Electric impedance tomography problem for surfaces with internal holes * *Supported by RFBR Grant 20-01 627A.

Abstract

Let (M, g) be a smooth compact Riemann surface with the multicomponent boundary . Let u = u f obey Δu = 0 in M, (the grounded holes) and v = v h obey Δv = 0 in M, (the isolated holes). Let and be the corresponding Dirichlet-to-Neumann map. The electric impedance tomography problem is to determine M from or . To solve it, an algebraic variant of the boundary control method is applied. The central role is played by the algebra of functions holomorphic on the manifold obtained by gluing two examples of M along . We show that is determined by (or ) up to isometric isomorphism. A relevant copy (M′, g′, Γ0′) of (M, g, Γ0) is constructed from the Gelfand spectrum of . By construction, this copy turns out to be conformally equivalent to (M, g, Γ0), obeys and provides a solution of the problem.