# American Institute of Mathematical Sciences

November  2011, 5(4): 745-773. doi: 10.3934/ipi.2011.5.745

## Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map

 1 University of Carthage, Department of Mathematics, Faculty of Sciences of Bizerte, 7021 Jarzouna Bizerte, Tunisia 2 Université Paris 13, CNRS, UMR 7539 LAGA, 99, avenue Jean-Baptiste Clément, F-93 430 Villetaneuse, France

Received  February 2011 Revised  September 2011 Published  November 2011

In this article we seek stability estimates in the inverse problem of determining the potential or the velocity in a wave equation in an anisotropic medium from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichlet-to-Neumann map associated to the wave equation. We prove in dimension $n\geq 2$ that the knowledge of the Dirichlet-to-Neumann map for the wave equation uniquely determines the electric potential and we prove Hölder-type stability in determining the potential. We prove similar results for the determination of velocities close to 1.
Citation: Mourad Bellassoued, David Dos Santos Ferreira. Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map. Inverse Problems and Imaging, 2011, 5 (4) : 745-773. doi: 10.3934/ipi.2011.5.745
##### References:
 [1] 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, Inventiones Math., 158 (2004), 261-321. doi: 10.1007/s00222-004-0371-6. [2] G. Alessandrini and J. Sylvester, Stability for multidimensional inverse spectral problem, Commun. PDE, 15 (1990), 711-736. doi: 10.1080/03605309908820705. [3] M. Belishev, Boundary control in reconstruction of manifolds and metrics (BC method), Inverse Problems, 13 (1997), R1-R45. doi: 10.1088/0266-5611/13/5/002. [4] M. Belishev and Y. V. Kurylev, To the reconstruction of a Riemannian manifold via its spectral data (BC- method), Commun. Partial Differ. Equations, 17 (1992), 767-804. doi: 10.1080/03605309208820863. [5] M. Bellassoued, Uniqueness and stability in determining the speed of propagation of second-order hyperbolic equation with variable coefficients, Applicable Analysis, 83 (2004), 983-1014. [6] M. Bellassoued, M. Choulli and M. Yamamoto, Stability estimate for an inverse wave equation and a multidimensional Borg-Levinson theorem, J. Diff. Equat., 247 (2009), 465-494. doi: 10.1016/j.jde.2009.03.024. [7] M. Bellassoued and M. Choulli, Stability estimate for an inverse problem for the magnetic Schrödinger equation from the Dirichlet-to-Neumann map, J. Funct. Anal., 258 (2010), 161-195. doi: 10.1016/j.jfa.2009.06.010. [8] M. Bellassoued, D. Jellali and M. Yamamoto, Lipschitz stability for a hyperbolic inverse problem by finite local boundary data, Applicable Analysis, 85 (2006), 1219-1243. doi: 10.1080/00036810600787873. [9] M. Bellassoued, D. Jellali and M. Yamamoto, Stability Estimate for the hyperbolic inverse boundary value problem by local Dirichlet-to-Neumann map, J. Math. Anal. Appl., 343 (2008), 1036-1046. doi: 10.1016/j.jmaa.2008.01.098. [10] A.-P. Calderón, On an inverse boundary value problem, in "Seminar on Numerical Analysis and its Applications to Continuum Physics" (Rio de Janeiro, 1980), Soc. Brasil. Mat., Rio de Janeiro, (1980), 65-73. [11] F. Cardoso and R. Mendoza, On the hyperbolic Dirichlet-to-Neumann functional, Comm. Partial Diff. Equations, 21 (1996), 1235-1252. [12] J. Cheng and G. Nakamura, Stability for the inverse potential problem by finite measurements on the boundary, Inverse Problems, 17 (2001), 273-280. [13] M. Choulli, "Une Introduction aux Problèmes Inverses Elliptiques et Paraboliques," Mathématiques et Applications, Vol. 65, Springer-Verlag, Berlin, 2009. [14] D. Dos Santos Ferreira, C. E. Kenig, M. Salo and G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems, Inventiones Math., 178 (2009), 119-171. doi: 10.1007/s00222-009-0196-4. [15] G. Eskin, A new approach to hyperbolic inverse problems, Inverse Problems, 22 (2006), 815-831. Available from: arXiv:math/0505452v3. [16] G. Eskin, Inverse hyperbolic problems with time-dependent coefficients, Comm. Partial Differential Equations, 32 (2007), 1737-1758. Available from: arXiv:math/0508161v2. [17] G. Eskin, Global uniqueness in the inverse scattering problem for the Schrödinger operator with external Yang-Mills potentials, Comm. Math. Phys., 222 (2001), 503-531. doi: 10.1007/s002200100522. [18] E. Hebey, "Sobolev Spaces on Riemannian Manifolds," Lecture Notes in Mathematics, 1635, Springer-Verlag, Berlin, 1996. [19] V. Isakov, An inverse hyperbolic problem with many boundary measurements, Comm. Part. Dif. Equations, 16 (1991), 1183-1195. doi: 10.1080/03605309108820794. [20] V. Isakov, "Inverse Problems for Partial Differential Equations," Applied Mathematical Sciences, 127, Springer-Verlag, New York, 1998. [21] V. Isakov and Z. Sun, Stability estimates for hyperbolic inverse problems with local boundary data, Inverse Problems, 8 (1992), 193-206. doi: 10.1088/0266-5611/8/2/003. [22] J. Jost, "Riemannian Geometry and Geometric Analysis," Universitext, Springer-Verlag, Berlin, 1995. [23] A. Katchalov, Y. Kurylev and M. Lassas, "Inverse Boundary Spectral Problems," Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123, Chapman & Hall/CRC, Boca Raton, FL, 2001. [24] Y. V. Kurylev and M. Lassas, Hyperbolic inverse problem with data on a part of the boundary, in "Differential Equations and Mathematical Physics"(Birmingham, AL, 1999), AMS/IP Stud. Adv. Math., 16, Amer. Math. Soc., Providence, RI, (2000), 259-272. [25] J.-L. Lions and E. Magenes, "Non-Homogenous Boundary Value Problems and Applications," Volumes I and II, Springer-Verlag, Berlin, 1972. [26] L. Pestov and G. Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid, Annals of Math., 161 (2005), 1093-1110. doi: 10.4007/annals.2005.161.1093. [27] Rakesh, Reconstruction for an inverse problem for the wave equation with constant velocity, Inverse Problems, 6 (1990), 91-98. doi: 10.1088/0266-5611/6/1/009. [28] Rakesh and W. Symes, Uniqueness for an inverse problems for the wave equation, Comm. Partial Diff. Equations, 13 (1988), 87-96. doi: 10.1080/03605308808820539. [29] A. Ramm and J. Sjöstrand, An inverse problem of the wave equation, Math. Z., 206 (1991), 119-130. doi: 10.1007/BF02571330. [30] V. Sharafutdinov, "Integral Geometry of Tensor Fields," Inverse and Ill-Posed Problems Series, VSP, Utrecht, 1994. [31] P. Stefanov and G. Uhlmann, Stability estimates for the hyperbolic Dirichlet-to-Neumann map in anisotropic media, J. Functional Anal., 154 (1998), 330-358. [32] P. Stefanov and G. Uhlmann, Stability estimates for the X-ray transform of tensor fields and boundary rigidity, Duke Math. J., 123 (2004), 445-467. [33] P. Stefanov and G. Uhlmann, Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map,, International Math. Research Notices, 2005 (): 1047.  doi: 10.1155/IMRN.2005.1047. [34] Z. Sun, On continous dependence for an inverse initial-boundary value problem for the wave equation, J. Math. Anal. App., 150 (1990), 188-204. doi: 10.1016/0022-247X(90)90207-V. [35] G. Uhlmann, Inverse boundary value problems and applications, in "Méthodes Semi-Classiques," Vol 1 (Nantes, 1991), Astérisque, 207 (1992), 153-221.

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##### References:
 [1] 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, Inventiones Math., 158 (2004), 261-321. doi: 10.1007/s00222-004-0371-6. [2] G. Alessandrini and J. Sylvester, Stability for multidimensional inverse spectral problem, Commun. PDE, 15 (1990), 711-736. doi: 10.1080/03605309908820705. [3] M. Belishev, Boundary control in reconstruction of manifolds and metrics (BC method), Inverse Problems, 13 (1997), R1-R45. doi: 10.1088/0266-5611/13/5/002. [4] M. Belishev and Y. V. Kurylev, To the reconstruction of a Riemannian manifold via its spectral data (BC- method), Commun. Partial Differ. Equations, 17 (1992), 767-804. doi: 10.1080/03605309208820863. [5] M. Bellassoued, Uniqueness and stability in determining the speed of propagation of second-order hyperbolic equation with variable coefficients, Applicable Analysis, 83 (2004), 983-1014. [6] M. Bellassoued, M. Choulli and M. Yamamoto, Stability estimate for an inverse wave equation and a multidimensional Borg-Levinson theorem, J. Diff. Equat., 247 (2009), 465-494. doi: 10.1016/j.jde.2009.03.024. [7] M. Bellassoued and M. Choulli, Stability estimate for an inverse problem for the magnetic Schrödinger equation from the Dirichlet-to-Neumann map, J. Funct. Anal., 258 (2010), 161-195. doi: 10.1016/j.jfa.2009.06.010. [8] M. Bellassoued, D. Jellali and M. Yamamoto, Lipschitz stability for a hyperbolic inverse problem by finite local boundary data, Applicable Analysis, 85 (2006), 1219-1243. doi: 10.1080/00036810600787873. [9] M. Bellassoued, D. Jellali and M. Yamamoto, Stability Estimate for the hyperbolic inverse boundary value problem by local Dirichlet-to-Neumann map, J. Math. Anal. Appl., 343 (2008), 1036-1046. doi: 10.1016/j.jmaa.2008.01.098. [10] A.-P. Calderón, On an inverse boundary value problem, in "Seminar on Numerical Analysis and its Applications to Continuum Physics" (Rio de Janeiro, 1980), Soc. Brasil. Mat., Rio de Janeiro, (1980), 65-73. [11] F. Cardoso and R. Mendoza, On the hyperbolic Dirichlet-to-Neumann functional, Comm. Partial Diff. Equations, 21 (1996), 1235-1252. [12] J. Cheng and G. Nakamura, Stability for the inverse potential problem by finite measurements on the boundary, Inverse Problems, 17 (2001), 273-280. [13] M. Choulli, "Une Introduction aux Problèmes Inverses Elliptiques et Paraboliques," Mathématiques et Applications, Vol. 65, Springer-Verlag, Berlin, 2009. [14] D. Dos Santos Ferreira, C. E. Kenig, M. Salo and G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems, Inventiones Math., 178 (2009), 119-171. doi: 10.1007/s00222-009-0196-4. [15] G. Eskin, A new approach to hyperbolic inverse problems, Inverse Problems, 22 (2006), 815-831. Available from: arXiv:math/0505452v3. [16] G. Eskin, Inverse hyperbolic problems with time-dependent coefficients, Comm. Partial Differential Equations, 32 (2007), 1737-1758. Available from: arXiv:math/0508161v2. [17] G. Eskin, Global uniqueness in the inverse scattering problem for the Schrödinger operator with external Yang-Mills potentials, Comm. Math. Phys., 222 (2001), 503-531. doi: 10.1007/s002200100522. [18] E. Hebey, "Sobolev Spaces on Riemannian Manifolds," Lecture Notes in Mathematics, 1635, Springer-Verlag, Berlin, 1996. [19] V. Isakov, An inverse hyperbolic problem with many boundary measurements, Comm. Part. Dif. Equations, 16 (1991), 1183-1195. doi: 10.1080/03605309108820794. [20] V. Isakov, "Inverse Problems for Partial Differential Equations," Applied Mathematical Sciences, 127, Springer-Verlag, New York, 1998. [21] V. Isakov and Z. Sun, Stability estimates for hyperbolic inverse problems with local boundary data, Inverse Problems, 8 (1992), 193-206. doi: 10.1088/0266-5611/8/2/003. [22] J. Jost, "Riemannian Geometry and Geometric Analysis," Universitext, Springer-Verlag, Berlin, 1995. [23] A. Katchalov, Y. Kurylev and M. Lassas, "Inverse Boundary Spectral Problems," Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123, Chapman & Hall/CRC, Boca Raton, FL, 2001. [24] Y. V. Kurylev and M. Lassas, Hyperbolic inverse problem with data on a part of the boundary, in "Differential Equations and Mathematical Physics"(Birmingham, AL, 1999), AMS/IP Stud. Adv. Math., 16, Amer. Math. Soc., Providence, RI, (2000), 259-272. [25] J.-L. Lions and E. Magenes, "Non-Homogenous Boundary Value Problems and Applications," Volumes I and II, Springer-Verlag, Berlin, 1972. [26] L. Pestov and G. Uhlmann, Two dimensional compact simple Riemannian manifolds are boundary distance rigid, Annals of Math., 161 (2005), 1093-1110. doi: 10.4007/annals.2005.161.1093. [27] Rakesh, Reconstruction for an inverse problem for the wave equation with constant velocity, Inverse Problems, 6 (1990), 91-98. doi: 10.1088/0266-5611/6/1/009. [28] Rakesh and W. Symes, Uniqueness for an inverse problems for the wave equation, Comm. Partial Diff. Equations, 13 (1988), 87-96. doi: 10.1080/03605308808820539. [29] A. Ramm and J. Sjöstrand, An inverse problem of the wave equation, Math. Z., 206 (1991), 119-130. doi: 10.1007/BF02571330. [30] V. Sharafutdinov, "Integral Geometry of Tensor Fields," Inverse and Ill-Posed Problems Series, VSP, Utrecht, 1994. [31] P. Stefanov and G. Uhlmann, Stability estimates for the hyperbolic Dirichlet-to-Neumann map in anisotropic media, J. Functional Anal., 154 (1998), 330-358. [32] P. Stefanov and G. Uhlmann, Stability estimates for the X-ray transform of tensor fields and boundary rigidity, Duke Math. J., 123 (2004), 445-467. [33] P. Stefanov and G. Uhlmann, Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map,, International Math. Research Notices, 2005 (): 1047.  doi: 10.1155/IMRN.2005.1047. [34] Z. Sun, On continous dependence for an inverse initial-boundary value problem for the wave equation, J. Math. Anal. App., 150 (1990), 188-204. doi: 10.1016/0022-247X(90)90207-V. [35] G. Uhlmann, Inverse boundary value problems and applications, in "Méthodes Semi-Classiques," Vol 1 (Nantes, 1991), Astérisque, 207 (1992), 153-221.
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