改进的求解方法𝐿²-四阶算子均匀化的近似

Pub Date : 2023-07-26 DOI:10.1090/spmj/1772

S. Pastukhova

{"title":"改进的求解方法𝐿²-四阶算子均匀化的近似","authors":"S. Pastukhova","doi":"10.1090/spmj/1772","DOIUrl":null,"url":null,"abstract":"<p>A divergent elliptic operator <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A Subscript epsilon\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>A</mml:mi>\n <mml:mi>ε</mml:mi>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">A_\\varepsilon</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of the fourth order with <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon\">\n <mml:semantics>\n <mml:mi>ε</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\varepsilon</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-periodic coefficients acting in the space <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R Superscript d\">\n <mml:semantics>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"double-struck\">R</mml:mi>\n </mml:mrow>\n <mml:mi>d</mml:mi>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">\\mathbb {R}^d</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is treated, where <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon\">\n <mml:semantics>\n <mml:mi>ε</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\varepsilon</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is a small parameter. For the resolvent <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper A Subscript epsilon Baseline plus 1 right-parenthesis Superscript negative 1\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msub>\n <mml:mi>A</mml:mi>\n <mml:mi>ε</mml:mi>\n </mml:msub>\n <mml:mo>+</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>−</mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n </mml:msup>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">(A_\\varepsilon +1)^{-1}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, approximations are constructed in the operator <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper L squared right-arrow upper L squared right-parenthesis\">\n <mml:semantics>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:msup>\n <mml:mi>L</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msup>\n <mml:mo stretchy=\"false\">→</mml:mo>\n <mml:msup>\n <mml:mi>L</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msup>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">{(L^2\\to L^2)}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-norm with remainder of order <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"epsilon cubed\">\n <mml:semantics>\n <mml:msup>\n <mml:mi>ε</mml:mi>\n <mml:mn>3</mml:mn>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">\\varepsilon ^3</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. The method of two-scale expansions with the use of smoothing is employed.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Improved resolvent 𝐿²-approximations in homogenization of fourth order operators\",\"authors\":\"S. Pastukhova\",\"doi\":\"10.1090/spmj/1772\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>A divergent elliptic operator <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper A Subscript epsilon\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mi>A</mml:mi>\\n <mml:mi>ε</mml:mi>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">A_\\\\varepsilon</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> of the fourth order with <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"epsilon\\\">\\n <mml:semantics>\\n <mml:mi>ε</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\varepsilon</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-periodic coefficients acting in the space <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper R Superscript d\\\">\\n <mml:semantics>\\n <mml:msup>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi>\\n </mml:mrow>\\n <mml:mi>d</mml:mi>\\n </mml:msup>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathbb {R}^d</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> is treated, where <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"epsilon\\\">\\n <mml:semantics>\\n <mml:mi>ε</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\varepsilon</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> is a small parameter. For the resolvent <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-parenthesis upper A Subscript epsilon Baseline plus 1 right-parenthesis Superscript negative 1\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:msub>\\n <mml:mi>A</mml:mi>\\n <mml:mi>ε</mml:mi>\\n </mml:msub>\\n <mml:mo>+</mml:mo>\\n <mml:mn>1</mml:mn>\\n <mml:msup>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo>−</mml:mo>\\n <mml:mn>1</mml:mn>\\n </mml:mrow>\\n </mml:msup>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">(A_\\\\varepsilon +1)^{-1}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, approximations are constructed in the operator <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"left-parenthesis upper L squared right-arrow upper L squared right-parenthesis\\\">\\n <mml:semantics>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:msup>\\n <mml:mi>L</mml:mi>\\n <mml:mn>2</mml:mn>\\n </mml:msup>\\n <mml:mo stretchy=\\\"false\\\">→</mml:mo>\\n <mml:msup>\\n <mml:mi>L</mml:mi>\\n <mml:mn>2</mml:mn>\\n </mml:msup>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">{(L^2\\\\to L^2)}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-norm with remainder of order <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"epsilon cubed\\\">\\n <mml:semantics>\\n <mml:msup>\\n <mml:mi>ε</mml:mi>\\n <mml:mn>3</mml:mn>\\n </mml:msup>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\varepsilon ^3</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. The method of two-scale expansions with the use of smoothing is employed.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2023-07-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/spmj/1772\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/spmj/1772","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 2

摘要

研究了一个具有ε \varepsilon周期系数的四阶发散椭圆算子A ε A_\varepsilon作用于空间rd \mathbb {R}^d，其中ε \varepsilon是一个小参数。对于解(A ε +1)−1 (A_\varepsilon +1)^{-1}，在算子(l2→l2) {(L^2\到L^2)} -范数中构造了近似，余数为ε 3 \varepsilon ^3阶。采用了双尺度展开式平滑法。

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Improved resolvent 𝐿²-approximations in homogenization of fourth order operators

A divergent elliptic operator A ε A_\varepsilon of the fourth order with ε \varepsilon -periodic coefficients acting in the space R d \mathbb {R}^d is treated, where ε \varepsilon is a small parameter. For the resolvent ( A ε + 1 ) − 1 (A_\varepsilon +1)^{-1} , approximations are constructed in the operator ( L 2 → L 2 ) {(L^2\to L^2)} -norm with remainder of order ε 3 \varepsilon ^3 . The method of two-scale expansions with the use of smoothing is employed.

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