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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 |
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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