American Institute of Mathematical Sciences

February  2007, 1(1): 107-134. doi: 10.3934/ipi.2007.1.107

Inverse boundary value problems in the horosphere - A link between hyperbolic geometry and electrical impedance tomography

 1 Institute of Mathematics, University of Tsukuba, Tsukuba, 305-8571, Japan

Received  August 2006 Revised  September 2006 Published  January 2007

We consider a boundary value problem for the Schrödinger operator $- \Delta + q(x)$ in a ball $\Omega : (x_1 + R)^2 + x_2^2 + (x_3 - r)^2 < r^2$, whose boundary we regard as a horosphere in the hyperbolic space $H^3$ realized in the upper half space $R^3_+$. Let $S = \{|x| = R, x_3 > 0\}$ be a hemisphere, which is generated by a family of geodesics in $H^3$. By imposing a suitable boundary condition on $\partial\Omega$ in terms of a pseudo-differential operator, we compute the integral mean of $q(x)$ over $S\cap\Omega$ from the local knowledge of the associated (generalized) Robin-to-Dirichlet map for $- \Delta + q(x)$ around $S\cap\partial\Omega$. The potential $q(x)$ is then reconstructed by virtue of the inverse Radon transform on hyperbolic space. If the support of $q(x)$ has a positive distance from $\partial\Omega$, one can construct this generalized Robin-to-Dirichlet map from the usual Dirichlet-to-Neumann map. These results explain the mathematical background of the well-known Barber-Brown algorithm in electrical impedance tomography.
Citation: Hiroshi Isozaki. Inverse boundary value problems in the horosphere - A link between hyperbolic geometry and electrical impedance tomography. Inverse Problems and Imaging, 2007, 1 (1) : 107-134. doi: 10.3934/ipi.2007.1.107
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