On the electric impedance tomography problem for nonorientable surfaces with internal holes

IF 0.7 4区 数学 Q2 MATHEMATICS
D. Korikov
{"title":"On the electric impedance tomography problem for nonorientable surfaces with internal holes","authors":"D. Korikov","doi":"10.1090/spmj/1778","DOIUrl":null,"url":null,"abstract":"Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper M comma g right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(M,g)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a compact (in general, nonorientable) surface with boundary <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"partial-differential upper M\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\"normal\">∂<!-- ∂ --></mml:mi> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\partial M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma 0\"> <mml:semantics> <mml:msub> <mml:mi mathvariant=\"normal\">Γ<!-- Γ --></mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding=\"application/x-tex\">\\Gamma _0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, …, <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma Subscript m minus 1\"> <mml:semantics> <mml:msub> <mml:mi mathvariant=\"normal\">Γ<!-- Γ --></mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>m</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding=\"application/x-tex\">\\Gamma _{m-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be connected components of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"partial-differential upper M\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\"normal\">∂<!-- ∂ --></mml:mi> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\partial M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"u equals u Superscript f Baseline left-parenthesis x right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>f</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">u=u^{f}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a solution to the problem <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Delta Subscript g Baseline u equals 0\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant=\"normal\">Δ<!-- Δ --></mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>g</mml:mi> </mml:mrow> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\Delta _{g}u=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=\"application/x-tex\">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"u vertical-bar Subscript normal upper Gamma 0 Baseline equals f\"> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:msub> <mml:mstyle scriptlevel=\"0\"> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">|</mml:mo> </mml:mrow> </mml:mstyle> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:msub> <mml:mi mathvariant=\"normal\">Γ<!-- Γ --></mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">u\\big |_{\\Gamma _0}=f</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"u vertical-bar Subscript normal upper Gamma Sub Subscript j Subscript Baseline equals 0\"> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:msub> <mml:mstyle scriptlevel=\"0\"> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">|</mml:mo> </mml:mrow> </mml:mstyle> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:msub> <mml:mi mathvariant=\"normal\">Γ<!-- Γ --></mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">u\\big |_{\\Gamma _j}=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"j equals 1\"> <mml:semantics> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">j=1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, …, <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m prime\"> <mml:semantics> <mml:msup> <mml:mi>m</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding=\"application/x-tex\">m’</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"partial-differential Subscript nu Baseline u vertical-bar Subscript normal upper Gamma Sub Subscript j Subscript Baseline equals 0\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant=\"normal\">∂<!-- ∂ --></mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>ν<!-- ν --></mml:mi> </mml:mrow> </mml:msub> <mml:mi>u</mml:mi> <mml:msub> <mml:mstyle scriptlevel=\"0\"> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">|</mml:mo> </mml:mrow> </mml:mstyle> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:msub> <mml:mi mathvariant=\"normal\">Γ<!-- Γ --></mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\partial _{\\nu }u\\big |_{\\Gamma _j}=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"j equals m prime plus 1\"> <mml:semantics> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>m</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">j=m’+1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, …, <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"m minus 1\"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">m-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"nu\"> <mml:semantics> <mml:mi>ν<!-- ν --></mml:mi> <mml:annotation encoding=\"application/x-tex\">\\nu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the outward normal. With this problem, one associates the DN map <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Lamda colon f right-arrow from bar partial-differential Subscript nu Baseline u Superscript f Baseline vertical-bar Subscript normal upper Gamma 0 Baseline\"> <mml:semantics> <mml:mrow> <mml:mi mathvariant=\"normal\">Λ<!-- Λ --></mml:mi> <mml:mo>:<!-- : --></mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=\"false\">↦<!-- ↦ --></mml:mo> <mml:msub> <mml:mi mathvariant=\"normal\">∂<!-- ∂ --></mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>ν<!-- ν --></mml:mi> </mml:mrow> </mml:msub> <mml:msup> <mml:mi>u</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>f</mml:mi> </mml:mrow> </mml:msup> <mml:msub> <mml:mstyle scriptlevel=\"0\"> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">|</mml:mo> </mml:mrow> </mml:mstyle> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:msub> <mml:mi mathvariant=\"normal\">Γ<!-- Γ --></mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\Lambda \\colon f\\mapsto \\partial _{\\nu }u^{f}\\big |_{\\Gamma _0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The purpose is to determine <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=\"application/x-tex\">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Lamda\"> <mml:semantics> <mml:mi mathvariant=\"normal\">Λ<!-- Λ --></mml:mi> <mml:annotation encoding=\"application/x-tex\">\\Lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. To this end, an algebraic version of the boundary control method is applied. The key instrument is the algebra <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"German upper A\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"fraktur\">A</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\mathfrak {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of functions holomorphic on the appropriate orientable double cover of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=\"application/x-tex\">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is proved that <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"German upper A\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"fraktur\">A</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\mathfrak {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is determined by <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Lamda\"> <mml:semantics> <mml:mi mathvariant=\"normal\">Λ<!-- Λ --></mml:mi> <mml:annotation encoding=\"application/x-tex\">\\Lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> up to isometric isomorphism. The spectrum of the algebra <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"German upper A\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"fraktur\">A</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\mathfrak {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> provides a relevant copy <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M prime\"> <mml:semantics> <mml:msup> <mml:mi>M</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding=\"application/x-tex\">M’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=\"application/x-tex\">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This copy is conformally equivalent to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding=\"application/x-tex\">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> while its DN map coincides with <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Lamda\"> <mml:semantics> <mml:mi mathvariant=\"normal\">Λ<!-- Λ --></mml:mi> <mml:annotation encoding=\"application/x-tex\">\\Lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":" 13","pages":"0"},"PeriodicalIF":0.7000,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"St Petersburg Mathematical Journal","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/spmj/1778","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

Let ( M , g ) (M,g) be a compact (in general, nonorientable) surface with boundary M \partial M and let Γ 0 \Gamma _0 , …, Γ m 1 \Gamma _{m-1} be connected components of M \partial M . Let u = u f ( x ) u=u^{f}(x) be a solution to the problem Δ g u = 0 \Delta _{g}u=0 in M M , u | Γ 0 = f u\big |_{\Gamma _0}=f , u | Γ j = 0 u\big |_{\Gamma _j}=0 , j = 1 j=1 , …, m m’ , ν u | Γ j = 0 \partial _{\nu }u\big |_{\Gamma _j}=0 , j = m + 1 j=m’+1 , …, m 1 m-1 , where ν \nu is the outward normal. With this problem, one associates the DN map Λ : f ν u f | Γ 0 \Lambda \colon f\mapsto \partial _{\nu }u^{f}\big |_{\Gamma _0} . The purpose is to determine M M from Λ \Lambda . To this end, an algebraic version of the boundary control method is applied. The key instrument is the algebra A \mathfrak {A} of functions holomorphic on the appropriate orientable double cover of M M . It is proved that A \mathfrak {A} is determined by Λ \Lambda up to isometric isomorphism. The spectrum of the algebra A \mathfrak {A} provides a relevant copy M M’ of M M . This copy is conformally equivalent to M M while its DN map coincides with Λ \Lambda .
内孔不可定向表面的电阻抗层析问题
设(M,g) (M,g)是一个边界为∂M的紧曲面(一般来说,是不可定向的) \partial M,设Γ 0 \Gamma _0,…,Γ m−1 \Gamma _{m-1} 是∂M的连通分量 \partial M。令u=u f (x) u=u^{f}(x)是问题的解Δ g u = 0 \Delta _{g}u=0 in M M, u | Γ 0 = f u\big |_{\Gamma _0}=f, u | Γ j = 0 u\big |_{\Gamma _j}=0, j=1 j=1,…,m ' m ',∂ν u | Γ j= 0 \partial _{\nu }你\big |_{\Gamma _j}=0, j=m ' +1 j=m ' +1,…,m−1 m-1,其中ν \nu 是外法线。对于这个问题,可以将DN映射Λ: f∈∂ν u f | Γ 0联系起来 \Lambda \colon f\mapsto \partial _{\nu }u^{f}\big |_{\Gamma _0} . 目的是从Λ中确定M \Lambda . 为此,应用了边界控制方法的代数版本。关键的工具是代数A \mathfrak {a} 在M M的适当可定向双盖上的函数全纯。证明了A \mathfrak {a} 由Λ决定 \Lambda 直到等距同构。代数A的谱 \mathfrak {a} 提供M ' M ' M的相关副本。此副本的保角等效于M M,而其DN映射与Λ一致 \Lambda .
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来源期刊
CiteScore
1.00
自引率
12.50%
发文量
52
审稿时长
>12 weeks
期刊介绍: This journal is a cover-to-cover translation into English of Algebra i Analiz, published six times a year by the mathematics section of the Russian Academy of Sciences.
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