On the electric impedance tomography problem for nonorientable surfaces with internal holes
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D. V. Korikov;
Translated by: the author - St. Petersburg Math. J. 34 (2023), 759-774
- DOI: https://doi.org/10.1090/spmj/1778
- Published electronically: November 9, 2023
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Abstract:
Let $(M,g)$ be a compact (in general, nonorientable) surface with boundary $\partial M$ and let $\Gamma _0$, …, $\Gamma _{m-1}$ be connected components of $\partial M$. Let $u=u^{f}(x)$ be a solution to the problem $\Delta _{g}u=0$ in $M$, $u\big |_{\Gamma _0}=f$, $u\big |_{\Gamma _j}=0$, $j=1$, …, $m’$, $\partial _{\nu }u\big |_{\Gamma _j}=0$, $j=m’+1$, …, $m-1$, where $\nu$ is the outward normal. With this problem, one associates the DN map $\Lambda \colon f\mapsto \partial _{\nu }u^{f}\big |_{\Gamma _0}$. The purpose is to determine $M$ from $\Lambda$.
To this end, an algebraic version of the boundary control method is applied. The key instrument is the algebra $\mathfrak {A}$ of functions holomorphic on the appropriate orientable double cover of $M$. It is proved that $\mathfrak {A}$ is determined by $\Lambda$ up to isometric isomorphism. The spectrum of the algebra $\mathfrak {A}$ provides a relevant copy $M’$ of $M$. This copy is conformally equivalent to $M$ while its DN map coincides with $\Lambda$.
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Bibliographic Information
- D. V. Korikov
- Affiliation: St. Petersburg Department of Steklov Mathematical Institute of the Russian Academy of Sciences, 27, Fontanka, 191023 St. Petersburg, Russia
- Email: thecakeisalie@list.ru
- Received by editor(s): October 13, 2021
- Published electronically: November 9, 2023
- Additional Notes: The work was supported by RFBR grant no. 20-01-00627-a
- © Copyright 2023 American Mathematical Society
- Journal: St. Petersburg Math. J. 34 (2023), 759-774
- MSC (2020): Primary 35R30; Secondary 46J15, 46J20, 30F15
- DOI: https://doi.org/10.1090/spmj/1778