Anisotropic inverse problems and Carleman estimates

David Dos Santos Ferreira

Séminaire Équations aux dérivées partielles (2007-2008)

  • Volume: 2007-2008, page 1-17

Abstract

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This note reports on recent results on the anisotropic Calderón problem obtained in a joint work with Carlos E. Kenig, Mikko Salo and Gunther Uhlmann [8]. The approach is based on the construction of complex geometrical optics solutions to the Schrödinger equation involving phases introduced in the work [12] of Kenig, Sjöstrand and Uhlmann in the isotropic setting. We characterize those manifolds where the construction is possible, and give applications to uniqueness for the corresponding anisotropic inverse problems in dimension n ≥ 3 .

How to cite

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Dos Santos Ferreira, David. "Anisotropic inverse problems and Carleman estimates." Séminaire Équations aux dérivées partielles 2007-2008 (2007-2008): 1-17. <http://eudml.org/doc/11183>.

@article{DosSantosFerreira2007-2008,
abstract = {This note reports on recent results on the anisotropic Calderón problem obtained in a joint work with Carlos E. Kenig, Mikko Salo and Gunther Uhlmann [8]. The approach is based on the construction of complex geometrical optics solutions to the Schrödinger equation involving phases introduced in the work [12] of Kenig, Sjöstrand and Uhlmann in the isotropic setting. We characterize those manifolds where the construction is possible, and give applications to uniqueness for the corresponding anisotropic inverse problems in dimension $n \ge 3$.},
author = {Dos Santos Ferreira, David},
journal = {Séminaire Équations aux dérivées partielles},
keywords = {anisotropic Calderón problem; geometrical optics solutions; Schrödinger equation; inverse problems; Carleman estimates},
language = {eng},
pages = {1-17},
publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Anisotropic inverse problems and Carleman estimates},
url = {http://eudml.org/doc/11183},
volume = {2007-2008},
year = {2007-2008},
}

TY - JOUR
AU - Dos Santos Ferreira, David
TI - Anisotropic inverse problems and Carleman estimates
JO - Séminaire Équations aux dérivées partielles
PY - 2007-2008
PB - Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2007-2008
SP - 1
EP - 17
AB - This note reports on recent results on the anisotropic Calderón problem obtained in a joint work with Carlos E. Kenig, Mikko Salo and Gunther Uhlmann [8]. The approach is based on the construction of complex geometrical optics solutions to the Schrödinger equation involving phases introduced in the work [12] of Kenig, Sjöstrand and Uhlmann in the isotropic setting. We characterize those manifolds where the construction is possible, and give applications to uniqueness for the corresponding anisotropic inverse problems in dimension $n \ge 3$.
LA - eng
KW - anisotropic Calderón problem; geometrical optics solutions; Schrödinger equation; inverse problems; Carleman estimates
UR - http://eudml.org/doc/11183
ER -

References

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  1. Yu. E. Anikonov, Some methods for the study of multidimensional inverse problems for differential equations, Nauka Sibirsk. Otdel, Novosibirsk (1978). MR504333
  2. K. Astala, M. Lassas, L. Päivärinta, Calderón’s inverse problem for anisotropic conductivity in the plane, Comm. Partial Differential Equations, 30 (2005), 207–224. Zbl1129.35483
  3. K. Astala, L. Päivärinta, Calderón’s inverse conductivity problem in the plane, Ann. of Math., 163 (2006), 265–299. Zbl1111.35004
  4. D. C. Barber, B. H. Brown, Progress in electrical impedance tomography, in Inverse problems in partial differential equations, edited by D. Colton, R. Ewing, and W. Rundell, SIAM, Philadelphia (1990), 151–164. Zbl0714.65095MR1046435
  5. R. M. Brown, G. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Comm. Partial Differential Equations, 22 (1997), 1009–1027. Zbl0884.35167MR1452176
  6. A. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics, Rio de Janeiro, Sociedade Brasileira de Matematica, (1980), 65–73. MR590275
  7. D. Dos Santos Ferreira, C. E. Kenig, J. Sjöstrand, G. Uhlmann, Determining a magnetic Schrödinger operator from partial Cauchy data, Comm. Math. Phys., 271 (2007), 467–488. Zbl1148.35096MR2287913
  8. D. Dos Santos Ferreira, C. E. Kenig, M. Salo, G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems, preprint (2008), arXiv:0803.3508. Zbl1181.35327
  9. C. Guillarmou, A. Sa Barreto, Inverse problems for Einstein manifolds, preprint (2007), arXiv:0710.1136. Zbl1229.58025
  10. L. Hörmander, The Analysis of Linear Partial Differential Operators III, Springer-Verlag, 1985. Zbl0601.35001MR781536
  11. H. Isozaki, Inverse spectral problems on hyperbolic manifolds and their applications to inverse boundary value problems in Euclidean space, Amer. J. Math., 126 (2004), 1261–1313. Zbl1069.35092MR2102396
  12. C. E. Kenig, J. Sjöstrand, G. Uhlmann, The Calderón problem with partial data, Ann. of Math., 165 (2007), 567–591. Zbl1127.35079MR2299741
  13. K. Knudsen, M. Salo, Determining non-smooth first order terms from partial boundary measurements, Inverse Problems and Imaging, 1 (2007), 349–369. Zbl1122.35152MR2282273
  14. R. Kohn, M. Vogelius, Identification of an unknown conductivity by means of measurements at the boundary, in Inverse Problems, edited by D. McLaughlin, SIAM-AMS Proc. No. 14, Amer. Math. Soc., Providence (1984), 113–123. Zbl0573.35084MR773707
  15. M. Lassas, M. Taylor, G. Uhlmann, The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary, Comm. Anal. Geom., 11 (2003), 207–221. Zbl1077.58012MR2014876
  16. M. Lassas, G. Uhlmann, On determining a Riemannian manifold from the Dirichlet-to-Neumann map, Ann. Sc. ENS, 34 (2001), 771–787. Zbl0992.35120MR1862026
  17. J. Lee, G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurement, Comm. Pure Appl. Math., 42 (1989), 1097–1112. Zbl0702.35036MR1029119
  18. W. Lionheart, Conformal uniqueness results in anisotropic electrical impedance imaging, Inverse Problems, 13 (1997), 125-134. Zbl0868.35140MR1435872
  19. R. G. Mukhometov, The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry (Russian), Dokl. Akad. Nauk SSSR, 232 (1977), 32–-35. Zbl0372.53034MR431074
  20. A. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math., 143 (1996), 71–96. Zbl0857.35135MR1370758
  21. G. Nakamura, Z. Sun, G. Uhlmann, Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field, Math. Ann., 303 (1995), 377–388. Zbl0843.35134MR1354996
  22. L. Päivärinta, A. Panchenko, G. Uhlmann, Complex geometric optics solutions for Lipschitz conductivities, Rev. Mat. Iberoamericana 19 (2003), 57–72. Zbl1055.35144MR1993415
  23. L. E. Payne, H. F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rat. Mech. Anal., 5 (1960), 286–292. Zbl0099.08402MR117419
  24. M. Salo, Inverse boundary value problems for the magnetic Schrödinger equation, J. Phys. Conf. Series, 73 (2007), 012020. 
  25. M. Salo, L. Tzou Carleman estimates and inverse problems for Dirac operators, preprint, 2007. 
  26. V. Sharafutdinov, Integral geometry of tensor fields, in Inverse and Ill-Posed Problems Series, VSP, Utrecht, 1994. Zbl0883.53004MR1374572
  27. V. Sharafutdinov, On emission tomography of inhomogeneous media, SIAM J. Appl. Math., 55 (1995), 707–718. Zbl0843.92013MR1331582
  28. Z. Sun, G. Uhlmann, Generic uniqueness for an inverse boundary value problem, Duke Math. J., 62 (1991), 131–155. Zbl0728.35132MR1104326
  29. Z. Sun, G. Uhlmann, Anisotropic inverse problems in two dimensions, Inverse Problems, 19 (2003), 1001–1010. Zbl1054.35139MR2024685
  30. J. Sylvester, An anisotropic inverse boundary value problem, Comm. Pure Appl. Math., 43 (1990), 201–232. Zbl0709.35102MR1038142
  31. J. Sylvester, G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153–169. Zbl0625.35078MR873380
  32. J. Sylvester, G. Uhlmann, Inverse boundary value problems at the boundary – continuous dependence, Comm. Pure Appl. Math., 41 (1988), 197–219. Zbl0632.35074MR924684

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