Y_f|{\mathcal {T}}_\sigma \Psi \\ G_{\omega ,\epsilon } \rightarrow |Y_f|{\mathcal {G}}\\ \end{array}\right. } \text{ in } L^2(\Omega )\hbox { weakly,} \end{aligned}$$

(2.10)

where ${\mathcal {K}}_0$ is defined in (2.9) with $\nu =0$, $|Y_f|$ is the volume of $Y_f$, and

$$\begin{aligned} {\mathcal {T}}_\sigma ={\left\{ \begin{array}{ll} \displaystyle \frac{1}{|Y_f|}\int _{Y}{\mathbf{T}}{\hbox {d}}y &{} \text{ if } \sigma =\infty ,\\ \displaystyle \frac{1}{|Y_f|}\bigg (\int _{Y_f}{\mathbf{T}}{\hbox {d}}y+\int _{Y_m}{\mathbf{T}}{\mathbb {W}}_\sigma {\hbox {d}}y\bigg ) &{} \text{ if } \sigma \in (0,\infty ),\\ \displaystyle \frac{1}{|Y_f|}\int _{Y_f}{\mathbf{T}}{\hbox {d}}y &{} \text{ if } \sigma =0. \end{array}\right. } \end{aligned}$$

(2.11)

See (2.7) for ${\mathbb {W}}_\sigma$. The $\Psi$ in (2.10) satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} -\nabla \cdot ({\mathcal {K}}_0\nabla \Psi )+|Y_f|{\mathcal {T}}_\sigma \Psi =|Y_f|{\mathcal {G}}&{} \text{ in } \Omega ,\\ \Psi = 0 &{} \text{ on } \partial \Omega .\\ \end{array}\right. } \end{aligned}$$

(2.12)

By Theorem 9.19 [16] and A6,

$$\begin{aligned} \Vert \Psi \Vert _{W^{3,n+\delta }(\Omega )}\le c\Vert {\mathcal {G}}\Vert _{W^{1,n+\delta }(\Omega )}, \end{aligned}$$

(2.13)

where c is a constant depending on ${\mathcal {K}}_0,{\mathbf{M}},|Y_f|,\Omega$. Now for any $\omega ,\epsilon \in (0,1)$ and on any $\epsilon (Y_m+j)\subset \Omega _m^\epsilon$ for some $j\in {\mathbb {Z}}^n$, we consider

$$\begin{aligned} {\left\{ \begin{array}{ll} -\nabla \cdot \big (\omega ^2{\mathbf{K}}_\epsilon \nabla \phi ^{(j)}_{\omega ,\epsilon } \big )+{\mathbf{T}}_{\omega ,\epsilon }\phi ^{(j)}_{\omega ,\epsilon }=G_{\omega ,\epsilon }&{} \text{ in } \epsilon (Y_m+j),\\ \phi ^{(j)}_{\omega ,\epsilon }=\Psi &{} \text{ on } \epsilon (\partial Y_m+j), \end{array}\right. } \end{aligned}$$

(2.14)

where $\Psi$ is the solution of (2.12). By Lax–Milgram Theorem [16], A5–A6, and (2.13), the $\phi ^{(j)}_{\omega ,\epsilon }$ of (2.14) is solvable uniquely in $H^1(\epsilon (Y_m+j))$. By Theorem 8.24 and Theorem 8.29 [16], $\phi ^{(j)}_{\omega ,\epsilon }\in L^\infty (\epsilon (Y_m+j))$. Moreover,

Theorem 2.3

Suppose A1–A6 and $V_{\omega ,\epsilon }=0$ in (1.1). There is a constant $\omega _0\in (0,1)$ such that, for any $\omega \in (0,\omega _0)$ and $\epsilon \in (0,1)$,

• 1. if $\omega ,\epsilon \rightarrow 0$ and $\frac{\omega }{\epsilon }\rightarrow \infty$, the solutions of (1.1), (2.12), and (2.14) satisfy

  $$\begin{aligned}&\Vert \mathbf{E}_{\omega ,\epsilon }(\Psi _{\omega ,\epsilon }-\Psi )\Vert _{L^\infty (\Omega )} +\Vert \Psi _{\omega ,\epsilon }-\mathop {\mathop {\sum }_{j\in {\mathbb {Z}}^n}}\limits _{\epsilon (Y_m+j)\subset \Omega _m^\epsilon } \phi ^{(j)}_{\omega ,\epsilon }\Vert _{L^\infty (\Omega _m^\epsilon )}\\&\qquad \le c(\Vert G_{\omega ,\epsilon }-{\mathcal {G}}\Vert _{L^{n+\delta }(\Omega _f^\epsilon )}+\max \{\omega , \epsilon /\omega \}), \end{aligned}$$

• 2. if $\omega ,\epsilon \rightarrow 0$ and $\frac{\omega }{\epsilon }\rightarrow \sigma \in (0,\infty )$, the solutions of (1.1), (2.12), and (2.14) satisfy

  $$\begin{aligned}&\Vert \mathbf{E}_{\omega ,\epsilon }\big (\Psi _{\omega ,\epsilon }-({\mathcal {X}}_{\Omega _f^\epsilon }+{\mathbb {W}}_{\frac{\omega }{\epsilon },\epsilon ,0}{\mathcal {X}}_{\Omega _m^\epsilon })\Psi \big )\Vert _{L^\infty (\Omega )} +\Vert \Psi _{\omega ,\epsilon }-\mathop {\mathop {\sum }_{j\in {\mathbb {Z}}^n}}\limits _ {\epsilon (Y_m+j)\subset \Omega _m^\epsilon }\phi ^{(j)}_{\omega ,\epsilon }\Vert _{L^\infty (\Omega _m^\epsilon )}\\&\qquad \le c(\Vert G_{\omega ,\epsilon }-{\mathcal {G}}\Vert _{L^{n+\delta }(\Omega _f^\epsilon )}+\max \{\omega ,\epsilon ,|\omega ^2/(\sigma \epsilon )^2-1|\}), \end{aligned}$$

• 3. if $\omega ,\epsilon \rightarrow 0$ and $\frac{\omega }{\epsilon }\rightarrow 0$, the solutions of (1.1) and (2.12) satisfy

  $$\begin{aligned}&\Vert \Psi _{\omega ,\epsilon }-({\mathcal {X}}_{\Omega _f^\epsilon }+{\mathbb {W}}_{\frac{\omega }{\epsilon },\epsilon ,0}{\mathcal {X}}_{\Omega _m^\epsilon })\Psi \Vert _{L^{n+\delta }(\Omega )}\\&\qquad \le c(\Vert G_{\omega ,\epsilon }-{\mathcal {G}}\Vert _{L^{n+\delta }(\Omega _f^\epsilon )}+\max \{\omega ,\epsilon ,|\omega \epsilon ^{-1}\ln (\omega \epsilon ^{-1})|^{\frac{1}{2}}\}), \end{aligned}$$

where c is a constant independent of $\omega , \epsilon$.

Theorem 2.3, based on Theorem 2.1, is a convergence result for (1.1) in strongly coupled case. Note that $L^\infty$ convergence estimate is obtained for $\frac{\omega }{\epsilon }\rightarrow \sigma >0$ case and that only $L^{n+\delta }$ convergence estimate is available for $\frac{\omega }{\epsilon }\rightarrow 0$ case. Next we present convergence estimates for the solutions of (1.1) in weakly coupled case (that is, Theorems 2.4 and 2.5).

Besides A1–A3, A4’, and A6, if

• A7. $\mathbf{E}_{1/\omega ,\epsilon }{\mathbf{T}}_{\omega ,\epsilon }(x)={\mathbf{P}}(\frac{x}{\epsilon })\ge 0$   and   ${\mathbf{P}}\in L^\infty _{per}({\mathbb {R}}^n)\cap C^{0,\alpha }(\overline{{\mathbb {A}}_f})$ for some $\alpha >0$,

the solution of (1.1) with $V_{\omega ,\epsilon }=0$ satisfies $\Vert \mathbf{E}_{\omega ,\epsilon }\nabla \Psi _{\omega ,\epsilon },{\mathbf{T}}_{\omega ,\epsilon }^{1/2}\Psi _{\omega ,\epsilon }\Vert _{L^2(\Omega )} \le c$ (independent of $\omega$, $\epsilon$). By compactness principle [3, 17],

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathbf{E}_{\omega ^2,\epsilon }{\mathbf{K}}_\epsilon \nabla \Psi _{\omega ,\epsilon } \rightarrow {\mathcal {K}}_0\nabla \Psi \\ {\mathbf{T}}_{\omega ,\epsilon } \Psi _{\omega ,\epsilon } \rightarrow |Y_f|\breve{{\mathcal {T}}} \Psi \\ G_{\omega ,\epsilon } \rightarrow |Y_f|{\mathcal {G}}\\ \end{array}\right. }\quad \text{ in } L^2(\Omega ) \hbox { weakly as } \omega ,\epsilon \rightarrow 0, \end{aligned}$$

(2.15)

where $\displaystyle \breve{{\mathcal {T}}} \bigg (=\frac{1}{|Y_f|}\int _{Y_f}{\mathbf{P}}(y){\hbox {d}}y\bigg )$ is a constant vector and ${\mathcal {K}}_0$ is defined in (2.9) with $\nu =0$. Similar to (2.12)–(2.13), the $\Psi$ in (2.15) satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} -\nabla \cdot ({\mathcal {K}}_0\nabla \Psi )+|Y_f|\breve{{\mathcal {T}}} \Psi =|Y_f|{\mathcal {G}}&{} \text{ in } \Omega ,\\ \Psi = 0 &{} \text{ on } \partial \Omega ,\\ \Vert \Psi \Vert _{W^{3,n+\delta }(\Omega )}\le c\Vert {\mathcal {G}}\Vert _{W^{1,n+\delta }(\Omega )},\\ \end{array}\right. } \end{aligned}$$

(2.16)

where $|Y_f|$ is the volume of $Y_f$ and c is a constant depending on ${\mathcal {K}}_0,{\mathbf{M}},|Y_f|, \Omega$.

We have the following result:

Theorem 2.4

Assume A1–A3, A4’, A6–A7, and $V_{\omega ,\epsilon }=0$ in (1.1). There is a $\omega _0\in (0,1)$ such that if $\omega <\omega _0$, then the solutions of (1.1), (2.16), and (2.14) with $\Psi$ obtained from (2.16) satisfy

$$\begin{aligned}&\Vert \mathbf{E}_{\omega ^{3/2},\epsilon }\Psi _{\omega ,\epsilon }-\Psi \Vert _{L^\infty (\Omega )} +\Vert \Psi _{\omega ,\epsilon }-\mathop {\mathop {\sum }_{j\in {\mathbb {Z}}^n}}\limits _ {\epsilon (Y_m+j)\subset \Omega _m^\epsilon }\phi ^{(j)}_{\omega ,\epsilon }\Vert _{L^\infty (\Omega _m^\epsilon )}\\&\qquad \le c(\Vert G_{\omega ,\epsilon }-{\mathcal {G}}\Vert _{L^{n+\delta }(\Omega _f^\epsilon )}+\max \{\omega ,\epsilon \}), \end{aligned}$$

where c is a constant independent of $\omega ,\epsilon$. See (2.16) for ${\mathcal {G}}$.

Under A1–A3 and A6, the solution of (1.1) with $V_{\omega ,\epsilon }={\mathbf{T}}_{\omega ,\epsilon }=0$ satisfies $\Vert \mathbf{E}_{\omega ,\epsilon }\nabla \Psi _{\omega ,\epsilon }\Vert _{L^2(\Omega )} \le c$ (independent of $\omega$, $\epsilon$). By compactness principle [3, 17],

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathbf{E}_{\omega ^2,\epsilon }{\mathbf{K}}_\epsilon \nabla \Psi _{\omega ,\epsilon } \rightarrow {\mathcal {K}}_0\nabla \Psi \\ G_{\omega ,\epsilon } \rightarrow |Y_f|{\mathcal {G}}\\ \end{array}\right. }\quad \text{ in } L^2(\Omega ) \hbox { weakly as } \omega ,\epsilon \rightarrow 0, \end{aligned}$$

(2.17)

where ${\mathcal {K}}_0$ is defined in (2.9) with $\nu =0$. The $\Psi$ in (2.17) satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} -\nabla \cdot ({\mathcal {K}}_0\nabla \Psi )=|Y_f|{\mathcal {G}}&{} \text{ in } \Omega ,\\ \Psi = 0 &{} \text{ on } \partial \Omega ,\\ \Vert \Psi \Vert _{W^{3,n+\delta }(\Omega )}\le c\Vert {\mathcal {G}}\Vert _{W^{1,n+\delta }(\Omega )},\\ \end{array}\right. } \end{aligned}$$

(2.18)

where $|Y_f|$ is the volume of $Y_f$ and c is a constant depending on ${\mathcal {K}}_0,|Y_f|, \Omega$. We also have

Theorem 2.5

Assume A1–A3, A6, and $V_{\omega ,\epsilon }={\mathbf{T}}_{\omega ,\epsilon }=0$ in (1.1). There is a $\omega _0\in (0,1)$ such that if $\omega <\omega _0$, the solutions of (1.1), (2.18), and (2.14) with ${\mathbf{T}}_{\omega ,\epsilon }=0$ and $\Psi$ from (2.18) satisfy

$$\begin{aligned}&\Vert \mathbf{E}_{\omega ,\epsilon }\Psi _{\omega ,\epsilon }-\Psi \Vert _{L^\infty (\Omega )} +\Vert \Psi _{\omega ,\epsilon }-\mathop {\mathop {\sum }_{j\in {\mathbb {Z}}^n}} \limits _{\epsilon (Y_m+j)\subset \Omega _m^\epsilon }\phi ^{(j)}_{\omega ,\epsilon }\Vert _{L^\infty (\Omega _m^\epsilon )}\\&\qquad \le c(\Vert G_{\omega ,\epsilon }-{\mathcal {G}}\Vert _{L^{n+\delta }(\Omega _f^\epsilon )}+\max \{\omega ,\epsilon \}), \end{aligned}$$

where c is a constant independent of $\omega ,\epsilon$. See (2.17) for ${\mathcal {G}}$.

Theorems 2.3, 2.4, 2.5 imply if $\omega ,\epsilon ,\Vert G_{\omega ,\epsilon }-{\mathcal {G}}\Vert _{L^{n+\delta }(\Omega _f^\epsilon )}$ are small enough, the homogenized solution $\Psi$ of (2.12) or (2.16) or (2.18) is a good approximation of the solution of (1.1) in the connected subregion $\Omega _f^\epsilon$, but the $\Psi$ may not be a good approximation of the solution of (1.1) in the disconnected subset $\Omega _m^\epsilon$. In the disconnected subset $\Omega _m^\epsilon$, the solution of (1.1) can be approximated by the solution of (2.14). One also notes that $\sum _{\mathop {j\in {\mathbb {Z}}^n}\limits _{\epsilon (Y_m+j)\subset \Omega _m^\epsilon }}\phi ^{(j)}_{\omega ,\epsilon }$ is obtained by solving mutually independent local problems.

3 A priori uniform estimates for interface problems

Let $\Gamma (x-y)$ denote the fundamental solution of the Laplace equation in ${\mathbb {R}}^n$; see §6.2 [14]. Define a single-layer and a double-layer potentials as, for any smooth function $\varphi$ on the boundary $\partial Y_m$ of $Y_m$,

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle {\mathcal {S}}_{\partial Y_m}(\varphi )(x)\equiv \int _{\partial Y_m}\Gamma (x-y)\varphi (y) {\hbox {d}}y\\ \displaystyle {\mathcal {L}}_{\partial Y_m}(\varphi )(x)\equiv \int _{\partial Y_m}\nabla _y \Gamma (x-y)\vec {\mathbf{n}}_y\ \varphi (y) {\hbox {d}}y\\ \end{array}\right. } \text{ for } x\in \partial Y_m, \end{aligned}$$

where $\vec {\mathbf{n}}_y$ is the unit vector outward normal to $\partial Y_m$. By tracing the argument of Lemma 4.1 [27], we know

Lemma 3.1

For any $p\in (1,\infty )$, $i\in \{0, 1\}$, and $\alpha \in (0, 1)$, the linear operators

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {S}}_{\partial Y_m}: W^{i-\frac{1}{p},p}(\partial Y_m)\rightarrow W^{i+1-\frac{1}{p},p}(\partial Y_m)\\ {\mathcal {L}}_{\partial Y_m}: W^{i+1-\frac{1}{p},p}(\partial Y_m)\rightarrow W^{i+2-\frac{1}{p},p}(\partial Y_m)\\ {\mathcal {S}}_{\partial Y_m}: C^{1,\alpha }(\partial Y_m)\rightarrow C^{2,\alpha }(\partial Y_m)\\ {\mathcal {L}}_{\partial Y_m}: C^{1,\alpha }(\partial Y_m)\rightarrow C^{2,\alpha }(\partial Y_m)\\ \end{array}\right. } \end{aligned}$$

are bounded; the operator $I-\ell {\mathcal {L}}_{\partial Y_m}$ is continuously invertible in $W^{i+1-\frac{1}{p},p}(\partial Y_m)$ and in $C^{2,\alpha }(\partial Y_m)$ for $\ell \in [-2,2]$; there is a constant c independent of $\ell$ so that

$$\begin{aligned} {\left\{ \begin{array}{ll} \Vert \varphi \Vert _{W^{i+1-\frac{1}{p},p}(\partial Y_m)}\le c\Vert (I-\ell {\mathcal {L}}_{\partial Y_m})(\varphi )\Vert _{W^{i+1-\frac{1}{p},p}(\partial Y_m)} &{} \text{ for } \varphi \in W^{i+1-\frac{1}{p},p}(\partial Y_m),\\ \Vert \varphi \Vert _{C^{2,\alpha }(\partial Y_m)}\le c\Vert (I-\ell {\mathcal {L}}_{\partial Y_m})(\varphi )\Vert _{C^{2,\alpha }(\partial Y_m)} &{} \text{ for } \varphi \in C^{2,\alpha }(\partial Y_m),\\ \end{array}\right. } \end{aligned}$$

where I is the identity operator.

By A1, let us assume

$$\begin{aligned} {\left\{ \begin{array}{ll} Y_m\subset {\mathbf{D}}_1\subset {\mathbf{D}}_2\subset Y,\\ \min \{dist(Y_m, \partial {\mathbf{D}}_1), dist({\mathbf{D}}_1,\partial {\mathbf{D}}_2), dist({\mathbf{D}}_2,\partial Y)\}>0.\\ \end{array}\right. } \end{aligned}$$

(3.1)

Lemma 3.2

Under A1–A2, $\omega \in (0,1]$, $p\in (n,6)$, ${\mathbf{M}}>0$, and ${\mathbb {P}}_\omega (x)\in [0,{\mathbf{M}}]$ for all $x\in Y$, any solution of

$$\begin{aligned} -\nabla \cdot ({\mathbb {E}}_{\omega ^2}{\mathbf{K}}\nabla U_\omega +Q_\omega )+{\mathbb {E}}_\omega {\mathbb {P}}_\omega U_\omega =F_\omega \qquad \text{ in } Y \end{aligned}$$

(3.2)

satisfies

$$\begin{aligned}&\Vert {\mathbb {E}}_{\omega ^{3/2}}U_\omega \Vert _{W^{1,p}({\mathbf{D}}_1\setminus \overline{Y_m})\cap W^{1,p}(Y_m)} \le c(\Vert U_\omega \Vert _{L^2(Y_f)} +\Vert {\mathbb {E}}_{1/\omega }Q_\omega \Vert _{L^{2}(Y)} \nonumber \\& + \Vert {\mathbb {E}}_{1/\omega }F_\omega \Vert _{H^{-1}(Y)} +\Vert {\mathbb {E}}_{1/\sqrt{\omega }}Q_\omega \Vert _{L^{p}(Y)} +\Vert {\mathbb {E}}_{1/\sqrt{\omega }}F_\omega \Vert _{W^{-1,p}(Y)} ), \end{aligned}$$

(3.3)

where c is a constant independent of $\omega$. See Sect. 2 for ${\mathbb {E}}_{\nu }$.

Proof

Let $p\in (n,6)$ and c denote a constant independent of $\omega$.

Step 1: Assume $Q_\omega \in W^{1,p}_0(Y_f)\cap W^{1,p}_0(Y_m)$, $F_\omega \in L^p(Y)$, and consider

$$\begin{aligned} {\left\{ \begin{array}{ll} -\nabla \cdot ({\mathbb {E}}_{\omega ^2}{\mathbf{K}}\nabla \varphi _\omega +Q_\omega )+{\mathbb {E}}_\omega {\mathbb {P}}_\omega \varphi _\omega =F_\omega &{} \text{ in } {\mathbf{D}}_2,\\ \varphi _\omega =0 &{} \text{ on } \partial {\mathbf{D}}_2.\\ \end{array}\right. } \end{aligned}$$

(3.4)

The unique existence of a solution of (3.4) in $H^1({\mathbf{D}}_2)$ is known by Lax–Milgram Theorem [16]. By Theorem 7.26 and Poincaré inequality [16],

(3.5)

By (3.4)–(3.5) and [22], we have

$$\begin{aligned} \Vert \varphi _\omega \Vert _{W^{1,p}({\mathbf{D}}_2\setminus {\mathbf{D}}_1)} \le c{\mathcal {I}}_\omega . \end{aligned}$$

(3.6)

See (3.1) for ${\mathbf{D}}_1$. Let $\widehat{\varphi }$ in $Y_m$ be the solution of

$$\begin{aligned}&{\left\{ \begin{array}{ll} -\nabla \cdot (\omega ^2\widehat{{\mathbf{k}}}\nabla \widehat{\varphi }+\omega ^2({\mathbf{K}}-\widehat{{\mathbf{k}}})\nabla \varphi _\omega +Q_\omega ) =F_\omega -\omega {\mathbb {P}}_\omega \varphi _\omega &{} \text{ in } Y_m,\\ \widehat{\varphi }|_{\partial Y_m}=0, \\ \end{array}\right. } \end{aligned}$$

(3.7)

and $\widehat{\varphi }$ in ${\mathbf{D}}_2\setminus \overline{Y_m}$ be the solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} -\nabla \cdot (\widehat{{\mathbf{K}}}\nabla \widehat{\varphi }+({\mathbf{K}}-\widehat{{\mathbf{K}}})\nabla \varphi _\omega +Q_\omega )= F_\omega -{\mathbb {P}}_\omega \varphi _\omega &{} \text{ in } {\mathbf{D}}_2{\setminus }\overline{Y_m},\\ \widehat{\varphi }|_{\partial ({\mathbf{D}}_2{\setminus }\overline{Y_m})}=0, \\ \end{array}\right. } \end{aligned}$$

(3.8)

where $\widehat{{\mathbf{K}}}, \widehat{{\mathbf{k}}}$ are two constants in the interval $(\min _{Y}{\mathbf{K}}, \max _{Y}{\mathbf{K}})$. By [22] and (3.5),

$$\begin{aligned}& {\left\{ \begin{array}{ll} \omega ^2\widehat{{\mathbf{k}}}\Vert \widehat{\varphi }\Vert _{W^{1,p}(Y_m)}\le c\big (\omega ^{1/2}{\mathcal {I}}_\omega +\omega ^2\Vert ({\mathbf{K}}-\widehat{{\mathbf{k}}})\nabla \varphi _\omega \Vert _{L^{p}(Y_m)}\big ),\\ \widehat{{\mathbf{K}}}\Vert \widehat{\varphi }\Vert _{W^{1,p}({\mathbf{D}}_2{\setminus }\overline{Y_m})} \le c \big ({\mathcal {I}}_\omega +\Vert ({\mathbf{K}}-\widehat{{\mathbf{K}}})\nabla \varphi _\omega \Vert _{L^{p}({\mathbf{D}}_2\setminus \overline{Y_m})} \big ). \end{array}\right. } \end{aligned}$$

(3.9)

If we define $\check{\varphi }\equiv \varphi _\omega -\widehat{\varphi }$ in ${\mathbf{D}}_2$, then (3.4) and (3.7)–(3.8) imply

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta \check{\varphi }=0 &{} \text{ in } {\mathbf{D}}_2{\setminus }\partial Y_m,\\ \lfloor \check{\varphi }\rfloor =0 &{} \text{ on } \partial Y_m,\\ \lfloor \check{{\mathbb {E}}}\nabla \check{\varphi }\rfloor \cdot \vec {{\mathbf{n}}}_y ={\mathcal {F}}/\widehat{{\mathbf{K}}} &{} \text{ on } \partial Y_m,\\ \check{\varphi }=0 &{} \text{ on } \partial {\mathbf{D}}_2, \end{array}\right. } \end{aligned}$$

(3.10)

where $\check{{\mathbb {E}}}\equiv {\left\{ \begin{array}{ll} \omega ^2 \widehat{{\mathbf{k}}}/\widehat{{\mathbf{K}}} &{} \text{ in } Y_m\\ 1 &{} \text{ in } Y_f \end{array}\right. }$ and $\vec {\mathbf{n}}_y$ is the unit vector outward normal to $\partial Y_m$. See (2.1) for (3.10)$_{2,3}$. Since $Q_\omega \in W^{1,p}_0(Y_f)\cap W^{1,p}_0(Y_m)$,

$$\begin{aligned}&{\mathcal {F}}\equiv \big (\omega ^2 \widehat{{\mathbf{k}}}\nabla \widehat{\varphi }_{,-} -\widehat{{\mathbf{K}}}\nabla \widehat{\varphi }_{,+} +\omega ^2({\mathbf{K}}-\widehat{{\mathbf{k}}})\nabla \varphi _{\omega ,-}-({\mathbf{K}}-\widehat{{\mathbf{K}}})\nabla \varphi _{\omega ,+}\big )\cdot \vec {{\mathbf{n}}}_y|_{\partial Y_m}. \end{aligned}$$

By (3.9),

$$\begin{aligned}&\Vert {\mathcal {F}}\Vert _{W^{-\frac{1}{p},p}(\partial Y_m)}\le c\big ({\mathcal {I}}_\omega +\omega ^2\Vert {\mathbf{K}}-\widehat{{\mathbf{k}}}\Vert _{L^{\infty }(Y_m)}\Vert \varphi _\omega \Vert _{W^{1,p}(Y_m)}\nonumber \\&+\Vert {\mathbf{K}}-\widehat{{\mathbf{K}}}\Vert _{L^{\infty }({\mathbf{D}}_2\setminus \overline{Y_m})}\Vert \varphi _\omega \Vert _{W^{1,p}({\mathbf{D}}_2\setminus \overline{Y_m})}\big ). \end{aligned}$$

(3.11)

By Green’s formula, (3.10), and Theorem 6.5.1 [14], we see that

$$\begin{aligned} {\left\{ \begin{array}{ll} \check{\varphi }/2+{\mathcal {L}}_{\partial Y_m}(\check{\varphi }) ={\mathcal {S}}_{\partial Y_m}(\partial _{{\mathbf{n}}_y} \check{\varphi }_{,-}|_{\partial Y_m}) \\ \check{\varphi }/2-{\mathcal {L}}_{\partial Y_m}(\check{\varphi }) =-{\mathcal {S}}_{\partial Y_m}(\partial _{{\mathbf{n}}_y} \check{\varphi }_{,+}|_{\partial Y_m})+{\mathcal {S}}_{\partial {\mathbf{D}}_2}(\partial _{{\mathbf{n}}_y} \check{\varphi }|_{\partial {\mathbf{D}}_2})\\ \end{array}\right. } \quad \text{ on } \partial Y_m, \end{aligned}$$

where $\partial _{{\mathbf{n}}_y}\check{\varphi }|_{\partial {\mathbf{D}}_2}$ is the normal derivative of $\check{\varphi }$ on $\partial {\mathbf{D}}_2$. So we have

$$\begin{aligned} \left( I- \frac{2(1-\check{\omega })}{1+\check{\omega }}{\mathcal {L}}_{\partial Y_m}\right) \check{\varphi }=\frac{2\ {\mathcal {S}}_{\partial {\mathbf{D}}_2}(\partial _{{\mathbf{n}}_y}\check{\varphi }|_{\partial {\mathbf{D}}_2})}{1+\check{\omega }} -\frac{2\ {\mathcal {S}}_{\partial Y_m}({\mathcal {F}})}{(1+\check{\omega })\widehat{{\mathbf{K}}}}\quad \text{ on } \partial Y_m, \end{aligned}$$

(3.12)

where $\check{\omega }\equiv \omega ^2 \widehat{{\mathbf{k}}}/\widehat{{\mathbf{K}}}$. Then (3.6), (3.9), (3.12), and Lemma 3.1 imply

$$\begin{aligned} {\left\{ \begin{array}{ll} \Vert \check{\varphi }\Vert _{W^{1-\frac{1}{p},p}(\partial Y_m)} \le c (\Vert {\mathcal {F}}\Vert _{W^{-\frac{1}{p},p}(\partial Y_m)}+ \Vert \partial _{{\mathbf{n}}_y}\check{\varphi }\Vert _{W^{-\frac{1}{p},p}(\partial {\mathbf{D}}_2)}),\\ \Vert \partial _{{\mathbf{n}}_y}\check{\varphi }\Vert _{W^{-\frac{1}{p},p}(\partial {\mathbf{D}}_2)} \le c\big ({\mathcal {I}}_\omega +\Vert ({\mathbf{K}}-\widehat{{\mathbf{K}}})\nabla \varphi _\omega \Vert _{L^{p}({\mathbf{D}}_2\setminus \overline{Y_m})} \big ). \end{array}\right. } \end{aligned}$$

(3.13)

(3.10)–(3.11) and (3.13) imply

$$\begin{aligned}&\Vert \check{\varphi }\Vert _{W^{1,p}({\mathbf{D}}_2\setminus \overline{Y_m})\cap W^{1,p}(Y_m)} \le c\big ({\mathcal {I}}_\omega +\omega ^2\Vert {\mathbf{K}}-\widehat{{\mathbf{k}}}\Vert _{L^{\infty }(Y_m)}\Vert \varphi _\omega \Vert _{W^{1,p}(Y_m)}\\&+\Vert {\mathbf{K}}-\widehat{{\mathbf{K}}}\Vert _{L^{\infty }({\mathbf{D}}_2\setminus \overline{Y_m})}\Vert \varphi _\omega \Vert _{W^{1,p}({\mathbf{D}}_2\setminus \overline{Y_m})} \big ). \end{aligned}$$

Together with (3.9), we obtain

$$\begin{aligned}&\Vert {\mathbb {E}}_{\omega ^{3/2}}\varphi _\omega \Vert _{W^{1,p}({\mathbf{D}}_2\setminus \overline{Y_m})\cap W^{1,p}(Y_m)} \le c\big ({\mathcal {I}}_\omega +\omega ^{3/2}\Vert {\mathbf{K}}-\widehat{{\mathbf{k}}}\Vert _{L^{\infty }(Y_m)}\Vert \varphi _\omega \Vert _{W^{1,p}(Y_m)} \\& +\Vert {\mathbf{K}}-\widehat{{\mathbf{K}}}\Vert _{L^{\infty }({\mathbf{D}}_2\setminus \overline{Y_m})}\Vert \varphi _\omega \Vert _{W^{1,p}({\mathbf{D}}_2\setminus \overline{Y_m})} \big ). \end{aligned}$$

By A2, we obtain

$$\begin{aligned} \Vert {\mathbb {E}}_{\omega ^{3/2}}\varphi _\omega \Vert _{W^{1,p}({\mathbf{D}}_2\setminus \overline{Y_m})\cap W^{1,p}(Y_m)}\le c{\mathcal {I}}_\omega . \end{aligned}$$

(3.14)

Step 2: Note $W^{1,p}_0(Y_f)$ (resp. $W^{1,p}_0(Y_m)$) is dense in $L^p(Y_f)$ (resp. $L^p(Y_m)$) as well as $L^p(Y)$ is dense in $W^{-1,p}(Y)$. By a limiting argument, we see that if $Q_\omega \in L^p(Y)$ and $F_\omega \in W^{-1,p}(Y)$, the solution of (3.4) still satisfies (3.14).

Step 3: Let $\eta$ be a smooth function satisfying $\eta \in C_0^\infty ({\mathbf{D}}_2)$, $\eta \in [0,1]$, $\eta =1$ in ${\mathbf{D}}_1$, $\Vert \nabla \eta \Vert _{W^{1,\infty }({\mathbf{D}}_2)}\le c$. Multiply (3.2) by $\eta$ to obtain

$$\begin{aligned} {\left\{ \begin{array}{ll} -\nabla \cdot ({\mathbb {E}}_{\omega ^2}{\mathbf{K}}\nabla (U_\omega \eta )-{\mathbf{K}}U_\omega \nabla \eta +Q_\omega \eta ) +{\mathbb {E}}_\omega {\mathbb {P}}_\omega U_\omega \eta \\ =F_\omega \eta -({\mathbf{K}}\nabla U_\omega +Q_\omega )\nabla \eta &{} \text{ in } {\mathbf{D}}_2,\\ U_\omega \eta =0 &{} \text{ on } \partial {\mathbf{D}}_2. \end{array}\right. } \end{aligned}$$

By the result of Step 2, we have

$$\begin{aligned}&\Vert {\mathbb {E}}_{\omega ^{3/2}}U_\omega \Vert _{W^{1,p}({\mathbf{D}}_1\setminus \overline{Y_m})\cap W^{1,p}(Y_m)} \le c(\Vert U_\omega \Vert _{L^p({\mathbf{D}}_2\setminus {\mathbf{D}}_1)} +\Vert {\mathbb {E}}_{1/\sqrt{\omega }}Q_\omega \Vert _{L^{p}(Y)} \nonumber \\& +\Vert {\mathbb {E}}_{1/\sqrt{\omega }}F_\omega \Vert _{W^{-1,p}(Y)} +\Vert {\mathbb {E}}_{1/\omega }Q_\omega \Vert _{L^{2}(Y)}+ \Vert {\mathbb {E}}_{1/\omega }F_\omega \Vert _{H^{-1}(Y)}). \end{aligned}$$

(3.15)

Let $\tilde{\eta }$ be another smooth function satisfying $\tilde{\eta }\in C_0^\infty (Y)$, $\tilde{\eta }\in [0,1]$, $\tilde{\eta }=1$ in ${\mathbf{D}}_2$, $\Vert \nabla \tilde{\eta }\Vert _{W^{1,\infty }(Y)}\le c$. Multiply (3.2) by $\tilde{\eta }$ and then use energy method to get

$$\begin{aligned} \Vert U_\omega \Vert _{L^p({\mathbf{D}}_2\setminus {\mathbf{D}}_1)}\le c(\Vert U_\omega \Vert _{L^2(Y_f)} +\Vert {\mathbb {E}}_{1/\omega }Q_\omega \Vert _{L^{2}(Y)}+ \Vert {\mathbb {E}}_{1/\omega }F_\omega \Vert _{H^{-1}(Y)}). \end{aligned}$$

Together with (3.15), we obtain (3.3).

Modifying the argument for Lemma 3.2 and employing Lemma 3.1, we see

Lemma 3.3

Under A1–A2, $\omega \in (0,1]$, and $p\in (n,\infty )$, any solution of

$$\begin{aligned} -\nabla \cdot ({\mathbb {E}}_{\omega ^2}{\mathbf{K}}\nabla U_\omega +Q_\omega )=F_\omega \qquad \text{ in } Y \end{aligned}$$

satisfies

$$\begin{aligned}& \Vert {\mathbb {E}}_{\omega ^i} U_\omega \Vert _{W^{1,p}({\mathbf{D}}_1\setminus \overline{Y_m})\cap W^{1,p}(Y_m)}\nonumber \\&\qquad \le c(\Vert U_\omega \Vert _{L^2(Y_f)} +\Vert {\mathbb {E}}_{\omega ^{i-2}}Q_\omega \Vert _{L^{p}(Y)} +\Vert {\mathbb {E}}_{\omega ^{i-2}} F_\omega \Vert _{W^{-1,p}(Y)}),\nonumber \\& \Vert U_\omega \Vert _{W^{2,p}({\mathbf{D}}_1\setminus \overline{Y_m})\cap W^{2,p}(Y_m)}\nonumber \\&\qquad \le c(\Vert U_\omega \Vert _{L^2(Y_f)} +\Vert {\mathbb {E}}_{\omega ^{-2}}Q_\omega \Vert _{W^{1,p}(Y_f)\cap W^{1,p}(Y_m)} +\Vert {\mathbb {E}}_{\omega ^{-2}} F_\omega \Vert _{L^{p}(Y)}),\nonumber \\&\Vert {\mathbb {E}}_{\omega ^i} U_\omega \Vert _{C^{2,\alpha }(\overline{{\mathbf{D}}_1\setminus Y_m})\cap C^{2,\alpha }(\overline{Y_m})} \le c(\Vert U_\omega \Vert _{L^2(Y_f)} +\Vert {\mathbb {E}}_{\omega ^{i-2}}Q_\omega \Vert _{C^{1,\alpha }(\overline{Y_f})\cap C^{1,\alpha }(\overline{Y_m})}\nonumber \\&\qquad +\Vert {\mathbb {E}}_{\omega ^{i-2}}F_\omega \Vert _{C^{0,\alpha }(\overline{Y_f})\cap C^{0,\alpha }(\overline{Y_m})}), \end{aligned}$$

(3.16)

where $i\in \{0,1\}$, $\alpha \in (0,1)$, and c is a constant independent of $\omega$. See (3.1) for ${\mathbf{D}}_1$.

Under A1–A2 and $\nu \in (0,1]$, the solution of (2.5) satisfies, by Lemma 3.3,

$$\begin{aligned} \Vert {{\mathbb {X}}}^{(i)}_\nu \Vert _{C^2(\overline{Y_f})\cap C^2(\overline{Y_m})} \le c, \end{aligned}$$

(3.17)

where c is independent of $\nu$. Under A1–A2, the solution of (2.6) satisfies, by Theorem 6.30 [16],

$$\begin{aligned} \Vert {{\mathbb {X}}}^{(i)}_0\Vert _{C^2(\overline{Y_f})} \le c, \end{aligned}$$

(3.18)

where c is a constant. By (3.17) and (3.18), it is not difficult to see that there are positive constants $\omega _0, \ell _5, \ell _6$ such that the symmetric positive definite matrix ${\mathcal {K}}_\nu$ for $\nu \in [0,1]$ in (2.9) satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} \ell _5I\le {\mathcal {K}}_\nu \le \ell _6I,\\ |{\mathcal {K}}_\nu -{\mathcal {K}}_0|\le c\nu \quad \text{ where } c \hbox { is independent of } \nu .\\ \end{array}\right. } \end{aligned}$$

(3.19)

Define a part of boundary of Y by $\partial \widetilde{Y}_n\equiv \{y\in \partial Y| y=(y_1,y_2,\cdots ,y_{n-1}, 0)\}$ and consider the following problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -\nabla \cdot ({\mathbb {E}}_{\omega ^2}{\mathbf{K}}\nabla U_\omega +Q_\omega )+{\mathbb {E}}_{\omega }{\mathbb {P}}_\omega U_\omega =F_\omega &{} \text{ in } Y,\\ U_\omega =U_{b_\omega } &{} \text{ on } \partial \widetilde{Y}_n\text{. }\\ \end{array}\right. } \end{aligned}$$

(3.20)

Let $Y_m\subset {\mathbf{D}}_3\subset Y$ satisfy $\min \{dist(Y_m, \partial {\mathbf{D}}_3), dist({\mathbf{D}}_3,\partial Y\setminus \partial \widetilde{Y}_n)\}>0.$ By an analogous argument as that for Lemma 3.2, we see

Lemma 3.4

Under A1–A2, $\omega \in (0,1]$, $p\in (n,6)$, ${\mathbf{M}}>0$, and ${\mathbb {P}}_\omega (x)\in [0,{\mathbf{M}}]$ for all $x\in Y$, any solution of (3.20) satisfies

$$\begin{aligned}& \Vert {\mathbb {E}}_{\omega ^{3/2}}U_\omega \Vert _{W^{1,p}({\mathbf{D}}_3\setminus \overline{Y_m})\cap W^{1,p}(Y_m)} \le c(\Vert U_{b_\omega }\Vert _{W^{1,p}(Y_f)}+\Vert U_\omega \Vert _{L^2(Y_f)}\nonumber \\& +\Vert {\mathbb {E}}_{1/\omega }Q_\omega \Vert _{L^{2}(Y)} +\Vert {\mathbb {E}}_{1/\omega }F_\omega \Vert _{H^{-1}(Y)} + \Vert {\mathbb {E}}_{1/\sqrt{\omega }}Q_\omega \Vert _{L^p(Y)} +\Vert {\mathbb {E}}_{1/\sqrt{\omega }}F_\omega \Vert _{W^{-1,p}(Y)}), \end{aligned}$$

where c is a constant independent of $\omega$.

Under A1–A2, $\omega \in (0,1]$, and $p\in (n,\infty )$, any solution of (3.20) with ${\mathbb {P}}_\omega =0$ satisfies

$$\begin{aligned}& \Vert {\mathbb {E}}_\omega U_\omega \Vert _{W^{1,p}({\mathbf{D}}_3\setminus \overline{Y_m})\cap W^{1,p}(Y_m)} \le c(\Vert U_{b_\omega }\Vert _{W^{1,p}(Y_f)}+\Vert U_\omega \Vert _{L^2(Y_f)} \nonumber \\& +\Vert {\mathbb {E}}_{1/\omega }Q_\omega \Vert _{L^{p}(Y)} +\Vert {\mathbb {E}}_{1/\omega }F_\omega \Vert _{W^{-1,p}(Y)}), \end{aligned}$$

where c is a constant independent of $\omega$.

One example below shows that the second-order derivatives of the solution of (1.1) may not be bounded uniformly in $\omega ,\epsilon$ in the high-permeability subregion $\Omega _f^\epsilon$.

Remark 3.1

Assume that $B_1(0)\subset \Omega (\omega )$ and $\eta$ is a bell-shaped smooth function satisfying $\eta \in C^\infty _0(B_1(0))$, $\eta \in [0,1]$, and $\eta (x)=1$ in $B_{1/2}(0)$. Employ (2.5), $\eta$, and ${{\mathbb {X}}}^{(1)}_{\omega ,\omega }$ for $\omega \in (0,1)$ to obtain

$$\begin{aligned} {\left\{ \begin{array}{ll} -\nabla \cdot \big (\mathbf{E}_{\omega ^2,\omega }{\mathbf{K}}_\omega (\nabla (\eta {{\mathbb {X}}}_{\omega ,\omega }^{(1)})-{{\mathbb {X}}}_{\omega ,\omega }^{(1)}\nabla \eta +\eta \vec {e}_1)\big )\\ = -\mathbf{E}_{\omega ^2,\omega }{\mathbf{K}}_\omega (\nabla {{\mathbb {X}}}_{\omega ,\omega }^{(1)}+ \vec {e}_1)\nabla \eta &{} \text{ in } \Omega ,\\ \eta {{\mathbb {X}}}_{\omega ,\omega }^{(1)}=0 &{} \text{ on } \partial \Omega ,\\ \end{array}\right. } \end{aligned}$$

where $\vec {e}_1$ is a unit vector in the first coordinate direction. By (3.17), we see that

$$\begin{aligned} \Vert {{\mathbb {X}}}_{\omega ,\omega }^{(1)}\nabla \eta - \eta \vec {e}_1\Vert _{W^{1,\infty }(B_1(0))} +\Vert (\nabla {{\mathbb {X}}}_{\omega ,\omega }^{(1)}+ \vec {e}_1)\nabla \eta \Vert _{L^\infty (B_1(0))} \end{aligned}$$

is bounded uniformly in $\omega$, but $\Vert \eta {{\mathbb {X}}}_{\omega ,\omega }^{(1)}\Vert _{W^{2,p}(B_1(0)\cap \Omega _f^\omega )}$ for $p\in [1,\infty ]$ is not bounded uniformly in $\omega$.

4 Uniform Hölder estimate

A1–A2 are assumed in this section. We shall derive uniform Hölder estimates for non-uniform elliptic equations, that is, Theorem 2.1 and Theorem 2.2. The Hölder estimate in the interior region is considered in Sect. 4.1, and the estimate around the boundary is in Sect. 4.2.

4.1 Interior estimate

For convenience, we let $\overline{B_1(0)}\subset \Omega$.

Lemma 4.1

For any $\delta ,{\mathbf{M}}>0$, there are $\theta _1,\theta _2\in (0,1)$ (depending on $\delta ,{\mathbf{M}},{\mathbf{K}},Y_f$) with $\theta _1<\theta ^2_2$ and there is a $\omega _0\in (0,1)$ (depending on $\theta _1,\theta _2,\delta ,{\mathbf{M}},{\mathbf{K}}$) such that if

$$\begin{aligned} -\nabla \cdot (\mathbf{E}_{\omega ^2,\nu }{\mathbf{K}}_{\nu }\nabla {\mathbb {U}}_{\omega ,\nu } + {\mathbb {Q}}_{\omega ,\nu })+\mathbf{E}_{\omega ,\nu }{\mathbb {P}}_{\omega ,\nu } {\mathbb {U}}_{\omega ,\nu }={\mathbb {F}}_{\omega ,\nu } \qquad \text{ in } B_1(0), \end{aligned}$$

(4.1)

and if

$$\begin{aligned} {\left\{ \begin{array}{ll} \omega ,\nu \in (0, \omega _0),\quad \theta \in [\theta _1,\theta _2],\quad {\mathbb {P}}_{\omega ,\nu }(x)\in [0,{\mathbf{M}}] \text{ for } \text{ all } x\in B_1(0),\\ \max \{\Vert \mathbf{E}_{\omega ,\nu }{\mathbb {U}}_{\omega ,\nu } \Vert _{L^2(B_1(0))}, \Vert \omega _0^{-1}{\mathbb {Q}}_{\omega ,\nu }{\mathcal {X}}_{\Omega _f^\nu } +\omega ^{-1}{\mathbb {Q}}_{\omega ,\nu }{\mathcal {X}}_{\Omega _m^\nu }\Vert _{L^{n+\delta }(B_1(0))},\\ \omega ^{-1}_0\Vert {\mathbb {F}}_{\omega ,\nu }{\mathcal {X}}_{\Omega _f^\nu }+\omega ^{-1}\max \{\omega , \nu \} {\mathbb {F}}_{\omega ,\nu }{\mathcal {X}}_{\Omega _m^\nu } \Vert _{L^{n+\delta }(B_1(0))} \} \le 1,\\ \end{array}\right. } \end{aligned}$$

(4.2)

then

(4.3)

where $\mu \equiv \frac{\delta }{n+\delta }$. See Sect. 2 for $(\Pi _\nu {\mathbb {U}}_{\omega ,\nu }|_{\Omega _f^\nu })_{0,\theta }$, the average value of the extension function $\Pi _\nu {\mathbb {U}}_{\omega ,\nu }|_{\Omega _f^\nu }$ in $B_\theta (0)$.

Proof

Consider the following problem

$$\begin{aligned} -\nabla \cdot ({\mathcal {K}}_{0}\nabla {\mathbb {U}})+{\mathbb {P}}{\mathbb {U}}=0 \qquad \text{ in } B_{3/4}(0), \end{aligned}$$

(4.4)

where ${\mathcal {K}}_{0}$ is defined in (2.9) and ${\mathbb {P}}(x)\in [0,{\mathbf{M}}]$ for $x\in B_{3/4}(0)$. Any solution ${\mathbb {U}}$ of (4.4) satisfies, by Theorem 9.11 [16] and (3.19),

$$\begin{aligned} \Vert {\mathbb {U}}\Vert _{C^{1,\alpha }(\overline{B_{1/2}(0)})}\le c\Vert {\mathbb {U}}\Vert _{L^2(B_{3/4}(0))}, \end{aligned}$$

where $\alpha \in (0,1)$ and c only depends on ${\mathcal {K}}_0,{\mathbf{M}}$. If $\check{\mu }$ satisfies $\mu < \check{\mu } <1$, then, by Theorem 1.2 in page 70 [15],

(4.5)

for $\theta$ (depending on $\delta ,{\mathcal {K}}_0,{\mathbf{M}}$) sufficiently small. Let us fix $\theta _1,\theta _2\in (0,\frac{1}{2})$ so that $\theta _1<\theta _2^2$ and (4.5) holds for any $\theta \in [\theta _1,\theta _2]$.

Now we claim (4.3)$_1$. If not, there is a sequence $\{\theta _{\omega ,\nu },{\mathbb {P}}_{\omega ,\nu },{\mathbb {U}}_{\omega ,\nu }, {\mathbb {Q}}_{\omega ,\nu }, {\mathbb {F}}_{\omega ,\nu }\}$ satisfying (4.1) and, as $\omega ,\nu \rightarrow 0,$

(4.6)

By energy method and A2, there is a constant c independent of $\omega ,\nu$ such that

$$\begin{aligned} \Vert {\mathbb {U}}_{\omega ,\nu } \Vert _{H^1(B_{4/5}(0)\cap \Omega _f^\nu )} +\Vert \omega \nabla {\mathbb {U}}_{\omega ,\nu }, \omega ^{1/2}{\mathbb {P}}_{\omega ,\nu }^{1/2} {\mathbb {U}}_{\omega ,\nu }\Vert _{L^2(B_{4/5}(0)\cap \Omega _m^\nu )} \le c. \end{aligned}$$

By compactness principle and by tracing the proof of Theorem 2.3 [3], we can extract a subsequence (same notation for subsequence) such that

$$\begin{aligned}&{\left\{ \begin{array}{ll} \Pi _\nu {\mathbb {U}}_{\omega ,\nu }|_{\Omega _f^\nu } \rightarrow {\mathbb {U}}&{} \text{ in } L^2(B_{3/4}(0)) \hbox { strongly} \\ \mathbf{E}_{\omega ^2,\nu }{\mathbf{K}}_\nu \nabla {\mathbb {U}}_{\omega ,\nu }\rightarrow {\mathcal {K}}_{0}\nabla {\mathbb {U}}&{} \text{ in } L^2(B_{3/4}(0)) \hbox { weakly}\\ \mathbf{E}_{\omega ,\nu }{\mathbb {P}}_{\omega ,\nu }{\mathbb {U}}_{\omega ,\nu }\rightarrow {\mathbb {P}}{\mathbb {U}}&{} \text{ in } L^2(B_{3/4}(0)) \hbox { weakly}\\ \end{array}\right. }\quad \text{ as } \omega ,\nu \rightarrow 0, \end{aligned}$$

(4.7)

where ${\mathbb {P}}(x)\in [0, {\mathbf{M}}]$ for all $x\in B_{3/4}(0)$, ${\mathcal {K}}_{0}$ is a constant symmetric positive definite matrix, $\ell _5\le {\mathcal {K}}_{0}\le \ell _6$, and $\ell _5,\ell _6$ are positive constants (see (2.9) and (3.19)). The ${\mathbb {U}}$ in (4.7) satisfies (4.4). Equations (4.5)–(4.7) then imply

If $\theta _2$ is small enough, then we get contradiction. Therefore, we prove (4.3)$_1$.

Set $\zeta \equiv \theta ^{-\mu }(\Pi _\nu {\mathbb {U}}_{\omega ,\nu }|_{\Omega _f^\nu }-(\Pi _\nu {\mathbb {U}}_{\omega ,\nu }|_{\Omega _f^\nu })_{0,\theta })$ and $\eta \equiv \theta ^{-\mu }({\mathbb {U}}_{\omega ,\nu }-(\Pi _\nu {\mathbb {U}}_{\omega ,\nu }|_{\Omega _f^\nu })_{0,\theta })$. (4.1) implies, for any smooth function $\varphi$ with support in $\nu (Y_m+j)\subset B_\theta (0)\cap \Omega _m^\nu$ for some $j\in {\mathbb {Z}}^n$,

$$\begin{aligned}&\omega ^2\int _{\nu (Y_m+j)} (\eta -\zeta ) \nabla \cdot ({\mathbf{K}}_\nu \nabla \varphi )-\omega \int _{\nu (Y_m+j)}{\mathbb {P}}_{\omega ,\nu }(\eta -\zeta )\varphi \nonumber \\&=\int _{\nu (Y_m+j)} \big (\omega ^2{\mathbf{K}}_\nu \nabla \zeta +\theta ^{-\mu } {\mathbb {Q}}_{\omega ,\nu }\big )\nabla \varphi +\theta ^{-\mu } (\omega {\mathbb {P}}_{\omega ,\nu }\Pi _\nu {\mathbb {U}}_{\omega ,\nu }|_{\Omega _f^\nu } - {\mathbb {F}}_{\omega ,\nu }) \varphi . \end{aligned}$$

(4.8)

If $\varphi$ is the solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} -\omega ^2\nabla \cdot ({\mathbf{K}}_\nu \nabla \varphi )+\omega {\mathbb {P}}_{\omega ,\nu }\varphi =\zeta -\eta &{} \text{ in } \nu (Y_m+j),\\ \varphi =0 &{} \text{ on } \nu (\partial Y_m+j),\\ \end{array}\right. } \end{aligned}$$

(4.9)

then $c_1\nu ^{-1}\omega ^2\Vert \varphi \Vert _{L^2(\nu (Y_m+j))}\le \omega ^2\Vert \nabla \varphi \Vert _{L^2(\nu (Y_m+j))}\le c_2\nu \Vert \eta -\zeta \Vert _{L^2(\nu (Y_m+j))},$ where $c_1, c_2$ are independent of $\nu$. (4.8) and (4.9) imply

$$\begin{aligned}&\int _{\nu (Y_m+j)}\omega ^2 |\eta -\zeta |^2 \le c\int _{\nu (Y_m+j)}\omega ^2\nu ^2|\nabla \zeta |^2\nonumber \\&+c\int _{\nu (Y_m+j)}\nu ^2\theta ^{-2\mu }(|\omega ^{-1}{\mathbb {Q}}_{\omega ,\nu }|^2 +\nu ^2{\mathbb {P}}^2_{\omega ,\nu }\big |\Pi _\nu {\mathbb {U}}_{\omega ,\nu }|_{\Omega _f^\nu }\big |^2+|\nu \omega ^{-1}{\mathbb {F}}_{\omega ,\nu }|^2). \end{aligned}$$

(4.10)

Summing (4.10) over all $\nu (Y_m+j)\subset B_\theta (0)\cap \Omega _m^\nu$ for $j\in {\mathbb {Z}}^n$, we obtain (4.3)$_2$ if $\omega _0$ is small enough. $\square$

Lemma 4.2

For any $\delta \in (0,3)$ and ${\mathbf{M}}>0$, there are $\theta _1,\theta _2\in (0,1)$ (depending on $\delta ,{\mathbf{M}},{\mathbf{K}},Y_f$) with $\theta _1<\theta ^2_2$ and there is a $\omega _0>0$ (depending on $\theta _1,\theta _2,\delta ,{\mathbf{M}},{\mathbf{K}}$) such that if

$$\begin{aligned} -\nabla \cdot (\mathbf{E}_{\omega ^2,\epsilon }{\mathbf{K}}_{\epsilon }\nabla U_{\omega ,\epsilon } + Q_{\omega ,\epsilon })+\gamma \mathbf{E}_{\omega ,\epsilon }{\mathbf{P}}_{\omega ,\epsilon } U_{\omega ,\epsilon }=F_{\omega ,\epsilon } \qquad \text{ in } B_1(0), \end{aligned}$$

(4.11)

and if $\omega ,\epsilon \in (0, \omega _0)$, $\theta \in [\theta _1,\theta _2]$, $\gamma \in [0, 1]$, ${\mathbf{P}}_{\omega ,\epsilon }(x)\in [0,{\mathbf{M}}]$ for all $x\in B_1(0)$, and k satisfying $\epsilon /\theta ^k\le \omega _0$, then

(4.12)

where $\mu \equiv \frac{\delta }{n+\delta }$ and

Proof

Let c denote a constant independent of $\omega ,\epsilon ,\gamma$. This proof is done by induction. For $k=1$, we define ${\mathbb {U}}_{\omega ,\epsilon }\equiv \frac{U_{\omega ,\epsilon }}{J_{\omega ,\epsilon }}$, ${\mathbb {Q}}_{\omega ,\epsilon }\equiv \frac{Q_{\omega ,\epsilon }}{J_{\omega ,\epsilon }}$, ${\mathbb {F}}_{\omega ,\epsilon }\equiv \frac{F_{\omega ,\epsilon }}{J_{\omega ,\epsilon }}$, ${\mathbb {P}}_{\omega ,\epsilon }\equiv \gamma {\mathbf{P}}_{\omega ,\epsilon }$. Then they satisfy (4.1) and (4.2) with $\nu =\epsilon$. By Lemma 4.1,

This implies (4.12) for $k=1$. By energy method and A2, any solution of (4.11) satisfies

$$\begin{aligned}&\Vert U_{\omega ,\epsilon }\Vert _{H^1(B_{4/5}(0)\cap \Omega _f^\epsilon )} \le c(\Vert \mathbf{E}_{\omega ,\epsilon } U_{\omega ,\epsilon } \Vert _{L^2(B_1(0))}+ \Vert \mathbf{E}_{1/\omega ,\epsilon } Q_{\omega ,\epsilon } \Vert _{L^2(B_1(0))}\nonumber \\&\qquad +\Vert F_{\omega ,\epsilon } \Vert _{L^2(B_1(0))}+\epsilon \omega ^{-1}\Vert F_{\omega ,\epsilon } \Vert _{L^2(B_1(0)\cap \Omega ^\epsilon _m)})\equiv c\hat{\mathcal {I}}. \end{aligned}$$

By Theorem 7.26 [16] and Remark 2.1,

$$\begin{aligned} \big \Vert \Pi _\epsilon U_{\omega ,\epsilon }|_{\Omega _f^\epsilon }\big \Vert _{L^{\frac{2n}{n-2}}(B_{4/5}(0))}\le c\hat{\mathcal {I}}. \end{aligned}$$

(4.13)

Suppose (4.12) holds for some k satisfying $\epsilon /\theta ^k\le \omega _0$, we define, in $B_1(0)\setminus \partial \Omega _m^\epsilon /\theta ^k$,

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathbb {U}}_{\omega ,\epsilon /\theta ^k}(x)\equiv J_{\omega ,\epsilon }^{-1}\theta ^{-k\mu } \big (U_{\omega ,\epsilon }(\theta ^kx)-(\Pi _\epsilon U_{\omega ,\epsilon }|_{\Omega _f^\epsilon })_{0,\theta ^k}\big ),\\ {\mathbb {Q}}_{\omega ,\epsilon /\theta ^k}(x)\equiv J_{\omega ,\epsilon }^{-1}\theta ^{k(1-\mu )} Q_{\omega ,\epsilon }(\theta ^kx),\\ {\mathbb {F}}_{\omega ,\epsilon /\theta ^k}(x)\equiv J_{\omega ,\epsilon }^{-1}\theta ^{k(2-\mu )} \big ( F_{\omega ,\epsilon }(\theta ^kx)-\gamma \mathbf{E}_{\omega ,\epsilon /\theta ^k}(x){\mathbf{P}}_{\omega ,\epsilon }(\theta ^kx)(\Pi _\epsilon U_{\omega ,\epsilon }|_{\Omega _f^\epsilon })_{0,\theta ^k}\big ),\\ {\mathbb {P}}_{\omega ,\epsilon /\theta ^k}(x)\equiv \theta ^{2k}\gamma {\mathbf{P}}_{\omega ,\epsilon }(\theta ^kx). \end{array}\right. } \end{aligned}$$

Then they satisfy

$$\begin{aligned} -\nabla \cdot (\mathbf{E}_{\omega ^2,\epsilon /\theta ^k}{\mathbf{K}}_{\epsilon /\theta ^k}\nabla {\mathbb {U}}_{\omega ,\epsilon /\theta ^k} + {\mathbb {Q}}_{\omega ,\epsilon /\theta ^k})+\mathbf{E}_{\omega ,\epsilon /\theta ^k}{\mathbb {P}}_{\omega ,\epsilon /\theta ^k}{\mathbb {U}}_{\omega ,\epsilon /\theta ^k}={\mathbb {F}}_{\omega ,\epsilon /\theta ^k} \quad \text{ in } B_1(0). \end{aligned}$$

By triangle inequality,

$$\begin{aligned} {\left\{ \begin{array}{ll} \Vert {\mathbb {F}}_{\omega ,\epsilon /\theta ^k} \Vert _{L^{n+\delta }(B_1(0)\cap \Omega _f^{\epsilon }/\theta ^k)}\\ \le \frac{\theta ^k}{J_{\omega ,\epsilon }} \Vert F_{\omega ,\epsilon } \Vert _{L^{n+\delta }(B_{\theta ^k}(0)\cap \Omega _f^\epsilon )} +\frac{\theta ^{k(3-\mu -n/2)}{\mathbf{M}}}{J_{\omega ,\epsilon }}\big \Vert \Pi _\epsilon U_{\omega ,\epsilon }|_{\Omega _f^\epsilon }\big \Vert _{L^{2n/(n-2)}(B_{\theta ^k}(0))},\\ \Vert {\mathbb {F}}_{\omega ,\epsilon /\theta ^k}\Vert _{L^{n+\delta }(B_1(0)\cap \Omega _m^{\epsilon }/\theta ^k)}\\ \le \frac{\theta ^k}{J_{\omega ,\epsilon }} \Vert F_{\omega ,\epsilon } \Vert _{L^{n+\delta }(B_{\theta ^k}(0)\cap \Omega _m^\epsilon )} +\frac{\omega \theta ^{k(3-\mu -n/2)}{\mathbf{M}}}{J_{\omega ,\epsilon }}\big \Vert \Pi _\epsilon U_{\omega ,\epsilon }|_{\Omega _f^\epsilon }\big \Vert _{L^{2n/(n-2)}(B_{\theta ^k}(0))}. \end{array}\right. } \end{aligned}$$

By induction, (4.13), and small $\theta$,

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathbb {P}}_{\omega ,\epsilon /\theta ^k}(x)\in [0,{\mathbf{M}}]\quad \text{ for } \text{ all } x\in B_1(0),\\ \Vert \mathbf{E}_{\omega ,\epsilon /\theta ^k}{\mathbb {U}}_{\omega ,\epsilon /\theta ^k}\Vert _{L^2(B_1(0))} \le 1, \\ \Vert \omega _0^{-1} {\mathbb {Q}}_{\omega ,\epsilon /\theta ^k}{\mathcal {X}}_{\Omega _f^\epsilon /\theta ^k}+\omega ^{-1}{\mathbb {Q}}_{\omega ,\epsilon /\theta ^k} {\mathcal {X}}_{\Omega _m^\epsilon /\theta ^k}\Vert _{L^{n+\delta }(B_1(0))}\le 1, \\ \omega _0^{-1}\Vert {\mathbb {F}}_{\omega ,\epsilon /\theta ^k}{\mathcal {X}}_{\Omega _f^\epsilon /\theta ^k}+\omega ^{-1}\max \{\omega ,\epsilon /\theta ^k \} {\mathbb {F}}_{\omega ,\epsilon /\theta ^k}{\mathcal {X}}_{\Omega _m^\epsilon /\theta ^k}\Vert _{L^{n+\delta }(B_1(0))}\le 1. \end{array}\right. } \end{aligned}$$

By Lemma 4.1 (take $\nu =\epsilon /\theta ^k$), we obtain

(4.14)

Note, by Remark 2.1,

(4.15)

Equations (4.14)–(4.15) imply the inequality (4.12) for $k+1$ case. $\square$

Lemma 4.3

For any $\delta \in (0,3)$ and ${\mathbf{M}}>0$, there is a $\omega _*\in (0,1)$ (depending on $\delta ,{\mathbf{M}},{\mathbf{K}},Y_f$) such that if $\omega ,\epsilon \in (0,\omega _*)$, $\gamma \in [0,1]$, and ${\mathbf{P}}_{\omega ,\epsilon }(x)\in [0,{\mathbf{M}}]$ for all $x\in B_1(0)$, then any solution of (4.11) satisfies

$$\begin{aligned}& [U_{\omega ,\epsilon }]_{C^{0,\mu }(\overline{B_{1/2}(0)\cap \Omega _f^\epsilon })} +\mathop {\mathop {\sup }_{j\in {\mathbb {Z}}^n}}\limits _{\epsilon (Y_m+j)\subset B_{1/2}(0)\cap \Omega _m^\epsilon } \omega ^{3/2}[U_{\omega ,\epsilon }]_{C^{0,\mu }(\epsilon (\overline{Y_m}+j))} \le c J_{\omega ,\epsilon }, \end{aligned}$$

(4.16)

where c is a constant independent of $\omega ,\epsilon ,\gamma$. See Lemma 4.2 for $\mu , J_{\omega ,\epsilon }$.

Proof

Let $\theta _1,\theta _2,\omega _0$ be same as those in Lemma 4.2, define $\omega _*\equiv \omega _0\theta _2/2$, and let $\omega ,\epsilon \le \omega _*$. Denote by c a constant independent of $\omega ,\epsilon ,\gamma$. Because of $\theta _1<\theta _2^2$, for any $r\in [ \epsilon /\omega _0,\theta _2]$, there are $\theta \in [\theta _1,\theta _2]$ and $k\in \mathbb {N}$ satisfying $r=\theta ^k$. Lemma 4.2 implies, for any $r\in [ \epsilon /\omega _0,\theta _2]$,

(4.17)

Now we define, in $B_{2/\omega _0}(0)\setminus \partial \Omega _m^\epsilon /\epsilon$,

$$\begin{aligned}&{\left\{ \begin{array}{ll} {\mathbb {U}}_{\omega ,1}(x)\equiv J_{\omega ,\epsilon }^{-1}\epsilon ^{-\mu } \big (U_{\omega ,\epsilon }(\epsilon x)-(\Pi _\epsilon U_{\omega ,\epsilon }|_{\Omega _f^\epsilon })_{0,{2\epsilon }/{\omega _0}}\big ),\\ {\mathbb {Q}}_{\omega ,1}(x)\equiv J_{\omega ,\epsilon }^{-1} \epsilon ^{1-\mu } Q_{\omega ,\epsilon }(\epsilon x),\\ {\mathbb {F}}_{\omega ,1}(x)\equiv J_{\omega ,\epsilon }^{-1}\epsilon ^{2-\mu }\big (F_{\omega ,\epsilon }(\epsilon x) -\gamma \mathbf{E}_{\omega ,1}(x){\mathbf{P}}_{\omega ,\epsilon }(\epsilon x) (\Pi _\epsilon U_{\omega ,\epsilon }|_{\Omega _f^\epsilon })_{0,{2\epsilon }/{\omega _0}}\big ), \\ {\mathbb {P}}_{\omega ,1}\equiv \epsilon ^2\gamma {\mathbf{P}}_{\omega ,\epsilon }(\epsilon x). \end{array}\right. } \end{aligned}$$

Then they satisfy

$$\begin{aligned}& -\nabla \cdot (\mathbf{E}_{\omega ^2,1}{\mathbf{K}}\nabla {\mathbb {U}}_{\omega ,1} +{\mathbb {Q}}_{\omega ,1})+\mathbf{E}_{\omega ,1}{\mathbb {P}}_{\omega ,1}{\mathbb {U}}_{\omega ,1} ={\mathbb {F}}_{\omega ,1} \quad \text{ in } B_{\frac{2}{\omega _0}}(0). \end{aligned}$$

Take $r= \frac{2\epsilon }{\omega _0}$ in (4.17) to get

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathbb {P}}_{\omega ,1}\in [0,{\mathbf{M}}] \quad \text{ for } \text{ all } x\in B_{2/\omega _0}(0), \\ \Vert \mathbf{E}_{\omega ,1}{\mathbb {U}}_{\omega ,1}\Vert _{L^2(B_{2/\omega _0}(0))} +\Vert \mathbf{E}_{1/\omega ,1}{\mathbb {Q}}_{\omega ,1}, \mathbf{E}_{1/\omega ,1}{\mathbb {F}}_{\omega ,1}\Vert _{L^{n+\delta }(B_{2/\omega _0}(0))} \le c.\\ \end{array}\right. } \end{aligned}$$

By (3.3) of Lemma 3.2,

$$\begin{aligned}{}[{\mathbb {U}}_{\omega ,1}]_{C^{0,\mu }(\overline{B_{1/\omega _0}(0)\cap \Omega _f^\epsilon /\epsilon })} +\omega ^{3/2}[{\mathbb {U}}_{\omega ,1}]_{C^{0,\mu }(\overline{B_{1/\omega _0}(0)\cap \Omega _m^\epsilon /\epsilon })} \le c. \end{aligned}$$

(4.18)

(4.18) implies that (4.17)$_1$ also holds for $r\le \epsilon /\omega _0$. So (4.17)$_1$ holds for $r\le \theta _2$. Next we shift the origin of the coordinate system to any point $z\in B_{1/2}(0)$ and repeat above argument to see that (4.17)$_1$ with 0 replaced by any $z\in B_{1/2}(0)$ also holds for $r\in (0,\theta _2)$. Together with Theorem 1.2 in page 70 [15], we obtain the Hölder estimate of $\Pi _\epsilon U_{\omega ,\epsilon }$ in $B_{1/2}(0)$. Hölder estimate of $U_{\omega ,\epsilon }$ in $\epsilon (\overline{Y_m}+j)\subset B_{1/2}(0)\cap \overline{\Omega ^\epsilon _m}$ is from (4.18). So (4.16) is proved. $\square$

Remark 4.1

Let $\omega _*$ be same as that in Lemma 4.3. By (3.3) of Lemma 3.2 with $p=n+\delta$, we know that if $\delta \in (0,3)$, ${\mathbf{M}}>0$, $\epsilon \in [\omega _*,1]$, $\omega \in (0,\omega _*)$, $\gamma \in [0,1]$, and ${\mathbf{P}}_{\omega ,\epsilon }(x)\in [0,{\mathbf{M}}]$ for all $x\in B_1(0)$, any solution of (4.11) satisfies (4.16). Together with Lemma 4.3, we know that any solution of (4.11) satisfies (4.16) if $\delta \in (0,3)$, ${\mathbf{M}}>0$, $\epsilon \in (0,1)$, $\omega \in (0,\omega _*)$, $\gamma \in [0,1]$, and ${\mathbf{P}}_{\omega ,\epsilon }(x)\in [0,{\mathbf{M}}]$ for all $x\in B_1(0)$.

Let us consider the solutions of (4.11) with ${\mathbf{P}}_{\omega ,\epsilon }=0$. By tracing the arguments of Lemma 4.2, Lemma 4.3, and Remark 4.1 as well as employing (3.16) of Lemma 3.3, then we have

Lemma 4.4

For any $\delta >0$, there is a $\omega _*>0$ (depending on $\delta ,{\mathbf{K}},Y_f$) such that if $\omega \in (0,\omega _*)$ and $\epsilon \in (0,1)$, then any solution of (4.11) with ${\mathbf{P}}_{\omega ,\epsilon }=0$ satisfies

$$\begin{aligned} {[}U_{\omega ,\epsilon }]_{C^{0,\mu }(\overline{B_{1/2}(0)\cap \Omega _f^\epsilon })} +\mathop {\mathop {\sup }_{j\in {\mathbb {Z}}^n}}\limits _{\epsilon (Y_m+j)\subset B_{1/2}(0)\cap \Omega _m^\epsilon } \omega [U_{\omega ,\epsilon }]_{C^{0,\mu }(\epsilon (\overline{Y_m}+j))} \le c J_{\omega ,\epsilon }, \end{aligned}$$

where c is a constant independent of $\omega ,\epsilon$. See Lemma 4.2 for $\mu , J_{\omega ,\epsilon }$.

4.2 Boundary estimate

In this subsection, we assume $0\in \partial \Omega$. By A1, there is a $C^{2,1}$ function $\rho :{\mathbb {R}}^{n-1}\rightarrow {\mathbb {R}}$ satisfying

$$\begin{aligned} {\left\{ \begin{array}{ll} \rho (0)=|\nabla \rho (0)|=0,\\ B_1(0)\cap \Omega /t= B_1(0)\cap \{(x',x_n)\in {\mathbb {R}}^n |\ t x_n > \rho (t x') \} &{} \text{ if } t\in (0,1].\\ \end{array}\right. } \end{aligned}$$

If $t=0$, we define $B_1(0)\cap \Omega /t\equiv B_1(0)\cap \{(x',x_n)\in {\mathbb {R}}^n|\ x_n > 0 \}$. Set

$$\begin{aligned} {\mathcal {E}}_{\nu ,\epsilon ,t}\equiv {\left\{ \begin{array}{ll} 1 &{} \text{ in } \Omega _f^\epsilon /t\\ \nu &{} \text{ in } \Omega _m^\epsilon /t\\ \end{array}\right. }\qquad \text{ for } t\in (0,1]. \end{aligned}$$

(4.19)

Lemma 4.5

For any $\delta ,{\mathbf{M}}>0$, there are $\tilde{\theta }_1,\tilde{\theta }_2\in (0,1)$ (depending on $\delta ,{\mathbf{M}},{\mathbf{K}},Y_f$, $\Omega$) satisfying $\tilde{\theta }_1<\tilde{\theta }_2^2$ and there is a $\tilde{\omega }_0>0$ (depending on $\tilde{\theta }_1,\tilde{\theta }_2,\delta ,{\mathbf{M}},{\mathbf{K}},\Omega$) satisfying $\tilde{\omega }_0 < \omega _0$ ($\omega _0$ is that in Lemma 4.1) such that if

$$\begin{aligned}&{\left\{ \begin{array}{ll} -\nabla \cdot ({\mathcal {E}}_{\omega ^2,\epsilon ,s}{\mathbf{K}}_{\epsilon /s}\nabla {\mathbb {U}}_{\omega ,\epsilon ,s} + {\mathbb {Q}}_{\omega ,\epsilon ,s})\\ +{\mathcal {E}}_{\omega ,\epsilon ,s}{\mathbb {P}}_{\omega ,\epsilon ,s} {\mathbb {U}}_{\omega ,\epsilon ,s} ={\mathbb {F}}_{\omega ,\epsilon ,s} &{} \text{ in } B_1(0)\cap \Omega /s,\\ {\mathbb {U}}_{\omega ,\epsilon ,s}=0 &{} \text{ on } B_1(0)\cap \partial \Omega /s,\\ \end{array}\right. } \end{aligned}$$

(4.20)

and if

$$\begin{aligned} {\left\{ \begin{array}{ll} \omega ,\frac{\epsilon }{s} \in (0, \tilde{\omega }_0),\ s\in (0,1],\ \tilde{\theta }\in [\tilde{\theta }_1,\tilde{\theta }_2], \ {\mathbb {P}}_{\omega ,\epsilon ,s}(x)\in [0,{\mathbf{M}}] \text{ for } x\in B_1(0)\cap \Omega /s,\\ \max \{ \Vert {\mathcal {E}}_{\omega ,\epsilon ,s} {\mathbb {U}}_{\omega ,\epsilon ,s} \Vert _{L^2(B_1(0)\cap \Omega /s)}, \Vert \frac{1}{\tilde{\omega }_0}{\mathbb {Q}}_{\omega ,\epsilon ,s}{\mathcal {X}}_{\Omega _f^\epsilon /s}+\frac{1}{\omega } {\mathbb {Q}}_{\omega ,\epsilon ,s}{\mathcal {X}}_{\Omega _m^\epsilon /s}\Vert _{L^{n+\delta }(B_1(0))}, \\ \frac{1}{\tilde{\omega }_0}\Vert {\mathbb {F}}_{\omega ,\epsilon ,s}{\mathcal {X}}_{\Omega _f^\epsilon /s}+\frac{1}{\omega } \max \{\omega ,\frac{\epsilon }{s} \}{\mathbb {F}}_{\omega ,\epsilon ,s}{\mathcal {X}}_{\Omega _m^\epsilon /s}\Vert _{L^{n+\delta }(B_1(0))} \}\le 1, \end{array}\right. } \end{aligned}$$

then

(4.21)

where $\mu \equiv \frac{\delta }{n+\delta }$.

Proof

Consider the following problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -\nabla \cdot ({\mathcal {K}}_{0}\nabla {\mathbb {U}})+{\mathbb {P}}{\mathbb {U}}=0 &{} \text{ in } B_{3/4}(0)\cap \Omega /t, \\ {\mathbb {U}}=0 &{} \text{ on } B_{3/4}(0)\cap \partial \Omega /t,\\ \end{array}\right. } \end{aligned}$$

(4.22)

where $t\in [0,1]$, ${\mathbb {P}}(x)\in [0,{\mathbf{M}}]$ for $x\in B_{3/4}(0)\cap \Omega /t$, and ${\mathcal {K}}_0$ is defined in (2.9). Any solution ${\mathbb {U}}$ of (4.22) satisfies, by Theorem 9.13 [16] and (3.19),

$$\begin{aligned} \Vert {\mathbb {U}}\Vert _{C^{1,\alpha }(\overline{B_{1/2}(0)\cap \Omega /t})}\le c\Vert {\mathbb {U}}\Vert _{L^2(B_{3/4}(0)\cap \Omega /t)}, \end{aligned}$$

(4.23)

where $\alpha \in (0,1)$ and c is a constant depending on ${\mathbf{M}},{\mathcal {K}}_0,\Omega$ but independent of t. If $\check{\mu }$ satisfies $\mu < \check{\mu } <1$, by (4.23),

(4.24)

for small $\tilde{\theta }$ (depending on $\delta ,{\mathbf{M}},{\mathcal {K}}_0,\Omega$). Fix $\tilde{\theta }_1,\tilde{\theta }_2\in (0,\frac{1}{2})$ such that $\tilde{\theta }_1<\tilde{\theta }_2^2$ and (4.24) holds for any $\tilde{\theta }\in [\tilde{\theta }_1,\tilde{\theta }_2]$.

We claim (4.21)$_1$. If not, there is a sequence $\{s_{\omega ,\epsilon },\tilde{\theta }_{\omega ,\epsilon },{\mathbb {P}}_{\omega ,\epsilon ,s_{\omega ,\epsilon }}, {\mathbb {U}}_{\omega ,\epsilon ,s_{\omega ,\epsilon }}$, ${\mathbb {Q}}_{\omega ,\epsilon ,s_{\omega ,\epsilon }}$, ${\mathbb {F}}_{\omega ,\epsilon ,s_{\omega ,\epsilon }}\}$ satisfying (4.20) and, as $\omega ,\epsilon \rightarrow 0$,

(4.25)

By energy method and A2, there is a constant c independent of $\omega ,\epsilon ,s_{\omega ,\epsilon }$ such that

$$\begin{aligned}&\Vert {\mathbb {U}}_{\omega ,\epsilon ,s_{\omega ,\epsilon }}\Vert _{H^1(B_{4/5}(0)\cap \Omega _f^\epsilon /s_{\omega ,\epsilon })}\\&\qquad +\Vert \omega \nabla {\mathbb {U}}_{\omega ,\epsilon ,s_{\omega ,\epsilon }}, \omega ^{1/2}{\mathbb {P}}_{\omega ,\epsilon ,s_{\omega ,\epsilon }}^{1/2} {\mathbb {U}}_{\omega ,\epsilon ,s_{\omega ,\epsilon }}\Vert _{L^2(B_{4/5}(0)\cap \Omega _m^\epsilon /s_{\omega ,\epsilon })} \le c. \end{aligned}$$

By compactness principle and by tracing the proof of Theorem 2.3 [3], we can extract a subsequence (same notation for subsequence) such that, as $\omega ,\epsilon /s_{\omega ,\epsilon } \rightarrow 0$,

$$\begin{aligned}&{\left\{ \begin{array}{ll} \Pi _{\epsilon /s_{\omega ,\epsilon }} {\mathbb {U}}_{\omega ,\epsilon ,s_{\omega ,\epsilon }}|_{\Omega _f^\epsilon /s_{\omega ,\epsilon }} \rightarrow {\mathbb {U}}&{} \text{ in } L^2(B_{3/4}(0)\cap \Omega /s_*) \hbox { strongly,} \\ {\mathcal {E}}_{\omega ^2,\epsilon ,s_{\omega ,\epsilon }}{\mathbf{K}}_{\epsilon /s_{\omega ,\epsilon }} \nabla {\mathbb {U}}_{\omega ,\epsilon ,s_{\omega ,\epsilon }} \rightarrow {\mathcal {K}}_{0}\nabla {\mathbb {U}}&{} \text{ in } L^2(B_{3/4}(0)\cap \Omega /s_*) \hbox { weakly,}\\ {\mathcal {E}}_{\omega ,\epsilon ,s_{\omega ,\epsilon }}{\mathbb {P}}_{\omega ,\epsilon ,s_{\omega ,\epsilon }}{\mathbb {U}}_{\omega ,\epsilon ,s_{\omega ,\epsilon }} \rightarrow {\mathbb {P}}{\mathbb {U}}&{} \text{ in } L^2(B_{3/4}(0)\cap \Omega /s_*) \hbox { weakly,}\\ \end{array}\right. } \end{aligned}$$

(4.26)

where ${\mathcal {K}}_{0}$ is a constant symmetric positive definite matrix, $\ell _5\le {\mathcal {K}}_{0}\le \ell _6$ (see (2.9) and (3.19)), and ${\mathbb {P}}(x)\in [0,{\mathbf{M}}]$ for $x\in B_{3/4}(0)\cap \Omega /s_*$. In (4.26), function ${\mathbb {U}}$ satisfies (4.22) with $t=s_*$. By (4.24)–(4.25), we conclude

(4.27)

But (4.27) is impossible if we take $\tilde{\theta }_2$ small enough. Therefore, there is a $\tilde{\omega }_0$ such that (4.21)$_1$ holds for $\omega ,\epsilon /s\le \tilde{\omega }_0$. Clearly, $\tilde{\omega }_0$ can be chosen so that $\tilde{\omega }_0< \omega _0$ (see Lemma 4.1 for $\omega _0$). The proof of (4.21)$_2$ is similar to that of (4.3)$_2$, so we skip it. $\square$

Lemma 4.6

For any $\delta ,{\mathbf{M}}>0$, there are $\tilde{\theta }_1,\tilde{\theta }_2\in (0,1)$ (depending on $\delta ,{\mathbf{M}},{\mathbf{K}},Y_f$, $\Omega$) satisfying $\tilde{\theta }_1<\tilde{\theta }_2^2$ and there is a $\tilde{\omega }_0>0$ (depending on $\tilde{\theta }_1,\tilde{\theta }_2,\delta ,{\mathbf{M}},{\mathbf{K}},\Omega$) satisfying $\tilde{\omega }_0 < \omega _0$ ($\omega _0$ is that in Lemma 4.2) such that if

$$\begin{aligned} {\left\{ \begin{array}{ll} -\nabla \cdot (\mathbf{E}_{\omega ^2,\epsilon }{\mathbf{K}}_{\epsilon }\nabla U_{\omega ,\epsilon } + Q_{\omega ,\epsilon })+\mathbf{E}_{\omega ,\epsilon }{\mathbf{P}}_{\omega ,\epsilon } U_{\omega ,\epsilon }=F_{\omega ,\epsilon } &{} \text{ in } B_1(0)\cap \Omega ,\\ U_{\omega ,\epsilon }=0 &{} \text{ on } B_1(0)\cap \partial \Omega ,\\ \end{array}\right. } \end{aligned}$$

(4.28)

and if $\omega ,\epsilon \in (0,\tilde{\omega }_0)$, $\tilde{\theta }\in [\tilde{\theta }_1,\tilde{\theta }_2]$, ${\mathbf{P}}_{\omega ,\epsilon }(x)\in [0,{\mathbf{M}}]$ for $x\in B_1(0)\cap \Omega$, and k satisfying $\epsilon /{\tilde{\theta }^k}\le \tilde{\omega }_0$, then

(4.29)

where $\tilde{J}_{\omega ,\epsilon }\equiv \Vert \mathbf{E}_{\omega ,\epsilon } U_{\omega ,\epsilon }\Vert _{L^{2}(B_1(0)\cap \Omega )} +\omega ^{-1}\Vert Q_{\omega ,\epsilon },\frac{1}{\tilde{\omega }_0}\max \{\omega ,\epsilon \} F_{\omega ,\epsilon }\Vert _{L^{n+\delta }(B_1(0)\cap \Omega _m^\epsilon )} +\frac{1}{\tilde{\omega }_0}\Vert Q_{\omega ,\epsilon },F_{\omega ,\epsilon } \Vert _{L^{n+\delta }(B_1(0)\cap \Omega ^\epsilon _f)}$ and $\mu \equiv \frac{\delta }{n+\delta }$.

Proof

The proof is similar to that of Lemma 4.2 and is done by induction on k. For $k=1$, (4.29) is deduced from Lemma 4.5 with $s=1$. Suppose (4.29) holds for some k with $\epsilon /{\tilde{\theta }^k}\le \tilde{\omega }_0$, we define

$$\begin{aligned}&{\left\{ \begin{array}{ll} {\mathbb {U}}_{\omega ,\epsilon ,\tilde{\theta }^k}(x)\equiv \tilde{J}_{\omega ,\epsilon }^{-1}\tilde{\theta }^{-k\mu } U_{\omega ,\epsilon }(\tilde{\theta }^kx)\\ {\mathbb {Q}}_{\omega ,\epsilon ,\tilde{\theta }^k}(x)\equiv \tilde{J}_{\omega ,\epsilon }^{-1}\tilde{\theta }^{k(1-\mu )} Q_{\omega ,\epsilon }(\tilde{\theta }^kx) \\ {\mathbb {F}}_{\omega ,\epsilon ,\tilde{\theta }^k}(x)\equiv \tilde{J}_{\omega ,\epsilon }^{-1}\tilde{\theta }^{k(2-\mu )} F_{\omega ,\epsilon }(\tilde{\theta }^kx)\\ {\mathbb {P}}_{\omega ,\epsilon ,\tilde{\theta }^k}(x)\equiv \tilde{\theta }^{2k}{\mathbf{P}}_{\omega ,\epsilon }(\tilde{\theta }^kx)\\ \end{array}\right. } \text{ in } B_1(0)\setminus \partial \Omega _m^\epsilon /\tilde{\theta }^k. \end{aligned}$$

Then they satisfy

$$\begin{aligned} {\left\{ \begin{array}{ll} -\nabla \cdot ({\mathcal {E}}_{\omega ^2,\epsilon ,\tilde{\theta }^k}{\mathbf{K}}_{\epsilon /\tilde{\theta }^k}\nabla {\mathbb {U}}_{\omega ,\epsilon ,\tilde{\theta }^k} + {\mathbb {Q}}_{\omega ,\epsilon ,\tilde{\theta }^k})\\ +{\mathcal {E}}_{\omega ,\epsilon ,\tilde{\theta }^k}{\mathbb {P}}_{\omega ,\epsilon ,\tilde{\theta }^k}{\mathbb {U}}_{\omega ,\epsilon ,\tilde{\theta }^k} ={\mathbb {F}}_{\omega ,\epsilon ,\tilde{\theta }^k} &{} \text{ in } B_1(0)\cap \Omega /\tilde{\theta }^k,\\ {\mathbb {U}}_{\omega ,\epsilon ,\tilde{\theta }^k}=0 &{} \text{ on } B_1(0)\cap \partial \Omega /\tilde{\theta }^k.\\ \end{array}\right. } \end{aligned}$$

Following the argument of Lemma 4.2 and employing Lemma 4.5 with $s=\tilde{\theta }^k$, we obtain (4.29) with $k+1$ in place of k. $\square$

Lemma 4.7

For any $\delta \in (0,3)$ and ${\mathbf{M}}>0$, there is a $\tilde{\omega }_*\in (0,1)$ (depending on $\delta ,{\mathbf{M}},{\mathbf{K}},Y_f,\Omega$) such that if $\omega ,\epsilon \le \tilde{\omega }_*$ and ${\mathbf{P}}_{\omega ,\epsilon }(x)\in [0,{\mathbf{M}}]$ for $x\in B_1(0)\cap \Omega$, then any solution of (4.28) satisfies

$$\begin{aligned}& [U_{\omega ,\epsilon }]_{C^{0,\mu }(\overline{B_{1/2}(0)\cap \Omega _f^\epsilon })} +\mathop {\mathop {\sup }_{j\in {\mathbb {Z}}^n}}\limits _{\epsilon (Y_m+j)\subset B_{1/2}(0)\cap \Omega _m^\epsilon } \omega ^{3/2}[U_{\omega ,\epsilon }]_{C^{0,\mu }(\epsilon (\overline{Y_m}+j))} \le c\hat{J}_{\omega ,\epsilon }, \end{aligned}$$

(4.30)

where $\mu \equiv \frac{\delta }{n+\delta }$; c is a constant independent of $\omega ,\epsilon$; $\hat{J}_{\omega ,\epsilon }$ is defined as

$$\begin{aligned} \hat{J}_{\omega ,\epsilon }\equiv & {} \frac{3}{\tilde{\omega }_*}(\Vert \mathbf{E}_{\omega ,\epsilon } U_{\omega ,\epsilon }\Vert _{L^{2}(B_1(0)\cap \Omega )} +\Vert Q_{\omega ,\epsilon },F_{\omega ,\epsilon } \Vert _{L^{n+\delta }(B_1(0)\cap \Omega ^\epsilon _f)}\nonumber \\&+\,\omega ^{-1}\Vert Q_{\omega ,\epsilon },\max \{\omega ,\epsilon \} F_{\omega ,\epsilon }\Vert _{L^{n+\delta }(B_1(0)\cap \Omega _m^\epsilon )}). \end{aligned}$$

(4.31)

Proof

Let $\tilde{\theta }_1,\tilde{\theta }_2,\tilde{\omega }_0,\tilde{J}_{\omega ,\epsilon }$ be same as those in Lemma 4.6, set $\tilde{\omega }_*\equiv \min \{\tilde{\omega }_0\tilde{\theta }_2/3, \omega _*\}$ where $\omega _*$ is the one in Lemma 4.3, and let $\omega ,\epsilon \le \tilde{\omega }_*$. Denote by c a constant independent of $\omega ,\epsilon$. By energy method and A2, any solution of (4.28) satisfies

$$\begin{aligned} \Vert U_{\omega ,\epsilon }\Vert _{H^1(B_{3/4}(0)\cap \Omega _f^\epsilon )}\le & {} c(\Vert \mathbf{E}_{\omega ,\epsilon }U_{\omega ,\epsilon } \Vert _{L^2(B_1(0)\cap \Omega )}+ \Vert \mathbf{E}_{1/\omega ,\epsilon } Q_{\omega ,\epsilon } \Vert _{L^2(B_1(0)\cap \Omega )}\nonumber \\&+\,\Vert F_{\omega ,\epsilon } \Vert _{L^2(B_1(0)\cap \Omega )}+\epsilon \omega ^{-1}\Vert F_{\omega ,\epsilon } \Vert _{L^2(B_1(0)\cap \Omega ^\epsilon _m)}) \equiv c\check{{\mathcal {I}}}. \end{aligned}$$

By Theorem 7.26 [16] and Remark 2.1,

$$\begin{aligned} \big \Vert \Pi _\epsilon U_{\omega ,\epsilon }|_{\Omega _f^\epsilon }\big \Vert _{L^{\frac{2n}{n-2}}(B_{3/4}(0)\cap \Omega )}\le c\check{\mathcal {I}}. \end{aligned}$$

(4.32)

For any $x \in B_{\tilde{\theta }_2/3}(0)\cap \Omega _f^{\epsilon }$, define $\eta (x)\equiv |x-x_0|$ where $x_0\in \partial \Omega$ satisfying $|x-x_0|=\min _{y\in \partial \Omega }|x-y|$. Then we have either case (1) $\eta (x)> \frac{2\epsilon }{3\tilde{\omega }_0}$ or case (2) $\eta (x)\le \frac{2\epsilon }{3\tilde{\omega }_0}$.

Let us consider case (1). Because of $\tilde{\theta }_1<\tilde{\theta }_2^2$, for any $r \in [ \epsilon /\tilde{\omega }_0,\tilde{\theta }_2]$, there are $\tilde{\theta }\in [\tilde{\theta }_1,\tilde{\theta }_2]$ and $k\in \mathbb {N}$ satisfying $r=\tilde{\theta }^k$. Since $\eta (x)\in [ \frac{2\epsilon }{3\tilde{\omega }_0},\frac{\tilde{\theta }_2}{3}]$, by Lemma 4.6,

So, for $s\in [\frac{\eta (x)}{2},\frac{\tilde{\theta }_2}{3}]$,

(4.33)

Next we shift the coordinate system such that x is located at the origin and we define, in $B_1(x)\setminus \partial \Omega _m^\epsilon /\eta (x)$,

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathbb {U}}_{\omega ,\epsilon ,\eta (x)}(y)\equiv \frac{U_{\omega ,\epsilon }(\eta (x)y)-(\Pi _\epsilon U_{\omega ,\epsilon }|_{\Omega _f^\epsilon })_{x,\eta (x)}}{\hat{J}_{\omega ,\epsilon }\ \eta ^{\mu }(x)},\\ {\mathbb {Q}}_{\omega ,\epsilon ,\eta (x)}(y)\equiv \frac{ Q_{\omega ,\epsilon }(\eta (x)y)}{\hat{J}_{\omega ,\epsilon }\ \eta ^{\mu -1}(x)},\\ {\mathbb {F}}_{\omega ,\epsilon ,\eta (x)}(y)\equiv \frac{F_{\omega ,\epsilon }(\eta (x)y) -\mathbf{E}_{\omega ,\epsilon /\eta (x)}(y){\mathbf{P}}_{\omega ,\epsilon }(\eta (x)y)(\Pi _\epsilon U_{\omega ,\epsilon }|_{\Omega _f^\epsilon })_{x,\eta (x)}}{\hat{J}_{\omega ,\epsilon }\ \eta ^{\mu -2}(x)}, \\ {\mathbb {P}}_{\omega ,\epsilon ,\eta (x)}(y)\equiv {\mathbf{P}}_{\omega ,\epsilon }(\eta (x)y).\\ \end{array}\right. } \end{aligned}$$

See (4.31) for $\hat{J}_{\omega ,\epsilon }$. Then these functions satisfy

$$\begin{aligned}& -\nabla \cdot (\mathbf{E}_{\omega ^2,\epsilon /\eta (x)}{\mathbf{K}}_{\epsilon /\eta (x)}\nabla {\mathbb {U}}_{\omega ,\epsilon ,\eta (x)}+{\mathbb {Q}}_{\omega ,\epsilon ,\eta (x)})\nonumber \\&\qquad +|\eta (x)|^2\mathbf{E}_{\omega ,\epsilon /\eta (x)}{\mathbb {P}}_{\omega ,\epsilon ,\eta (x)} {\mathbb {U}}_{\omega ,\epsilon ,\eta (x)} = {\mathbb {F}}_{\omega ,\epsilon ,\eta (x)} \text{ in } B_1(x). \end{aligned}$$

(4.34)

Take $s=\eta (x)<1$ in (4.33) to see, by (4.32),

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathbb {P}}_{\omega ,\epsilon ,\eta (x)}(y)\in [0,{\mathbf{M}}] \text{ for } \text{ all } y\in B_1(x).\\ \tilde{\omega }_0^{-1}(\Vert \mathbf{E}_{\omega ,\epsilon /\eta (x)}{\mathbb {U}}_{\omega ,\epsilon ,\eta (x)}\Vert _{L^{2}(B_1(x))} +\Vert {\mathbb {Q}}_{\omega ,\epsilon ,\eta (x)},{\mathbb {F}}_{\omega ,\epsilon ,\eta (x)}\Vert _{L^{n+\delta }(B_1(x)\cap \Omega _f^{\epsilon }/\eta (x))}\\ +\omega ^{-1}\Vert {\mathbb {Q}}_{\omega ,\epsilon ,\eta (x)},\max \{\omega ,\epsilon /\eta (x)\} {\mathbb {F}}_{\omega ,\epsilon ,\eta (x)}\Vert _{L^{n+\delta }(B_1(x)\cap \Omega _m^{\epsilon }/\eta (x))}) \le c.\\ \end{array}\right. } \end{aligned}$$

Apply Lemma 4.3 to (4.34) to obtain

$$\begin{aligned}&[{\mathbb {U}}_{\omega ,\epsilon ,\eta (x)}]_{C^{0,\mu }(\overline{B_{1/2}(x)\cap \Omega _f^\epsilon /\eta (x)})}\nonumber \\&+\mathop {\mathop {\sup }_{j\in {\mathbb {Z}}^n}}\limits _{\frac{\epsilon }{\eta (x)}(Y_m+j)\subset B_{1/2}(0)\cap \Omega _m^\epsilon /\eta (x)} \omega ^{3/2}[{\mathbb {U}}_{\omega ,\epsilon ,\eta (x)}]_{C^{0,\mu }(\frac{\epsilon }{\eta (x)}(\overline{Y_m}+j))}\le c. \end{aligned}$$

(4.35)

Which implies

(4.36)

Next we consider case (2). Because of $\tilde{\theta }_1<\tilde{\theta }_2^2$, for any $r \in [ \epsilon /\tilde{\omega }_0,\tilde{\theta }_2]$, there are $\tilde{\theta }\in [\tilde{\theta }_1,\tilde{\theta }_2]$ and $k\in \mathbb {N}$ satisfying $r=\tilde{\theta }^k$. By Lemma 4.6,

(4.37)

This implies, for $s \in [\frac{\epsilon }{3\tilde{\omega }_0},\frac{\tilde{\theta }_2}{3}]$,

(4.38)

Again we shift the coordinate system such that x is located at the origin. Define, in $(B_{ 1/\tilde{\omega }_0}(x)\cap \Omega /\epsilon )\setminus \partial \Omega _m^\epsilon /\epsilon$,

$$\begin{aligned}&{\left\{ \begin{array}{ll} {\mathbb {U}}_{\omega ,1}(y)\equiv \hat{J}_{\omega ,\epsilon }^{-1}\epsilon ^{-\mu } \big (U_{\omega ,\epsilon }(\epsilon y)-(\Pi _\epsilon U_{\omega ,\epsilon }|_{\Omega ^\epsilon _f})_{x,{\epsilon }/{\tilde{\omega }_0}}\big ),\\ {\mathbb {Q}}_{\omega ,1} (y)\equiv \hat{J}_{\omega ,\epsilon }^{-1}\epsilon ^{1-\mu } Q_{\omega ,\epsilon }(\epsilon y),\\ {\mathbb {F}}_{\omega ,1} (y)\equiv \hat{J}_{\omega ,\epsilon }^{-1}\epsilon ^{2-\mu }(F_{\omega ,\epsilon }(\epsilon y) -{\mathcal {E}}_{\omega ,\epsilon ,\epsilon }(y){\mathbf{P}}_{\omega ,\epsilon }(\epsilon y)(\Pi _\epsilon U_{\omega ,\epsilon }|_{\Omega ^\epsilon _f})_{x,{\epsilon }/{\tilde{\omega }_0}}),\\ {\mathbb {P}}_{\omega ,1}(y)\equiv \epsilon ^2{\mathbf{P}}_{\omega ,\epsilon }(\epsilon y),\\ \end{array}\right. } \end{aligned}$$

and define

$$\begin{aligned} {\mathbb {U}}_{b_{\omega }}\equiv -\hat{J}_{\omega ,\epsilon }^{-1}\epsilon ^{-\mu } (\Pi _\epsilon U_{\omega ,\epsilon }|_{\Omega ^\epsilon _f})_{x,{\epsilon }/{\tilde{\omega }_0}} \qquad \text{ in } B_{\frac{1}{\tilde{\omega }_0}}(x)\cap \Omega /\epsilon . \end{aligned}$$

See (4.19) for ${\mathcal {E}}_{\omega ,\epsilon ,\epsilon }$.

By (4.37)$_1$, ${\mathbb {U}}_{b_{\omega }}$ is a constant independent of $\omega ,\epsilon$. Then these functions satisfy

$$\begin{aligned} {\left\{ \begin{array}{ll} -\nabla \cdot ({\mathcal {E}}_{\omega ^2,\epsilon ,\epsilon }{\mathbf{K}}\nabla {\mathbb {U}}_{\omega ,1}+{\mathbb {Q}}_{\omega ,1}) +{\mathcal {E}}_{\omega ,\epsilon ,\epsilon }{\mathbb {P}}_{\omega ,1} {\mathbb {U}}_{\omega ,1}= {\mathbb {F}}_{\omega ,1} &{} \text{ in } B_{\frac{1}{\tilde{\omega }_0}}(x)\cap \Omega /\epsilon , \\ {\mathbb {U}}_{\omega ,1}={\mathbb {U}}_{b_{\omega }} &{} \text{ on } B_{\frac{1}{\tilde{\omega }_0}}(x)\cap \partial \Omega /\epsilon . \\ \end{array}\right. } \end{aligned}$$

Take $s= \frac{\epsilon }{\tilde{\omega }_0}$ in (4.38) to see, by (4.37)$_1$,

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathbb {P}}_{\omega ,1}\in [0,{\mathbf{M}}] \text{ for } \text{ all } x\in B_{1/\tilde{\omega }_0}(x)\cap \Omega /\epsilon ,\\ \Vert {\mathcal {E}}_{\omega ,\epsilon ,\epsilon }{\mathbb {U}}_{\omega ,1}\Vert _{L^2(B_{1/\tilde{\omega }_0}(x)\cap \Omega /\epsilon )} +\Vert {\mathcal {E}}_{1/\omega ,\epsilon ,\epsilon }{\mathbb {Q}}_{\omega ,1},{\mathcal {E}}_{1/\omega ,\epsilon ,\epsilon }{\mathbb {F}}_{\omega ,1}\Vert _{L^{n+\delta }(B_{ 1/\tilde{\omega }_0}(x)\cap \Omega /\epsilon )} \\ +\Vert {\mathbb {U}}_{b_{\omega }}\Vert _{W^{1,n+\delta }(B_{1/\tilde{\omega }_0}(x)\cap \Omega /\epsilon )} \le c.\\ \end{array}\right. } \end{aligned}$$

By Lemma 3.4,

$$\begin{aligned}{}[{\mathbb {U}}_{\omega ,1}]_{C^{0,\mu }(\overline{B_{1/2\tilde{\omega }_0}(x)\cap \Omega _f^\epsilon /\epsilon })} + \omega ^{3/2}[{\mathbb {U}}_{\omega ,1}]_{C^{0,\mu }(\overline{B_{1/2\tilde{\omega }_0}(x)\cap \Omega _m^\epsilon /\epsilon })} \le c. \end{aligned}$$

(4.39)

(4.39) imply (4.38)$_1$ holds for $s \le \frac{\epsilon }{2\tilde{\omega }_0}$.

The Hölder estimate of $\Pi _\epsilon U_{\omega ,\epsilon }$ follows from (4.33)$_1$, (4.36), (4.38)$_1$, (4.39), and Theorem 1.2 in page 70 [15]. The Hölder estimate of $U_{\omega ,\epsilon }$ in $\epsilon (\overline{Y_m}+j)\subset B_{1/2}(0)\cap \overline{\Omega ^\epsilon _m}$ is from (4.35) and (4.39). $\square$

Remark 4.2

Let $\tilde{\omega }_*$ be same as that in Lemma 4.7. By Lemma 3.4 with $p=n+\delta$, we know that if $\delta \in (0,3)$, ${\mathbf{M}}>0$, $\epsilon \in [\tilde{\omega }_*,1]$, $\omega \in (0,\tilde{\omega }_*)$, and ${\mathbf{P}}_{\omega ,\epsilon }(x)\in [0,{\mathbf{M}}]$ for all $x\in B_1(0)\cap \Omega$, any solution of (4.28) satisfies (4.30). Together with Lemma 4.7, any solution of (4.28) satisfies (4.30) if $\delta \in (0,3)$, ${\mathbf{M}}>0$, $\epsilon \in (0,1)$, $\omega \in (0,\tilde{\omega }_*)$, and ${\mathbf{P}}_{\omega ,\epsilon }(x)\in [0,{\mathbf{M}}]$ for all $x\in B_1(0)\cap \Omega$.

Let us consider the solutions of (4.28) with ${\mathbf{P}}_{\omega ,\epsilon }=0$. By tracing the arguments of Lemma 4.7 and Remark 4.2 and employing Lemma 3.4, then we have

Lemma 4.8

For any $\delta >0$, there is a $\tilde{\omega }_*\in (0,1)$ (depending on $\delta ,{\mathbf{K}},Y_f,\Omega$) such that, if $\omega \in (0,\tilde{\omega }_*)$ and $\epsilon \in (0,1)$, then any solution of (4.28) with ${\mathbf{P}}_{\omega ,\epsilon }=0$ satisfies

$$\begin{aligned}{}[U_{\omega ,\epsilon }]_{C^{0,\mu }(\overline{B_{1/2}(0)\cap \Omega _f^\epsilon })} +\mathop {\mathop {\sup }_{j\in {\mathbb {Z}}^n}}\limits _{\epsilon (Y_m+j)\subset B_{1/2}(0)\cap \Omega _m^\epsilon } \omega [U_{\omega ,\epsilon }]_{C^{0,\mu }(\epsilon (\overline{Y_m}+j))} \le c\hat{J}_{\omega ,\epsilon }, \end{aligned}$$

where c is a constant independent of $\omega ,\epsilon$. See Lemma 4.7 for $\mu , \hat{J}_{\omega ,\epsilon }$.

By energy method, partition of unity, Remark 4.1, Remark 4.2, Lemma 4.4, Lemma 4.8, and Poincaré inequality [16], we conclude

Lemma 4.9

Under A1–A2, for any $\delta \in (0,3)$ and ${\mathbf{M}}>0$, there is a constant $\tilde{\omega }_*\in (0,1)$ (depending on $\delta ,{\mathbf{M}},{\mathbf{K}},Y_f,\Omega$) such that if

$$\begin{aligned} {\left\{ \begin{array}{ll} -\nabla \cdot (\mathbf{E}_{\omega ^2,\epsilon }{\mathbf{K}}_{\epsilon }\nabla U_{\omega ,\epsilon } + Q_{\omega ,\epsilon })+\mathbf{E}_{\omega ,\epsilon }{\mathbf{P}}_{\epsilon }U_{\omega ,\epsilon }=F_{\omega ,\epsilon } &{} \text{ in } \Omega ,\\ U_{\omega ,\epsilon }=0 &{} \text{ on } \partial \Omega ,\\ \end{array}\right. } \end{aligned}$$

and if $\omega \in (0,\tilde{\omega }_*)$, $\epsilon \in (0,1)$, and ${\mathbf{P}}_{\omega ,\epsilon }(x)\in [0,{\mathbf{M}}]$ for all $x\in \Omega$, then

$$\begin{aligned}&[U_{\omega ,\epsilon }]_{C^{0,\mu }(\overline{\Omega ^\epsilon _f})} + \mathop {\mathop {\sup }_{j\in {\mathbb {Z}}^n}}\limits _{\epsilon (Y_m+j)\subset \Omega _m^\epsilon } \omega ^\lambda [U_{\omega ,\epsilon }]_{C^{0,\mu }(\epsilon (\overline{Y_m}+j))}\nonumber \\&\qquad \le c(\Vert Q_{\omega ,\epsilon }, F_{\omega ,\epsilon }\Vert _{L^{n+\delta }(\Omega ^\epsilon _f)} +\omega ^{-1}\Vert Q_{\omega ,\epsilon },\max \{\omega ,\epsilon \} F_{\omega ,\epsilon }\Vert _{L^{n+\delta }(\Omega ^\epsilon _m)}), \end{aligned}$$

where $\mu \equiv \frac{\delta }{n+\delta }$ and c is a positive constant independent of $\omega ,\epsilon$. Here $\lambda$ is $\frac{3}{2}$ if ${\mathbf{P}}_{\omega ,\epsilon }\not =0$ and is 1 if ${\mathbf{P}}_{\omega ,\epsilon }=0$.

Under A1–A4, we multiply (1.1) by $|\Psi _{\omega ,\epsilon }|^{q-2}\Psi _{\omega ,\epsilon }$ for $q>2$ and integrate over $\Omega$ to obtain

$$\begin{aligned}&\Vert \Psi _{\omega ,\epsilon }\Vert _{L^q(\Omega )} \le c\Vert \mathbf{E}_{1/\omega ,\epsilon }V_{\omega ,\epsilon }, G_{\omega ,\epsilon }\Vert _{L^q(\Omega )}, \end{aligned}$$

(4.40)

where c is independent of $\omega ,\epsilon$. Then we write (1.1) as

$$\begin{aligned} {\left\{ \begin{array}{ll} -\nabla \cdot (\mathbf{E}_{\omega ^2,\epsilon }{\mathbf{K}}_{\epsilon }\nabla \Psi _{\omega ,\epsilon }+V_{\omega ,\epsilon }) =G_{\omega ,\epsilon }-{\mathbf{T}}_{\omega ,\epsilon }\Psi _{\omega ,\epsilon } &{} \text{ in } \Omega ,\\ \Psi _{\omega ,\epsilon }=0 &{} \text{ on } \partial \Omega .\\ \end{array}\right. } \end{aligned}$$

Theorem 2.1 follows by energy method, (4.40) for $q=n+\delta$, and Lemma 4.9 for ${\mathbf{P}}_{\omega ,\epsilon }=0$. Theorem 2.2 is a direct consequence of energy method and Lemma 4.9.

5 Convergence estimates

In this section, we prove Theorems 2.3, 2.4, 2.5. For each $\nu \in (0,1)$ and $i_1,i_2\in \{1,2,\cdots ,n\}$, we find ${\mathbb {Y}}^{(i_1,i_2)}_\nu \in H^1_{per}({\mathbb {R}}^n)$ satisfying

$$\begin{aligned}&{\left\{ \begin{array}{ll} \nabla \cdot \big ({\mathbb {E}}_{\nu ^2}{\mathbf{K}}(\nabla {\mathbb {Y}}_\nu ^{(i_1,i_2)}+{{\mathbb {X}}}_\nu ^{(i_2)}\vec {e}_{i_1})\big )\\ \qquad =\frac{{\mathcal {K}}_\nu ^{(i_1,i_2)}}{|Y_f|}{\mathcal {X}}_{Y_f}-{\mathbb {E}}_{\nu ^2}{\mathbf{K}}(\partial _{i_1}{{\mathbb {X}}}_\nu ^{(i_2)}+\delta _{i_1,i_2}) &{} \text{ in } Y,\\ \displaystyle \int _{Y_f} {\mathbb {Y}}_\nu ^{(i_1,i_2)}(y) {\hbox {d}}y=0,\\ \end{array}\right. } \end{aligned}$$

(5.1)

where $\vec {e}_{i_1}$ is a unit vector in the $i_1$th coordinate direction, $\delta _{i_1,i_2}$ is 1 if $i_1=i_2$ and is 0 if $i_1\not =i_2$, and ${\mathcal {K}}_\nu ^{(i_1,i_2)}$ is the $(i_1,i_2)$th component of ${\mathcal {K}}_\nu$. See (2.5) for ${{\mathbb {X}}}^{(i)}_{\nu }$ and (2.9) for ${\mathcal {K}}_\nu$. By Lax–Milgram Theorem [16], A1–A2, and (3.17), ${\mathbb {Y}}_\nu ^{(i_1,i_2)}$ is uniquely solvable. By Lemma 3.3,

$$\begin{aligned} \Vert {\mathbb {Y}}_\nu ^{(i_1,i_2)}\Vert _{C^2(\overline{Y_f})\cap C^2(\overline{Y_m})}\le c, \end{aligned}$$

(5.2)

where c is a constant independent of $\nu$. Define $n\times n$ matrices ${\mathbb {Y}}_\nu \equiv ({\mathbb {Y}}_\nu ^{(i_1,i_2)})$ and ${\mathbb {Y}}_{\nu ,s}(x)\equiv s^2{\mathbb {Y}}_\nu (\frac{x}{s})$ for $\nu ,s\in (0,1)$.

5.1 Proof of Theorem 2.3

A1–A6 are assumed. This subsection consists of two parts. The first part is for $\omega ,\epsilon \rightarrow 0$, $\frac{\omega }{\epsilon }\rightarrow \infty$, and the second part is for $\omega ,\epsilon \rightarrow 0, \frac{\omega }{\epsilon }\rightarrow \sigma \in [0,\infty )$.

5.1.1 Part 1: $\omega ,\epsilon \rightarrow 0, \frac{\omega }{\epsilon }\rightarrow \infty$.

For each $\nu \in (0,1)$, we find $\widetilde{\mathbb {W}}_\nu \in H^1_{per}({\mathbb {R}}^n)$ satisfying

$$\begin{aligned}&{\left\{ \begin{array}{ll} \nabla \cdot ({\mathbb {E}}_{\nu ^2}{\mathbf{K}}\nabla \widetilde{\mathbb {W}}_\nu )={\mathbf{T}}-{\mathcal {T}}_\infty {\mathcal {X}}_{Y_f} &{} \text{ in } Y,\\ \displaystyle \int _{Y_f} \widetilde{\mathbb {W}}_\nu (y) {\hbox {d}}y=0.\\ \end{array}\right. } \end{aligned}$$

(5.3)

See (2.11) for ${\mathcal {T}}_\infty$. By Lax–Milgram Theorem [16], $\widetilde{\mathbb {W}}_\nu$ is uniquely solvable. By Lemma 3.3 and A5,

$$\begin{aligned} \Vert {\mathbb {E}}_{\nu }\widetilde{\mathbb {W}}_\nu \Vert _{C^2(\overline{Y_f})\cap C^2(\overline{Y_m})}\le c/\nu , \end{aligned}$$

(5.4)

where c is independent of $\nu$. Define $\widetilde{\mathbb {W}}_{\nu ,s}(x)\equiv s^2\widetilde{\mathbb {W}}_\nu (\frac{x}{s})$ for $\nu ,s\in (0,1)$.

Let $\Psi _{\omega ,\epsilon }$ be the solution of (1.1) with $V_{\omega ,\epsilon }=0$, $\Psi$ be the solution of (2.12), and

$$\begin{aligned} \varphi _{\omega ,\epsilon }\equiv \Psi _{\omega ,\epsilon }-\Psi -\widetilde{\mathbb {W}}_{\omega ,\epsilon }\Psi -{{\mathbb {X}}}_{\omega ,\epsilon }\nabla \Psi -{\mathbb {Y}}_{\omega ,\epsilon }\nabla ^2 \Psi \qquad \text{ in } \Omega . \end{aligned}$$

See (2.5) for ${{\mathbb {X}}}_{\omega ,\epsilon }$ and (5.1) for ${\mathbb {Y}}_{\omega ,\epsilon }$. By (2.12)–(2.13), (3.17), (3.19), and (5.1)–(5.4),

$$\begin{aligned} {\left\{ \begin{array}{ll} -\nabla \cdot \big (\mathbf{E}_{\omega ^2,\epsilon }{\mathbf{K}}_\epsilon \big (\nabla \varphi _{\omega ,\epsilon }+\widetilde{\mathbb {W}}_{\omega ,\epsilon }\nabla \Psi +{\mathbb {Y}}_{\omega ,\epsilon }\nabla ^3 \Psi \big )\big )+{\mathbf{T}}_{\omega ,\epsilon }\varphi _{\omega ,\epsilon }\\ ={\mathcal {O}}_1(\omega +\epsilon /\omega )+G_{\omega ,\epsilon }-{\mathcal {G}}{\mathcal {X}}_{\Omega _f^\epsilon } &{} \text{ in } \Omega ,\\ \varphi _{\omega ,\epsilon }={\mathcal {O}}_2(\epsilon +\epsilon ^2/\omega ^2) &{} \text{ on } \partial \Omega ,\\ \end{array}\right. } \end{aligned}$$

where ${\mathcal {O}}_1(\nu )$ denotes a function satisfying $\Vert {\mathcal {O}}_1(\nu )\Vert _{L^{n+\delta }(\Omega )}\le c\nu$ and ${\mathcal {O}}_2(\nu )$ denotes a function satisfying $\Vert {\mathcal {O}}_2(\nu )\Vert _{L^{\infty }(\Omega )}\le c\nu$ for some constant c independent of $\omega ,\epsilon$. See (2.12) for ${\mathcal {G}}$. Decompose $\varphi _{\omega ,\epsilon }$ as $\varphi _{\omega ,\epsilon }=\widehat{\varphi }_{\omega ,\epsilon }+\check{\varphi }_{\omega ,\epsilon }$, where $\widehat{\varphi }_{\omega ,\epsilon }$ satisfies

$$\begin{aligned}&{\left\{ \begin{array}{ll} -\nabla \cdot \big (\mathbf{E}_{\omega ^2,\epsilon }{\mathbf{K}}_\epsilon \big (\nabla \widehat{\varphi }_{\omega ,\epsilon }+\widetilde{\mathbb {W}}_{\omega ,\epsilon }\nabla \Psi +{\mathbb {Y}}_{\omega ,\epsilon }\nabla ^3 \Psi \big )\big )+{\mathbf{T}}_{\omega ,\epsilon }\widehat{\varphi }_{\omega ,\epsilon }\\ ={\mathcal {O}}_1(\omega +\epsilon /\omega ) +G_{\omega ,\epsilon }-{\mathcal {G}}{\mathcal {X}}_{\Omega _f^\epsilon } &{} \text{ in } \Omega ,\\ \widehat{\varphi }_{\omega ,\epsilon }= 0 &{} \text{ on } \partial \Omega ,\\ \end{array}\right. } \end{aligned}$$

(5.5)

and $\check{\varphi }_{\omega ,\epsilon }$ satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} -\nabla \cdot (\mathbf{E}_{\omega ^2,\epsilon }{\mathbf{K}}_\epsilon \nabla \check{\varphi }_{\omega ,\epsilon })+{\mathbf{T}}_{\omega ,\epsilon }\check{\varphi }_{\omega ,\epsilon }=0 &{} \text{ in } \Omega ,\\ \check{\varphi }_{\omega ,\epsilon } ={\mathcal {O}}_2(\epsilon +\epsilon ^2/\omega ^2) &{} \text{ on } \partial \Omega .\\ \end{array}\right. } \end{aligned}$$

(5.6)

By Theorem 2.1, A6, (2.13), (5.2), and (5.26), the solution of (5.5) satisfies

$$\begin{aligned}&\Vert \mathbf{E}_{\omega ,\epsilon }\widehat{\varphi }_{\omega ,\epsilon }\Vert _{L^\infty (\Omega )} \le c(\max \{\omega ,\epsilon /\omega \}+\Vert G_{\omega ,\epsilon }-{\mathcal {G}}\Vert _{L^{n+\delta }(\Omega _f^\epsilon )}), \end{aligned}$$

(5.7)

where c is independent of $\omega ,\epsilon$. By Theorem 8.1 [16], the solution of (5.6) satisfies

$$\begin{aligned} \Vert \check{\varphi }_{\omega ,\epsilon }\Vert _{L^\infty (\Omega )} =\Vert \check{\varphi }_{\omega ,\epsilon }\Vert _{L^\infty (\partial \Omega )}\le c(\epsilon +\epsilon ^2/\omega ^2), \end{aligned}$$

(5.8)

where c is a constant independent of $\omega ,\epsilon$.

From (5.7) and (5.8), we see that the difference between the solution of (1.1) with $V_{\omega ,\epsilon }=0$ and the solution of (2.12) satisfies

$$\begin{aligned}& \Vert \mathbf{E}_{\omega ,\epsilon }(\Psi _{\omega ,\epsilon }-\Psi )\Vert _{L^\infty (\Omega )} \le c(\max \{\omega ,\epsilon /\omega \}+\Vert G_{\omega ,\epsilon }-{\mathcal {G}}\Vert _{L^{n+\delta }(\Omega _f^\epsilon )}), \end{aligned}$$

(5.9)

where c is a constant independent of $\omega ,\epsilon$. Now let us consider (2.14). We note that the solution of (1.1) with $V_{\omega ,\epsilon }=0$ and the solution of (2.14) satisfy, for any $\epsilon (Y_m+j)\subset \Omega ^\epsilon _m$ and $j\in {\mathbb {Z}}^n$,

$$\begin{aligned} {\left\{ \begin{array}{ll} -\omega ^{2}\nabla \cdot \big ({\mathbf{K}}_\epsilon \nabla (\Psi _{\omega ,\epsilon }-\phi ^{(j)}_{\omega ,\epsilon }) \big )+{\mathbf{T}}_{\omega ,\epsilon }(\Psi _{\omega ,\epsilon }-\phi ^{(j)}_{\omega ,\epsilon })=0&{} \text{ in } \epsilon (Y_m+j),\\ \Psi _{\omega ,\epsilon }-\phi ^{(j)}_{\omega ,\epsilon }=\Psi _{\omega ,\epsilon }-\Psi &{} \text{ on } \partial \epsilon (Y_m+j). \end{array}\right. } \end{aligned}$$

By (5.9) and Theorem 8.1 [16], we conclude

$$\begin{aligned} \Vert \Psi _{\omega ,\epsilon }-\mathop {\mathop {\sum }_{j\in {\mathbb {Z}}^n}}\limits _{\epsilon (Y_m+j)\subset \Omega _m^\epsilon }\phi ^{(j)}_{\omega ,\epsilon }\Vert _{L^\infty (\Omega _m^\epsilon )}\le c(\max \{\omega ,\epsilon /\omega \}+\Vert G_{\omega ,\epsilon }-{\mathcal {G}}\Vert _{L^{n+\delta }(\Omega _f^\epsilon )}), \end{aligned}$$

(5.10)

where c is independent of $\omega ,\epsilon$. (5.9)–(5.10) imply Theorem 2.3 for $\omega ,\epsilon \rightarrow 0, \frac{\omega }{\epsilon }\rightarrow \infty$.

5.1.2 Part 2: $\omega ,\epsilon \rightarrow 0, \frac{\omega }{\epsilon }\rightarrow \sigma \in [0,\infty )$.

For $\nu \in (0,1)$, $\beta \in (0,\infty )$, and $i_1,i_2\in \{1,\cdots ,n\}$, we find $\widetilde{{\mathbb {X}}}^{(i_1)}_{\nu ,\beta },\widetilde{\mathbb {Y}}^{(i_1,i_2)}_{\nu ,\beta }\in H_{per}^1({\mathbb {R}}^n)$ satisfying

$$\begin{aligned}&{\left\{ \begin{array}{ll} \nabla \cdot \left( {\mathbb {E}}_{\nu ^2}{\mathbf{K}}\big (\nabla \widetilde{{\mathbb {X}}}^{(i_1)}_{\nu ,\beta }+ ({\mathcal {X}}_{Y_f}+{\mathbb {W}}_\beta {\mathcal {X}}_{Y_m})\vec {e}_{i_1}\big )\right) =0 &{} \text{ in } Y,\\ \displaystyle \int _{Y_f}\widetilde{{\mathbb {X}}}^{(i_1)}_{\nu ,\beta }(y) {\hbox {d}}y=0, \end{array}\right. } \end{aligned}$$

(5.11)

$$\begin{aligned}&{\left\{ \begin{array}{ll} \nabla \cdot \left( {\mathbb {E}}_{\nu ^2}{\mathbf{K}}(\nabla \widetilde{\mathbb {Y}}_{\nu ,\beta }^{(i_1,i_2)}+\widetilde{{\mathbb {X}}}_{\nu ,\beta }^{(i_2)}\vec {e}_{i_1})\right) \\ \qquad =\frac{\widetilde{\mathcal {K}}_{\nu ,\beta }^{(i_1,i_2)}}{|Y_f|}{\mathcal {X}}_{Y_f}-{\mathbb {E}}_{\nu ^2}{\mathbf{K}}\left( \partial _{i_1}\widetilde{{\mathbb {X}}}_{\nu ,\beta }^{(i_2)}+({\mathcal {X}}_{Y_f}+{\mathbb {W}}_\beta {\mathcal {X}}_{Y_m})\delta _{i_1,i_2}\right) &{} \text{ in } Y,\\ \displaystyle \int _{Y_f}\widetilde{\mathbb {Y}}_{\nu ,\beta }^{(i_1,i_2)}(y) {\hbox {d}}y=0, \end{array}\right. } \end{aligned}$$

(5.12)

where $\vec {e}_{i_1}, \delta _{i_1,i_2}$ are same as those in (5.1), ${\mathbb {W}}_\beta$ is from (2.7)–(2.8), and $\widetilde{\mathcal {K}}^{(i_1,i_2)}_{\nu ,\beta }$ is defined as

$$\begin{aligned} \widetilde{\mathcal {K}}^{(i_1,i_2)}_{\nu ,\beta }\equiv \int _{Y}{\mathbb {E}}_{\nu ^2}{\mathbf{K}}\left( \partial _{i_1}\widetilde{{\mathbb {X}}}_{\nu ,\beta }^{(i_2)}+({\mathcal {X}}_{Y_f}+{\mathbb {W}}_\beta {\mathcal {X}}_{Y_m})\delta _{i_1,i_2}\right) {\hbox {d}}y. \end{aligned}$$

(5.13)

$\widetilde{{\mathbb {X}}}^{(i_1)}_{\nu ,\beta },\widetilde{\mathbb {Y}}_{\nu ,\beta }^{(i_1,i_2)}$ in (5.11)–(5.12) are uniquely solvable by Lax–Milgram Theorem [16].

Lemma 5.1

Under A1–A2 and A4, the solution ${\mathbb {W}}_\beta$ of (2.7)–(2.8) satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathbb {W}}_\beta (x)\in [0,1] &{} \text{ for } \text{ any } x\in Y_m, \beta \in (0,\infty ),\\ \Vert {\mathbb {E}}_{\beta ^2}{\mathbb {W}}_\beta \Vert _{W^{2,p}(Y_f)\cap W^{2,p}(Y_m)}\le c &{} \text{ for } \text{ any } p\in (1,\infty ), \beta \in (0,\infty ),\\ \displaystyle \int _{Y_m}{\mathbf{T}}{\mathbb {W}}_\beta {\hbox {d}}y\le c\sqrt{|\beta \ln \beta |} &{} \text{ if } \beta \in (0,\beta ^*),\\ \end{array}\right. } \end{aligned}$$

(5.14)

where $\beta ^*$ is a constant depending on ${\mathbf{K}},{\mathbf{T}}$ and $diam|\Omega |$, and c is independent of $\beta$.

Proof

Corollary 3.2 [16] implies (5.14)$_1$. Theorems 9.11, 9.15 [16], extension method in Theorem 7.25 [16], and (5.14)$_1$ imply (5.14)$_2$. For any $x\in Y_m$, we set $\eta _x\equiv \min _{z\in \partial Y_m} |z-x|$ and $\xi _x\equiv \max _{z\in \partial Y_m} |z-x|$. Next we fix $t\in {\mathbb {R}}$ and $x\in Y_m$ as well as define $\varphi (y)\equiv \exp ((|y-x|^2-\eta _x^2)t)$ for $y\in Y_m$. Then $\varphi$ satisfies

$$\begin{aligned} \beta ^2\nabla \cdot ({\mathbf{K}}\nabla \varphi )-{\mathbf{T}}\varphi =\beta ^2\varphi (4t^2{\mathbf{K}}|y-x|^2+2t(y-x)\nabla {\mathbf{K}}+2t{\mathbf{K}}n-{\mathbf{T}}/\beta ^2). \end{aligned}$$

We find that there are $\beta ^*<1$ and $c^*>0$ (depending on ${\mathbf{K}},{\mathbf{T}},diam|\Omega |$ but not $x\in Y_m$) such that if $\beta \in (0,\beta ^*)$, $t= c^*/\beta$, and $x\in Y_m$, then

$$\begin{aligned} {\left\{ \begin{array}{ll} \beta ^2\nabla \cdot ({\mathbf{K}}\nabla \varphi )- {\mathbf{T}}\varphi \le 0 &{} \text{ in } B_{\xi _x}(x)\cap Y_m,\\ \varphi \ge 1 &{} \text{ in } Y_m\setminus B_{\eta _x}(x).\\ \end{array}\right. } \end{aligned}$$

(5.15)

Corollary 3.2 [16], (2.7), and (5.15) imply

$$\begin{aligned} 0\le {\mathbb {W}}_\beta \le \varphi \qquad \text{ on } Y_m. \end{aligned}$$

(5.16)

For any $\nu >0$, define $Y_m(\nu )\equiv \{y\in Y_m|\min _{z\in \partial Y_m}|z-y|\ge \nu \}$. By (5.16), it is easy to see that if $\beta \in (0,\beta ^*)$, then, for any $x\in Y_m(\sqrt{2|\beta \ln \beta |/c^*})$,

$$\begin{aligned} 0\le {\mathbb {W}}_\beta \le \beta \qquad \text{ on } B_{\eta _x/2}(x). \end{aligned}$$

(5.17)

So if $\beta \in (0,\beta ^*)$, $\displaystyle \int _{Y_m}{\mathbf{T}}{\mathbb {W}}_\beta {\hbox {d}}y\le c \sqrt{|\beta \ln \beta |}$ by (5.14)$_1$ and (5.17). $\square$

By Lemma 3.3, Lemma 5.1, and energy method, there is a $\omega _0\in (0,1)$ such that, for any $\nu \in (0,\omega _0)$ and $\beta \in (0,\infty )$,

$$\begin{aligned} {\left\{ \begin{array}{ll} \Vert \widetilde{{\mathbb {X}}}_{\nu ,\beta }^{(i_1)}\Vert _{W^{1,p}(Y_f)\cap W^{1,p}(Y_m)}, \Vert \widetilde{\mathbb {Y}}_{\nu ,\beta }^{(i_1,i_2)}\Vert _{W^{2,p}(Y_f)\cap W^{2,p}(Y_m)}\le c,\\ \displaystyle \Vert \widetilde{{\mathbb {X}}}^{(i_1)}_{\nu ,\beta },\widetilde{\mathbb {Y}}_{\nu ,\beta }^{(i_1,i_2)}\Vert _{C^{2,\alpha }(\overline{Y_f})\cap C^{2,\alpha }(\overline{Y_m})} \le \frac{c}{\beta ^2},\\ \displaystyle |\widetilde{\mathcal {K}}^{(i_1,i_2)}_{\nu ,\beta }-{\mathcal {K}}_0^{(i_1,i_2)}|\le c(\frac{\nu }{\beta }+\nu ^2) \text{ as } \nu \rightarrow 0,\\ |{\mathcal {T}}_\tau -{\mathcal {T}}_\sigma |\le {\left\{ \begin{array}{ll} c|\frac{\tau ^2}{\sigma ^2}-1| &{} \text{ if } \tau \rightarrow \sigma \in (0,\infty ),\\ c\sqrt{|\tau \ln \tau |} &{} \text{ if } \tau \rightarrow \sigma = 0,\\ \end{array}\right. } \end{array}\right. } \end{aligned}$$

(5.18)

where $p\in (n,\infty )$, $\alpha \in (0,1)$, ${\mathcal {K}}_0^{(i_1,i_2)}$ is the $(i_1,i_2)$th component of ${\mathcal {K}}_0$ (see (2.9)), and c is a constant independent of $\nu ,\beta ,\tau ,\sigma$. See (5.13) for $\widetilde{\mathcal {K}}^{(i_1,i_2)}_{\nu ,\beta }$, and see (2.11) for ${\mathcal {T}}_\sigma$. Define $\widetilde{{\mathbb {X}}}_{\omega ,\beta ,\epsilon }(x)\equiv \epsilon \widetilde{{\mathbb {X}}}_{\omega ,\beta }(\frac{x}{\epsilon })$ $\widetilde{\mathbb {Y}}_{\omega ,\beta }\equiv (\widetilde{\mathbb {Y}}_{\omega ,\beta }^{(i_1,i_2)})$ and $\widetilde{\mathbb {Y}}_{\omega ,\beta ,\epsilon }(x)\equiv \epsilon ^2\widetilde{\mathbb {Y}}_{\omega ,\beta }(\frac{x}{\epsilon })$ for $\beta \in (0,\infty ), \omega ,\epsilon \in (0,1), i_1,i_2\in {\mathbb {Z}}$.

Let $\Psi _{\omega ,\epsilon }$ be the solution of (1.1) with $V_{\omega ,\epsilon }=0$, $\Psi$ be the solution of (2.12), and

$$\begin{aligned} \Phi _{\omega ,\epsilon }\equiv \Psi _{\omega ,\epsilon }-({\mathcal {X}}_{\Omega _f^\epsilon }+{\mathbb {W}}_{\frac{\omega }{\epsilon },\epsilon ,0}{\mathcal {X}}_{\Omega _m^\epsilon })\Psi -{\mathbb {W}}_{\frac{\omega }{\epsilon },\epsilon ,2}\Psi {\mathcal {X}}_{\Omega _f^\epsilon } -\widetilde{{\mathbb {X}}}_{\omega ,\frac{\omega }{\epsilon },\epsilon }\nabla \Psi -\widetilde{\mathbb {Y}}_{\omega ,\frac{\omega }{\epsilon },\epsilon }\nabla ^2 \Psi \end{aligned}$$

in $\Omega$. See remark after (2.8) for ${\mathbb {W}}_{\frac{\omega }{\epsilon },\epsilon ,i}$. By (2.7)–(2.8), (5.11)–(5.12), (5.18), and Lemma 5.1 with $\beta =\frac{\omega }{\epsilon }$, we obtain

$$\begin{aligned}&{\left\{ \begin{array}{ll} -\nabla \cdot \big (\mathbf{E}_{\omega ^2,\epsilon }{\mathbf{K}}_\epsilon \big (\nabla \Phi _{\omega ,\epsilon } +{\mathbb {W}}_{\frac{\omega }{\epsilon },\epsilon ,2}\nabla \Psi {\mathcal {X}}_{\Omega _f^\epsilon }+\widetilde{\mathbb {Y}}_{\omega ,\frac{\omega }{\epsilon },\epsilon }\nabla ^3 \Psi \big )\big )\\ +{\mathbf{T}}_{\omega ,\epsilon }\Phi _{\omega ,\epsilon } =G_{\omega ,\epsilon }+\frac{\widetilde{\mathcal {K}}_{\omega ,\omega /\epsilon }}{|Y_f|}\nabla ^2\Psi {\mathcal {X}}_{\Omega _f^\epsilon } -{\mathcal {T}}_{\frac{\omega }{\epsilon }}\Psi {\mathcal {X}}_{\Omega _f^\epsilon }+{\mathcal {O}}_1(\epsilon ) &{} \text{ in } \Omega \setminus \partial \Omega _m^\epsilon ,\\ \lfloor \mathbf{E}_{\omega ^2,\epsilon }{\mathbf{K}}_\epsilon \big (\nabla \Phi _{\omega ,\epsilon }+ {\mathbb {W}}_{\frac{\omega }{\epsilon },\epsilon ,2}\nabla \Psi {\mathcal {X}}_{\Omega _f^\epsilon }+\widetilde{\mathbb {Y}}_{\omega ,\frac{\omega }{\epsilon },\epsilon }\nabla ^3 \Psi \big )\rfloor \cdot \vec {{\mathbf{n}}}^\epsilon =0 &{} \text{ on } \partial \Omega _m^\epsilon ,\\ \lfloor \Phi _{\omega ,\epsilon } \rfloor = -{\mathbb {W}}_{\frac{\omega }{\epsilon },\epsilon ,2}\Psi {\mathcal {X}}_{\Omega _f^\epsilon } &{} \text{ on } \partial \Omega _m^\epsilon ,\\ \Phi _{\omega ,\epsilon }={\mathcal {O}}_2(\epsilon ) &{} \text{ on } \partial \Omega ,\\ \end{array}\right. } \end{aligned}$$

(5.19)

where ${\mathcal {O}}_1(\nu ), {\mathcal {O}}_2(\nu )$ are same as those in Part 1. See (2.2) for (5.19)$_{2,3}$. Let us define

$$\begin{aligned} \varphi _{\omega ,\epsilon }\equiv {\left\{ \begin{array}{ll} \Phi _{\omega ,\epsilon }+{\mathbb {W}}_{\frac{\omega }{\epsilon },\epsilon ,2}\Psi {\mathcal {X}}_{\Omega _f^\epsilon } &{} \text{ on } \Omega _f^\epsilon ,\\ \Phi _{\omega ,\epsilon } &{} \text{ on } \Omega _m^\epsilon .\\ \end{array}\right. } \end{aligned}$$

By (2.12)–(2.13), (5.18), and Lemma 5.1, $\varphi _{\omega ,\epsilon }$ satisfies

$$\begin{aligned}&{\left\{ \begin{array}{ll} -\nabla \cdot \big (\mathbf{E}_{\omega ^2,\epsilon }{\mathbf{K}}_\epsilon \big (\nabla \varphi _{\omega ,\epsilon } +{\mathcal {O}}_1(\epsilon )\big )\big ) +{\mathbf{T}}_{\omega ,\epsilon }\varphi _{\omega ,\epsilon } =G_{\omega ,\epsilon }-{\mathcal {G}}{\mathcal {X}}_{\Omega _f^\epsilon }+\tilde{{\mathcal {X}}} &{} \text{ in } \Omega ,\\ \varphi _{\omega ,\epsilon }={\mathcal {O}}_2(\epsilon ) &{} \text{ on } \partial \Omega ,\\ \end{array}\right. } \end{aligned}$$

where

$$\begin{aligned} \tilde{{\mathcal {X}}}\equiv {\left\{ \begin{array}{ll} {\mathcal {O}}_1(\epsilon +|\frac{\omega ^2}{\sigma ^2\epsilon ^2}-1|) &{} \text{ if } \frac{\omega }{\epsilon }\rightarrow \sigma \in (0,\infty ),\\ {\mathcal {O}}_1(\epsilon +|\frac{\omega }{\epsilon }\ln \frac{\omega }{\epsilon }|^{\frac{1}{2}}) &{} \text{ if } \frac{\omega }{\epsilon }\rightarrow 0.\\ \end{array}\right. } \end{aligned}$$

We write the $\varphi _{\omega ,\epsilon }$ as $\varphi _{\omega ,\epsilon }=\check{\varphi }_{\omega ,\epsilon }+\widehat{\varphi }_{\omega ,\epsilon }$, where $\check{\varphi }_{\omega ,\epsilon }$ satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} -\nabla \cdot (\mathbf{E}_{\omega ^2,\epsilon }{\mathbf{K}}_\epsilon \nabla \check{\varphi }_{\omega ,\epsilon })+{\mathbf{T}}_{\omega ,\epsilon }\check{\varphi }_{\omega ,\epsilon }=0 &{} \text{ in } \Omega ,\\ \check{\varphi }_{\omega ,\epsilon } ={\mathcal {O}}_2(\epsilon ) &{} \text{ on } \partial \Omega ,\\ \end{array}\right. } \end{aligned}$$

(5.20)

and $\widehat{\varphi }_{\omega ,\epsilon }$ satisfies

$$\begin{aligned}&{\left\{ \begin{array}{ll} -\nabla \cdot \big (\mathbf{E}_{\omega ^2,\epsilon }{\mathbf{K}}_\epsilon \big (\nabla \widehat{\varphi }_{\omega ,\epsilon } +{\mathcal {O}}_1(\epsilon ) \big )\big ) +{\mathbf{T}}_{\omega ,\epsilon }\widehat{\varphi }_{\omega ,\epsilon } =G_{\omega ,\epsilon }-{\mathcal {G}}{\mathcal {X}}_{\Omega _f^\epsilon }+\tilde{{\mathcal {X}}} &{} \text{ in } \Omega ,\\ \widehat{\varphi }_{\omega ,\epsilon }= 0 &{} \text{ on } \partial \Omega .\\ \end{array}\right. } \end{aligned}$$

(5.21)

By Theorem 8.1 [16], the solution of (5.20) satisfies

$$\begin{aligned} \Vert \check{\varphi }_{\omega ,\epsilon }\Vert _{L^\infty (\Omega )} =\Vert \check{\varphi }_{\omega ,\epsilon }\Vert _{L^\infty (\partial \Omega )}\le c\epsilon , \end{aligned}$$

(5.22)

where c is independent of $\omega ,\epsilon$. Next we consider (5.21) for $\frac{\omega }{\epsilon }\rightarrow \sigma \in (0,\infty )$ and $\frac{\omega }{\epsilon }\rightarrow 0$ separately.

Case 1: $\frac{\omega }{\epsilon }\rightarrow \sigma \in (0,\infty )$. By Theorem 2.1 and A6, the solution of (5.21) satisfies

$$\begin{aligned} \Vert \mathbf{E}_{\omega ,\epsilon }\widehat{\varphi }_{\omega ,\epsilon }\Vert _{L^\infty (\Omega )} \le c(\max \{\omega ,\epsilon ,|\omega ^2/(\sigma \epsilon )^2-1|\}+\Vert G_{\omega ,\epsilon }-{\mathcal {G}}\Vert _{L^{n+\delta }(\Omega _f^\epsilon )}), \end{aligned}$$

(5.23)

where c is independent of $\omega ,\epsilon$. Employing (5.22)–(5.23) and modifying the argument of Part 1, we obtain Theorem 2.3 for $\omega ,\epsilon \rightarrow 0, \frac{\omega }{\epsilon }\rightarrow \sigma \in (0,\infty )$ case.

Case 2: $\frac{\omega }{\epsilon }\rightarrow 0$. If $\widehat{\varphi }_{\omega ,\epsilon }$ is the solution of (5.21), we multiply (5.21) by $|\widehat{\varphi }_{\omega ,\epsilon }|^{n+\delta -2}\widehat{\varphi }_{\omega ,\epsilon }$ and integrate by part to see, by A6,

$$\begin{aligned} \Vert \widehat{\varphi }_{\omega ,\epsilon }\Vert _{L^{n+\delta }(\Omega )} \le c(\max \{\omega ,\epsilon ,|\frac{\omega }{\epsilon }\ln \frac{\omega }{\epsilon }|^{\frac{1}{2}}\}+\Vert G_{\omega ,\epsilon }-{\mathcal {G}}\Vert _{L^{n+\delta }(\Omega _f^\epsilon )}), \end{aligned}$$

(5.24)

where c is independent of $\omega ,\epsilon$. By (5.22) and (5.24), we obtain Theorem 2.3 for $\omega ,\epsilon \rightarrow 0, \frac{\omega }{\epsilon }\rightarrow 0$ case.

5.2 Proof of Theorem 2.4

The proof is similar to that of Theorem 2.3. Let us assume that A1–A3, A4’, A6, and A7 hold. For each $\nu \in (0,1)$, we find $\breve{{\mathbb {W}}}_\nu \in H^1_{per}({\mathbb {R}}^n)$ satisfying

$$\begin{aligned}&{\left\{ \begin{array}{ll} \nabla \cdot ({\mathbb {E}}_{\nu ^2}{\mathbf{K}}\nabla \breve{{\mathbb {W}}}_\nu )=({\mathbb {E}}_{\nu }{\mathbf{P}}-\breve{{\mathcal {T}}}) {\mathcal {X}}_{Y_f} &{} \text{ in } Y,\\ \displaystyle \int _{Y_f}\breve{\mathbb {W}}_\nu (y) {\hbox {d}}y=0.\\ \end{array}\right. } \end{aligned}$$

(5.25)

See A7 for ${\mathbf{P}}$, and (2.15) for constant $\breve{{\mathcal {T}}}$. By Lax–Milgram Theorem [16], Lemma 3.3, and A7, $\breve{{\mathbb {W}}}_\nu$ is uniquely solvable and

$$\begin{aligned} \Vert \breve{{\mathbb {W}}}_\nu \Vert _{C^2(\overline{Y_f})\cap C^2(\overline{Y_m})}\le c, \end{aligned}$$

(5.26)

where c is a constant independent of $\nu$. Define $\breve{{\mathbb {W}}}_{\nu ,s}(x)\equiv s^2\breve{{\mathbb {W}}}_\nu (\frac{x}{s})$ for $\nu ,s\in (0,1)$.

Let $\Psi _{\omega ,\epsilon }$ be the solution of (1.1) with $V_{\omega ,\epsilon }=0$, $\Psi$ be the solution of (2.16), and

$$\begin{aligned} \varphi _{\omega ,\epsilon }\equiv \Psi _{\omega ,\epsilon }-\Psi -\breve{{\mathbb {W}}}_{\omega ,\epsilon }\Psi -{{\mathbb {X}}}_{\omega ,\epsilon }\nabla \Psi -{\mathbb {Y}}_{\omega ,\epsilon }\nabla ^2 \Psi \qquad \text{ in } \Omega . \end{aligned}$$

See (2.5) for ${{\mathbb {X}}}_{\omega ,\epsilon }$ and (5.1) for ${\mathbb {Y}}_{\omega ,\epsilon }$. By (2.16), (3.17), (3.19), (5.1)–(5.2), and (5.25)–(5.26), we obtain

$$\begin{aligned}&{\left\{ \begin{array}{ll} -\nabla \cdot \big (\mathbf{E}_{\omega ^2,\epsilon }{\mathbf{K}}_\epsilon \big (\nabla \varphi _{\omega ,\epsilon }+{\mathcal {O}}_1(\epsilon ^2) \big )\big )+{\mathbf{T}}_{\omega ,\epsilon }\varphi _{\omega ,\epsilon }\\ ={\mathcal {O}}_1(\epsilon +\omega ){\mathcal {X}}_{\Omega _f^\epsilon }+{\mathcal {O}}_1(\omega ){\mathcal {X}}_{\Omega _m^\epsilon }+(G_{\omega ,\epsilon }-{\mathcal {G}}){\mathcal {X}}_{\Omega _f^\epsilon } &{} \text{ in } \Omega ,\\ \varphi _{\omega ,\epsilon }={\mathcal {O}}_2(\epsilon ) &{} \text{ on } \partial \Omega ,\\ \end{array}\right. } \end{aligned}$$

where ${\mathcal {O}}_1(\nu ), {\mathcal {O}}_2(\nu )$ are defined as those in Part 1. See (2.15) for ${\mathcal {G}}$. Modifying the argument of Part 1 in Sect. 5.1 and employing Theorem 2.2, we obtain Theorem 2.4.

5.3 Proof of Theorem 2.5

We assume A1–A3 and A6. Let $\Psi _{\omega ,\epsilon }$ be the solution of (1.1) with $V_{\omega ,\epsilon }={\mathbf{T}}_{\omega ,\epsilon }=0$, $\Psi$ be the solution of (2.18), and define

$$\begin{aligned} \varphi _{\omega ,\epsilon }\equiv \Psi _{\omega ,\epsilon }-\Psi -{{\mathbb {X}}}_{\omega ,\epsilon }\nabla \Psi -{\mathbb {Y}}_{\omega ,\epsilon }\nabla ^2 \Psi \qquad \text{ in } \Omega . \end{aligned}$$

See (2.5) for ${{\mathbb {X}}}_{\omega ,\epsilon }$ and (5.1) for ${\mathbb {Y}}_{\omega ,\epsilon }$. By (2.18), (3.17), (3.19), and (5.2), $\varphi _{\omega ,\epsilon }$ satisfies

$$\begin{aligned}&{\left\{ \begin{array}{ll} -\nabla \cdot \big (\mathbf{E}_{\omega ^2,\epsilon }{\mathbf{K}}_\epsilon \big (\nabla \varphi _{\omega ,\epsilon }+{\mathcal {O}}_1(\epsilon ^2) \big )\big )\\ ={\mathcal {O}}_1(\epsilon +\omega ){\mathcal {X}}_{\Omega _f^\epsilon }+{\mathcal {O}}_1(\omega ){\mathcal {X}}_{\Omega _m^\epsilon }+(G_{\omega ,\epsilon }-{\mathcal {G}}){\mathcal {X}}_{\Omega _f^\epsilon } &{} \text{ in } \Omega ,\\ \varphi _{\omega ,\epsilon }={\mathcal {O}}_2(\epsilon ) &{} \text{ on } \partial \Omega .\\ \end{array}\right. } \end{aligned}$$

Modifying the argument of Part 1 in Sect. 5.1 and employing Theorem 2.2, we obtain Theorem 2.5.