Advanced Search
Article Contents
Article Contents

On the stable recovery of a metric from the hyperbolic DN map with incomplete data

Abstract Related Papers Cited by
  • We show that given two hyperbolic Dirichlet to Neumann maps associated to two Riemannian metrics of a Riemannian manifold with boundary which coincide near the boundary are close then the lens data of the two metrics is the same. As a consequence, we prove uniqueness of recovery a conformal factor (sound speed) locally under some conditions on the latter.
    Mathematics Subject Classification: Primary: 35R30.


    \begin{equation} \\ \end{equation}
  • [1]

    M. Bellassoued and D. D. Santos Ferreira, Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map, Inverse Probl. Imaging, 5 (2011), 745-773.doi: 10.3934/ipi.2011.5.745.


    M. I. Belishev, An approach to multidimensional inverse problems for the wave equation, Dokl. Akad. Nauk SSSR, 297 (1987), 524-527.


    M. I. Belishev and Y. V. Kurylev, To the reconstruction of a Riemannian manifold via its spectral data (BC-method), Comm. Partial Differential Equations, 17 (1992), 767-804.doi: 10.1080/03605309208820863.


    G. Bao and H. Zhang, Sensitivity analysis of an inverse problem for the wave equation with caustics, J. Amer. Math. Soc., 27 (2014), 953-981.doi: 10.1090/S0894-0347-2014-00787-6.


    C. Croke and P. Herreros, Lens rigidity with trapped geodesics in two dimensions, Asian Journal of Mathematics, 20 (2016), 47-58, http://www.math.upenn.edu/~ccroke/dvi-papers/SurfaceLensRigidity.pdf.doi: 10.4310/AJM.2016.v20.n1.a3.


    C. Croke, Boundary and lens rigidity of finite quotients, Proc. Amer. Math. Soc., 133 (2005), 3663-3668 (electronic).doi: 10.1090/S0002-9939-05-07927-X.


    _______, Scattering rigidity with trapped geodesics, Ergodic Theory Dynam. Systems, 34 (2014), 826-836.doi: 10.1017/etds.2012.164.


    A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 123, Chapman & Hall/CRC, Boca Raton, FL, 2001.doi: 10.1201/9781420036220.


    I. Lasiecka, J.-L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192.


    S. Liu and L. Oksanen, A lipschitz stable reconstruction formula for the inverse problem for the wave equation, Trans. Amer. Math. Soc., 368 (2016), 319-335.doi: 10.1090/tran/6332.


    M. Lassas, V. Sharafutdinov and G. Uhlmann, Semiglobal boundary rigidity for Riemannian metrics, Math. Ann., 325 (2003), 767-793.doi: 10.1007/s00208-002-0407-4.


    C. Montalto, Stable determination of a simple metric, a covector field and a potential from the hyperbolic Dirichlet-to-Neumann map, Comm. Partial Differential Equations, 39 (2014), 120-145.doi: 10.1080/03605302.2013.843429.


    R. G. Muhometov, On a problem of reconstructing Riemannian metrics, Sibirsk. Mat. Zh., 22 (1981), 119-135, 237.


    L. Pestov and G. Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid, Ann. of Math. (2), 161 (2005), 1093-1110.doi: 10.4007/annals.2005.161.1093.


    M. Salo, Stability for solutions of wave equations with $C^{1,1}$ coefficients, Inverse Probl. Imaging, 1 (2007), 537-556.doi: 10.3934/ipi.2007.1.537.


    P. Stefanov and G. Uhlmann, Boundary rigidity and stability for generic simple metrics, J. Amer. Math. Soc., 18 (2005), 975-1003.doi: 10.1090/S0894-0347-05-00494-7.


    _______, Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map, Int. Math. Res. Not., 17 (2005), 1047-1061.


    _______, Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds, J. Differential Geom., 82 (2009), 383-409.


    _______, Thermoacoustic tomography with variable sound speed, Inverse Problems, 25 (2009), 075011, 16pp.doi: 10.1088/0266-5611/25/7/075011.


    _______, Thermoacoustic tomography arising in brain imaging, Inverse Problems, 27 (2011), 045004, 26pp.doi: 10.1088/0266-5611/27/4/045004.


    P. Stefanov, G. Uhlmann and A. Vasy, Boundary rigidity with partial data, J. Amer. Math. Soc., 29 (2016), 299-332.doi: 10.1090/jams/846.


    P. Stefanov and Y. Yang, The inverse problem for the Dirichlet-to-Neumann map on Lorentzian manifolds, arXiv:1607.08690, 2016.


    D. Tataru, Unique continuation for solutions to PDE's; between Hörmander's theorem and Holmgren's theorem, Comm. Partial Differential Equations, 20 (1995), 855-884.doi: 10.1080/03605309508821117.


    M. Zworski, Semiclassical Analysis, Graduate Studies in Mathematics, vol. 138, American Mathematical Society, Providence, RI, 2012.doi: 10.1090/gsm/138.

  • 加载中

Article Metrics

HTML views() PDF downloads(91) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint