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

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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

Mourad Bellassoued ^1, and David Dos Santos Ferreira ^2,

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

Abstract
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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.

Keywords: Dirichlet-to-Neumann map., Stability estimates, Hyperbolic inverse problem.

Mathematics Subject Classification: Primary: 58J32, 35L05; Secondary: 58J5.

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.

show all references

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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