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.
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References
Anikonov, Yu.E.: Some Methods for the Study of Multidimensional Inverse Problems for Differential Equations. Nauka Sibirsk, Otdel, Novosibirsk (1978)
Astala, K., Päivärinta, L.: Calderón’s inverse conductivity problem in the plane. Ann. Math. 163, 265–299 (2006)
Astala, K., Lassas, M., Päivärinta, L.: Calderón’s inverse problem for anisotropic conductivity in the plane. Commun. Partial Differ. Equ. 30, 207–224 (2005)
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)
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)
Bukhgeim, A.L.: Recovering a potential from Cauchy data in the two-dimensional case. J. Inverse Ill-posed Probl. 16, 19–34 (2008)
Dimassi, M., Sjöstrand, J.: Spectral Asymptotics in the Semi-classical Limit. Cambridge University Press, Cambridge (1999)
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)
Eisenhart, L.: Riemannian Geometry, 2nd edn. Princeton University Press, Princeton (1949)
Guillarmou, C., Sa Barreto, A.: Inverse problems for Einstein manifolds. Inverse Probl. Imaging 3, 1–15 (2009)
Hörmander, L.: The Analysis of Linear Partial Differential Operators III–IV. Springer, Berlin (1985)
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)
Jost, J.: Riemannian Geometry and Geometric Analysis. Springer, Berlin (2002)
Kenig, C.E., Sjöstrand, J., Uhlmann, G.: The Calderón problem with partial data. Ann. Math. 165, 567–591 (2007)
Knudsen, K., Salo, M.: Determining non-smooth first order terms from partial boundary measurements. Inverse Probl. Imaging 1, 349–369 (2007)
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)
Lassas, M., Uhlmann, G.: On determining a Riemannian manifold from the Dirichlet-to-Neumann map. Ann. Sci. Éc. Norm. Super. 34, 771–787 (2001)
Lassas, M., Taylor, M., Uhlmann, G.: The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary. Commun. Anal. Geom. 11, 207–221 (2003)
Lebeau, G., Robbiano, L.: Contrôle exact de l’équation de la chaleur. Commun. Partial Differ. Equ. 20, 335–356 (1995)
Lebeau, G., Robbiano, L.: Stabilisation de l’équation des ondes par le bord. Duke Math. J. 86, 465–491 (1997)
Lee, J., Uhlmann, G.: Determining anisotropic real-analytic conductivities by boundary measurements. Commun. Pure Appl. Math. 42, 1097–1112 (1989)
Lionheart, W.: Conformal uniqueness results in anisotropic electrical impedance imaging. Inverse Probl. 13, 125–134 (1997)
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)
Nachman, A.: Global uniqueness for a two-dimensional inverse boundary value problem. Ann. Math. 143, 71–96 (1996)
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)
Payne, L.E., Weinberger, H.F.: An optimal Poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5, 286–292 (1960)
Petersen, P.: Riemannian Geometry. Springer, Berlin (1998)
Salo, M.: Inverse boundary value problems for the magnetic Schrödinger equation. J. Phys. Conf. Ser. 73, 012020 (2007)
Salo, M., Tzou, L.: Carleman estimates and inverse problems for Dirac operators. Math. Ann. 344, 161–184 (2009)
Sharafutdinov, V.: Integral geometry of tensor fields. In: Inverse and Ill-Posed Problems Series. VSP, Utrecht (1994)
Sharafutdinov, V.: On emission tomography of inhomogeneous media. SIAM J. Appl. Math. 55, 707–718 (1995)
Sun, Z., Uhlmann, G.: Generic uniqueness for an inverse boundary value problem. Duke Math. J. 62, 131–155 (1991)
Sun, Z., Uhlmann, G.: Anisotropic inverse problems in two dimensions. Inverse Probl. 19, 1001–1010 (2003)
Sylvester, J.: An anisotropic inverse boundary value problem. Commun. Pure Appl. Math. 43, 201–232 (1990)
Sylvester, J., Uhlmann, G.: A global uniqueness theorem for an inverse boundary value problem. Ann. Math. 125, 153–169 (1987)
Sylvester, J., Uhlmann, G.: Inverse boundary value problems at the boundary—continuous dependence. Commun. Pure Appl. Math. 41, 197–219 (1988)
Taylor, M.: Pseudodifferential Operators. Princeton University Press, Princeton (1981)
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Dos Santos Ferreira, D., Kenig, C.E., Salo, M. et al. Limiting Carleman weights and anisotropic inverse problems. Invent. math. 178, 119–171 (2009). https://doi.org/10.1007/s00222-009-0196-4
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DOI: https://doi.org/10.1007/s00222-009-0196-4