Electric impedance tomography problem for surfaces with internal holes*Supported by RFBR Grant 20-01 627A. (20th September 2021) - Badanin, A V; Belishev, M I; Korikov, D V
Abstract: Let ( M, g ) be a smooth compact Riemann surface with the multicomponent boundary Γ = Γ 0 ∪ Γ 1 ∪ ⋯ ∪ Γ m ≕ Γ 0 ∪ Γ . Let u = u f obey Δ u = 0 in M, u | Γ 0 = f, u | Γ = 0 (the grounded holes) and v = v h obey Δ v = 0 in M, v | Γ 0 = h, ∂ ν v | Γ = 0 (the isolated holes). Let Λ g gr : f ↦ ∂ ν u f | Γ 0 and Λ g is : h ↦ ∂ ν v h | Γ 0 be the corresponding Dirichlet-to-Neumann map. The electric impedance tomography problem is to determine M from Λ g gr or Λ g is . To solve it, an algebraic variant of the boundary control method is applied. The central role is played by the algebra A of functions holomorphic on the manifold obtained by gluing two examples of M along Γ . We show that A is determined by Λ g gr (or Λ g is ) up to isometric isomorphism. A relevant copy ( M ′, g ′, Γ0 ′) of ( M, g, Γ0 ) is constructed from the Gelfand spectrum of A . By construction, this copy turns out to be conformally equivalent to ( M, g, Γ0 ), obeys Γ 0 ′ = Γ 0, Λ g ′ gr = Λ g gr, Λ g ′ is = Λ g is and provides a solution of the problem.