zbMATH — the first resource for mathematics

Identification of small inhomogeneities of extreme conductivity by boundary measurements: A theorem on continuous dependence. (English) Zbl 0684.35087
The authors consider the inverse problem to determine perturbations of the coefficient \(\gamma\) (x) in the differential equation \(\nabla \cdot (\gamma \nabla u)=0\) in \(\Omega <{\mathbb{R}}^ n\), \(n\geq 2\), from measurements of u and \(\gamma\) (\(\partial u/\partial \nu)\) on the boundary \(\partial \Omega\). They treat the cases where the perturbed coefficients are either infinite or zero on a finite (but unknown) number of small (order \(\epsilon)\) subregions of \(\Omega\). In the first part they prove an asymptotic formula of the field \(u_{\epsilon}\) with respect to \(\epsilon\). In the second part this result is used to prove a Lipschitz-continuous dependence estimate of the function which maps - for fixed Neumann data - the Dirichlet data onto the parameters describing \(\gamma\).
Reviewer: A.Kirsch

35R30 Inverse problems for PDEs
78A30 Electro- and magnetostatics
35J25 Boundary value problems for second-order elliptic equations
Full Text: DOI
[1] G. Alessandrini, An identification problem for an elliptic equation in two variables, Univ. of Florence, Technical Report, 1986. · Zbl 0662.35118
[2] G. Alessandrini, Stable determination of conductivity by boundary measurements, IMA Tech. Report, 1987.
[3] 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. · Zbl 0084.30402
[4] 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.
[5] H. Bellout & A. Friedman, Identification problem in potential theory, Archive Rational Mech. Anal., 101 (1988), 143–160. · Zbl 0659.35102
[6] Proceedings of the EEC workshop on electrical impedance imaging, Sheffield, England, 1986. B. H. Brown editor.
[7] H. O. Cordes, Über die Bestimmtheit der Lösungen elliptischer Differentialgleichungen durch Anfangsvorhaben, Nachr. Akad. Wiss. Goettingen Math.-Phys. Kl. IIa (1956), 239–258. · Zbl 0074.08002
[8] A. Friedman, Detection of mines by electric measurements, SIAM J. Appl. Math. 47 (1987), 201–212. · Zbl 0636.35084
[9] A. Friedman, & B. Gustafsson, Identification of the conductivity coefficient in an elliptic equation, SIAM J. Math. Anal., 18 (1987), 777–787. · Zbl 0644.35091
[10] D. G. Gisser, D. Isaacson & J. C. Newell, Electric current computet tomography and eigenvalues I, Preprint, 1987. · Zbl 0723.35083
[11] 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.
[12] R. Kohn & M. Vogelius, Determining conductivity by boundary measurements, Comm. Pure Appl. Math. 37 (1984), 289–298. · Zbl 0586.35089
[13] R. Kohn & M. Vogelius, Determining conductivity by boundary measurements II. Interior results, Comm. Pure Appl. Math. 38 (1985), 643–667. · Zbl 0595.35092
[14] R. Kohn & M. Vogelius, in preparation.
[15] I.-J. Lee, Determining conductivity by boundary measurements: some numerical results, Univ. of Maryland Tech. Report, 1988.
[16] N. G. Meyers & J. Serrin, The exterior Dirichlet problem for second order elliptic partial differential equations, J. Math. Mech. 9 (1960), 513–538. · Zbl 0094.29701
[17] K. Miller, Stabilized numerical analytic prolongation with poles, SIAM J. Appl. Math. 18 (1970), 346–363. · Zbl 0211.19303
[18] S. Ozawa, Spectra of domains with small spherical Neumann boundary, J. Fac. Sci. Univ. Tokyo, Sect. IA 30 (1983), pp. 259–277. · Zbl 0541.35061
[19] M. Schiffer & G. Szegö, Virtual mass and polarization, Trans. Amer. Math. Soc. 67 (1949), pp. 130–205. · Zbl 0035.11803
[20] J. Sylvester & G. Uhlmann, A uniqueness theorem for an inverse boundary value problem in electrical prospection, Comm. Pure Appl. Math. 39 (1986), 91–112. · Zbl 0611.35088
[21] J. Sylvester & G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Annals of Math. 125 (1987), 153–169. · Zbl 0625.35078
[22] J. Sylvester & G. Uhlmann, Inverse boundary value problems at the boundary –continuous dependence, Comm. Pure Appl. Math., 41 (1988), 197–219. · Zbl 0632.35074
[23] A. Wexler, B. Fry & M. R. Neumann, Impedance-computed tomography algorithm and system, Appl. Optics 24 (1985), 3985–3992.
[24] 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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.