Boundary rigidity with partial data

Stefanov, Plamen¹ ; Uhlmann, Gunther² ; Vasy, Andras³

¹ Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
² Department of Mathematics, University of Washington, Seattle, Washington 98195 and Department of Mathematics, University of Helsinki, Finland FI-00014
³ Department of Mathematics, Stanford University, Stanford, California 94305

Journal of the American Mathematical Society, Tome 29 (2016) no. 2, pp. 299-332

Résumé

We study the boundary rigidity problem with partial data consisting of determining locally the Riemannian metric of a Riemannian manifold with boundary from the distance function measured at pairs of points near a fixed point on the boundary. We show that one can recover uniquely and in a stable way a conformal factor near a strictly convex point where we have the information. In particular, this implies that we can determine locally the isotropic sound speed of a medium by measuring the travel times of waves joining points close to a convex point on the boundary. The local results lead to a global lens rigidity uniqueness and stability result assuming that the manifold is foliated by strictly convex hypersurfaces.

Détail

DOI : 10.1090/jams/846

Bibliographie

[1] Bao, Gang, Zhang, Hai Sensitivity analysis of an inverse problem for the wave equation with caustics J. Amer. Math. Soc. 2014 953 981

[2] Besson, G., Courtois, G., Gallot, S. Entropies et rigidités des espaces localement symétriques de courbure strictement négative Geom. Funct. Anal. 1995 731 799

[3] Burago, Dmitri, Ivanov, Sergei Boundary rigidity and filling volume minimality of metrics close to a flat one Ann. of Math. (2) 2010 1183 1211

[4] Croke, Christopher Scattering rigidity with trapped geodesics Ergodic Theory Dynam. Systems 2014 826 836

[5] Creager, K. C. Anisotropy of the inner core from differential travel times of the phases PKP and PKIPK Nature 1992

[6] Croke, Christopher B. Rigidity for surfaces of nonpositive curvature Comment. Math. Helv. 1990 150 169

[7] Croke, Christopher B. Rigidity and the distance between boundary points J. Differential Geom. 1991 445 464

[8] Croke, Christopher B. Rigidity theorems in Riemannian geometry 2004 47 72

[9] Croke, Christopher B., Dairbekov, Nurlan S. Lengths and volumes in Riemannian manifolds Duke Math. J. 2004 1 14

[10] Croke, Christopher B., Kleiner, Bruce Conjugacy and rigidity for manifolds with a parallel vector field J. Differential Geom. 1994 659 680

[11] Dairbekov, Nurlan S. Integral geometry problem for nontrapping manifolds Inverse Problems 2006 431 445

[12] Guillarmou, C. Lens rigidity for manifolds with hyperbolic trapped set

[13] Gromov, Mikhael Filling Riemannian manifolds J. Differential Geom. 1983 1 147

[14] Herglotz, G. Über die Elastizitaet der Erde bei Beruecksichtigung ihrer variablen Dichte Z. Math. Phys. 1905

[15] Hã¶Rmander, L. The analysis of linear partial differential operators 1983

[16] Ivanov, Sergei Volume comparison via boundary distances 2010 769 784

[17] Krishnan, Venkateswaran P. A support theorem for the geodesic ray transform on functions J. Fourier Anal. Appl. 2009 515 520

[18] Krishnan, Venkateswaran P., Stefanov, Plamen A support theorem for the geodesic ray transform of symmetric tensor fields Inverse Probl. Imaging 2009 453 464

[19] Lassas, Matti, Sharafutdinov, Vladimir, Uhlmann, Gunther Semiglobal boundary rigidity for Riemannian metrics Math. Ann. 2003 767 793

[20] Melrose, Richard B. Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces 1994 85 130

[21] Michel, Ren㩠Sur la rigidité imposée par la longueur des géodésiques Invent. Math. 1981/82 71 83

[22] Muhometov, R. G. The reconstruction problem of a two-dimensional Riemannian metric, and integral geometry Dokl. Akad. Nauk SSSR 1977 32 35

[23] Muhometov, R. G. On a problem of reconstructing Riemannian metrics Sibirsk. Mat. Zh. 1981

[24] Muhometov, R. G., Romanov, V. G. On the problem of finding an isotropic Riemannian metric in an 𝑛-dimensional space Dokl. Akad. Nauk SSSR 1978 41 44

[25] Parenti, Cesare Operatori pseudo-differenziali in 𝑅ⁿ e applicazioni Ann. Mat. Pura Appl. (4) 1972 359 389

[26] Paternain, Gabriel P., Salo, Mikko, Uhlmann, Gunther Tensor tomography on surfaces Invent. Math. 2013 229 247

[27] Paternain, Gabriel P., Salo, Mikko, Uhlmann, Gunther The attenuated ray transform for connections and Higgs fields Geom. Funct. Anal. 2012 1460 1489

[28] Paternain, Gabriel P., Salo, Mikko, Uhlmann, Gunther Tensor tomography: progress and challenges Chinese Ann. Math. Ser. B 2014 399 428

[29] Pestov, Leonid, Uhlmann, Gunther Two dimensional compact simple Riemannian manifolds are boundary distance rigid Ann. of Math. (2) 2005 1093 1110

[30] Pestov, L. N., Sharafutdinov, V. A. Integral geometry of tensor fields on a manifold of negative curvature Sibirsk. Mat. Zh. 1988

[31] Ranjan, Akhil, Shah, Hemangi Convexity of spheres in a manifold without conjugate points Proc. Indian Acad. Sci. Math. Sci. 2002 595 599

[32] Sharafutdinov, Vladimir, Skokan, Michal, Uhlmann, Gunther Regularity of ghosts in tensor tomography J. Geom. Anal. 2005 499 542

[33] Sharafutdinov, V. A. Integral geometry of tensor fields 1994 271

[34] Sharafutdinov, V. A. Integral geometry of a tensor field on a surface of revolution Sibirsk. Mat. Zh. 1997

[35] Sharafutdinov, V. A. A problem in integral geometry in a nonconvex domain Sibirsk. Mat. Zh. 2002 1430 1442

[36] Sharafutdinov, Vladimir Variations of Dirichlet-to-Neumann map and deformation boundary rigidity of simple 2-manifolds J. Geom. Anal. 2007 147 187

[37] ŠUbin, M. A. Pseudodifferential operators in 𝑅ⁿ Dokl. Akad. Nauk SSSR 1971 316 319

[38] Stefanov, Plamen Microlocal approach to tensor tomography and boundary and lens rigidity Serdica Math. J. 2008 67 112

[39] Stefanov, Plamen, Uhlmann, Gunther Rigidity for metrics with the same lengths of geodesics Math. Res. Lett. 1998 83 96

[40] Stefanov, Plamen, Uhlmann, Gunther Stability estimates for the X-ray transform of tensor fields and boundary rigidity Duke Math. J. 2004 445 467

[41] Stefanov, Plamen, Uhlmann, Gunther Boundary rigidity and stability for generic simple metrics J. Amer. Math. Soc. 2005 975 1003

[42] Stefanov, Plamen, Uhlmann, Gunther Boundary and lens rigidity, tensor tomography and analytic microlocal analysis 2008 275 293

[43] Stefanov, Plamen, Uhlmann, Gunther Integral geometry on tensor fields on a class of non-simple Riemannian manifolds Amer. J. Math. 2008 239 268

[44] Stefanov, Plamen, Uhlmann, Gunther Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds J. Differential Geom. 2009 383 409

[45] Stefanov, Plamen, Uhlmann, Gunther Recovery of a source term or a speed with one measurement and applications Trans. Amer. Math. Soc. 2013 5737 5758

[46] Stefanov, Plamen, Uhlmann, Gunther The geodesic X-ray transform with fold caustics Anal. PDE 2012 219 260

[47] Stefanov, P., Uhlmann, G. Multi-wave methods via ultrasound

[48] Triebel, Hans Interpolation theory, function spaces, differential operators 1978 528

[49] Uhlmann, G., Vasy, A. The inverse problem for the local geodesic ray transform

[50] Vargo, James A proof of lens rigidity in the category of analytic metrics Math. Res. Lett. 2009 1057 1069

[51] Wiechert, E., Zoeppritz, K. Über Erdbebenwellen Nachr. Koenigl. Geselschaft Wiss. Göttingen 1907

[52] Zhou, H. The inverse problem for the local ray transform