{"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":null,"pages":null},"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 Γ0\Gamma _0, …, Γm−1\Gamma _{m-1} be connected components of ∂M\partial M. Let u=uf(x)u=u^{f}(x) be a solution to the problem Δgu=0\Delta _{g}u=0 in MM, u|Γ0=fu\big |_{\Gamma _0}=f, u|Γj=0u\big |_{\Gamma _j}=0, j=1j=1, …, m′m’, ∂νu|Γj=0\partial _{\nu }u\big |_{\Gamma _j}=0, j=m′+1j=m’+1, …, m−1m-1, where ν\nu is the outward normal. With this problem, one associates the DN map Λ:f↦∂νuf|Γ0\Lambda \colon f\mapsto \partial _{\nu }u^{f}\big |_{\Gamma _0}. The purpose is to determine MM 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 MM. 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 MM. This copy is conformally equivalent to MM while its DN map coincides with Λ\Lambda.
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.