Duke Mathematical Journal

Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets

Matti Lassas and Lauri Oksanen

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We consider the inverse problem to determine a smooth compact Riemannian manifold with boundary (M,g) from a restriction ΛS,R of the Dirichlet-to-Neumann operator for the wave equation on the manifold. Here S and R are open sets in M and the restriction ΛS,R corresponds to the case where the Dirichlet data is supported on R+×S and the Neumann data is measured on R+×R. In the novel case where SR=, we show that ΛS,R determines the manifold (M,g) uniquely, assuming that the wave equation is exactly controllable from the set of sources S. Moreover, we show that the exact controllability can be replaced by the Hassell–Tao condition for eigenvalues and eigenfunctions, that is,

λjCνϕjL2(S)2,j=1,2,,

where λj are the Dirichlet eigenvalues and where (ϕj)j=1 is an orthonormal basis of the corresponding eigenfunctions.

Article information

Source
Duke Math. J., Volume 163, Number 6 (2014), 1071-1103.

Dates
First available in Project Euclid: 11 April 2014

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1397223296

Digital Object Identifier
doi:10.1215/00127094-2649534

Mathematical Reviews number (MathSciNet)
MR3192525

Zentralblatt MATH identifier
1375.35634

Subjects
Primary: 35R30: Inverse problems
Secondary: 35R01: Partial differential equations on manifolds [See also 32Wxx, 53Cxx, 58Jxx]

Citation

Lassas, Matti; Oksanen, Lauri. Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets. Duke Math. J. 163 (2014), no. 6, 1071--1103. doi:10.1215/00127094-2649534. https://projecteuclid.org/euclid.dmj/1397223296


Export citation

References

  • [1] R. Alexander and S. Alexander, Geodesics in Riemannian manifolds-with-boundary, Indiana Univ. Math. J. 30 (1981), 481–488.
  • [2] M. Anderson, A. Katsuda, Y. Kurylev, M. Lassas, and M. Taylor, Boundary regularity for the Ricci equation, geometric convergence, and Gel’fand’s inverse boundary problem, Invent. Math. 158 (2004), 261–321.
  • [3] K. Astala and L. Päivärinta, Calderón’s inverse conductivity problem in the plane, Ann. of Math. (2) 163 (2006), 265–299.
  • [4] K. Astala, L. Päivärinta, and M. Lassas, Calderón’s inverse problem for anisotropic conductivity in the plane, Comm. Partial Differential Equations 30 (2005), 207–224.
  • [5] C. Bardos, G. Lebeau, and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim. 30 (1992), 1024–1065.
  • [6] M. I. Belishev, An approach to multidimensional inverse problems for the wave equation (in Russian), Dokl. Akad. Nauk SSSR 297, no. 3 (1987), 524–527; English translation in Soviet Math. Dokl. 36, no. 3 (1988), 481–484.
  • [7] M. I. Belishev, Recent progress in the boundary control method, Inverse Problems 23 (2007), R1–R67.
  • [8] M. I. Belishev and Y. Kurylev, To the reconstruction of a Riemannian manifold via its spectral data (BC-method), Comm. Partial Differential Equations 17 (1992), 767–804.
  • [9] A. S. Blagoveščenskiĭ, A one-dimensional inverse boundary value problem for a second order hyperbolic equation, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 15 (1969), 85–90.
  • [10] A. S. Blagoveščenskiĭ, The inverse boundary value problem of the theory of wave propagation in an anisotropic medium, Trudy Mat. Inst. Steklov. 115 (1971), 39–56. (errata insert), 1971.
  • [11] A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl. 16 (2008), 19–33.
  • [12] N. Burq and P. Gérard, Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes, C. R. Math. Acad. Sci. Paris Sér. I 325 (1997), 749–752.
  • [13] A.-P. Calderón, “On an inverse boundary value problem” in Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), Sociedade Brasileira de Matemática, Rio de Janeiro, 1980, 65–73.
  • [14] I. Chavel, Riemannian Geometry: A Modern Introduction, 2nd ed., Cambridge Stud. Adv. Math. 98, Cambridge Univ. Press, Cambridge, 2006.
  • [15] D. Dos Santos Ferreira, C. E. Kenig, M. Salo, and G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems, Invent. Math. 178 (2009), 119–171.
  • [16] C. Guillarmou and L. Tzou, Calderón inverse problem with partial data on Riemann surfaces, Duke Math. J. 158 (2011), 83–120.
  • [17] A. Hassell and T. Tao, Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions, Math. Res. Lett. 9 (2002), 289–305. Correction, Math. Res. Lett. 17 (2010), 793–794.
  • [18] T. Helin, M. Lassas, and L. Oksanen, An inverse problem for the wave equation with one measurement and the pseudorandom source, Anal. PDE 5 (2012), 887–912.
  • [19] G. Henkin and V. Michel, On the explicit reconstruction of a Riemann surface from its Dirichlet-Neumann operator, Geom. Funct. Anal. 17 (2007), 116–155.
  • [20] G. Henkin and V. Michel, Inverse conductivity problem on Riemann surfaces, J. Geom. Anal. 18 (2008), 1033–1052.
  • [21] G. Henkin and R. G. Novikov, On the reconstruction of conductivity of a bordered two-dimensional surface in $\mathbb{R}^{3}$ from electrical current measurements on its boundary, J. Geom. Anal. 21 (2011), 543–587.
  • [22] G. Henkin and M. Santacesaria, On an inverse problem for anisotropic conductivity in the plane, Inverse Problems 26 (2010), art. ID 095011.
  • [23] O. Y. Imanuvilov, G. Uhlmann, and M. Yamamoto, The Calderón problem with partial data in two dimensions, J. Amer. Math. Soc. 23 (2010), 655–691.
  • [24] O. Y. Imanuvilov, G. Uhlmann, and M. Yamamoto, Determination of second-order elliptic operators in two dimensions from partial Cauchy data, Proc. Natl. Acad. Sci. USA 108 (2011), 467–472.
  • [25] O. Y. Imanuvilov, G. Uhlmann, and M. Yamamoto, Inverse boundary value problem by measuring Dirichlet data and Neumann data on disjoint sets, Inverse Problems 27 (2011), art. ID 085007.
  • [26] H. Isozaki, Y. Kurylev, and M. Lassas, Forward and inverse scattering on manifolds with asymptotically cylindrical ends, J. Funct. Anal. 258 (2010), 2060–2118.
  • [27] A. Katchalov and Y. Kurylev, Multidimensional inverse problem with incomplete boundary spectral data, Comm. Partial Differential Equations 23 (1998), 55–95.
  • [28] A. Katchalov, Y. Kurylev, and M. Lassas, Inverse Boundary Spectral Problems, Chapman Hall/CRC Monogr. Surv. Pure Appl. Math. 123, Chapman & Hall/CRC, Boca Raton, Fla., 2001.
  • [29] C. E. Kenig, J. Sjöstrand, and G. Uhlmann, The Calderón problem with partial data, Ann. of Math. (2) 165 (2007), 567–591.
  • [30] M. G. Kreĭn, Determination of the density of a nonhomogeneous symmetric cord by its frequency spectrum, Doklady Akad. Nauk SSSR (N.S.) 76 (1951), 345–348.
  • [31] I. Lasiecka, J.-L. Lions, and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl. (9) 65 (1986), 149–192.
  • [32] M. Lassas and L. Oksanen, An inverse problem for a wave equation with sources and observations on disjoint sets, Inverse Problems 26 (2010), art. ID 085012.
  • [33] M. Lassas, M. Taylor, and G. Uhlmann, The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary, Comm. Anal. Geom. 11 (2003), 207–221.
  • [34] M. Lassas and G. Uhlmann, On determining a Riemannian manifold from the Dirichlet-to-Neumann map, Ann. Sci. Éc. Norm. Supér. (4) 34 (2001), 771–787.
  • [35] J. M. Lee, Riemannian Manifolds: An Introduction to Curvature, Grad. Texts in Math. 176, Springer, New York, 1997.
  • [36] J. M. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math. 42 (1989), 1097–1112.
  • [37] A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. (2) 143 (1996), 71–96.
  • [38] L. Oksanen, Solving an inverse problem for the wave equation by using a minimization algorithm and time-reversed measurements, Inverse Probl. Imaging 5 (2011), 731–744.
  • [39] Rakesh, Characterization of transmission data for Webster’s horn equation, Inverse Problems 16 (2000), L9–L24.
  • [40] Rakesh and P. Sacks, Uniqueness for a hyperbolic inverse problem with angular control on the coefficients, J. Inverse Ill-Posed Probl. 19 (2011), 107–126.
  • [41] P. Stefanov and G. Uhlmann, Integral geometry on tensor fields on a class of non-simple Riemannian manifolds, Amer. J. Math. 130 (2008), 239–268.
  • [42] J. Sylvester, An anisotropic inverse boundary value problem, Comm. Pure Appl. Math. 43 (1990), 201–232.
  • [43] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2) 125 (1987), 153–169.
  • [44] 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.
  • [45] D. Tataru, Unique continuation for operators with partially analytic coefficients, J. Math. Pures Appl. (9) 78 (1999), 505–521.
  • [46] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, Cambridge, 1944.