Electric Impedance Tomography problem for surfaces with internal holes

Abstract

Let $(M,g)$ be a smooth compact Riemann surface with the multicomponent boundary $Γ=Γ_0\cupΓ_1\cup\dots\cupΓ_m=:Γ_0\cup\tildeΓ$. Let $u=u^f$ obey $Δu=0$ in $M$, $u|_{Γ_0}=f,\,\,u|_{\tildeΓ}=0$ (the grounded holes) and $v=v^h$ obey $Δv=0$ in $M$, $v|_{Γ_0}=h,\,\,\partial_νv|_{\tildeΓ}=0$ (the isolated holes). Let $Λ_{g}^{\rm gr}: f\mapsto\partial_νu^f|_{Γ_{0}}$ and $Λ_{g}^{\rm is}: h\mapsto\partial_νv^h|_{Γ_{0}}$ be the corresponding DN-maps. The EIT problem is to determine $M$ from $Λ_{g}^{\rm gr}$ or $Λ_{g}^{\rm is}$. 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 $\tildeΓ$. We show that this algebra is determined by $Λ_{g}^{\rm gr}$ (or $Λ_{g}^{\rm is}$) up to isometric isomorphism. Its Gelfand spectrum (the set of characters) plays the role of the material for constructing a relevant copy $(M',g',Γ_{0}')$ of $(M,g,Γ_{0})$. This copy is conformally equivalent to the original, provides $Γ_{0}'=Γ_{0},\,\,Λ_{g'}^{\rm gr}=Λ_{g}^{\rm gr},\,\,Λ_{g'}^{\rm is}=Λ_{g}^{\rm is}$, and thus solves the problem.