Uniform regularity estimates in homogenization theory of elliptic system with lower order terms

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• 作者：Qiang Xu ^{xuqiang09@lzu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Elliptic systems ; Homogenization ; Uniform regularity estimates ; Green's matrix
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 June 2016
• 年：2016
• 卷：438
• 期：2
• 页码：1066-1107
• 全文大小：795 K

文摘

In this paper, we extend the uniform regularity estimates obtained by M. Avellaneda and F. Lin in [3] and [6] to the more general second order elliptic systems in divergence form class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16001359&_mathId=si1.gif&_user=111111111&_pii=S0022247X16001359&_rdoc=1&_issn=0022247X&md5=709b099b895908631f5de423771d9fdf" title="Click to view the MathML source">{L_ε,ε>0}class="mathContainer hidden">class="mathCode">$\{{\mathcal{L}}_{\epsilon },\epsilon >0\}$, with rapidly oscillating periodic coefficients. We establish not only sharp class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16001359&_mathId=si2.gif&_user=111111111&_pii=S0022247X16001359&_rdoc=1&_issn=0022247X&md5=fa1fe4d31e810200061f64fadd6d62f3" title="Click to view the MathML source">W^1,pclass="mathContainer hidden">class="mathCode">$W^{1,p}$ estimates, Hölder estimates, Lipschitz estimates and non-tangential maximal function estimates for the Dirichlet problem on a bounded class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16001359&_mathId=si3.gif&_user=111111111&_pii=S0022247X16001359&_rdoc=1&_issn=0022247X&md5=41a35eec86cd4585570c087b756357e7" title="Click to view the MathML source">C^1,ηclass="mathContainer hidden">class="mathCode">$C^{1,\eta }$ domain, but also a sharp class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16001359&_mathId=si4.gif&_user=111111111&_pii=S0022247X16001359&_rdoc=1&_issn=0022247X&md5=4c3ec2baca85d000a5050d23c36a7659" title="Click to view the MathML source">O(ε)class="mathContainer hidden">class="mathCode">$O(\epsilon )$ convergence rate in class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16001359&_mathId=si5.gif&_user=111111111&_pii=S0022247X16001359&_rdoc=1&_issn=0022247X&md5=d1d24e5fc92acf1ddaa15b073166f89b">class="imgLazyJSB inlineImage" height="18" width="47" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16001359-si5.gif">class="mathContainer hidden">class="mathCode">$H_0^1(\mathrm{\Omega })$ by virtue of the Dirichlet correctors. Moreover, we define the Green's matrix associated with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16001359&_mathId=si201.gif&_user=111111111&_pii=S0022247X16001359&_rdoc=1&_issn=0022247X&md5=2f3b3dbbc6f0708ab1a98921ed25501f" title="Click to view the MathML source">L_εclass="mathContainer hidden">class="mathCode">${\mathcal{L}}_{\epsilon }$ and obtain its decay estimates. We remark that the well known compactness methods are not employed here, instead we construct the transformations (1.11) to make full use of the results in [3] and [6].