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On the EIT problem for nonorientable surfaces

  • M. I. Belishev EMAIL logo und D. V. Korikov
Veröffentlicht/Copyright: 18. Dezember 2020

Abstract

Let (Ω,g) be a smooth compact two-dimensional Riemannian manifold with boundary and let Λg:fνu|Ω be its DN map, where u obeys Δgu=0 in Ω and u|Ω=f. The Electric Impedance Tomography Problem is to determine Ω from Λg. A criterion is proposed that enables one to detect (via Λg) whether Ω is orientable or not. The algebraic version of the BC-method is applied to solve the EIT problem for the Moebius band. The main instrument is the algebra of holomorphic functions on the double covering 𝕄 of M, which is determined by Λg up to an isometric isomorphism. Its Gelfand spectrum (the set of characters) plays the role of the material for constructing a relevant copy (M,g) of (M,g). This copy is conformally equivalent to the original, provides M=M, Λg=Λg, and thus solves the problem.

Award Identifier / Grant number: 20-01 627-À

Award Identifier / Grant number: 18-01-00269A

Funding statement: M. I. Belishev was supported by RFBR, grant Nos. 20-01 627-À and 18-01-00269A.

Acknowledgements

The authors are very much grateful to A. F. Vakulenko for useful discussions on the subject of the paper. His advices allowed us to significantly simplify the proof of the basic Theorem 2.3.

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Received: 2020-09-16
Accepted: 2020-11-04
Published Online: 2020-12-18
Published in Print: 2021-06-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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