Skip to main content
SpringerLink
Log in

Inverse Boundary Problems in Two Dimensions

  • Chapter
Function Spaces, Differential Operators and Nonlinear Analysis

Abstract

In this paper we survey some of the recent progress on inverse boundary problems in two dimensions. The common theme is the use of inverse scattering for a ∂ ∂ type system in two dimensions.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Reference

  1. G. Alessandrini, Stable determination of conductivity by boundary measure-ments, Appl. Anal. 27 (1988), no. 1–3, 153–172.

    Article  MathSciNet  MATH  Google Scholar 

  2. J.A. Barcelx, T. Barcelx, and A. Ruiz, Stability of the inverse conductivity problem in the plane for less regular conductivities, J. Differential Equations 173 (2001), no. 2, 231–270.

    Article  MathSciNet  MATH  Google Scholar 

  3. R. Beals and R. Coifman, The spectral problem for the Davey-Stewartson and Ishimori hierarchies inNonlinear evolution equations: Integrability and spectral methodsManchester University Press (1988), 15–23.

    Google Scholar 

  4. R.M. Brown, Estimates for a scattering map associated to a two-dimensional first order system, to appear in J. Nonlinear Science.

    Google Scholar 

  5. R.M. Brown and G. Uhlmann, Uniqueness in the inverse conductivity problem with less regular conductivities in two dimensions, unpublished, 1996.

    Google Scholar 

  6. R.M. Brown and G. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Comm. in PDE 22 (1997), 1009–1027.

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Calderon, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), Soc. Brasil. Mat., Rio de Janeiro, 1980, 65–73.

    Google Scholar 

  8. M. Cheney, D. Isaacson, and J. Newell, Electrical impedance tomography, SIAM Rev. 41 (1999), 85–101.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Cheng and M. Yamamoto, Determination of two convection terms from Dirichlet to Neumann map in two-dimensional case, University of Tokyo, Graduate School of Mathematical Sciences Technical Note UTMS, 98–31.

    Google Scholar 

  10. J. Cheng and M Yamamoto, The global uniqueness for determining two con-vection coefficients from Dirichlet to Neumann map in two dimensions, Inverse Problems 16 (2000), L25–L30.

    Article  MathSciNet  Google Scholar 

  11. R. Coifman and Y. Meyer, Au dela, des Operateurs pseudo-differentiels, Vol. 57 of Asterisque, Societe Mathematique de France, 1978.

    Google Scholar 

  12. H.L. Cycon, R. Froese, W. Kirsch, and B. Simon, Schrodinger operators with applications to quantum mechanics and global geometry, Texts and Monographs in Physics, Springer-Verlag, 1987.

    Google Scholar 

  13. E. Francini, Recovering a complex coefficient in a planar domain from the Dirichlet-to-Neumann map, Inverse Problems 16 (2000), 107–119.

    Article  MathSciNet  MATH  Google Scholar 

  14. H. Kang, A uniqueness theorem for an inverse boundary value problem in two dimensions, to appear in J. Math. Anal. Appl.

    Google Scholar 

  15. H. Kang and G Uhlmann, Inverse problems for the Pauli Hamiltonian in two dimensions, preprint.

    Google Scholar 

  16. R. Kohn and M. Vogelius, Determining conductivity by boundary measure-ments, Comm. Pure Appl. Math. 37 (1984), 289–298.

    Article  MathSciNet  MATH  Google Scholar 

  17. K. Knudsen, A. Tamasan, Reconstruction of less regular conductivities in the plane, MSRI preprint series, Berkeley, 2001.

    Google Scholar 

  18. L. Liu, Stability estimates for the two-dimensional inverse conductivity prob-lem, Ph.D. thesis, Department of Mathematics, University of Rochester, New York, 1997.

    Google Scholar 

  19. N.I. Muskhelishvili, Singular integral equations. Boundary problems of function theory and their application to mathematical physics, P. Noordhoff N. V., Groningen, 1953.

    MATH  Google Scholar 

  20. A.I. Nachman, Reconstructions from boundary measurements, Ann. of Math. 128 (1988), 531–576.

    Article  MathSciNet  MATH  Google Scholar 

  21. A.I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. 143 (1996), 71–96.

    Article  MathSciNet  MATH  Google Scholar 

  22. G. Nakamura, Z. Sun, and G Uhlmann, Global Identifiability for an inverse problem for the Schrodinger equation in a magnetic field, Math. Ann. 303 (1995), 377–388.

    Article  MathSciNet  MATH  Google Scholar 

  23. G. Nakamura and G. Uhlmann, Identification of Lame parameters by bound-ary observations, American J. of Math. 115 (1993), 1161–1187.

    Article  MathSciNet  MATH  Google Scholar 

  24. S. Siltanen, J. Mueller, and D. Isaacson, An implementation of the recon-struction algorithm of A. Nachman for the 2D inverse conductivity problem, Inverse Problems 16 (2000), 681–699.

    Article  MathSciNet  MATH  Google Scholar 

  25. Z. Sun, An inverse boundary value problem for Schrodinger operator with vector potentials, Trans. of AMS 338 (1993), 953–971.

    MATH  Google Scholar 

  26. Z. Sun, An inverse boundary value problem for Schrodinger operator with vector potential in two dimensions, Comm. in PDE 18 (1993), 83–124.

    Article  MATH  Google Scholar 

  27. J. Sylvester and G. Uhlmann, A uniqueness theorem for an inverse boundary value problem in electrical prospection, Comm. Pure Appl. Math. 39 (1986), 91–112.

    Article  MathSciNet  MATH  Google Scholar 

  28. J. Sylvester and G. Uhlmann, Inverse boundary value problems at the boundary-continuous dependence, Comm. Pure Appl. Math. 41 (1988), 197–219.

    Article  MathSciNet  Google Scholar 

  29. Z. Sun and G. Uhlmann, Generic uniqueness for an inverse boundary value problem, Duke Math. J. 62 (1991), 131–155.

    Article  MathSciNet  MATH  Google Scholar 

  30. Z. Sun, Recovery of singularities for formally determined inverse problems, Comm. Math. Phys. 153 (1993), 431–445.

    Article  Google Scholar 

  31. L. Sung, An inverse scattering transform for the Davey-Stewartson II equations. I, J. Math. Anal. Appl. 183 (1994), 121–154.

    Article  MathSciNet  MATH  Google Scholar 

  32. L. Sung, An inverse scattering transform for the Davey-Stewartson II equations. II, J. Math. Anal. Appl. 183 (1994), 289–325.

    Article  MathSciNet  MATH  Google Scholar 

  33. L. Sung, An inverse scattering transform for the Davey-Stewartson II equations. III, J. Math. Anal. Appl. 183 (1994), 477–494

    Article  MathSciNet  MATH  Google Scholar 

  34. G. Uhlmann, Developments in inverse problems since Calderon’s foundational paper inHarmonic analysis and partial differential equations(Chicago, IL, 1996), Univ. Chicago Press, Chicago, IL, 1999, pp. 295–345.

    Google Scholar 

  35. I.N. Vekua, Generalized analytic functions, Pergamon Press, London, 1962.

    MATH  Google Scholar 

  36. M.S. Zhdanov and G.V. Keller, The geolectric methods in geophysical exploration, Methods in Geochemistry and Geophysics, 31, Elsevier (1994).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Washington, Box 354350, Seattle, WA, 98195, USA

    Gunther Uhlmann

Authors
  1. Gunther Uhlmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute of Mathematics, Friedrich-Schiller-University, Jena, 07740, Germany

    Dorothee Haroske , Thomas Runst  & Hans-Jürgen Schmeisser ,  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer Basel AG

About this chapter

Cite this chapter

Uhlmann, G. (2003). Inverse Boundary Problems in Two Dimensions. In: Haroske, D., Runst, T., Schmeisser, HJ. (eds) Function Spaces, Differential Operators and Nonlinear Analysis. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8035-0_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8035-0_9

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9414-2

  • Online ISBN: 978-3-0348-8035-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation