• DOI:10.1093/imrn/rnr046
• Corpus ID: 119580440

Gel’fand–Calderón’s Inverse Problem for Anisotropic Conductivities on Bordered Surfaces in ℝ3

@article{Henkin2010GelfandCaldernsIP,
  title={Gel’fand–Calder{\'o}n’s Inverse Problem for Anisotropic Conductivities on Bordered Surfaces in ℝ3},
  author={Gennadi M. Henkin and Matteo Santacesaria},
  journal={International Mathematics Research Notices},
  year={2010},
  volume={2012},
  pages={781-809}
}

• G. Henkin, Matteo Santacesaria
• Published 3 June 2010
• Mathematics
• International Mathematics Research Notices

Let $X$ be a smooth bordered surface in $\real^3$ with smooth boundary and $\hat \sigma$ a smooth anisotropic conductivity on $X$. If the genus of $X$ is given, then starting from the Dirichlet-to-Neumann operator $\Lambda_{\hat \sigma}$ on $\partial X$, we give an explicit procedure to find a unique Riemann surface $Y$ (up to a biholomorphism), an isotropic conductivity $\sigma$ on $Y$ and the boundary values of a quasiconformal diffeomorphism $F: X \to Y$ which transforms $\hat \sigma$ into $\sigma$.
As a corollary we obtain the following uniqueness result: if $\sigma_1, \sigma_2$ are two smooth anisotropic conductivities on $X$ with $\Lambda_{\sigma_1}= \Lambda_{\sigma_2}$, then there exists a smooth diffeomorphism $\Phi: \bar X \to \bar X$ which transforms $\sigma_1$ into $\sigma_2$.