设$\Omega $是${\Bbb R}^n$中边界Lipschitz光滑的有界开区域, $f\in L^{\infty}(\Omega )$, 我们在权Sobolev空间$X=W^{1, p}_0(\mu^{\varepsilon }_1, \mu^{\varepsilon }_2, \Omega )$里考虑下述边值问题的均匀化

$ (P_{\varepsilon})~ \left\{ \begin{array}{ll} -\mbox{div}a(\frac{x}{\varepsilon},u^{\varepsilon},\nabla u^{\varepsilon}) +g(\frac{x}{\varepsilon},u^{\varepsilon}) =f(x), \quad &x\in \Omega,\\[2mm] u^{\varepsilon}(x)=0,\qquad \qquad &x\in \partial{\Omega}. \end{array} \right. $

这个模型有几个简单的例子, 比如

$-\nabla \left(a\left(\frac{x}{\varepsilon }\right)\nabla u\right) + b\left(\frac{x}{\varepsilon }\right) u=f,$

(1.1)

以及

$-\nabla \left(A\left(\frac{x}{\varepsilon }\right)|u|^r |\nabla u|^{p-2}\nabla u\right) + v_0\left(\frac{x}{\varepsilon }\right)|u|^{p-2}u =f.$

(1.2)

这篇文章中, 我们通过联合权Sobolev空间与调和分析中的经典地补偿紧性办法来研究退化椭圆方程$(P_{\varepsilon })$的均匀化, 其中$(P_{\varepsilon })$的退化性是指它的方程满足下面假设条件$(H_2)$和$(H_5)$.

首先是对于每个$\varepsilon $问题$(P_{\varepsilon })$的存在性, 读者可以查阅文献[6, 13-14].

当$\varepsilon $趋于零时, 对于问题$(P_{\varepsilon })$非退化也就是满足强制条件

$a(y, \alpha , \lambda)\lambda \geq |\lambda|^p {~~}(p>1)$

(1.3)

时的渐近行为已有大量的研究, 比如文献[1, 4]等.而且, 在退化椭圆方程的情形, 文献[2]在更强的条件下有一个初步的结果, 另外相关结果也参考文献[5, 7].对于抛物方程问题的相关研究可以参考文献[3, 8-10].事实上, 偏微分方程的均匀化理论一直是一个热门的问题, 可以参看Kenig, Lin和Shen最近的工作[11-12].

在这个文章中我们将用字母$C$各种可能与$\varepsilon $无关的正常数.

本文的主要假设为

$(H_1)$向量函数$a(y, \alpha , \lambda)=(a_1, a_2, \cdots, a_n): \mathbb{R}^n\times \mathbb{R}\times \mathbb{R}^n \rightarrow \mathbb{R}^n$是Carathéodory类函数, 也就是说, $a$关于$y$可测, 关于$(\alpha , \lambda)$连续.

$(H_2)$对于$ Y=(0, 1)^n$, $a(y, \alpha , \lambda)$关于变量$y$是以$Y$为周期的.

$(H_3)$对任意$\lambda\in \mathbb{R}^n, \alpha \in \mathbb{R}, $存在$p>1$, $C>0$, 使得

$\begin{eqnarray*} % number to remove numbering (before each equation) &&a(y,\alpha,\lambda)\lambda \geq \mu_1(y)|\lambda|^p, \\ && |a(y,\alpha,\lambda)|\leq C \mu_1(y)(1+|\lambda|^{p-1}+|\alpha|^{p-1}), \end{eqnarray*}$

其中正函数$\mu_1(y)$是$Y$ -周期$A_p$ Muckenhoupt权(见定义2.1).

$(H_4)$对任意的$\lambda_1, \lambda_2\in \mathbb{R}^n$以及$\lambda_1 \neq \lambda_2$,

$ [a(y, \alpha , \lambda_1)-a(y, \alpha , \lambda_2)](\lambda_1-\lambda_2)>0. $

$(H_5)$对于$\alpha _1, \alpha _2\in R$, $\forall r\in(0, 1)$, 存在正常数$\beta>0, $

$\begin{eqnarray*} &&|a(y, \alpha_1,\lambda)-a(y,\alpha_2,\lambda)| \\ &\leq& \beta\mu_1(y)|\alpha_1-\alpha_2|^r(1+ |\alpha_1|^{p-1-r}+|\alpha_2|^{p-1-r} +|\lambda|^{p-1-r}). \end{eqnarray*}$

$(H_6)$设连续函数$g(y, \alpha ):{\Bbb R}^n \times {\Bbb R}\rightarrow {\Bbb R}$满足下列条件$(p>1, \beta>0, 0＜r\leq 1)$

$ \forall \ \alpha \in {\Bbb R}, g(y, \alpha , )\alpha \geq \mu_2(y)|\alpha |^p, $

$ |g(y, \alpha )|\leq C \mu_2(y)(1+|\alpha |^{p-1}), $

$ \forall \ \alpha _1, \alpha _2\in {\Bbb R}, {~~} [g(y, \alpha _1)-g(y, \alpha _2)](\alpha _1-\alpha _2)>0, $

$ |g(y, \alpha _1, )-g(y, \alpha _2)|\leq \beta\mu_2(y)|\alpha _1-\alpha _2|^r, $

正函数$\mu_2(y)$是$Y$ -周期$A_p$ Muckenhoupt权(见定义2.1).

现在我们定义一些泛函空间来研究我们的问题, 假设$\mu_1$, $\mu_2$是$\Omega $上的正连续可测函数, 满足

$\mu_i\ \mbox{ 以及 }\ \mu^{-\frac1{p-1}}_i\in L^{1}_{loc}({\Omega }), i=1, 2,$

(1.4)

其中$2 \leq p＜ \infty$.

对每个开集$\Omega \subset {\Bbb R}^n$, 定义

$L^p(\mu_2, \Omega )=\{u|u \in L^1_{loc}(\Omega ), {~~} u\mu_2^{1/p}\in L^p(\Omega )\},$

(1.5)

$W^{1, p}(\mu_1, \mu_2, \Omega )= \{u|{~}\mu_1^{1/p}\nabla u\in (L^p(\Omega ))^n, {~~} u\in L^p(\mu_2, {\Omega })\}.$

(1.6)

容易证明$W^{1, p}(\mu_1\mu_2, \Omega )$赋予范数

$\|u\|_{W^{1, p}(\mu_1, \mu_2, \Omega )}=\left(\int_{\Omega }(\mu_2|u|^p+\mu_1|\nabla u|^p) {\rm d}x \right)^{\frac1 p}$

(1.7)

是自反可分的Banach空间.我们代表${W^{1, p}_0(\mu_1, \mu_2, \Omega )}$为$C^{\infty}_0(\Omega )$类函数在$W^{1, p}(\mu_1, \mu_2, \Omega )$的完备化, $V^*$为$V$的对偶空间, 以及$ V=W^{1, p}_0(\mu_1, \mu_2, \Omega )$.

定义1.1  对每个$\varepsilon >0$, 函数$u^{\varepsilon }$被称为问题$(P_{\varepsilon })$的弱解是指$u^{\varepsilon }\in V=W^{1, p}_0(\mu_1^{\varepsilon }, \mu_2^{\varepsilon }, \Omega )$且$u^{\varepsilon }$在分布意义下满足$(P_{\varepsilon })$, 即

$\begin{equation} \left\{ \begin{array}{ll} u\in V\mbox{ 使得 }\\[2mm] \displaystyle \int_{\Omega} \left[a(\frac{x}{\varepsilon},u^{\varepsilon},\nabla u^{\varepsilon})\nabla v +g(\frac{x}{\varepsilon},u^{\varepsilon})v\right]{\rm d}x = \int_{\Omega}fv {\rm d}x, \quad \forall v \in V,\\[2mm] u(x)=0, \qquad x\in \partial{\Omega}, \end{array} \right. \end{equation}$

(1.8)

本文主要定理是:

定理1.1  如果$(H_1)-(H_6)$被满足, 对每个${\varepsilon }>0$, 问题$(P_{\varepsilon })$存在唯一解$u^{\varepsilon }$, 对于$u\in W^{1, p}_0(\Omega )$满足下述边值问题

$ (P_0)\ \left\{ \begin{array}{ll} -\mbox{div}A(u, \nabla u)+G(u) =f(x), {~~~~~}& x\in \Omega , \\ u(x)=0,&x\in \partial {\Omega }. \end{array} \right. $

则, 当$\varepsilon \rightarrow 0$, 下述结论成立

$u^{\varepsilon }\rightharpoonup u {~~~~}\mbox{ 弱 } {~~} W^{1, 1}_0({\Omega })),$

(1.9)

$a_{\varepsilon }(\frac{x}{\varepsilon }, u^{\varepsilon }, \nabla u^{\varepsilon }) \rightharpoonup A(u, \nabla u) {~~~~} \mbox{ 弱 } {~~} L^1(\Omega _T),$

(1.10)

$g(\frac{x}{\varepsilon }, u^{\varepsilon }) \rightharpoonup G(u) {~~~~} \mbox{ 弱 } {~~} L^1(\Omega ),$

(1.11)

期中算子$A: {\Bbb R}\times {\Bbb R}^n\rightarrow \mathbb{R}^n$及$G(u)$被定义为

$A(\alpha , \lambda)=\int_Y a(y, \alpha , \lambda+\nabla{\Phi}^{\alpha }_{\lambda}(y)){\rm d}y, \quad \quad G(u)=\int_Y g(y, u){\rm d}y.$

(1.12)

同时, ${\Phi}^{\alpha }_{\lambda}$满足下列问题

$\left\{ \begin{array}{lcr} \int_Y a(y, \alpha , \lambda+\nabla{\Phi}^{\alpha }_{\lambda}(y))\varphi {\rm d}y =0, {~~} \forall \varphi \in W^{1, p}_{_{per}}(\mu_1, \mu_2, Y), \\[2mm] {\Phi}^{\alpha }_{\lambda}\in W^{1, p}_{_{per}}(\mu_1, \mu_2, Y), \end{array} \right.$

(1.13)

其中$W^{1, p}_{_{per}}(\mu_1, \mu_2, Y)$代表$W^{1, p}(\mu_1, \mu_2, Y)$中一些$Y$的两边具有同样的trace的函数类.

定义2.1^[5]  设$p>1, K\geq 1$且$\mu$是${\Bbb R}^n$中的权(即, $\mu$满足$(1.4)$式), $\mu$属于$A_p(K)$类是指:对${\Bbb R}^n$中的各面平行于坐标平面的任意矩形$Q$, 有

$\left({\rlap{-} \smallint }_{Q}\mu{\rm d}y\right) \left({\rlap{-} \smallint }_{Q}\mu^{-\frac1{p-1}}{\rm d}y\right)^{p-1} \leq K,$

(2.1)

其中$|Q|$代表$Q$的Lebesgue测度, ${\rlap{-} \smallint }_{Q}\mu{\rm d}y=\frac{1}{|Q|}\int_{Q}\mu{\rm d}y$, 记$A_p:=\bigcup_{K\geq 1}A_p(K)$.

引理2.1^[5]  设$p>1$, $K\geq 1$, 然后存在两个正常数$\delta =\delta (n, p, K)$和$C=C(n, p, K)$对${\Bbb R}^n$中的各面平行于坐标平面的任意矩形$Q$, $\mu \in A_p(K)$, 使得

$\left({\rlap{-} \smallint }_Q\mu^{1+\delta }{\rm d}x\right)^{1/(\delta +1)}\leq C {\rlap{-} \smallint }_Q\mu{\rm d}x,$

(2.2)

$\left({\rlap{-} \smallint }_{Q}\mu^{-(1+\delta )/(p-1)}{\rm d}x\right)^{1/(1+\delta )} \leq C {\rlap{-} \smallint }_{Q}\mu^{-1/(p-1)}{\rm d}x.$

(2.3)

引理2.2^[10]  对所有$(\alpha , \lambda)\in {\Bbb R}\times {\Bbb R}^n$, 存在正常数$C_1=C_1(n, p), C_2=C_2(n, p)$, 使得

$|A(\alpha , \lambda)|\leq C_1(1+|\alpha |^{p-1}+|\lambda|^{p-1}),$

(2.4)

$A(\alpha , \lambda)\lambda\geq C_2|\lambda|^p.$

(2.5)

引理2.3^[10]  向量函数$A(\alpha , \lambda):{\Bbb R}\times {\Bbb R}^n \rightarrow {\Bbb R}^n $是连续的且满足, 对任意的$\alpha , \lambda_1, \lambda_2 $, $\lambda_1 \neq \lambda_2, $有

$[A(\alpha , \lambda_1)-A(\alpha , \lambda_2)](\lambda_1-\lambda_2)>0 .$

(2.6)

证  用$u^{\varepsilon }$数乘以$(P_{\varepsilon })$方程两边并积分得

$\int_{\Omega } a(\frac{x}{\varepsilon }, u^{\varepsilon }, \nabla u^{\varepsilon })\nabla u^{\varepsilon }+ \int_{\Omega }g(\frac{x}{\varepsilon }, u^{\varepsilon })u^{\varepsilon } {\rm d}x =\int_{\Omega }f\mbox{~}u^{\varepsilon }{\rm d}x.$

(3.1)

由条件$(H_3$)和$(H_5), $有

$\int_{\Omega }\mu_1^{\varepsilon }|\nabla u^{\varepsilon }|^p {\rm d}x+\int_{\Omega }\mu_2^{\varepsilon }|u^{\varepsilon }|^p {\rm d}x \leq\int_{\Omega }f\mbox{~}u^{\varepsilon }{\rm d}x,$

(3.2)

其中$\mu^{\varepsilon }_i=\mu_i(\frac{x}{\varepsilon }) i=1, 2.$

由于$f\in L^{\infty}(\Omega )$, 我们有

$\begin{equation} \int_{\Omega}\left(\mu_1^{\varepsilon}|\nabla u^{\varepsilon}|^p + \mu_2^{\varepsilon}|u^{\varepsilon}|^p \right){\rm d}x \leq C \qquad \mbox{$(C$与$ \varepsilon$ 无关).} \end{equation}$

(3.3)

在另外一方面, 我们能说明存在一个与$\varepsilon $无关的正常数$C$使得

$\int_{\Omega }|a(\frac{x}{\varepsilon }, u^{\varepsilon }, \nabla u^{\varepsilon })|^{p'} \left(\mu_1^{\varepsilon } \right)^{-1/(p-1)} {\rm d}x\leq C,$

(3.4)

其中$\frac1p+\frac{1}{p'}=1$.

由$(H_2)$, Hölder不等式和Poincaré不等式, 可得

$\begin{array}{l} \int_\Omega | a(\frac{x}{\varepsilon },{u^\varepsilon },\nabla {u^\varepsilon }){|^{p'}}{\left( {\mu _1^\varepsilon } \right)^{ - 1/(p - 1)}}{\rm{d}}x \le \int_\Omega {\mu _1^\varepsilon } {(1 + |{u^\varepsilon }{|^{p - 1}} + |\nabla {u^\varepsilon }{|^{p - 1}})^{p'}}{\rm{d}}x\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \le C\int_\Omega {\mu _1^\varepsilon } (|{u^\varepsilon }{|^p} + |\nabla {u^\varepsilon }{|^p}){\rm{d}}x\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \le C\int_\Omega {\mu _1^\varepsilon } |\nabla {u^\varepsilon }{|^p}{\rm{d}}x. \end{array}$

(3.5)

然后(3.4)式来自于(3.3)和(3.5)式.

类似地, 由(3.3)式, 可得

$\int _{\Omega }|g(\frac{x}{\varepsilon }, u^{\varepsilon })|^{p'} \left(\mu_2^{\varepsilon } \right)^{-1/(p-1)} {\rm d}x\leq C.$

(3.6)

为了证明$\{u^{\varepsilon }\}, \{a^{\varepsilon }\}$以及$\{g^{\varepsilon }\}$的收敛性, 我们首先证明对于$\varepsilon \in (0, 1)$, 存在$\sigma >0$使得$\{u^{\varepsilon }\}$在$W^{1, 1+\sigma}_0(\Omega ))$, $\{a^{\varepsilon }\}$以及$\{g^{\varepsilon }\}$在$(L^{1+\sigma}(\Omega ))^n$一致有界.

用Hölder不等式, 有

$\begin{eqnarray} &&\int _{ \Omega} (|u^{\varepsilon}|^{1+\sigma} +|\nabla u^{\varepsilon}| ^{1+\sigma}){\rm d}x \\ & \leq& \left(\int_{\Omega}|u^{\varepsilon}|^p \mu_2^{\varepsilon}{\rm d}x \right)^{(1+\sigma)/p} \left(\int_{\Omega} (\mu_2^{\varepsilon})^{-(1+\sigma)/(p-(1+\sigma))} {\rm d}x \right)^{1-(1+\sigma)/p} \\ && +\left(\int_{\Omega}| \nabla u^{\varepsilon}|^p\mu_1^{\varepsilon}{\rm d}x \right)^{(1+\sigma)/p} \left(\int_{\Omega} (\mu_1^{\varepsilon})^{-(1+\sigma)/(p-(1+\sigma))} {\rm d}x \right)^{1-(1+\sigma)/p} \\ \label{2.13} &\leq& C\left(\int_{\Omega} (\mu_1^{\varepsilon})^{-(1+\sigma)/(p-(1+\sigma))} {\rm d}x\right)^{1-(1+\sigma)/p}+\left(\int_{\Omega} (\mu_2^{\varepsilon})^{-(1+\sigma)/(p-(1+\sigma))} {\rm d}x\right)^{1-(1+\sigma)/p},~~ \end{eqnarray}$

(3.7)

然后, 通过选择$0＜\sigma＜1$使得$\frac{1+\sigma}{p-(1+\sigma)}=\frac{1+\delta }{p-1}$, 其中$\delta $在引理2.1所定义, 再从文献[7]和(3.14)式, 存在正常数$C$使得

$\int_{\Omega } \left(|u^{\varepsilon }|^{1+\sigma} +|\nabla u^{\varepsilon }|^{1+\sigma}\right){\rm d}x\leq C.$

(3.8)

类似地, 由(3.4)和(3.6)式, 可得

$\begin{equation} \|a^{\varepsilon}\|_{(L^{1+\sigma}(\Omega))^n} \leq C,\qquad \|g^{\varepsilon}\|_{L^{1+\sigma}(\Omega)}\leq C. \end{equation}$

(3.9)

因此, 存在$u^*\in W^{1, 1+\sigma}_0(\Omega )), a_0\in$ $(L^{1+\sigma}(\Omega ))^n, g_0\in L^{1+\sigma}(\Omega )$使得

$\begin{equation} u^{\varepsilon}\rightharpoonup u^* \qquad \mbox{ 弱 }\ W^{1,1+\sigma}_0(\Omega)), \end{equation}$

(3.10)

$\begin{equation} a(\frac{x}{\varepsilon},u^{\varepsilon},\nabla u^{\varepsilon}) \rightharpoonup a_0 \qquad \mbox{ 弱 }\ (L^{1+ \sigma}(\Omega))^n. \end{equation}$

(3.11)

$\begin{equation} g(\frac{x}{\varepsilon},u^{\varepsilon}) \rightharpoonup g_0 \qquad \mbox{ 弱 }\ L^{1+ \sigma}(\Omega). \end{equation}$

(3.12)

另外, 由Sobolev嵌入定理以及(3.10)式有

$\begin{equation} u^{\varepsilon}\rightarrow u^* \qquad \mbox{ 强 }\ L^{1+\sigma}(\Omega). \end{equation}$

(3.13)

为了完成这个定理的证明, 只需证明

${\rm ⅰ)}~u^*\in W^{1, p}_0(\Omega ); {\rm ⅱ)} a_0=A(u^*, \nabla u^*); {\rm ⅲ)} g_0=G(u)\ \mbox{几乎处处 $\Omega .$}$

(3.14)

事实上, 由$(P_0)$解的唯一性以及(3.14)式, 容易有

$\begin{equation} u=u^* \qquad \mbox{几乎处处 $ \Omega.$} \end{equation}$

(3.15)

让我们首先证明(3.14)式的ⅰ), 这需要表明

$ \nabla u^*\in (L^p(\Omega ))^n. $

由Hölder不等式, 对于每个$\psi \in C_0^0(\Omega )$

$\begin{eqnarray} \int_{\Omega} |\nabla u^{\varepsilon}| |\psi| {\rm d}x &\leq& \Big(\int _{\Omega} |\nabla u^{\varepsilon}|^p \mu_1^{\varepsilon} {\rm d}x\Big)^{1/p} \Big(\int_{\Omega} (\mu_1^{\varepsilon})^{-1/(p-1)}|\psi|^{p'} {\rm d}x\Big)^{1/p'} \\ \label{2.22} &\leq& C \Big(\int _{\Omega} (\mu_1^{\varepsilon})^{-1/(p-1)}|\psi|^{p'} {\rm d}x \Big)^{1/p'}. \end{eqnarray}$

(3.16)

取(3.16)式极限有

$ \int _{\Omega }|\nabla u^*| |\psi| {\rm d}x\leq C\|\psi\|_{L^{p'}(\Omega ) }, \forall{~} \psi \in C_0^0(\Omega ), $

类似地, 由(3.4})和(3.6)式, 有

$\begin{equation} a_0 \in(L^{p'}(\Omega))^n, \qquad g_0 \in L^{p'}(\Omega). \end{equation}$

(3.17)

再用$(H_6)$, (1.12)式和(3.13)式, 则

$\begin{eqnarray} && \int_{\Omega}\left[g(\frac{x}{\varepsilon},u^{\varepsilon}) -G(u)\right]\varphi(x){\rm d}x \\ &=&\int_{\Omega}\left[g(\frac{x}{\varepsilon},u^{\varepsilon}) -g(\frac{x}{\varepsilon},u)\right]\varphi(x){\rm d}x+ \int_{\Omega}\left[g(\frac{x}{\varepsilon},u)-G(u)\right]\varphi(x){\rm d}x \\ & \leq& C \int_{\Omega} \left|u^{\varepsilon}-u\right|^r\varphi(x){\rm d}x +\int_{\Omega} \left[g(\frac{x}{\varepsilon},u)-G(u)\right]\varphi(x){\rm d}x \rightarrow 0,~~\forall \varphi(x)\in L^{1+\sigma}(\Omega). \end{eqnarray}$

(3.18)

注意到$0＜\sigma＜1$,

$\begin{equation} g(\frac{x}{\varepsilon},u^{\varepsilon}) \rightharpoonup G(u) \qquad \mbox{ 弱 }\ L^{1+ \sigma}(\Omega). \end{equation}$

(3.19)

由(3.12)和(3.19)式, 我们获得(3.14)式的ⅲ).

下面我们证明(3.14)式的ⅱ).我们将充分利用文献[10]的技巧, 对$k\in N$, 取${\Bbb R}^{n}$的边长为$2^{-k}$矩形分解$\{Q_{i, k}\}$, 同时定义

$ [\Omega](i,k)= Q_{i,k}\bigcap\Omega \qquad I_k=\{i:| [\Omega](i,k)|\neq 0\}. $

显然, $I_k$是有限集, 对于函数$u$, 记

$ \langle u \rangle_{i, k}= |[\Omega ](i, k)|^{-1} \int _{[\Omega ](i, k)} u(x) {\rm d}x, $

以及${\Large \chi_{_{i, k}} }$代表$[\Omega ](i, k)$的特征函数.

由引理2.3知道$A(\alpha , \lambda)$是连续的, 结果对每个$\lambda\in {\Bbb R}^n$, 有

$ \lim\limits_{k \rightarrow \infty} \sum \limits_{i\in I_k} \chi_{i,k}A(\langle u \rangle_{i,k},\lambda)=A(\alpha,\lambda)\qquad \mbox{ 几乎处处 }\ \Omega . $

容易明白

$\begin{equation} A_k(x)\stackrel{\mathit{\boldsymbol{}}}{=} \sum \limits_{i\in I_k} \chi_{i,k}A(\langle u \rangle_{i,k},\lambda)\rightarrow A(\alpha,\lambda)\qquad \mbox{强 }\ L^{1+\sigma}(\Omega ). \end{equation}$

(3.20)

同时

$\begin{equation} \sum \limits_{i\in I_k} \chi_{i,k}|u-\langle u \rangle_{i,k}|^p\rightarrow 0 \qquad \mbox{ 强 }\ L^{1}(\Omega ). \end{equation}$

(3.21)

对$i, k\in N$, $\lambda\in {\Bbb R}^n$, $\alpha =\langle u \rangle_{i, k}$, 设${\Phi}^{\lambda}_{i, k}$是(1.13)式的解.记

$ W^{\lambda}_{i, k}(y)=\lambda y+ {\Phi}^{\lambda}_{i, k}(y) $

且

$ \omega^{\varepsilon }_{i, k}(x)=\varepsilon W^{\lambda}_{i, k}(x/\varepsilon )=\lambda x+\varepsilon {\Phi}^{\lambda}_{i, k}(x/\varepsilon ), $

因为${\Phi}^{\lambda}_{i, k}$是(1.13)式的解, 有

$ \int_Y a(y, \alpha , \nabla W^{\lambda}_{i, k}(y))(\nabla W^{\lambda}_{i, k}-\lambda) {\rm d}y=0, $

再由$(H_2)$,

$ \int_Y \mu_1(y) |\nabla W^{\lambda}_{i, k}|^p {\rm d}y\leq \int_Y a(y, \alpha , \nabla W^{\lambda}_{i, k}(y))\lambda {\rm d}y\leq C|A(\alpha , \lambda)||\lambda|. $

对于$\lambda\in {\Bbb R}^n$, 由(2.4)式

$ \int_Y \mu_1(y) |\nabla W^{\lambda}_{i, k}|^p {\rm d}y \leq C(1+|\alpha |^{p-1}+|\lambda|^{p-1})|\lambda| \leq C(|\alpha |^{p-1}|\lambda|+|\lambda|^p). $

同理可得, 存在常数$C>0$和$\sigma>0$使得

$\int_Y |\nabla W^{\lambda}_{i, k}|^{1+\sigma}{\rm d}y\leq C,$

(3.22)

然后, 类似于(3.10)和(3.13)式, 有

$\begin{equation} \omega^{\varepsilon}_{i,k}(x)\rightharpoonup \lambda x \stackrel{\mathit{\boldsymbol{}}}{=}\omega \qquad\mbox{ 强 }\ L^{1+\sigma}(\Omega), \end{equation}$

(3.23)

$\begin{equation} \nabla \omega^{\varepsilon}_{i,k}\rightharpoonup \lambda=\nabla\omega \qquad\mbox{ 弱 }\ L^{1+\sigma}(\Omega ). \end{equation}$

(3.24)

从(1.13)式, 对$\varepsilon >0$, 有

$\begin{equation} \mbox{div}a(x/\varepsilon,\langle u \rangle_{i,k},\nabla \omega^{\varepsilon}_{i,k}) =0\qquad \mbox{在$ C_0^0(\Omega )$中.} \end{equation}$

(3.25)

设

$ E_k=\{\psi\in C_0^0(\Omega ):\psi=0\ \mbox{ 在某个领域内}\ \bigcup\limits_{i\in I_k}\partial [ \Omega ](i, k)\}, $

且

$ \omega^{\varepsilon }_k(x)=\sum\limits_{i\in I_k}\chi_{i, k}(x)\omega^{\varepsilon }_{i, k}(x), $

$ d^{\varepsilon }_{k}(x)=\sum\limits_{i\in I_k}\chi_{i, k}(x)\omega^{\varepsilon }_{i, k}(x) a(x/\varepsilon , \langle u \rangle_{i, k}, \nabla \omega^{\varepsilon }_{i, k}). $

由(3.24)式有

$\begin{equation} d^{\varepsilon}_{k}(x) \rightharpoonup \sum\limits_{i\in I_k}A(\langle u \rangle_{i,k},\lambda) \qquad \mbox{ 弱 }\ L^{p'}(\Omega). \end{equation}$

(3.26)

在另外一方面, 我们能获得一个类似于文献[5, 定理2.3]的权补偿紧性结果.

因此, 对于$\psi_k\in E_k$和(3.11), (3.13), (3.23), (3.24)式, 有

$\begin{eqnarray} &&\lim\limits_{\varepsilon\rightarrow 0}\int _{\Omega} [a(x/\varepsilon ,u^{\varepsilon},\nabla u^{\varepsilon})- d^{\varepsilon}_{k}](\nabla u^{\varepsilon}-\nabla \omega^{\varepsilon}_k)\psi_k {\rm d}x \\ & =&\int _{\Omega} (a_0-A_k)(\nabla u^*-\nabla \omega)\psi_k {\rm d}x, \qquad \forall\psi_k \in E_k. \end{eqnarray}$

(3.27)

另一方面,

$\begin{eqnarray} &&\int_{\Omega} |a(\frac{x}{\varepsilon},u^{\varepsilon},\nabla u^{\varepsilon}) -\sum\limits_{i\in I_k}\chi_{i,k}a(\frac{x}{\varepsilon},\langle u^* \rangle_{i,k},\nabla u^{\varepsilon})| {\rm d}x \\ &\leq& \int_{\Omega} \Big[ \Big| a(\frac{x}{\varepsilon}, u^{\varepsilon},\nabla u^{\varepsilon}) - a(\frac{x}{\varepsilon}, u^*, \nabla u^{\varepsilon}) \Big| \\ && + \Big|a(\frac{x}{\varepsilon}, u^*,\nabla u^{\varepsilon}) -\sum\limits_{i\in I_k}\chi_{i,k} a(\frac{x}{\varepsilon},\langle u^* \rangle_{i,k},\nabla u^{\varepsilon})\Big|\Big] {\rm d}x \\ \label{2.34} & =&H+G. \end{eqnarray}$

(3.28)

由$(H_5)$和Hölder不等式, 有

$\begin{eqnarray} H&\leq& \beta\int_{\Omega} \mu_1^{{\varepsilon}}|u^{\varepsilon}-u^*|^r(1+ |u^{\varepsilon}|^{p-1-r}+|u^*|^{p-1-r} +|\nabla u^{\varepsilon}|^{p-1-r}) {\rm d}x \\ &\leq& C\left(\int_{\Omega_T}\mu_{1}^{\varepsilon}|\nabla u^{\varepsilon}|^p {\rm d}x\right)^{\frac{p-1-r}{p}} \left( \int_{\Omega_T}\mu_1^{{\varepsilon}} \big|u^{\varepsilon}-u^*\big|^{\frac{rp}{1+r}} {\rm d}x\right)^{\frac{1+r}{p}}, \\ \label{2.35} & \leq &C \left(\int_{\Omega} \mu_1^{{\varepsilon}} \big|u^{\varepsilon}-u^*\big|^{\frac{rp}{1+r}} {\rm d}x\right)^{\frac{1+r}{p}}. \end{eqnarray}$

(3.29)

类似地, 有

$G \leq C\bigg( \sum\limits_{i\in I_k}\int _{\Omega _T(i, k)}\hspace{-3mm}\mu_1^{{\varepsilon }}|u^*-\langle u^* \rangle_{i, k}|^{\frac{rp}{1+r}}{\rm d}x \bigg)^{\frac{1+r}{p}}.$

(3.30)

再由(1.9), (3.21), (3.29), (3.30)式以及$(H_5)$, 对任意的非负函数$\psi\in C_0^0(\Omega )$, 有

$\int_{\Omega } \bigg |a-\sum\limits_{i\in I_k}\chi_{i, k}a(x/\varepsilon , \langle u \rangle_{i, k}, \nabla u^{\varepsilon })\bigg| {\rm d}x\rightarrow 0,$

(3.31)

$\int_{\Omega } (a_0-A)(\nabla u^*-\nabla \omega)\psi {\rm d}x \geq 0.$

(3.32)

再用Minty技巧, 对任意$\xi \in {\Bbb R}^n$, 有

$[a_0(x)-A(u^*(x), \nabla u^*(x))]\xi\geq 0,$

(3.33)

这暗含(3.14)式的ⅱ)成立, 定理1.1的证明完成.