Gel'fand-Calderón's inverse problem for anisotropic conductivities on bordered surfaces in $\mathbb{R}^3$

Details Gel'fand-Calderón's inverse problem for anisotropic conductivities on bordered surfaces in $\mathbb{R}^3$ Gel'fand-Calderón's inverse problem for anisotropic conductivities on bordered surfaces in $\mathbb{R}^3$

2010-06-03

arxiv.org/abs/1006.0647v2

Int Math Res Notices (2012) 2012 (4): 781-809

Physics

Mathematical Physics

21 pages, no figures; added corrections in Theorem 5.1

Scientific paper

10.1093/imrn/rnr046

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 $\sigma_2$.

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