Convergence rates in homogenization of p-Laplace equations

Abstract

This paper is concerned with homogenization of p-Laplace equations with rapidly oscillating periodic coefficients. The main difficulty of this work is due to the nonlinear structure in this field concerning p-Laplace equations itself. Utilizing the layer and co-layer type estimates as well as homogenization techniques, we establish the desired error estimates. As a consequence, we obtain the rates of convergence for solutions in $W_{0}^{1,p}$ as well as $L^{p}$. Meanwhile, our convergence rate results do not involve the higher derivatives any more. This may be viewed as rather surprising. The novelty of this work is that it seems to find a new analysis method in quantitative homogenization.

1 Introduction

In this paper, we shall establish the rates of convergence for p-Laplace equations with rapidly oscillating periodic coefficients. More precisely, let Ω be a bounded Lipschitz domain in $\mathbb{R} ^{n}$. Suppose that $u_{\varepsilon }\in W^{1,p}(\varOmega )$, for any $1\leq p<\infty$, is a weak solution to the following problem:

$$ \textstyle\begin{cases} L_{\varepsilon }u_{\varepsilon }=-\operatorname{div} (A(x/\varepsilon ) \vert \triangledown u_{\varepsilon } \vert ^{p-2}\triangledown u_{\varepsilon } ) =F& \mbox{in } \varOmega , \\ u_{\varepsilon } =f& \mbox{on }\partial \varOmega . \end{cases} $$

(1.1)

Throughout this paper, the summation convention is used. We assume that the matrix $A(y)=(a_{ij}(y))$ with $1\leq i$, $j\leq n$, is real, bounded measurable, and satisfies the following conditions.

• Periodicity conditions: for any $y\in \mathbb{R}^{n}$ and $Y=[0,1)^{n}\simeq \mathbb{R}^{n}/\mathbb{Z}^{n}$,

  $$ A(y+Y)=A(y). $$

  (1.2)

• Coerciveness and growth conditions: there exists a $\lambda >0$, for any $y\in \mathbb{R}^{n}$ and $\xi ,\xi '\in \mathbb{R}^{n}$,

  $$\begin{aligned} \lambda \bigl( \vert \xi \vert + \bigl\vert \xi ' \bigr\vert \bigr)^{p-2} \bigl\vert \xi -\xi ' \bigr\vert ^{2}&\leq \bigl\langle A(y) \vert \xi \vert ^{p-2}\xi -A(y) \bigl\vert \xi ' \bigr\vert ^{p-2}\xi ',\xi -\xi '\bigr\rangle \\ &\leq \frac{1}{ \lambda }\bigl( \vert \xi \vert + \bigl\vert \xi ' \bigr\vert \bigr)^{p-2} \bigl\vert \xi -\xi ' \bigr\vert ^{2}. \end{aligned}$$

  (1.3)

• Smoothness conditions: with $1/p+1/p'=1$,

  $$ F\in W^{-1,p'}(\varOmega ), \qquad f\in W^{1,p}( \partial \varOmega ). $$

  (1.4)

It is well known that the solution $u_{\varepsilon }\rightharpoonup u _{0}$ weakly in $W^{1,p}(\varOmega )$, as $\varepsilon \rightarrow 0$, where $u_{0}$ is the solution to the homogenized problem

$$ \textstyle\begin{cases} L_{0}u_{0} =-\operatorname{div} (Q \vert \triangledown u_{0} \vert ^{p-2} \triangledown u_{0} ) =F & \mbox{in } \varOmega , \\ u_{0} =f& \mbox{on }\partial \varOmega . \end{cases} $$

(1.5)

The Q is a constant matrix, defined by

$$ Q= \int _{Y} \bigl[A(y) \bigl\vert \triangledown \chi (y)+1 \bigr\vert ^{p-2}\bigl(\triangledown \chi (y)+1\bigr) \bigr]\,dy, $$

(1.6)

where the corrector $\chi (y)$ satisfies the following cell problem:

$$ \textstyle\begin{cases} \operatorname{div} [A(y) \vert \triangledown \chi (y)+1 \vert ^{p-2}(\triangledown \chi +1) ]=0 & \mbox{in } Y, \\ \int _{Y}\chi (y)\,dy=0. \end{cases} $$

(1.7)

Recently, there has been published much classical work about convergence of solutions for linear operators in homogenization with the various settings. In 2011, Gérard and Masmoudi [4] obtained the $L^{2}$ convergence rate for the boundary layers Neumann problems. In 2012, Kenig, Lin and Shen [7] obtained $L^{2}$ as well as $H^{\frac{1}{2}}$ convergence rates for the elliptic oscillating operators. In 2013, Aleksanyan, Shahgholian and Sjölin [1, 2] proved pointwise as well as $L^{p}$ estimates for fixed operators and oscillating Dirichlet boundary data. In 2014, Kenig, Lin and Shen [8] established $W^{k,p}$ convergence rates, via the asymptotic behavior of the Green or Neumann functions. In 2015, the first author [24] obtained the pointwise as well as $W^{1,p}$ convergence rates for fixed operators and oscillating Neumann boundary data. In 2015, Gu [5] also proved convergence rates in $L^{2}$ and $H^{1}$ for linear Stokes systems. In 2016, Shen [18] proved the $L^{2}$ convergence rate for the mixed Dirichlet–Neumann boundary value problems. In 2018, Niu and Xu [11] got the $L^{2}$ convergence rate for 2mth-order equations with periodic oscillating coefficients.

The nonlinear operators case in homogenization have also been studied extensively. Piat and Deferanceschi [15] have obtained convergence weakly in $W^{1,p}$ for the quasi-linear monotone operator. Pastukhova [14] considered nonlinear equations of monotone type with multiscale coefficients, and established the $L^{2}$ convergence rate. Recently, Wang, Xu and Zhao [21] studied the quasilinear elliptic equations and obtained an error estimate in $L^{2}$. We refer the reader to see [3, 6, 10, 16, 23] and their references for more results about nonlinear problems in homogenization.

The motivation for studying this paper is inspired by the problems raised by Wang, Xu and Zhao in [22] for the p-Laplace type equations. The aim of the paper is to obtain the accurate convergence rates of solutions for the classical p-Laplace equations with rapidly oscillating periodic coefficients. Thanks to the layer and co-layer type estimates, we could handle the different ingredients in the integral by energy methods. The similar procedures could be found in [9] or [12], which were used to analyze the spatial and mechanical properties for solutions of reflecting the microstructure of the materials.

The following are the main results of this paper.

Theorem 1

Let Ω be a bounded Lipschitz domain in $\mathbb{R}^{n}$. Suppose that $u_{\varepsilon }\in W^{1,p}(\varOmega )$ and $u_{0}\in W^{1,q}(\varOmega )$, with $q>p\geq 4$, are the weak solutions of the problems (1.1) and (1.5), respectively. Then, under the assumptions (1.2)–(1.4), there exists a constant C such that

$$ \bigl\Vert u_{\varepsilon }-u_{0}-\varepsilon \chi T_{\varepsilon }( \eta _{\varepsilon }\triangledown u_{0}) \bigr\Vert _{W_{0}^{1,p}(\varOmega )}\leq C\varepsilon ^{\frac{1}{p}-\frac{1}{q}} \Vert \triangledown u_{0} \Vert _{L^{q}(\varOmega )}, $$

where $T_{\varepsilon }$ is the smoothing operator, and $\eta _{\varepsilon }$ is a cut-off function.

Theorem 2

Under the same conditions as Theorem 1, then there exists a constant C, with some $q>p\geq 4$, such that

$$ \Vert u_{\varepsilon }-u_{0} \Vert _{L^{p}(\varOmega )}\leq C \varepsilon ^{\frac{1}{p}-\frac{1}{q}} \Vert \triangledown u_{0} \Vert _{L^{q}(\varOmega )}. $$

The astute reader may have already noticed that our convergence rate result in Theorem 1, which do not involve the higher derivatives any more. This may be viewed as rather surprising, even though in the linear case. The novelty of this work is that it seems to find a new analysis method, which depends on the layer and co-layer type estimates, in quantitative homogenization. To the best of our knowledge, there are few contributions in the field concerning p-Laplace equations in homogenization.

The rest of the paper is organized as follows. Section 2 contains some basic notations and useful propositions which will play crucial roles to obtain convergence rates. In Sect. 3, we show that the solution $u_{\varepsilon }$ of p-Laplace equations is convergent to the solution $u_{0}$ of the corresponding homogenized problems, this is based on the energy method as well as using homogenization tools.

2 Preliminaries

We begin by specifying our notations.

Let $B_{r}(x)$ denote an open ball with center x and radius r. $\varOmega _{\varepsilon }=\{x\in \varOmega : \operatorname{dist}(x,\partial \varOmega )> \varepsilon \}$, we also call it the co-layer part of Ω, associated with so-called layer part is denoted by $\varOmega \setminus \varOmega _{\varepsilon }$. Set $\eta _{\varepsilon }\in C_{0}^{\infty }( \varOmega )$ is a cut-off function, satisfying $\eta _{\varepsilon }=1$ in $\varOmega _{\varepsilon }$, $\eta _{\varepsilon }=0$ outside $\varOmega _{\varepsilon }$ and $|\triangledown \eta _{\varepsilon }|\leq C/ \varepsilon$. In the whole paper, we use C to denote positive constant which may vary in different formulas.

Proposition 2.1

Let $F=(F_{1},F_{2},\ldots ,F_{n})\in L^{p}(Y)$. Suppose that $\int _{Y}F_{j}(y)\,dy=0$ and $\operatorname{div}F(y)=0$ in Y. Then there exists $\varPhi _{ij}\in W^{1,p}(Y)$ such that $\varPhi _{ij}=-\varPhi _{ji}$ and $F_{j}=\frac{\partial \varPhi _{ij}}{\partial y_{i}}$.

This proposition is well known. It is called the technique of flux correctors. The linear operator case is well known (see for example [7], Lemma 3.1). Let $f_{j}\in W^{2,p}(Y)$ be the solution to the cell problem $\triangle f_{j}=F_{j}$ in Y. Then we could define $\varPhi _{ij}(y)=\frac{\partial }{\partial y_{i}}[f_{j}(y)]-\frac{\partial }{\partial y_{j}}[f_{i}(y)]$. From an energy estimate, we may get the desired properties.

Recently, the smoothing operator was introduced by Suslina in [19, 20]. Meanwhile, applying the smoothing operator to get error estimates was first established by Shen in [17]. Next, we will introduce the smoothing operator and its properties. This work is to extend the usage of smoothing operator to the case of p-Laplace equations, of independent interest itself.

Definition 2.2

Fix $\psi \in C^{\infty }_{0}(B_{1}(0))$ such that $\psi \geq 0$ and $\int _{\mathbb{R}^{n}}\psi \,dx=1$. Define operator $T_{\varepsilon }$ on $L^{2}$ as

$$ T_{\varepsilon }(u) (x)=u\ast \psi _{\varepsilon }= \int _{\mathbb{R}^{n}}u(x-y) \psi _{\varepsilon }(y)\,dy, $$

where $\psi _{\varepsilon }(x)=\varepsilon ^{-n}\psi (x/\varepsilon )$. We call it the smoothing operator.

Proposition 2.3

Let $u_{0}\in W^{1,p}(\varOmega )$ and a periodic function $f\in L^{p}(Y)$, for some $1< p<\infty$. Then we have

$$ \bigl\Vert f(\cdot /\varepsilon )T_{\varepsilon }(u_{0}) \bigr\Vert _{L^{p}(\varOmega )}\leq C \Vert f \Vert _{L^{p}(Y)} \Vert u_{0} \Vert _{L^{p}(\varOmega )} $$

and

$$ \bigl\Vert u_{0}-T_{\varepsilon }(\triangledown u_{0}) \bigr\Vert _{L^{p}(\varOmega _{\varepsilon })} \leq C\varepsilon \Vert \triangledown u_{0} \Vert _{L^{p}(\varOmega _{\varepsilon })}. $$

These estimates could be proved by Fubini’s theorem and Hölder’s inequality. We refer the reader to [13, 17] or [18] for the detailed proof.

The main interest of the present work is to attempt to find a new approach to analyzing the error estimates for homogenization problems. Fortunately, it may be a new way to derive rates of convergence, via the co-layer and layer type estimates.

Proposition 2.4

(Co-layer and layer type estimates)

If $u_{0}\in W^{1,p}( \varOmega )$ for some $q>p>1$, then we have estimates

$$\begin{aligned}& \int _{\varOmega \setminus \varOmega _{\varepsilon }} \vert \triangledown u _{0} \vert ^{p}\,dx\leq C\varepsilon ^{1-\frac{p}{q}} \biggl( \int _{\varOmega } \vert \triangledown u_{0} \vert ^{q}\,dx \biggr)^{\frac{p}{q}}, \\& \int _{\varOmega _{\varepsilon }} \bigl\vert \triangledown ^{2} u_{0} \bigr\vert ^{2} \vert \cdot \triangledown u_{0} \vert ^{p-2}\,dx\leq C \varepsilon ^{-1-\frac{p}{q}} \biggl( \int _{\varOmega } \vert \triangledown u _{0} \vert ^{q}\,dx \biggr)^{\frac{p}{q}}, \end{aligned}$$

and

$$ \int _{\varOmega _{\varepsilon }} \bigl\vert \triangledown ^{2} u_{0} \bigr\vert ^{p} \,dx \leq C\varepsilon ^{1-\frac{p}{q}-p} \biggl( \int _{\varOmega } \vert \triangledown u_{0} \vert ^{q}\,dx \biggr)^{\frac{p}{q}}. $$

These estimates will play crucial roles to obtain convergence rates in the present paper, and they also do not involve the higher derivatives any more. These estimates could be derived by regularity estimates, and they may be found in [22].

3 Proofs of theorems

The goal of this section is to establish $W_{0}^{1,p}$ and $L^{p}$ convergence rates of solutions for the p-Laplace equations in homogenization.

Set the first-order approximation term

$$ v_{\varepsilon }= u_{0}+\varepsilon \chi T_{\varepsilon }( \eta _{\varepsilon }\triangledown u_{0}). $$

We find that

$$ \triangledown v_{\varepsilon }=\triangledown u_{0}+ \triangledown \chi T_{\varepsilon }(\eta _{\varepsilon }\triangledown u_{0})+ \varepsilon \chi T_{\varepsilon }\bigl(\eta _{\varepsilon }\triangledown ^{2} u_{0}\bigr)+ \varepsilon \chi T_{\varepsilon }( \triangledown \eta _{\varepsilon } \triangledown u_{0}). $$

In view of the fact that, for any $\varphi \in C_{0}^{\infty }(\varOmega )$,

$$ \int _{\varOmega }A(x/\varepsilon ) \vert \triangledown u_{\varepsilon } \vert ^{p-2} \triangledown u_{\varepsilon } \cdot \triangledown \varphi \,dx= \int _{\varOmega }Q \vert \triangledown u_{0} \vert ^{p-2}\triangledown u_{0} \cdot \triangledown \varphi \,dx, $$

we obtain

$$\begin{aligned}& \int _{\varOmega } \bigl[A(x/\varepsilon ) \vert \triangledown u_{\varepsilon } \vert ^{p-2} \triangledown u_{\varepsilon } -A(x/\varepsilon ) \vert \triangledown v_{ \varepsilon } \vert ^{p-2}\triangledown v_{\varepsilon } \bigr] \cdot \triangledown \varphi \,dx \\& \quad = \int _{\varOmega } \bigl[Q \vert \triangledown u_{0} \vert ^{p-2}\triangledown u_{0}-Q \bigl\vert T _{\varepsilon }(\eta _{\varepsilon }\triangledown u_{0}) \bigr\vert ^{p-2}T_{ \varepsilon }(\eta _{\varepsilon }\triangledown u_{0}) \bigr]\cdot \triangledown \varphi \,dx \\& \qquad {}+ \int _{\varOmega } \bigl[Q \bigl\vert T_{\varepsilon }(\eta _{\varepsilon }\triangledown u_{0}) \bigr\vert ^{p-2}T_{\varepsilon }(\eta _{\varepsilon }\triangledown u_{0})-A(x/ \varepsilon ) \bigl\vert T_{\varepsilon }(\eta _{\varepsilon }\triangledown u_{0}) \bigr\vert ^{p-2} \\& \qquad {}\cdot T _{\varepsilon }(\eta _{\varepsilon }\triangledown u_{0}) \vert \triangledown \chi +1 \vert ^{p-2}(\triangledown \chi +1) \bigr]\cdot \triangledown \varphi \,dx \\& \qquad {}+ \int _{\varOmega } \bigl[A(x/\varepsilon ) \bigl\vert T_{\varepsilon }( \eta _{\varepsilon }\triangledown u_{0}) \bigr\vert ^{p-2}T_{\varepsilon }( \eta _{\varepsilon }\triangledown u_{0}) \vert \triangledown \chi +1 \vert ^{p-2}( \triangledown \chi +1) \\& \qquad {}-A(x/\varepsilon ) \vert \triangledown v_{\varepsilon } \vert ^{p-2}\triangledown v_{\varepsilon } \bigr] \cdot \triangledown \varphi \,dx \\& \quad \doteq I_{1}+I_{2}+I_{3}. \end{aligned}$$

(3.1)

To estimate $I_{1}$, we note Proposition 2.3, and Proposition 2.4 for the co-layer and layer type estimates, showing that

$$\begin{aligned} \vert I_{1} \vert \leq& C \int _{\varOmega } \bigl\vert \triangledown u_{0}-T_{\varepsilon }( \eta _{\varepsilon }\triangledown u_{0}) \bigr\vert \bigl( \vert \triangledown u_{0} \vert + \bigl\vert T_{\varepsilon }(\eta _{\varepsilon }\triangledown u_{0}) \bigr\vert \bigr)^{p-2}\cdot \vert \triangledown \varphi \vert \,dx \\ \leq& C \int _{\varOmega \setminus \varOmega _{\varepsilon }} \vert \triangledown u _{0} \vert ^{p-1} \vert \triangledown \varphi \vert \,dx+C \int _{\varOmega _{\varepsilon }} \bigl\vert \triangledown u_{0}-T_{\varepsilon }( \eta _{\varepsilon }\triangledown u_{0}) \bigr\vert \vert \triangledown u_{0} \vert ^{p-2} \vert \triangledown \varphi \vert \,dx \\ \leq& C \biggl( \int _{\varOmega \setminus \varOmega _{\varepsilon }} \vert \triangledown u_{0} \vert ^{p}\,dx \biggr)^{1-\frac{1}{p}}+C \int _{\varOmega _{\varepsilon }} \bigl\vert \eta _{\varepsilon }\triangledown u_{0}-T_{\varepsilon }( \eta _{\varepsilon }\triangledown u_{0}) \bigr\vert \vert \triangledown u_{0} \vert ^{p-2} \vert \triangledown \varphi \vert \,dx \\ \leq& C \biggl( \int _{\varOmega \setminus \varOmega _{\varepsilon }} \vert \triangledown u_{0} \vert ^{p}\,dx \biggr)^{1-\frac{1}{p}}+C \biggl( \int _{\varOmega _{\varepsilon }} \bigl\vert \eta _{\varepsilon }\triangledown u_{0}-T_{\varepsilon }( \eta _{\varepsilon }\triangledown u_{0}) \bigr\vert ^{\frac{p}{p-1}} \vert \triangledown u_{0} \vert ^{\frac{p(p-2)}{p-1}}\,dx \biggr)^{1-\frac{1}{p}} \\ \leq& C \biggl( \int _{\varOmega \setminus \varOmega _{\varepsilon }} \vert \triangledown u_{0} \vert ^{p}\,dx \biggr)^{1-\frac{1}{p}}+C\varepsilon \biggl( \int _{ \varOmega _{\varepsilon }} \bigl\vert \triangledown ^{2} u_{0} \bigr\vert ^{\frac{p}{p-1}} \vert \triangledown u_{0} \vert ^{\frac{p(p-2)}{p-1}}\,dx \biggr)^{1-1/p} \\ \leq& C \biggl( \int _{\varOmega \setminus \varOmega _{\varepsilon }} \vert \triangledown u_{0} \vert ^{p}\,dx \biggr)^{1-\frac{1}{p}}+C\varepsilon \biggl( \int _{ \varOmega _{\varepsilon }} \bigl\vert \triangledown ^{2} u_{0} \bigr\vert ^{2} \vert \triangledown u _{0} \vert ^{p-2}\,dx \biggr)^{\frac{1}{2}} \biggl( \int _{\varOmega } \vert \triangledown u_{0} \vert ^{p}\,dx \biggr)^{\frac{p-2}{2p}} \\ \leq& C\varepsilon ^{(1-\frac{1}{p})(1-\frac{p}{q})} \biggl( \int _{\varOmega } \vert \triangledown u_{0} \vert ^{q}\,dx \biggr)^{\frac{p-1}{q}}+C \varepsilon ^{\frac{1}{2}-\frac{p}{2q}} \biggl( \int _{\varOmega } \vert \triangledown u_{0} \vert ^{q}\,dx \biggr)^{\frac{p}{2q}} \biggl( \int _{\varOmega } \vert \triangledown u_{0} \vert ^{q}\,dx \biggr)^{\frac{p-2}{2q}} \\ \leq& C\varepsilon ^{\frac{1}{2}(1-\frac{p}{q})} \biggl( \int _{\varOmega } \vert \triangledown u_{0} \vert ^{q}\,dx \biggr)^{\frac{p-1}{q}} \end{aligned}$$

(3.2)

for some $q>p\geq 2$, where we have used the Hölder inequality.

Next, we shall estimate $I_{2}$. Let

$$ F(y,\xi )=Q \bigl\vert T_{\varepsilon }(\xi ) \bigr\vert ^{p-2}T_{\varepsilon }(\xi )-A(y) \bigl\vert T _{\varepsilon }( \xi ) \bigr\vert ^{p-2}T_{\varepsilon }(\xi ) \bigl\vert \chi (y)+1 \bigr\vert ^{p-2}\bigl( \triangledown \chi (y)+1\bigr). $$

Note that $F(\cdot ,\xi )$ is a periodic function with the first variable and satisfies the conditions of Proposition 2.1. Then there exists $\varPhi (\cdot ,\xi )$, such that $\varPhi _{ij}=-\varPhi _{ji}$, and

$$ Q \bigl\vert T_{\varepsilon }(\xi ) \bigr\vert ^{p-2}T_{\varepsilon }( \xi )-A(y) \bigl\vert T_{\varepsilon }(\xi ) \bigr\vert ^{p-2}T_{\varepsilon }( \xi ) \bigl\vert \chi (y)+1 \bigr\vert ^{p-2}\bigl(\triangledown \chi (y)+1\bigr)=\operatorname{div}_{y} \varPhi (y,\xi ). $$

Thus, it gives

$$\begin{aligned} I_{2} =& \int _{\varOmega }F(x,x/\varepsilon )\cdot \triangledown \varphi \,dx \\ =& \int _{\varOmega }\operatorname{div}_{y} \varPhi (y,\xi ) \cdot \triangledown \varphi \,dx \\ =& \int _{\varOmega }\frac{\partial }{\partial y_{i}} \bigl( \varPhi _{ij}(y, \xi ) \bigr)\cdot \frac{\partial \varphi }{\partial x_{j}}\,dx \\ =&- \int _{\varOmega }\frac{\partial }{\partial x_{i}} \bigl( \varepsilon \varPhi _{ij}(x,x/\varepsilon ) \bigr)\cdot \frac{\partial \varphi }{ \partial x_{j}}\,dx+ \int _{\varOmega }\frac{\partial }{\partial y_{i}} \bigl( \varPhi _{ij}(y,\xi ) \bigr)\cdot \frac{\partial \varphi }{\partial x _{j}}\,dx \\ &{}+ \int _{\varOmega }\varepsilon \varPhi _{ij}(x,x/\varepsilon )\cdot \frac{ \partial \varphi }{\partial x_{i}\partial x_{j}}\,dx \\ =& \int _{\varOmega } \biggl[\frac{\partial }{\partial y_{i}} \bigl( \varPhi _{ij}(y,\xi ) \bigr)-\frac{\partial }{\partial x_{i}} \bigl( \varepsilon \varPhi _{ij}(x,x/\varepsilon ) \bigr) \biggr]\cdot \frac{\partial \varphi }{\partial x_{j}} \,dx, \end{aligned}$$

where we have used the divergence theorem and the antisymmetry of $\varPhi _{ij}$.

As a result, using Proposition 2.4 again, we get

$$\begin{aligned} \vert I_{2} \vert \leq& C \int _{\varOmega } \bigl\vert \triangledown _{y} \varPhi (y,\xi )-\triangledown _{x} \varPhi (x,x/\varepsilon ) \bigr\vert \cdot \vert \triangledown \varphi \vert \,dx \\ \leq& C\varepsilon \int _{\varOmega _{\varepsilon }} \bigl\vert T_{\varepsilon }\bigl( \triangledown ^{2} u_{0}\bigr) \bigr\vert \cdot \vert \triangledown u_{0} \vert ^{p-2}\cdot \vert \triangledown \varphi \vert \,dx \\ \leq& C\varepsilon \biggl( \int _{\varOmega } \vert \triangledown u_{0} \vert ^{p}\,dx \biggr) ^{1-\frac{2}{p}} \biggl( \int _{\varOmega _{\varepsilon }} \bigl\vert T_{\varepsilon }\bigl( \triangledown ^{2} u_{0}\bigr) \bigr\vert ^{p}\,dx \biggr)^{\frac{1}{p}} \\ \leq& C\varepsilon \biggl( \int _{\varOmega } \vert \triangledown u_{0} \vert ^{q}\,dx \biggr) ^{\frac{p-2}{q}} \biggl( \int _{\varOmega _{\varepsilon }} \bigl\vert \triangledown ^{2} u_{0} \bigr\vert ^{p}\,dx \biggr)^{\frac{1}{p}} \\ \leq& C\varepsilon ^{\frac{1}{p}-\frac{1}{q}} \biggl( \int _{\varOmega } \vert \triangledown u_{0} \vert ^{q}\,dx \biggr)^{\frac{p-1}{q}} \end{aligned}$$

(3.3)

for some $q>p$.

For $I_{3}$, it follows that

$$\begin{aligned}& \vert I_{3} \vert \\& \quad \leq C \int _{\varOmega } \bigl\vert (\triangledown \chi +1)T_{\varepsilon }( \eta _{\varepsilon }\triangledown u_{0})- \triangledown v_{\varepsilon } \bigr\vert \bigl( \bigl\vert (\triangledown \chi +1)T_{\varepsilon }(\eta _{\varepsilon }\triangledown u_{0}) \bigr\vert + \vert \triangledown v_{\varepsilon } \vert \bigr) ^{p-2}\cdot \vert \triangledown \varphi \vert \,dx \\& \quad \leq C \int _{\varOmega } \bigl\vert \triangledown u_{0}-T_{\varepsilon }( \eta _{\varepsilon }\triangledown u_{0}) \bigr\vert \bigl( \vert \triangledown u_{0} \vert + \bigl\vert (\triangledown \chi +1) T_{\varepsilon }(\eta _{\varepsilon }\triangledown u_{0}) \bigr\vert \bigr)^{p-2}\cdot \vert \triangledown \varphi \vert \,dx \\& \qquad {} +C \int _{\varOmega } \bigl\vert \triangledown u_{0}-T_{\varepsilon }( \eta _{\varepsilon }\triangledown u_{0}) \bigr\vert \\& \qquad {}\cdot \bigl( \bigl\vert \triangledown \chi T_{\varepsilon }(\eta _{\varepsilon } \triangledown u_{0}) \bigr\vert + \bigl\vert \varepsilon \chi T_{\varepsilon }\bigl(\eta _{\varepsilon }\triangledown ^{2} u_{0}\bigr) \bigr\vert + \bigl\vert \varepsilon \chi T_{\varepsilon }(\triangledown \eta _{\varepsilon }\triangledown u_{0}) \bigr\vert \bigr)^{p-2} \vert \triangledown \varphi \vert \,dx \\& \qquad {}+ C \int _{\varOmega } \bigl\vert \varepsilon \chi T_{\varepsilon } \bigl(\eta _{\varepsilon }\triangledown ^{2} u_{0} \bigr)+\varepsilon \chi T_{\varepsilon }(\triangledown \eta _{\varepsilon } \triangledown u_{0}) \bigr\vert \bigl( \bigl\vert (\triangledown \chi +1)T_{\varepsilon }(\eta _{\varepsilon }\triangledown u_{0}) \bigr\vert + \vert \triangledown v_{\varepsilon } \vert \bigr)^{p-2} \\& \qquad {}\cdot \vert \triangledown \varphi \vert \,dx \\& \quad \doteq I_{31}+I_{32}+I_{33}. \end{aligned}$$

(3.4)

Here, we divide the estimate into three ingredients.

Similar to the estimate of $I_{1}$, we have

$$\begin{aligned} \vert I_{31} \vert \leq& C \biggl( \int _{\varOmega \setminus \varOmega _{\varepsilon }} \vert \triangledown u_{0} \vert ^{p}\,dx \biggr)^{1-\frac{1}{p}}+C\varepsilon \biggl( \int _{ \varOmega _{\varepsilon }} \bigl\vert \triangledown ^{2} u_{0} \bigr\vert ^{2} \vert \triangledown u _{0} \vert ^{p-2}\,dx \biggr)^{\frac{1}{2}} \biggl( \int _{\varOmega } \vert \triangledown u_{0} \vert ^{p}\,dx \biggr)^{\frac{p-2}{2p}} \\ \leq& C\varepsilon ^{\frac{1}{2}(1-\frac{p}{q})} \biggl( \int _{\varOmega } \vert \triangledown u_{0} \vert ^{q}\,dx \biggr)^{\frac{p-1}{q}}. \end{aligned}$$

(3.5)

Next, we proceed to deal with $I_{32}$:

$$\begin{aligned} \vert I_{32} \vert \leq& C \int _{\varOmega } \bigl\vert \triangledown u_{0}-T_{\varepsilon }( \eta _{\varepsilon }\triangledown u_{0}) \bigr\vert \bigl( \bigl\vert T_{\varepsilon }( \eta _{\varepsilon }\triangledown u_{0}) \bigr\vert + \bigl\vert \varepsilon T_{ \varepsilon }( \triangledown \eta _{\varepsilon }\triangledown u_{0}) \bigr\vert \bigr)^{p-2} \vert \triangledown \varphi \vert \,dx \\ &{}+C \int _{\varOmega } \bigl\vert \triangledown u_{0}-T_{\varepsilon }( \eta _{\varepsilon }\triangledown u_{0}) \bigr\vert \bigl\vert \varepsilon \chi T_{\varepsilon }\bigl( \eta _{\varepsilon }\triangledown ^{2} u_{0}\bigr) \bigr\vert ^{p-2} \vert \triangledown \varphi \vert \,dx \\ \leq& C \int _{\varOmega \setminus \varOmega _{\varepsilon }} \vert \triangledown u _{0} \vert ^{p-1} \vert \triangledown \varphi \vert \,dx+C \int _{\varOmega _{\varepsilon }} \bigl\vert \triangledown u_{0}-T_{\varepsilon }( \eta _{\varepsilon }\triangledown u_{0}) \bigr\vert \vert \triangledown u_{0} \vert ^{p-2} \vert \triangledown \varphi \vert \,dx \\ &{}+C \int _{\varOmega _{\varepsilon }} \vert \triangledown u_{0} \vert \cdot \bigl\vert \varepsilon \chi T_{\varepsilon }\bigl(\eta _{\varepsilon } \triangledown ^{2} u_{0}\bigr) \bigr\vert ^{p-2} \vert \triangledown \varphi \vert \,dx \\ \leq& C\varepsilon ^{\frac{1}{2}(1-\frac{p}{q})} \biggl( \int _{\varOmega } \vert \triangledown u_{0} \vert ^{q}\,dx \biggr)^{\frac{p-1}{q}}+ \biggl( \int _{\varOmega } \vert \triangledown u_{0} \vert ^{q}\,dx \biggr)^{\frac{1}{q}} \biggl( \int _{ \varOmega _{\varepsilon }} \bigl\vert \varepsilon \chi T_{\varepsilon } \bigl( \eta _{\varepsilon }\triangledown ^{2} u_{0} \bigr) \bigr\vert ^{p}\,dx \biggr)^{ 1- \frac{2}{p}} \\ \leq& C\varepsilon ^{\frac{1}{2}(1-\frac{p}{q})} \biggl( \int _{\varOmega } \vert \triangledown u_{0} \vert ^{q}\,dx \biggr)^{\frac{p-1}{q}}+ \biggl( \int _{\varOmega} \vert \triangledown u_{0} \vert ^{q}\,dx \biggr)^{\frac{1}{q}} \varepsilon ^{(1-\frac{p}{q})(1-\frac{2}{p})} \biggl( \int _{\varOmega } \vert \triangledown u_{0}\biggr) \vert ^{q}\,dx )^{\frac{p-2}{q}} \\ \leq& C\varepsilon ^{\frac{1}{2}(1-\frac{p}{q})} \biggl( \int _{\varOmega } \vert \triangledown u_{0} \vert ^{q}\,dx \biggr)^{\frac{p-1}{q}}, \end{aligned}$$

(3.6)

for some $q>p\geq 4$, where we have used Proposition 2.3 and Proposition 2.4.

Last, it remains to handle $I_{33}$:

$$\begin{aligned} \vert I_{33} \vert \leq& C\varepsilon \int _{\varOmega _{\varepsilon }} \bigl\vert T_{\varepsilon }\bigl( \triangledown ^{2} u_{0}\bigr)+ T_{\varepsilon }( \triangledown u_{0}) \bigr\vert \cdot \vert \triangledown u_{0} \vert ^{p-2} \vert \triangledown \varphi \vert \,dx \\ &{}+C\varepsilon ^{p-1} \int _{\varOmega _{\varepsilon }} \bigl\vert T_{\varepsilon }\bigl( \triangledown ^{2} u_{0}\bigr)+ T_{\varepsilon }( \triangledown u_{0}) \bigr\vert ^{p-1} \vert \triangledown \varphi \vert \,dx \\ \leq& C\varepsilon \biggl( \int _{\varOmega } \vert \triangledown u_{0} \vert ^{p}\,dx \biggr) ^{1-\frac{2}{p}} \biggl( \int _{\varOmega _{\varepsilon }} \bigl\vert T_{\varepsilon }\bigl( \triangledown ^{2} u_{0}\bigr)+ T_{\varepsilon }( \triangledown u_{0}) \bigr\vert ^{p}\,dx \biggr) ^{\frac{1}{p}} \\ &{}+C\varepsilon ^{p-1} \biggl( \int _{\varOmega _{\varepsilon }}\bigl( \bigl\vert T_{\varepsilon }\bigl( \triangledown ^{2} u_{0}\bigr) \bigr\vert + \bigl\vert T_{\varepsilon }(\triangledown u_{0}) \bigr\vert \bigr)^{p}\,dx \biggr) ^{1-\frac{1}{p}} \\ \leq& C\varepsilon \biggl( \int _{\varOmega } \vert \triangledown u_{0} \vert ^{q}\,dx \biggr) ^{\frac{p-2}{q}} \biggl[ \biggl( \int _{\varOmega } \vert \triangledown u_{0} \vert ^{q}\,dx \biggr) ^{\frac{1}{q}}+ \biggl( \int _{\varOmega _{\varepsilon }} \bigl\vert \triangledown ^{2} u _{0} \bigr\vert ^{p}\,dx \biggr)^{\frac{1}{p}} \biggr] \\ &{}+C\varepsilon ^{p-1} \biggl( \int _{\varOmega } \vert \triangledown u_{0} \vert ^{q}\,dx \biggr) ^{\frac{p-1}{q}}+C\varepsilon ^{p-1} \biggl( \int _{\varOmega _{\varepsilon }} \bigl\vert \triangledown ^{2} u_{0} \bigr\vert ^{p}\,dx \biggr)^{1-\frac{1}{p}} \\ \leq& C\bigl(\varepsilon +\varepsilon ^{\frac{1}{p}-\frac{1}{q}}+ \varepsilon ^{p-1}+\varepsilon ^{(p-1)(\frac{1}{p}-\frac{1}{q})}\bigr) \biggl( \int _{\varOmega } \vert \triangledown u_{0} \vert ^{q}\,dx \biggr)^{\frac{p-1}{q}} \\ \leq& C\varepsilon ^{\frac{1}{p}-\frac{1}{q}} \biggl( \int _{\varOmega } \vert \triangledown u_{0} \vert ^{q}\,dx \biggr)^{\frac{p-1}{q}}, \end{aligned}$$

(3.7)

for some $q>p\geq 4$.

This, together with (3.1)–(3.7), shows that, for some $q>p\geq 4$,

$$\begin{aligned}& \biggl\vert \int _{\varOmega } \bigl[A(x/\varepsilon ) \vert \triangledown u_{\varepsilon } \vert ^{p-2} \triangledown u_{\varepsilon } -A(x/\varepsilon ) \vert \triangledown v_{ \varepsilon } \vert ^{p-2}\triangledown v_{\varepsilon } \bigr] \cdot \triangledown \varphi \,dx \biggr\vert \\& \quad \leq C \varepsilon ^{\frac{1}{p}-\frac{1}{q}} \biggl( \int _{\varOmega } \vert \triangledown u_{0} \vert ^{q}\,dx \biggr)^{\frac{p-1}{q}}. \end{aligned}$$

(3.8)

Then let $\varphi =v_{\varepsilon }=u_{\varepsilon }-u_{0}-\varepsilon \chi T_{\varepsilon }(\eta _{\varepsilon }\triangledown u_{0})$. It gives

$$\begin{aligned}& \bigl\Vert \triangledown \bigl[u_{\varepsilon }-u_{0}- \varepsilon \chi T_{\varepsilon }(\eta _{\varepsilon }\triangledown u_{0}) \bigr] \bigr\Vert ^{p-1}_{L^{p}(\varOmega )} \\& \quad \leq C \biggl\vert \int _{\varOmega } \bigl[A(x/\varepsilon ) \vert \triangledown u_{\varepsilon } \vert ^{p-2} \triangledown u_{\varepsilon } -A(x/\varepsilon ) \vert \triangledown v_{ \varepsilon } \vert ^{p-2}\triangledown v_{\varepsilon } \bigr]\cdot \triangledown \varphi \,dx \biggr\vert \\& \quad \leq C\varepsilon ^{\frac{1}{p}-\frac{1}{q}} \biggl( \int _{\varOmega } \vert \triangledown u_{0} \vert ^{q}\,dx \biggr)^{\frac{p-1}{q}}. \end{aligned}$$

In view of the fact that $T_{\varepsilon }(\eta _{\varepsilon }\triangledown u_{0})=0$ in the $\varOmega \setminus \varOmega _{\varepsilon }$ and the Poincaré inequality, this completes the proof of Theorem 1.

It follows from Theorem 1 and Proposition 2.3, together with Minkowski’s inequality, that

$$\begin{aligned} \Vert u_{\varepsilon }-u_{0} \Vert _{L^{p}(\varOmega )} \leq& C \varepsilon \bigl\Vert \chi T_{\varepsilon }(\eta _{\varepsilon } \triangledown u_{0}) \bigr\Vert _{L^{p}(\varOmega )}+C \varepsilon ^{\frac{1}{p}-\frac{1}{q}} \Vert \triangledown u_{0} \Vert _{L^{q}(\varOmega )} \\ \leq& C\varepsilon \Vert \triangledown u_{0} \Vert _{L^{p}( \varOmega )}+C\varepsilon ^{\frac{1}{p}-\frac{1}{q}} \Vert \triangledown u_{0} \Vert _{L^{q}(\varOmega )} \\ \leq& C\varepsilon ^{\frac{1}{p}-\frac{1}{q}} \Vert \triangledown u _{0} \Vert _{L^{q}(\varOmega )}, \end{aligned}$$

(3.9)

with $q>p\geq 4$.

This completes the proof of Theorem 2.