Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence
Purchase on Springer.com
$39.95 / €34.95 / £29.95*
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.
We consider an electrostatic problem for a conductor consisting of finitely many small inhomogeneities of extreme conductivity, embedded in a spatially varying reference medium. Firstly we establish an asymptotic formula for the voltage potential in terms of the reference voltage potential, the location of the inhomogeneities and their geometry. Secondly we use this representation formula to prove a Lipschitz-continuous dependence estimate for the corresponding inverse problem. This estimate bounds the difference in the location and in certain geometric properties of two sets of inhomogeneities by the difference in the boundary voltage potentials corresponding to a fixed current distribution.
- G. Alessandrini, An identification problem for an elliptic equation in two variables, Univ. of Florence, Technical Report, 1986.
- G. Alessandrini, Stable determination of conductivity by boundary measurements, IMA Tech. Report, 1987.
- N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations and inequalities of second order, J. Math. Pures Appl. 36 (1957), 235–249.
- D. C. Barber & B. H. Brown, Recent developments in Applied Potential Tomography—APT. In Information Processing in Medical Imaging, ed. S. L. Bacharach, 106–121. Nijhoff 1986.
- H. Bellout & A. Friedman, Identification problem in potential theory, Archive Rational Mech. Anal., 101 (1988), 143–160.
- Proceedings of the EEC workshop on electrical impedance imaging, Sheffield, England, 1986. B. H. Brown editor.
- H. O. Cordes, Über die Bestimmtheit der Lösungen elliptischer Differentialgleichungen durch Anfangsvorhaben, Nachr. Akad. Wiss. Goettingen Math.-Phys. Kl. IIa (1956), 239–258.
- A. Friedman, Detection of mines by electric measurements, SIAM J. Appl. Math. 47 (1987), 201–212.
- A. Friedman, & B. Gustafsson, Identification of the conductivity coefficient in an elliptic equation, SIAM J. Math. Anal., 18 (1987), 777–787.
- D. G. Gisser, D. Isaacson & J. C. Newell, Electric current computet tomography and eigenvalues I, Preprint, 1987.
- R. E. Kleinman & T. B. A. Seniop, Rayleigh Scattering. Chap. 1 in, “Low and High Frequency Asymptotics”, V.K. Varadan and V. V. Varadan, Eds. Elsevier Science Publishers, 1986.
- R. Kohn & M. Vogelius, Determining conductivity by boundary measurements, Comm. Pure Appl. Math. 37 (1984), 289–298.
- R. Kohn & M. Vogelius, Determining conductivity by boundary measurements II. Interior results, Comm. Pure Appl. Math. 38 (1985), 643–667.
- R. Kohn & M. Vogelius, in preparation.
- I.-J. Lee, Determining conductivity by boundary measurements: some numerical results, Univ. of Maryland Tech. Report, 1988.
- N. G. Meyers & J. Serrin, The exterior Dirichlet problem for second order elliptic partial differential equations, J. Math. Mech. 9 (1960), 513–538.
- K. Miller, Stabilized numerical analytic prolongation with poles, SIAM J. Appl. Math. 18 (1970), 346–363.
- S. Ozawa, Spectra of domains with small spherical Neumann boundary, J. Fac. Sci. Univ. Tokyo, Sect. IA 30 (1983), pp. 259–277.
- M. Schiffer & G. Szegö, Virtual mass and polarization, Trans. Amer. Math. Soc. 67 (1949), pp. 130–205.
- J. Sylvester & G. Uhlmann, A uniqueness theorem for an inverse boundary value problem in electrical prospection, Comm. Pure Appl. Math. 39 (1986), 91–112.
- J. Sylvester & G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Annals of Math. 125 (1987), 153–169.
- J. Sylvester & G. Uhlmann, Inverse boundary value problems at the boundary —continuous dependence, Comm. Pure Appl. Math., 41 (1988), 197–219.
- A. Wexler, B. Fry & M. R. Neumann, Impedance-computed tomography algorithm and system, Appl. Optics 24 (1985), 3985–3992.
- T. J. Yorkey, J. G. Webster & W. J. Tompkins, Comparing reconstruction algorithms for electrical impedance tomography, IEEE Trans. Biomedical Eng. BME-34 (1987), 843–852.
- Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence
Archive for Rational Mechanics and Analysis
Volume 105, Issue 4 , pp 299-326
- Cover Date
- Print ISSN
- Online ISSN
- Additional Links