# 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}
}
• 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…
7 Citations
• Mathematics
• 2010
We prove that a potential $q$ can be reconstructed from the Dirichlet-to-Neumann map for the Schrodinger operator $-\Delta_g + q$ in a fixed admissible 3-dimensional Riemannian manifold $(M,g)$. We
• Mathematics
• 2011
We prove identification of coefficients up to gauge equivalence by Cauchy data at the boundary for elliptic systems on oriented compact surfaces with boundary or domains of $${\mathbb{C}}$$ . In the
• Mathematics
Annales Henri Poincaré
• 2013
We prove identification of coefficients up to gauge equivalence by Cauchy data at the boundary for elliptic systems on oriented compact surfaces with boundary or domains of
The global uniqueness for inverse boundary value problems of elliptic equations at fixed frequency in dimension n D 2 is quite particular and remained open for many years. Now these problems are well
The prominent mathematician Gennadi Markovich Henkin, an expert in complex analysis, mathematical physics, and applications of mathematics to economics, passed away in Paris on 19 January 2016 after
• V. Michel
• Mathematics
The Journal of Geometric Analysis
• 2019
In this article, we introduce a process to reconstruct a Riemann surface with boundary equipped with a linked conductivity tensor from its boundary and the Dirichlet–Neumann operator associated with

## References

SHOWING 1-10 OF 34 REFERENCES

• Mathematics
• 2009
On a fixed smooth compact Riemann surface with boundary $(M_0,g)$, we show that for the Schr\"odinger operator $\Delta +V$ with potential $V\in C^{1,\alpha}(M_0)$ for some $\alpha>0$, the
• Mathematics
• 2010
An electrical potential U on a bordered real surface X in ℝ3 with isotropic conductivity function σ>0 satisfies the equation d(σdcU)|X=0, where $d^{c}= i(\bar{ \partial }-\partial )$, \$d=\bar{
• Mathematics
• 2007
For any orientable compact surface with boundary, we compute the regularized determinant of the Dirichlet-to-Neumann (DN) map in terms of main value at 0 of a Ruelle zeta function using
We consider the impedance tomography problem for anisotropic conductivities. Given a bounded region Ω in space, a diffeomorphism Ψ from Ω to itself which restricts to the identity on ∂ Ω, and a
• Mathematics
• 2010
Let be a bounded domain with a smooth boundary and a smooth anisotropic conductivity on . Starting from the Dirichlet-to-Neumann operator on , we give an explicit procedure to find a unique (up to a
• Mathematics
• 2005
Abstract.This article gives a complex analysis lighting on the problem which consists in restoring a bordered connected riemaniann surface from its boundary and its Dirichlet–Neumann operator. The
We show that the coefficient -y(x) of the elliptic equation Vie (QyVu) = 0 in a two-dimensional domain is uniquely determined by the corresponding Dirichlet-to-Neumann map on the boundary, and give a
• Mathematics
• 2004
Abstract We study the inverse conductivity problem for an anisotropic conductivity σ ∈ L ∞ in bounded and unbounded domains. Also, we give applications of the results in the case when two sets of
• Mathematics
• 2011
1. Some Classical Theorems.- 1.1. The Riesz-Thorin Theorem.- 1.2. Applications of the Riesz-Thorin Theorem.- 1.3. The Marcinkiewicz Theorem.- 1.4. An Application of the Marcinkiewicz Theorem.- 1.5.