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 Mat. Sb. (N.S.), 1986, Volume 130(172), Number 2(6), Pages 265–274 (Mi msb1868)

This article is cited in 35 scientific papers (total in 35 papers)

Carleman estimates and inverse problems for second-order hyperbolic equations

A. Khaidarov

Abstract: The author considers the problem of finding the time-independent coefficients of a second order hyperbolic equation from the Cauchy data at the initial moment and on a part of the lateral surface of a cylindrical domain. Estimates of Hörmander's Carleman type are obtained. On the basis of these estimates the uniqueness of the extension of the Cauchy data is proved, as well as the uniqueness of recovering the time-independent coefficients of hyperbolic equations.
Bibliography: 9 titles.

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English version:
Mathematics of the USSR-Sbornik, 1987, 58:1, 267–277

Bibliographic databases:

UDC: 517.95
MSC: 35L15, 35L20, 35B45, 35R30
Received: 23.07.1984

Citation: A. Khaidarov, “Carleman estimates and inverse problems for second-order hyperbolic equations”, Mat. Sb. (N.S.), 130(172):2(6) (1986), 265–274; Math. USSR-Sb., 58:1 (1987), 267–277

Citation in format AMSBIB
\Bibitem{Kha86} \by A.~Khaidarov \paper Carleman estimates and inverse problems for second-order hyperbolic equations \jour Mat. Sb. (N.S.) \yr 1986 \vol 130(172) \issue 2(6) \pages 265--274 \mathnet{http://mi.mathnet.ru/msb1868} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=854975} \zmath{https://zbmath.org/?q=an:0656.35146} \transl \jour Math. USSR-Sb. \yr 1987 \vol 58 \issue 1 \pages 267--277 \crossref{https://doi.org/10.1070/SM1987v058n01ABEH003103} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Khaidarov A., “On the Estimates of Stability in Multidimensional Inverse Problems for Differential-Equations”, 303, no. 4, 1988, 803–806
2. M. I. Belishev, “Wave bases in multidimensional inverse problems”, Math. USSR-Sb., 67:1 (1990), 23–42
3. Masahiro Yamamoto, “Uniqueness and stability in multidimensional hyperbolic inverse problems”, Journal de Mathématiques Pures et Appliquées, 78:1 (1999), 65
4. Khaidarov A., Zaripov Z., “Uniqueness and Stability Solution of Inverse Problems for Differential Equations of the Second Order”, Dokl. Akad. Nauk, 368:3 (1999), 316–317
5. Oleg Yu Imanuvilov, Masahiro Yamamoto, Inverse Probl, 17:4 (2001), 717
6. Oleg Yu. Imanuvilov, Masahiro Yamamoto, “GLOBAL UNIQUENESS AND STABILITY IN DETERMINING COEFFICIENTS OF WAVE EQUATIONS”, Communications in Partial Differential Equations, 26:7-8 (2001), 1409
7. Oleg Yu Imanuvilov, Masahiro Yamamoto, “Determination of a coefficient in an acoustic equation with a single measurement”, Inverse Probl, 19:1 (2003), 157
8. Ching-Lung Lin, Jenn-Nan Wang, “Uniqueness in inverse problems for an elasticity system with residual stress by a single measurement”, Inverse Probl, 19:4 (2003), 807
9. Haidarov A., Shodiev D., “Uniqueness of Extension of Solutions of Differential Equations of the Second Order”, Ill-Posed and Non-Classical Problems of Mathematical Physics and Analysis, Proceedings, Inverse and Ill-Posed Problems Series, ed. Lavrentev M., Vsp Bv-C/O Brill Acad Publ, 2003, 199–205
10. Oleg Yu. Imanuvilov, Masahiro Yamamoto, “Carleman estimates for the non-stationary Lamé system and the application to an inverse problem”, ESAIM: COCV, 11:1 (2004), 1
11. Oleg Yu. Imanuvilov, Masahiro Yamamoto †, “Carleman estimate for a stationary isotropic Lamé system and the applications”, Applicable Analysis, 83:3 (2004), 243
12. O. Yu. Imanuvilov, M. Yamamoto, “Stability estimate in a Cauchy problem for a hyperbolic equation with variable coefficients”, j inv ill-posed problems, 13:6 (2005), 583
13. Arif Amirov, Masahiro Yamamoto, “Unique continuation and an inverse problem for hyperbolic equations across a general hypersurface”, J Phys Conf Ser, 12 (2005), 1
14. V. G. Romanov, M. Yamamoto, “On the determination of the sound speed and a damping coefficient by two measurements”, Applicable Analysis, 84:10 (2005), 1025
15. Shumin Li, Masahiro Yamamoto, “An inverse source problem for Maxwell's equations in anisotropic media”, Applicable Analysis, 84:10 (2005), 1051
16. Imanuvilov O., Yamamoto M., “Carleman Estimates for the Three-Dimensional Nonstationary Lame System and Application to an Inverse Problem”, Control Theory of Partial Differential Equations, Pure and Applied Mathematics : a Program of Monographs, Textbooks, and Lecture Notes, 242, eds. Imanuvilov O., Leugering G., Triggiani R., Zhang B., Chapman & Hall/Crc Press, 2005, 337–374
17. D. Jellali, “An inverse problem for the acoustic wave equation with finite sets of boundary data”, j inv ill-posed problems, 14:7 (2006), 665
18. Mourad Bellassoued, Masahiro Yamamoto, “Logarithmic stability in determination of a coefficient in an acoustic equation by arbitrary boundary observation”, Journal de Mathématiques Pures et Appliquées, 85:2 (2006), 193
19. Shumin Li, Masahiro Yamamoto, “An Inverse Problem for Maxwell's Equations in Anisotropic Media*”, Chin Ann Math Ser B, 28:1 (2007), 35
20. Alfredo Lorenzi, Francesca Messina, Vladimir G. Romanov, “Recovering a Lamé kernel in a viscoelastic system”, Applicable Analysis, 86:11 (2007), 1375
21. Mourad Bellassoued, Masahiro Yamamoto, “Lipschitz stability in determining density and two Lamé coefficients”, Journal of Mathematical Analysis and Applications, 329:2 (2007), 1240
22. Yuan G., Yamamoto M., “Lipschitz Stability in Inverse Problems for a Kirchhoff Plate Equation”, Asymptotic Anal., 53:1-2 (2007), 29–60
23. M. Bellassoued, D. Jellali, M. Yamamoto, “Lipschitz stability in an inverse problem for a hyperbolic equation with a finite set of boundary data”, Applicable Analysis, 87:10-11 (2008), 1105
24. Arif Amirov, Masahiro Yamamoto, “A timelike Cauchy problem and an inverse problem for general hyperbolic equations”, Applied Mathematics Letters, 21:9 (2008), 885
25. N. Baranibalan, K. Sakthivel, K. Balachandran, “Uniqueness and stability in inverse parabolic equations with memory”, Nonlinear Analysis: Hybrid Systems, 2:4 (2008), 1077
26. Ganghua Yuan, Masahiro Yamamoto, “Lipschitz stability in the determination of the principal part of a parabolic equation”, ESAIM COCV, 15:3 (2009), 525
27. Oleg Yu. Imanuvilov, Victor Isakov, Masahiro Yamamoto, “New realization of the pseudoconvexity and its application to an inverse problem”, GAPA, 88:5 (2009), 637
28. Oleg Yu. Imanuvilov, Masahiro Yamamoto, “An inverse problem and an observability inequality for the Lame system with stress boundary condition”, GAPA, 88:5 (2009), 711
29. Ganghua Yuan, Masahiro Yamamoto, “Carleman estimates for the Schrödinger equation and applications to an inverse problem and an observability inequality”, Chin Ann Math Ser B, 2010
30. Victor Isakov, “Carleman estimates for some anisotropic elasticity systems and applications”, EECT, 1:1 (2012), 141
31. Mourad Choulli, Oleg Yu. Imanuvilov, Jean-Pierre Puel, Masahiro Yamamoto, “Inverse source problem for linearized Navier–Stokes equations with data in arbitrary sub-domain”, Applicable Analysis, 2012, 1
32. Shumin Li, Masahiro Yamamoto, “An inverse problem for Maxwell's equations in isotropic and non-stationary media”, Applicable Analysis, 2012, 1
33. Klibanov M.V., “Carleman Estimates for Global Uniqueness, Stability and Numerical Methods for Coefficient Inverse Problems”, J. Inverse Ill-Posed Probl., 21:4 (2013), 477–560
34. Shumin Li, “Carleman estimates for second-order hyperbolic systems in anisotropic cases and applications. Part I: Carleman estimates”, Applicable Analysis, 2014, 1
35. Li Deng, “An inverse problem for the Schrödinger equation with variable coefficients and lower order terms”, Journal of Mathematical Analysis and Applications, 2015
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