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Communications in Mathematical Physics

, Volume 340, Issue 2, pp 639–659 | Cite as

Uniqueness in Calderón’s Problem for Conductivities with Unbounded Gradient

  • Boaz HabermanEmail author
Article

Abstract

We prove uniqueness in the inverse conductivity problem for uniformly elliptic conductivities in \({W^{s,p}(\Omega)}\), where \({\Omega \subset \mathbb{R}^{n}}\) is Lipschitz, \({3\leq n \leq 6}\), and s and p are such that \({ W^{s,p}(\Omega)\not \subset W^{1,\infty}(\Omega)}\). In particular, we obtain uniqueness for conductivities in \({W^{1,n}(\Omega)}\) (n = 3, 4). This improves on the result of the author and Tataru, who assumed that the conductivity is Lipschitz.

Keywords

Electric Impedance Tomography Fourier Multiplier Cauchy Data Unique Continuation Carleman Estimate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. AKS62.
    Aronszajn N., Krzywicki A., Szarski J.: A unique continuation theorem for exterior differential forms on Riemannian manifolds. Ark. Mat. 4(5), 417–453 (1962)zbMATHMathSciNetCrossRefGoogle Scholar
  2. Ale90.
    Alessandrini G.: Singular solutions of elliptic equations and the determination of conductivity by boundary measurements. J. Differ. Equ. 84(2), 252–272 (1990)zbMATHMathSciNetCrossRefADSGoogle Scholar
  3. Ale92.
    Alessandrini, G.: A simple proof of the unique continuation property for two dimensional elliptic equations in divergence form. Quaderni Matematici II Serie, vol. 276. Dipartimento di Scienze Matematiche, Trieste (1992)Google Scholar
  4. ALP11.
    Astala, K., Lassas, M., Päivärinta, L.: The borderlines of the invisibility and visibility for Calderon’s inverse problem (2011). arXiv:1109.2749 [math-ph]
  5. Bou93.
    Bourgain J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Geom. Funct. Anal. 3(2), 107–156 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  6. Bro96.
    Brown R.M.: Global uniqueness in the impedance-imaging problem for less regular conductivities. SIAM J. Math. Anal. 27(4), 1049 (1996)zbMATHMathSciNetCrossRefADSGoogle Scholar
  7. Bro03.
    Brown B.H.: Electrical impedance tomography (EIT): a review. J. Med. Eng. Technol. 27(3), 97–108 (2003)CrossRefADSGoogle Scholar
  8. Bro13.
    Brown R.M.: Recovering the conductivity at the boundary from the Dirichlet to Neumann map: a pointwise result. J. Inverse Ill-Posed Probl. 9(6), 567–574 (2013)Google Scholar
  9. Cal80.
    Alberto C.: On an inverse boundary value problem. Comput. Appl. Math. 25(2–3), 133–138 (1980)Google Scholar
  10. Cha90.
    Chanillo S.: A problem in electrical prospection and a n-dimensional Borg–Levinson theorem. Proc. Am. Math. Soc. 108(3), 761–767 (1990)zbMATHMathSciNetGoogle Scholar
  11. CR14.
    Caro, P., Rogers, K.: Global uniqueness for the Calderón problem with Lipschitz conductivities (2014). arXiv:1411.8001 [math]
  12. DSFKSU09.
    Ferreira D.D.S., Kenig C.E., Salo M., Uhlmann G.: Limiting Carleman weights and anisotropic inverse problems. Invent. Math. 178(1), 119–171 (2009)zbMATHMathSciNetCrossRefADSGoogle Scholar
  13. GLU03.
    Greenleaf A., Lassas M., Uhlmann G.: On nonuniqueness for Calderón’s inverse problem. Math. Res. Lett. 10(5), 685–693 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  14. HT13.
    Haberman B., Tataru D.: Uniqueness in Calderón’s problem with Lipschitz conductivities. Duke Math. J. 162(3), 497–516 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  15. KRS87.
    Kenig C.E., Ruiz A., Sogge C.D.: Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators. Duke Math. J. 55(2), 329–347 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  16. KSVW08.
    Kohn R.V., Shen H., Vogelius M.S., Weinstein M.I.: Cloaking via change of variables in electric impedance tomography. Inverse Probl. 24(1), 015016 (2008)MathSciNetCrossRefADSGoogle Scholar
  17. KT01.
    Koch H., Tataru D.: Carleman estimates and unique continuation for second-order elliptic equations with nonsmooth coefficients. Commun. Pure Appl. Math. 54(3), 339–360 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  18. KU14.
    Krupchyk, K., Uhlmann, G.: Uniqueness in an inverse boundary problem for a magnetic Schrödinger operator with a bounded magnetic potential. Commun. Math. Phys. 327(3), 993–1009 (2014)Google Scholar
  19. KV84.
    Kohn R., Vogelius M.: Determining conductivity by boundary measurements. Commun. Pure Appl. Math. 37(3), 289–298 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  20. Man98.
    Mandache N.: On a counterexample concerning unique continuation for elliptic equations in divergence form, mathematical physics. Anal. Geom. 1(3), 273–292 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  21. Mar87.
    Marschall J.: The trace of Sobolev–Slobodeckij spaces on Lipschitz domains. Manuscr. Math. 58(1–2), 47–65 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  22. Mil73.
    Miller K.: Nonunique continuation for uniformly parabolic and elliptic equations in selfadjoint divergence form with Hölder continuous coefficients. Bull. Am. Math. Soc. 79(2), 350–354 (1973)zbMATHCrossRefGoogle Scholar
  23. NS14.
    Nguyen, H.-M., Spirn, D.: Recovering a potential from Cauchy data via complex geometrical optics solutions (2014). arXiv:1403.2255 [math]
  24. Pli63.
    Pliś A.: On non-uniqueness in Cauchy problem for an elliptic second order differential equation. Bull. de l’Acad. Pol. des Sci. Sér. des Sci. Math. Astron. et Phys. 11, 95–100 (1963)zbMATHGoogle Scholar
  25. PPU03.
    Päivärinta L., Panchenko A., Uhlmann G.: Complex geometrical optics solutions for Lipschitz conductivities. Rev. Mat. Iberoam. 19(1), 57–72 (2003)zbMATHCrossRefGoogle Scholar
  26. ST09.
    Salo M., Tzou L.: Carleman estimates and inverse problems for Dirac operators. Math. Ann. 344(1), 161–184 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  27. Ste93.
    Stein, E.M: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. In: Princeton Mathematical Series, vol. 43. Princeton University Press, Princeton, With the assistance of T. S. Murphy (1993)Google Scholar
  28. SU87.
    Sylvester J., Uhlmann G.: A global uniqueness theorem for an inverse boundary value problem. Ann. Math. 125(1), 153–169 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  29. SU88.
    Sylvester J., Uhlmann G.: Inverse boundary value problems at the boundary—continuous dependence. Commun. Pure Appl. Math. 41(2), 197–219 (1988)MathSciNetCrossRefGoogle Scholar
  30. Tat96.
    Tataru D.: The \({{X}^s_\theta}\) spaces and unique continuation for solutions to the semilinear wave equation. Commun. Partial Differ. Equ. 21(5–6), 841–887 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  31. Tom75.
    Tomas P.A.: A restriction theorem for the Fourier transform. Bull. Am. Math. Soc. 81(2), 477–478 (1975)zbMATHMathSciNetCrossRefGoogle Scholar
  32. Wol92.
    Wolff T.: A property of measures in \({\mathbb{R}^{n}}\) and an application to unique continuation. Geom. Funct. Anal. 2(2), 225–284 (1992)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.University of ChicagoChicagoUSA

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