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Inventiones mathematicae

, Volume 178, Issue 1, pp 119–171 | Cite as

Limiting Carleman weights and anisotropic inverse problems

  • David Dos Santos Ferreira
  • Carlos E. KenigEmail author
  • Mikko Salo
  • Gunther Uhlmann
Article

Abstract

In this article we consider the anisotropic Calderón problem and related inverse problems. The approach is based on limiting Carleman weights, introduced in Kenig et al. (Ann. Math. 165:567–591, 2007) in the Euclidean case. We characterize those Riemannian manifolds which admit limiting Carleman weights, and give a complex geometrical optics construction for a class of such manifolds. This is used to prove uniqueness results for anisotropic inverse problems, via the attenuated geodesic ray transform. Earlier results in dimension n≥3 were restricted to real-analytic metrics.

Keywords

Manifold Riemannian Manifold Pseudodifferential Operator Principal Symbol Eikonal Equation 
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. 1.
    Anikonov, Yu.E.: Some Methods for the Study of Multidimensional Inverse Problems for Differential Equations. Nauka Sibirsk, Otdel, Novosibirsk (1978) Google Scholar
  2. 2.
    Astala, K., Päivärinta, L.: Calderón’s inverse conductivity problem in the plane. Ann. Math. 163, 265–299 (2006) zbMATHCrossRefGoogle Scholar
  3. 3.
    Astala, K., Lassas, M., Päivärinta, L.: Calderón’s inverse problem for anisotropic conductivity in the plane. Commun. Partial Differ. Equ. 30, 207–224 (2005) zbMATHCrossRefGoogle Scholar
  4. 4.
    Barber, D.C., Brown, B.H.: Progress in electrical impedance tomography. In: Colton, D., Ewing, R., Rundell, W. (eds.) Inverse Problems in Partial Differential Equations, pp. 151–164. SIAM, Philadelphia (1990) Google Scholar
  5. 5.
    Brown, R.M., Uhlmann, G.: Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions. Commun. Partial Differ. Equ. 22, 1009–1027 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bukhgeim, A.L.: Recovering a potential from Cauchy data in the two-dimensional case. J. Inverse Ill-posed Probl. 16, 19–34 (2008) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dimassi, M., Sjöstrand, J.: Spectral Asymptotics in the Semi-classical Limit. Cambridge University Press, Cambridge (1999) zbMATHGoogle Scholar
  8. 8.
    Dos Santos Ferreira, D., Kenig, C.E., Sjöstrand, J., Uhlmann, G.: Determining a magnetic Schrödinger operator from partial Cauchy data. Comm. Math. Phys. 271, 467–488 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Eisenhart, L.: Riemannian Geometry, 2nd edn. Princeton University Press, Princeton (1949) zbMATHGoogle Scholar
  10. 10.
    Guillarmou, C., Sa Barreto, A.: Inverse problems for Einstein manifolds. Inverse Probl. Imaging 3, 1–15 (2009) CrossRefGoogle Scholar
  11. 11.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators III–IV. Springer, Berlin (1985) Google Scholar
  12. 12.
    Isozaki, H.: Inverse spectral problems on hyperbolic manifolds and their applications to inverse boundary value problems in Euclidean space. Am. J. Math. 126, 1261–1313 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Jost, J.: Riemannian Geometry and Geometric Analysis. Springer, Berlin (2002) zbMATHGoogle Scholar
  14. 14.
    Kenig, C.E., Sjöstrand, J., Uhlmann, G.: The Calderón problem with partial data. Ann. Math. 165, 567–591 (2007) zbMATHCrossRefGoogle Scholar
  15. 15.
    Knudsen, K., Salo, M.: Determining non-smooth first order terms from partial boundary measurements. Inverse Probl. Imaging 1, 349–369 (2007) zbMATHMathSciNetGoogle Scholar
  16. 16.
    Kohn, R., Vogelius, M.: Identification of an unknown conductivity by means of measurements at the boundary. In: McLaughlin, D. (ed.) Inverse Problems. SIAM-AMS Proc., vol. 14, pp. 113–123. Am. Math. Soc., Providence (1984) Google Scholar
  17. 17.
    Lassas, M., Uhlmann, G.: On determining a Riemannian manifold from the Dirichlet-to-Neumann map. Ann. Sci. Éc. Norm. Super. 34, 771–787 (2001) zbMATHMathSciNetGoogle Scholar
  18. 18.
    Lassas, M., Taylor, M., Uhlmann, G.: The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary. Commun. Anal. Geom. 11, 207–221 (2003) zbMATHMathSciNetGoogle Scholar
  19. 19.
    Lebeau, G., Robbiano, L.: Contrôle exact de l’équation de la chaleur. Commun. Partial Differ. Equ. 20, 335–356 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Lebeau, G., Robbiano, L.: Stabilisation de l’équation des ondes par le bord. Duke Math. J. 86, 465–491 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Lee, J., Uhlmann, G.: Determining anisotropic real-analytic conductivities by boundary measurements. Commun. Pure Appl. Math. 42, 1097–1112 (1989) zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Lionheart, W.: Conformal uniqueness results in anisotropic electrical impedance imaging. Inverse Probl. 13, 125–134 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Mukhometov, R.G.: The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry. Dokl. Akad. Nauk SSSR 232, 32–35 (1977) (in Russian) MathSciNetGoogle Scholar
  24. 24.
    Nachman, A.: Global uniqueness for a two-dimensional inverse boundary value problem. Ann. Math. 143, 71–96 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Nakamura, G., Sun, Z., Uhlmann, G.: Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field. Math. Ann. 303, 377–388 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Payne, L.E., Weinberger, H.F.: An optimal Poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5, 286–292 (1960) zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Petersen, P.: Riemannian Geometry. Springer, Berlin (1998) zbMATHGoogle Scholar
  28. 28.
    Salo, M.: Inverse boundary value problems for the magnetic Schrödinger equation. J. Phys. Conf. Ser. 73, 012020 (2007) CrossRefGoogle Scholar
  29. 29.
    Salo, M., Tzou, L.: Carleman estimates and inverse problems for Dirac operators. Math. Ann. 344, 161–184 (2009) zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Sharafutdinov, V.: Integral geometry of tensor fields. In: Inverse and Ill-Posed Problems Series. VSP, Utrecht (1994) Google Scholar
  31. 31.
    Sharafutdinov, V.: On emission tomography of inhomogeneous media. SIAM J. Appl. Math. 55, 707–718 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Sun, Z., Uhlmann, G.: Generic uniqueness for an inverse boundary value problem. Duke Math. J. 62, 131–155 (1991) zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Sun, Z., Uhlmann, G.: Anisotropic inverse problems in two dimensions. Inverse Probl. 19, 1001–1010 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Sylvester, J.: An anisotropic inverse boundary value problem. Commun. Pure Appl. Math. 43, 201–232 (1990) zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Sylvester, J., Uhlmann, G.: A global uniqueness theorem for an inverse boundary value problem. Ann. Math. 125, 153–169 (1987) CrossRefMathSciNetGoogle Scholar
  36. 36.
    Sylvester, J., Uhlmann, G.: Inverse boundary value problems at the boundary—continuous dependence. Commun. Pure Appl. Math. 41, 197–219 (1988) CrossRefMathSciNetGoogle Scholar
  37. 37.
    Taylor, M.: Pseudodifferential Operators. Princeton University Press, Princeton (1981) zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • David Dos Santos Ferreira
    • 1
  • Carlos E. Kenig
    • 2
    Email author
  • Mikko Salo
    • 3
  • Gunther Uhlmann
    • 4
  1. 1.LAGA, MathématiqueUniversité Paris 13VilletaneuseFrance
  2. 2.Department of MathematicsUniversity of ChicagoChicagoUSA
  3. 3.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland
  4. 4.Department of MathematicsUniversity of WashingtonSeattleUSA

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