Uniform regularity estimates in homogenization theory of elliptic systems with lower order terms on the Neumann boundary problem

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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 ; Neumann boundary problem ; Convergence rates
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：15 October 2016
• 年：2016
• 卷：261
• 期：8
• 页码：4368-4423
• 全文大小：693 K

文摘

In this paper, we mainly employed the idea of the previous paper [34] to study the sharp uniform class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616301632&_mathId=si1.gif&_user=111111111&_pii=S0022039616301632&_rdoc=1&_issn=00220396&md5=2eaaf4d4f987be7a25867ef83e8360c9" title="Click to view the MathML source">W^1,pclass="mathContainer hidden">class="mathCode">$W^{1,p}$ estimates with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616301632&_mathId=si2.gif&_user=111111111&_pii=S0022039616301632&_rdoc=1&_issn=00220396&md5=83d257e85eeb91a3785fdd4186e771f3" title="Click to view the MathML source">1<p≤∞class="mathContainer hidden">class="mathCode">$1<p\le \infty$ for more general elliptic systems with the Neumann boundary condition on a bounded class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616301632&_mathId=si3.gif&_user=111111111&_pii=S0022039616301632&_rdoc=1&_issn=00220396&md5=fd3110bac11d9a19374414395b666eb4" title="Click to view the MathML source">C^1,ηclass="mathContainer hidden">class="mathCode">$C^{1,\eta }$ domain, arising in homogenization theory. Based on the skills developed by Z. Shen in [27] and by T. Suslina in  and , we also established the class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616301632&_mathId=si4.gif&_user=111111111&_pii=S0022039616301632&_rdoc=1&_issn=00220396&md5=a250880cf74bef1fc34cc1f1f81a1409" title="Click to view the MathML source">L²class="mathContainer hidden">class="mathCode">$L^2$ convergence rates on a bounded class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616301632&_mathId=si100.gif&_user=111111111&_pii=S0022039616301632&_rdoc=1&_issn=00220396&md5=b9f8acb213e64de452805a187949fb2a" title="Click to view the MathML source">C^1,1class="mathContainer hidden">class="mathCode">$C^{1,1}$ domain and a Lipschitz domain, respectively. Here we found a “rough” version of the first order correctors (see (1.12)), which can unify the proof in [27] and [32]. It allows us to skip the corresponding convergence results on class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616301632&_mathId=si224.gif&_user=111111111&_pii=S0022039616301632&_rdoc=1&_issn=00220396&md5=983dc1119d21a1b621beb27bbb1d0210" title="Click to view the MathML source">R^dclass="mathContainer hidden">class="mathCode">${\mathbb{R}}^d$ that are the preconditions in  and . Our results can be regarded as an extension of [23] developed by C. Kenig, F. Lin, Z. Shen, as well as of [32] investigated by T. Suslina.