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The Calderón problem with partial data in two dimensions


Authors: Oleg Yu. Imanuvilov, Gunther Uhlmann and Masahiro Yamamoto
Journal: J. Amer. Math. Soc. 23 (2010), 655-691
MSC (2010): Primary 35R30; Secondary 35Q60
Published electronically: February 16, 2010
MathSciNet review: 2629983
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Abstract: We prove for a two-dimensional bounded domain that the Cauchy data for the Schrödinger equation measured on an arbitrary open subset of the boundary uniquely determines the potential. This implies, for the conductivity equation, that if we measure the current fluxes at the boundary on an arbitrary open subset of the boundary produced by voltage potentials supported in the same subset, we can uniquely determine the conductivity. We use Carleman estimates with degenerate weight functions to construct appropriate complex geometrical optics solutions to prove the results.


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  • 1. V. Alekseev, V. Tikhomirov, and S. Fomin, Optimal Control, Consultants Bureau, New York, 1987. MR 924574 (89e:49002)
  • 2. K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Ann. of Math., 163 (2006), 265-299. MR 2195135 (2007b:30019)
  • 3. K. Astala, M. Lassas, and L. Päiväirinta, Calderón's inverse problem for anisotropic conductivity in the plane, Comm. Partial Differential Equations, 30 (2005), 207-224. MR 2131051 (2005k:35421)
  • 4. A. Böttcher and B. Silbermann, Analysis of Toeplitz Operators, Springer-Verlag, Berlin, 2006. MR 2223704 (2007k:47001)
  • 5. R. Brown and R. Torres, Uniqueness in the inverse conductivity problem for conductivities with $ 3/2$ derivatives in $ L^p, p>2n,$ J. Fourier Analysis Appl., 9 (2003), 1049-1056. MR 2026763 (2004k:35392)
  • 6. R. Brown and G. Uhlmann, Uniqueness in the inverse conductivity problem with less regular conductivities in two dimensions, Comm. Partial Differential Equations, 22 (1997), 1009-1027. MR 1452176 (98f:35155)
  • 7. A. Bukhgeim, Recovering the potential from Cauchy data in two dimensions, J. Inverse Ill-Posed Probl., 16 (2008), 19-34. MR 2387648 (2008m:30049)
  • 8. A. Bukhgeim and G. Uhlmann, Recovering a potential from partial Cauchy data, Comm. Partial Differential Equations, 27 (2002), 653-668. MR 1900557 (2003d:35262)
  • 9. A. P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and Its Applications to Continuum Physics, 65-73, Soc. Brasil. Mat., Río de Janeiro, 1980. MR 590275 (81k:35160)
  • 10. J. Cheng and M. Yamamoto, Determination of two convection coefficients from Dirichlet to Neumann map in the two-dimensional case, SIAM J. Math. Anal., 35 (2004), 1371-1393. MR 2083783 (2005g:35296)
  • 11. D. Dos Santos Ferreira, C. Kenig, J. Sjöstrand, and G. Uhlmann, Determining a magnetic Schrödinger operator from partial Cauchy data, Comm. Math. Phys., 271 (2007), 467-488. MR 2287913 (2008a:35044)
  • 12. D. Dos Santos Ferreira, C. Kenig, M. Salo, and G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems, Invent. Math., 178 (2009), 119-171. MR 2534094
  • 13. L. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 2000. MR 1625845 (99e:35001)
  • 14. A. Greenleaf, M. Lassas, and G. Uhlmann, The Calderón problem for conormal potentials, I: Global uniqueness and reconstruction, Comm. Pure Appl. Math., 56 (2003), 328-352. MR 1941812 (2003j:35324)
  • 15. H. Heck and J.-N. Wang, Stability estimates for the inverse boundary value problem by partial Cauchy data, Inverse Problems, 22 (2006), 1787-1796. MR 2261266 (2007g:35270)
  • 16. L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer-Verlag, Berlin, 1985. MR 0717035 (85g:35002a)
  • 17. V. Isakov, On uniqueness in the inverse conductivity problem with local data, Inverse Problems and Imaging, 1 (2007), 95-105. MR 2262748 (2007m:35273)
  • 18. H. Kang and G. Uhlmann, Inverse problems for the Pauli Hamiltonian in two dimensions, Journal of Fourier Analysis and Applications, 10 (2004), 201-215. MR 2054308 (2005d:81135)
  • 19. C. Kenig, J. Sjöstrand, and G. Uhlmann, The Calderón problem with partial data, Ann. of Math., 165 (2007), 567-591. MR 2299741 (2008k:35498)
  • 20. K. Knudsen, The Calderón problem with partial data for less smooth conductivities, Comm. Partial Differential Equations, 31 (2006), 57-71. MR 2209749 (2006k:35303)
  • 21. K. Knudsen and M. Salo, Determining nonsmooth first order terms from partial boundary measurements, Inverse Problems and Imaging, 1 (2007), 349-369. MR 2282273 (2008k:35500)
  • 22. R. Kohn and M. Vogelius, Determining conductivity by boundary measurements, Comm. Pure Appl. Math., 38 (1985), 643-667. MR 803253 (86k:35155)
  • 23. R. Kohn and M. Vogelius, Identification of an unknown conductivity by means of measurements at the boundary, in Inverse Problems, edited by D. McLaughlin, SIAM-AMS Proceedings, 14 (1984), 113-123. MR 773707
  • 24. A. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math., 143 (1996), 71-96. MR 1370758 (96k:35189)
  • 25. L. Päivärinta, A. Panchenko, and G. Uhlmann, Complex geometrical optics for Lipschitz conductivities, Revista Matematica Iberoamericana, 19 (2003), 57-72. MR 1993415 (2004f:35187)
  • 26. Z. Sun and G. Uhlmann, Anisotropic inverse problems in two dimensions, Inverse Problems, 19 (2003), 1001-1010. MR 2024685 (2004k:35415)
  • 27. J. Sylvester, An anisotropic inverse boundary value problem, Comm. Pure Appl. Math., 43 (1990), 201-232. MR 1038142 (90m:35202)
  • 28. J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169. MR 873380 (88b:35205)
  • 29. J. Sylvester and G. Uhlmann, Inverse boundary value problems at the boundary--continuous dependence, Comm. Pure Appl. Math., 41 (1988), 197-219. MR 924684 (89f:35213)
  • 30. L. Tzou, Stability estimates for coefficients of magnetic Schrödinger equation from full and partial measurements, Comm. Partial Differential Equations, 33 (2008), 1911-1952. MR 2475324
  • 31. G. Uhlmann, Commentary on Calderón's paper (29) ``On an inverse boundary value problem'', Selected Papers of A.P. Calderón, edited by Alexandra Bellow, Carlos Kenig, and Paul Malliavin, AMS (2008), 623-636. MR 2435340
  • 32. I. Vekua, Generalized Analytic Functions, Pergamon Press, Oxford, 1962. MR 0150320 (27:321)

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Additional Information

Oleg Yu. Imanuvilov
Affiliation: Department of Mathematics, Colorado State University, 101 Weber Building, Fort Collins, Colorado 80523
Email: oleg@math.colostate.edu

Gunther Uhlmann
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
Email: gunther@math.washington.edu

Masahiro Yamamoto
Affiliation: Department of Mathematical Sciences, University of Tokyo, Komaba, Meguro, Tokyo 153, Japan
Email: myama@ms.u-tokyo.ac.jp

DOI: https://doi.org/10.1090/S0894-0347-10-00656-9
Keywords: Calder\'on's problem, partial data, complex geometrical optics solutions
Received by editor(s): November 25, 2008
Published electronically: February 16, 2010
Additional Notes: The first author was partly supported by NSF grant DMS 0808130.
The second author was partly supported by the NSF and a Walker Family Endowed Professorship.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.