Electric Impedance Tomography problem for surfaces with internal holes

Abstract

Let $(M,g)$ be a smooth compact Riemann surface with the multicomponent boundary $\Gamma=\Gamma_0\cup\Gamma_1\cup\dots\cup\Gamma_m=:\Gamma_0\cup\tilde\Gamma$. Let $u=u^f$ obey $\Delta u=0$ in $M$, $u|_{\Gamma_0}=f,\,\,u|_{\tilde\Gamma}=0$ (the grounded holes) and $v=v^h$ obey $\Delta v=0$ in $M$, $v|_{\Gamma_0}=h,\,\,\partial_\nu v|_{\tilde\Gamma}=0$ (the isolated holes). Let $\Lambda_{g}^{\rm gr}: f\mapsto\partial_\nu u^f|_{\Gamma_{0}}$ and $\Lambda_{g}^{\rm is}: h\mapsto\partial_\nu v^h|_{\Gamma_{0}}$ be the corresponding DN-maps. The EIT problem is to determine $M$ from $\Lambda_{g}^{\rm gr}$ or $\Lambda_{g}^{\rm is}$. To solve it, an algebraic version of the BC-method is applied. The main instrument is the algebra of holomorphic functions on the ma\-ni\-fold ${\mathbb M}$, which is obtained by gluing two examples of $M$ along $\tilde{\Gamma}$. We show that this algebra is determined by $\Lambda_{g}^{\rm gr}$ (or $\Lambda_{g}^{\rm is}$) up to isometric isomorphism. Its Gelfand spectrum (the set of characters) plays the role of the material for constructing a relevant copy $(M',g',\Gamma_{0}')$ of $(M,g,\Gamma_{0})$. This copy is conformally equivalent to the original, provides $\Gamma_{0}'=\Gamma_{0},\,\,\Lambda_{g'}^{\rm gr}=\Lambda_{g}^{\rm gr},\,\,\Lambda_{g'}^{\rm is}=\Lambda_{g}^{\rm is}$, and thus solves the problem.