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arXiv · 2009.08367

On the EIT problem for nonorientable surfaces

M. I. Belishev·D. V. Korikov
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Abstract

Let $(\Omega,g)$ be a smooth compact two-dimensional Riemannian manifold with boundary, $\Lambda_g: f\mapsto \partial_\nu u|_{\partial\Omega}$ its DN map, where $u$ obeys $\Delta_g u=0$ in $\Omega$ and $u|_{\partial \Omega}=f$. The Electric Impedance Tomography problem is to determine $\Omega$ from $\Lambda_g$. A criterion is proposed that enables one to detect (via $\Lambda_g$) whether $\Omega$ 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 ${\mathbb M}$ of $M$, which is determined by $\Lambda_g$ 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 $\partial M'=\partial M,\,\,\Lambda_{g'}=\Lambda_g$, and thus solves the problem.

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M. I. Belishev, D. V. Korikov. 2020-09-17. On the EIT problem for nonorientable surfaces. https://arxiv.org/abs/2009.08367

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