Abstract
In this paper we demonstrate uniqueness of a transparent obstacle, of coefficients of rather general boundary transmission condition, and of a potential coefficient inside obstacle from partial Dirichlet-to Neumann map or from complete scattering data at fixed frequency. The proposed transmission problem includes in particular the isotropic elliptic equation with discontinuous conductivity coefficient. Uniqueness results are shown to be optimal. Hence the considered form can be viewed as a canonical form of isotropic elliptic transmission problems. Proofs use singular solutions of elliptic equations and complex geometrical optics. Determining an obstacle and boundary conditions (i.e. reflecting and transmitting properties of its boundary and interior) is of interest for acoustical and electromagnetic inverse scattering, for modeling fluid/structure interaction, and for defects detection.
Similar content being viewed by others
References
Alessandrini, G.: Singular Solutions of Elliptic Equations and the Determination of Conductivity by Boundary Measurements. J. Diff. Eq. 84, 252–273 (1990)
Alessandrini, G., Di Christo, M.: Stable determination of an inclusion by boundary measurements. SIAM J. Math. Anal. 37, 200–218 (2005)
Bal, G.: Reconstructions in impedance and optical tomography with singular interfaces. Inverse Problems 21, 113–131 (2005)
Belishev, M.: Boundary Control in Reconstruction of Manifolds and Metrics. Inverse Problems 13, R1–R45 (1997)
Bukhgeim, A., Uhlmann, G.: Recovering a Potential from Partial Cauchy Data. Comm. Part. Diff. Eq. 27, 653–668 (2002)
Colton, D., Kress, R.: Inverse acoustic and electromagnetic scattering theory. Springer-Verlag, New York (1998)
Colton, D., Kress, R.: Using fundamental solutions in inverse scattering. Inverse Problems 22, R49–R66 (2006)
Dos Santos Ferreira, D., Kenig, C., Sjöstrand, J., Uhlmann, G.: Determining a magnetic Schrd̈inger operator from partial Cauchy data. Commun. Math. Phys. 271, 467–488 (2007)
Eskin, G., Ralston, J.: Inverse scattering problem for the Schrd̈inger equation with magnetic potential at a fixed energy. Commun. Math. Phys. 173, 199–224 (1995)
Isakov, V.: On uniqueness of recovery of a discontinuous conductivity coefficient. Commun. Pure Appl. Math. 41, 865–877 (1988)
Isakov, V.: On uniqueness in the inverse transmission scattering problem. Comm. Part. Diff. Eq. 15, 1565–1587 (1990)
Isakov, V.: Inverse Problems for Partial Differential Equations. Springer-Verlag, New York (2006)
Isakov, V., Nachman, A.: Global uniqueness for a two-dimensional elliptic inverse problem. Trans. AMS 347, 3375–3391 (1995)
Kirsch, A., Kress, R.: Uniqueness in Inverse Obstacle Problems. Inverse Problems 9, 285–299 (1993)
Kirsch, A., Päivärinta, L.: On recovering obstacles inside inhomogeneites. Math. Meth. Appl. Sci. 21, 619–651 (1998)
Kohn, R., Vogelius, M.: Determining Conductivity by Boundary Measurements, Interior Results, II. Comm. Pure Appl. Math. 38, 643–667 (1985)
Katchalov, A., Kurylev, Y., Lassas, M.: Inverse Boundary Spectral Problems. Chapman and Hall-CRC, London (2000)
Kwon, K.: Identification of anisotropic anomalous region in inverse problems. Inverse Problems 20, 1117–1136 (2004)
Lax, P., Phillips, R.: Scattering Theory. Academic Press, New York-London (1989)
Ladyzhenskaya, O.A., Uraltseva, N.N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York-London (1969)
Majda, A., Taylor, M.: Inverse Scattering Problems for transparent obstacles, electromagnetic waves, and hyperbolic systems. Comm. Part. Diff. Eq. 2, 395–438 (1977)
McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge Univ. Press, Cambridge (2000)
Miranda, C.: Partial Differential Equations of Elliptic Type. Springer-Verlag, New-York-Berlin (1970)
Scharafutdinov, V.: Integral Geometry of Tensor Fields. Utrecht, VSP (1994)
Sylvester, J., Uhlmann, G.: Global Uniqueness Theorem for an Inverse Boundary Problem. Ann. Math. 125, 153–169 (1987)
Valdivia, N.: Uniqueness in Inverse Obstacle Scattering with Conductive Boundary Conditions. Applic. Anal. 83, 825–853 (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by B. Simon
Rights and permissions
About this article
Cite this article
Isakov, V. On Uniqueness in the General Inverse Transmisson Problem. Commun. Math. Phys. 280, 843–858 (2008). https://doi.org/10.1007/s00220-008-0485-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-008-0485-6