{"title":"关于分段局部周期算子的均匀化问题","authors":"N. N. Senik","doi":"10.1134/S1061920823020139","DOIUrl":null,"url":null,"abstract":"<p> We discuss homogenization of a strongly elliptic operator <span>\\(\\mathcal A^\\varepsilon=-\\operatorname{div}A(x,x/\\varepsilon_\\#)\\nabla\\)</span> on a bounded <span>\\(C^{1,1}\\)</span> domain in <span>\\(\\mathbb R^d\\)</span> with either Dirichlet or Neumann boundary condition. The function <span>\\(A\\)</span> is piecewise Lipschitz in the first variable and periodic in the second one, and the function <span>\\(\\varepsilon_\\#\\)</span> is identically equal to <span>\\(\\varepsilon_i(\\varepsilon)\\)</span> on each piece <span>\\(\\Omega_i\\)</span>, with <span>\\(\\varepsilon_i(\\varepsilon)\\to0\\)</span> as <span>\\(\\varepsilon\\to0\\)</span>. For <span>\\(\\mu\\)</span> in a resolvent set, we show that the resolvent <span>\\((\\mathcal A^\\varepsilon-\\mu)^{-1}\\)</span> converges, as <span>\\(\\varepsilon\\to0\\)</span>, in the operator norm on <span>\\(L_2(\\Omega)^n\\)</span> to the resolvent <span>\\((\\mathcal A^0-\\mu)^{-1}\\)</span> of the effective operator at the rate <span>\\(\\varepsilon_ {\\vee} \\)</span>, where <span>\\(\\varepsilon_ {\\vee} \\)</span> stands for the largest of <span>\\(\\varepsilon_i(\\varepsilon)\\)</span>. We also obtain an approximation for the resolvent in the operator norm from <span>\\(L_2(\\Omega)^n\\)</span> to <span>\\(H^1(\\Omega)^n\\)</span> with error of order <span>\\(\\varepsilon_ {\\vee} ^{1/2}\\)</span>. </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.7000,"publicationDate":"2023-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On Homogenization for Piecewise Locally Periodic Operators\",\"authors\":\"N. N. Senik\",\"doi\":\"10.1134/S1061920823020139\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p> We discuss homogenization of a strongly elliptic operator <span>\\\\(\\\\mathcal A^\\\\varepsilon=-\\\\operatorname{div}A(x,x/\\\\varepsilon_\\\\#)\\\\nabla\\\\)</span> on a bounded <span>\\\\(C^{1,1}\\\\)</span> domain in <span>\\\\(\\\\mathbb R^d\\\\)</span> with either Dirichlet or Neumann boundary condition. The function <span>\\\\(A\\\\)</span> is piecewise Lipschitz in the first variable and periodic in the second one, and the function <span>\\\\(\\\\varepsilon_\\\\#\\\\)</span> is identically equal to <span>\\\\(\\\\varepsilon_i(\\\\varepsilon)\\\\)</span> on each piece <span>\\\\(\\\\Omega_i\\\\)</span>, with <span>\\\\(\\\\varepsilon_i(\\\\varepsilon)\\\\to0\\\\)</span> as <span>\\\\(\\\\varepsilon\\\\to0\\\\)</span>. For <span>\\\\(\\\\mu\\\\)</span> in a resolvent set, we show that the resolvent <span>\\\\((\\\\mathcal A^\\\\varepsilon-\\\\mu)^{-1}\\\\)</span> converges, as <span>\\\\(\\\\varepsilon\\\\to0\\\\)</span>, in the operator norm on <span>\\\\(L_2(\\\\Omega)^n\\\\)</span> to the resolvent <span>\\\\((\\\\mathcal A^0-\\\\mu)^{-1}\\\\)</span> of the effective operator at the rate <span>\\\\(\\\\varepsilon_ {\\\\vee} \\\\)</span>, where <span>\\\\(\\\\varepsilon_ {\\\\vee} \\\\)</span> stands for the largest of <span>\\\\(\\\\varepsilon_i(\\\\varepsilon)\\\\)</span>. We also obtain an approximation for the resolvent in the operator norm from <span>\\\\(L_2(\\\\Omega)^n\\\\)</span> to <span>\\\\(H^1(\\\\Omega)^n\\\\)</span> with error of order <span>\\\\(\\\\varepsilon_ {\\\\vee} ^{1/2}\\\\)</span>. </p>\",\"PeriodicalId\":763,\"journal\":{\"name\":\"Russian Journal of Mathematical Physics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.7000,\"publicationDate\":\"2023-06-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Russian Journal of Mathematical Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://link.springer.com/article/10.1134/S1061920823020139\",\"RegionNum\":3,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Russian Journal of Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1134/S1061920823020139","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
On Homogenization for Piecewise Locally Periodic Operators
We discuss homogenization of a strongly elliptic operator \(\mathcal A^\varepsilon=-\operatorname{div}A(x,x/\varepsilon_\#)\nabla\) on a bounded \(C^{1,1}\) domain in \(\mathbb R^d\) with either Dirichlet or Neumann boundary condition. The function \(A\) is piecewise Lipschitz in the first variable and periodic in the second one, and the function \(\varepsilon_\#\) is identically equal to \(\varepsilon_i(\varepsilon)\) on each piece \(\Omega_i\), with \(\varepsilon_i(\varepsilon)\to0\) as \(\varepsilon\to0\). For \(\mu\) in a resolvent set, we show that the resolvent \((\mathcal A^\varepsilon-\mu)^{-1}\) converges, as \(\varepsilon\to0\), in the operator norm on \(L_2(\Omega)^n\) to the resolvent \((\mathcal A^0-\mu)^{-1}\) of the effective operator at the rate \(\varepsilon_ {\vee} \), where \(\varepsilon_ {\vee} \) stands for the largest of \(\varepsilon_i(\varepsilon)\). We also obtain an approximation for the resolvent in the operator norm from \(L_2(\Omega)^n\) to \(H^1(\Omega)^n\) with error of order \(\varepsilon_ {\vee} ^{1/2}\).
期刊介绍:
Russian Journal of Mathematical Physics is a peer-reviewed periodical that deals with the full range of topics subsumed by that discipline, which lies at the foundation of much of contemporary science. Thus, in addition to mathematical physics per se, the journal coverage includes, but is not limited to, functional analysis, linear and nonlinear partial differential equations, algebras, quantization, quantum field theory, modern differential and algebraic geometry and topology, representations of Lie groups, calculus of variations, asymptotic methods, random process theory, dynamical systems, and control theory.