Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-29T13:19:52.913Z Has data issue: false hasContentIssue false
  • English
  • Français

A general representation formula for boundary voltage perturbationscaused by internal conductivity inhomogeneities of low volume fraction

Published online by Cambridge University Press:  15 March 2003

Yves Capdeboscq  and
Michael S. Vogelius
Yves Capdeboscq
Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA. vogelius@math.rutgers.edu.
Michael S. Vogelius
Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA. vogelius@math.rutgers.edu.
Get access

Abstract

We establish an asymptotic representation formula for the steady state voltage perturbations caused by low volume fraction internal conductivity inhomogeneities. This formula generalizes and unifies earlier formulas derived for special geometries and distributions of inhomogeneities.

Keywords

Voltage perturbationsconductivity inhomogeneitieslow volume fraction.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 37 , Issue 1 , January 2003 , pp. 159 - 173
Copyright
© EDP Sciences, SMAI, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alessandrini, G., Rosset, E. and Seo, J.K., Optimal size estimates for the inverse conductivity problem with one measurement. Proc. Amer. Math. Soc. 128 (2000) 53-64. CrossRef
H. Ammari and H. Kang, High-order terms in the asymptotic expansions of the steady-state voltage potentials in the presence of conductivity inhomogeneities of small diameter. Preprint (2002).
H. Ammari and J.K. Seo, A new formula for the reconstruction of conductivity inhomogeneities. Preprint (2002).
Ammari, H., Moskow, S. and Vogelius, M.S., Boundary integral formulae for the reconstruction of electric and electromagnetic inhomogeneities of small volume. ESAIM Control Optim. Calc. Var. 9 (2003) 49-66. CrossRef
Beretta, E., Mukherjee, A. and Vogelius, M.S., Asymptotic formulas for steady state voltage potentials in the presence of conductivity imperfections of small area. Z. Angew. Math. Phys. 52 (2001) 543-572. CrossRef
E. Beretta, E. Francini and M.S. Vogelius, Asymptotic formulas for steady state voltage potentials in the presence of thin inhomogeneities. A rigorous error analysis. Preprint (2002).
M. Brühl, M. Hanke and M.S. Vogelius, A direct impedance tomography algorithm for locating small inhomogeneities. Numer. Math. (to appear).
Y. Capdeboscq and M.S. Vogelius, Optimal asymptotic estimates for the volume of internal inhomogeneities in terms of multiple boundary measurements. ESAIM: M2AN (to appear).
Cedio-Fengya, D.J., Moskow, S. and Vogelius, M.S., Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction. Inverse Problems 14 (1998) 553-595. CrossRef
Friedman, A. and Vogelius, M.S., Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence. Arch. Ration. Mech. Anal. 105 (1989) 299-326. CrossRef
D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Grundlehren der mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin, Heidelberg, New York (1983).
Kang, H., Seo, J.K. and Sheen, D., The inverse conductivity problem with one measurement: stability and estimation of size. SIAM J. Math. Anal. 28 (1997) 1389-1405. CrossRef
Kwon, O., Seo, J.K. and Yoon, J-R., A real time algorithm for the location search of discontinuous conductivities with one measurement. Comm. Pure Appl. Math. 55 (2002) 1-29. CrossRef
F. Murat and L. Tartar, H-Convergence, in Topics in the Mathematical Modelling of Composite Materials, A. Cherkaev and R.V. Kohn Eds., Progress in Nonlinear Differential Equations and Their Applications, Vol. 31, pp. 21-43. Birkhäuser, Boston, Basel, Berlin (1997).
G.C. Papanicolaou, Diffusion in random media, Surveys in Applied Mathematics, Vol. 1, Chap. 3, J.B. Keller, D.W. Mclaughlin and G.C. Papanicolaou Eds., Plenum Press, New York (1995).