Anisotropic inverse problems and Carleman estimates
Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2007-2008), Talk no. 8, 17 p.

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 n3.

     author = {Dos Santos Ferreira, David},
     title = {Anisotropic inverse problems and {Carleman} estimates},
     journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
     note = {talk:8},
     pages = {1--17},
     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {2007-2008},
     language = {en},
     url = {}
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Dos Santos Ferreira, David. Anisotropic inverse problems and Carleman estimates. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2007-2008), Talk no. 8, 17 p.

[1] Yu. E. Anikonov, Some methods for the study of multidimensional inverse problems for differential equations, Nauka Sibirsk. Otdel, Novosibirsk (1978). | MR

[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. | Zbl

[3] K. Astala, L. Päivärinta, Calderón’s inverse conductivity problem in the plane, Ann. of Math., 163 (2006), 265–299. | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR

[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. | MR

[8] D. Dos Santos Ferreira, C. E. Kenig, M. Salo, G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems, preprint (2008), arXiv:0803.3508.

[9] C. Guillarmou, A. Sa Barreto, Inverse problems for Einstein manifolds, preprint (2007), arXiv:0710.1136.

[10] L. Hörmander, The Analysis of Linear Partial Differential Operators III, Springer-Verlag, 1985. | MR | Zbl

[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. | MR | Zbl

[12] C. E. Kenig, J. Sjöstrand, G. Uhlmann, The Calderón problem with partial data, Ann. of Math., 165 (2007), 567–591. | MR | Zbl

[13] K. Knudsen, M. Salo, Determining non-smooth first order terms from partial boundary measurements, Inverse Problems and Imaging, 1 (2007), 349–369. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[16] M. Lassas, G. Uhlmann, On determining a Riemannian manifold from the Dirichlet-to-Neumann map, Ann. Sc. ENS, 34 (2001), 771–787. | Numdam | MR | Zbl

[17] J. Lee, G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurement, Comm. Pure Appl. Math., 42 (1989), 1097–1112. | MR | Zbl

[18] W. Lionheart, Conformal uniqueness results in anisotropic electrical impedance imaging, Inverse Problems, 13 (1997), 125-134. | MR | Zbl

[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. | MR | Zbl

[20] A. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math., 143 (1996), 71–96. | MR | Zbl

[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. | MR | Zbl

[22] L. Päivärinta, A. Panchenko, G. Uhlmann, Complex geometric optics solutions for Lipschitz conductivities, Rev. Mat. Iberoamericana 19 (2003), 57–72. | MR | Zbl

[23] L. E. Payne, H. F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rat. Mech. Anal., 5 (1960), 286–292. | MR | Zbl

[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. | MR | Zbl

[27] V. Sharafutdinov, On emission tomography of inhomogeneous media, SIAM J. Appl. Math., 55 (1995), 707–718. | MR | Zbl

[28] Z. Sun, G. Uhlmann, Generic uniqueness for an inverse boundary value problem, Duke Math. J., 62 (1991), 131–155. | MR | Zbl

[29] Z. Sun, G. Uhlmann, Anisotropic inverse problems in two dimensions, Inverse Problems, 19 (2003), 1001–1010. | MR | Zbl

[30] J. Sylvester, An anisotropic inverse boundary value problem, Comm. Pure Appl. Math., 43 (1990), 201–232. | MR | Zbl

[31] J. Sylvester, G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153–169. | MR | Zbl

[32] J. Sylvester, G. Uhlmann, Inverse boundary value problems at the boundary – continuous dependence, Comm. Pure Appl. Math., 41 (1988), 197–219. | MR | Zbl