# Conformal mapping and inverse conductivity problem with one measurement

ESAIM: Control, Optimisation and Calculus of Variations (2007)

- Volume: 13, Issue: 1, page 163-177
- ISSN: 1292-8119

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topDambrine, Marc, and Kateb, Djalil. "Conformal mapping and inverse conductivity problem with one measurement." ESAIM: Control, Optimisation and Calculus of Variations 13.1 (2007): 163-177. <http://eudml.org/doc/249929>.

@article{Dambrine2007,

abstract = {
This work deals with a two-dimensional inverse problem in the field of tomography. The geometry of an unknown inclusion has to be reconstructed from boundary measurements. In this paper, we extend previous results of R. Kress and his coauthors: the leading idea is to use the conformal mapping function as unknown. We establish an integrodifferential equation that the trace of the Riemann map solves. We write it as a fixed point equation and give conditions for contraction. We conclude with a series of numerical examples illustrating the performance of the method.
},

author = {Dambrine, Marc, Kateb, Djalil},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Inverse conductivity problem; conformal mapping.; inverse conductivity problem; conformal mapping},

language = {eng},

month = {2},

number = {1},

pages = {163-177},

publisher = {EDP Sciences},

title = {Conformal mapping and inverse conductivity problem with one measurement},

url = {http://eudml.org/doc/249929},

volume = {13},

year = {2007},

}

TY - JOUR

AU - Dambrine, Marc

AU - Kateb, Djalil

TI - Conformal mapping and inverse conductivity problem with one measurement

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2007/2//

PB - EDP Sciences

VL - 13

IS - 1

SP - 163

EP - 177

AB -
This work deals with a two-dimensional inverse problem in the field of tomography. The geometry of an unknown inclusion has to be reconstructed from boundary measurements. In this paper, we extend previous results of R. Kress and his coauthors: the leading idea is to use the conformal mapping function as unknown. We establish an integrodifferential equation that the trace of the Riemann map solves. We write it as a fixed point equation and give conditions for contraction. We conclude with a series of numerical examples illustrating the performance of the method.

LA - eng

KW - Inverse conductivity problem; conformal mapping.; inverse conductivity problem; conformal mapping

UR - http://eudml.org/doc/249929

ER -

## References

top- I. Akduman and R. Kress, Electrostatic imaging via conformal mapping, Inverse Problems18 (2002) 1659–1672.
- M. Dambrine and D. Kateb, Work in progress.
- E. Fabes, H. Kang and J.K. Seo, Inverse conductivity problem with one measurement: Error estimates and approximate identification for perturbed disks. SIAM J. Math. Anal.30 (1999) 699–720.
- G.M. Golutsin, Geometrische Funktionentheorie. Deutscher Verlag der Wissenschaften, Berlin (1957).
- H. Haddar and R. Kress, Conformal mappings and inverse boundary value problems. Inverse Problems21 (2005) 935–953.
- P. Henrici, Applied and computational complex analysis, Vol 1,3. John Wiley & Sons (1986).
- N.I. Muskhelishvili, Some basic problems of the mathematical theory of elasticity. Noordhoff, Groniningen (1953).
- M. Taylor, Partial Differential Equations, Vol. 1: Basic Theory. Applied Math. Sciences115, Springer-Verlag, New York (1996).

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