Gel'fand-Calderón's inverse problem for anisotropic conductivities on bordered surfaces in ℝ 3

Let X be a smooth bordered surface in ℝ 3 with a smooth boundary and σ̂ a smooth anisotropic conductivity on X. If the genus of X is given, then starting from the Dirichlet-to-Neumann operator Λ σ̂ on ∂X, we give an explicit procedure to find a unique Riemann surface Y (up to a biholomorphism), an isotropic conductivity σ on Y and a quasiconformal diffeomorphism F:X→Y which transforms σ̂ into σ.As a corollary, we obtain the following uniqueness result: if σ 1 and σ 2 are two smooth anisotropic conductivities on X with Λ σ1= Λ σ2, then there exists a smooth diffeomorphism Φ:X̄ → X̄ such that Φ|∂X=Id and Φ*σ 1=σ 2. © The Author(s) 2011.