On the electric impedance tomography problem for nonorientable surfaces with internal holes

Abstract

Let ( M , g ) (M,g) be a compact (in general, nonorientable) surface with boundary ∂ M \partial M and let Γ 0 \Gamma _0 , …, Γ m − 1 \Gamma _{m-1} be connected components of ∂ M \partial M . Let u = u f ( x ) u=u^{f}(x) be a solution to the problem Δ g u = 0 \Delta _{g}u=0 in M M , u | Γ 0 = f u\big |_{\Gamma _0}=f , u | Γ j = 0 u\big |_{\Gamma _j}=0 , j = 1 j=1 , …, m ′ m’ , ∂ ν u | Γ j = 0 \partial _{\nu }u\big |_{\Gamma _j}=0 , j = m ′ + 1 j=m’+1 , …, m − 1 m-1 , where ν \nu is the outward normal. With this problem, one associates the DN map Λ : f ↦ ∂ ν u f | Γ 0 \Lambda \colon f\mapsto \partial _{\nu }u^{f}\big |_{\Gamma _0} . The purpose is to determine M M from Λ \Lambda . To this end, an algebraic version of the boundary control method is applied. The key instrument is the algebra A \mathfrak {A} of functions holomorphic on the appropriate orientable double cover of M M . It is proved that A \mathfrak {A} is determined by Λ \Lambda up to isometric isomorphism. The spectrum of the algebra A \mathfrak {A} provides a relevant copy M ′ M’ of M M . This copy is conformally equivalent to M M while its DN map coincides with Λ \Lambda .