Carleman estimates for the non-stationary Lamé system and the application to an inverse problem

Oleg Yu. Imanuvilov; Masahiro Yamamoto

ESAIM: Control, Optimisation and Calculus of Variations (2005)

  • Volume: 11, Issue: 1, page 1-56
  • ISSN: 1292-8119

Abstract

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In this paper, we establish Carleman estimates for the two dimensional isotropic non-stationary Lamé system with the zero Dirichlet boundary conditions. Using this estimate, we prove the uniqueness and the stability in determining spatially varying density and two Lamé coefficients by a single measurement of solution over ( 0 , T ) × ω , where T > 0 is a sufficiently large time interval and a subdomain ω satisfies a non-trapping condition.

How to cite

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Imanuvilov, Oleg Yu., and Yamamoto, Masahiro. "Carleman estimates for the non-stationary Lamé system and the application to an inverse problem." ESAIM: Control, Optimisation and Calculus of Variations 11.1 (2005): 1-56. <http://eudml.org/doc/245104>.

@article{Imanuvilov2005,
abstract = {In this paper, we establish Carleman estimates for the two dimensional isotropic non-stationary Lamé system with the zero Dirichlet boundary conditions. Using this estimate, we prove the uniqueness and the stability in determining spatially varying density and two Lamé coefficients by a single measurement of solution over $(0,T) \times \omega $, where $T &gt; 0$ is a sufficiently large time interval and a subdomain $\omega $ satisfies a non-trapping condition.},
author = {Imanuvilov, Oleg Yu., Yamamoto, Masahiro},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Carleman estimate; Lamé system; inverse problem; uniqueness; stability; Lamé coefficients},
language = {eng},
number = {1},
pages = {1-56},
publisher = {EDP-Sciences},
title = {Carleman estimates for the non-stationary Lamé system and the application to an inverse problem},
url = {http://eudml.org/doc/245104},
volume = {11},
year = {2005},
}

TY - JOUR
AU - Imanuvilov, Oleg Yu.
AU - Yamamoto, Masahiro
TI - Carleman estimates for the non-stationary Lamé system and the application to an inverse problem
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2005
PB - EDP-Sciences
VL - 11
IS - 1
SP - 1
EP - 56
AB - In this paper, we establish Carleman estimates for the two dimensional isotropic non-stationary Lamé system with the zero Dirichlet boundary conditions. Using this estimate, we prove the uniqueness and the stability in determining spatially varying density and two Lamé coefficients by a single measurement of solution over $(0,T) \times \omega $, where $T &gt; 0$ is a sufficiently large time interval and a subdomain $\omega $ satisfies a non-trapping condition.
LA - eng
KW - Carleman estimate; Lamé system; inverse problem; uniqueness; stability; Lamé coefficients
UR - http://eudml.org/doc/245104
ER -

References

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  1. [1] 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. Zbl0786.93009MR1178650
  2. [2] M. Bellassoued, Distribution of resonances and decay of the local energy for the elastic wave equations. Comm. Math. Phys. 215 (2000) 375-408. Zbl0978.35077MR1799852
  3. [3] M. Bellassoued, Carleman estimates and decay rate of the local energy for the Neumann problem of elasticity. Progr. Nonlinear Differ. Equations Appl. 46 (2001) 15-36. Zbl0983.35031MR1839164
  4. [4] M. Bellassoued, Unicité et contrôle pour le système de Lamé. ESAIM: COCV 6 (2001) 561-592. Zbl1007.35006MR1872389
  5. [5] L. Baudouin and J.-P. Puel, Uniqueness and stability in an inverse problem for the Schrödinger equation. Inverse Problems 18 (2002) 1537-1554. Zbl1023.35091MR1955903
  6. [6] A.L. Bukhgeim, Introduction to the Theory of Inverse Problems. VSP, Utrecht (2000). 
  7. [7] A.L. Bukhgeim, J. Cheng, V. Isakov and M. Yamamoto, Uniqueness in determining damping coefficients in hyperbolic equations, in Analytic Extension Formulas and their Applications, Kluwer, Dordrecht (2001) 27-46. Zbl0994.35119MR1830375
  8. [8] A.L. Bukhgeim and M.V. Klibanov, Global uniqueness of a class of multidimensional inverse problems. Soviet Math. Dokl. 24 (1981) 244-247. Zbl0497.35082
  9. [9] T. Carleman, Sur un problème d’unicité pour les systèmes d’équations aux derivées partielles à deux variables independantes. Ark. Mat. Astr. Fys. 2B (1939) 1-9. Zbl0022.34201
  10. [10] B. Dehman and L. Robbiano, La propriété du prolongement unique pour un système elliptique. Le système de Lamé. J. Math. Pures Appl. 72 (1993) 475-492. Zbl0832.73012MR1239100
  11. [11] G. Duvaut and J.L. Lions, Inequalities in Mechanics and Physics. Springer-Verlag, Berlin (1976). Zbl0331.35002MR521262
  12. [12] Yu.V. Egorov, Linear Differential Equations of Principal Type. Consultants Bureau New York (1986). Zbl0669.35001MR872855
  13. [13] M. Eller, V. Isakov, G. Nakamura and D. Tataru, Uniqueness and stability in the Cauchy problem for Maxwell’s and the elasticity system, in Nonlinear Partial Differential Equations, Vol. 14, Collège de France Seminar, Elsevier-Gauthier Villars. Ser. Appl. Math. 31 (2002) 329-350. Zbl1038.35159
  14. [14] M.E. Gurtin, The Linear Theory of Elasticity, in Encyclopedia of Physics, Vol. VIa/2, Mechanics of Solids II, C. Truesdell Ed., Springer-Verlag, Berlin (1972). 
  15. [15] L. Hörmander, Linear Partial Differential Operators. Springer-Verlag, Berlin (1963). Zbl0108.09301MR404822
  16. [16] M. Ikehata, G. Nakamura and M. Yamamoto, Uniqueness in inverse problems for the isotropic Lamé system. J. Math. Sci. Univ. Tokyo 5 (1998) 627-692. Zbl0924.73069MR1675236
  17. [17] O. Imanuvilov, Controllability of parabolic equations. Mat. Sbornik 6 (1995) 109-132. Zbl0845.35040MR1349016
  18. [18] O. Imanuvilov, On Carleman estimates for hyperbolic equations. Asymptotic Analysis (2002) 32 185-220. Zbl1050.35046MR1993649
  19. [19] O. Imanuvilov, V. Isakov and M. Yamamoto, An inverse problem for the dynamical Lamé system with two sets of boundary data. Commun. Pure Appl. Math. 56 (2003) 1366-1382. Zbl1044.35105MR1980857
  20. [20] O. Imanuvilov, V. Isakov and M. Yamamoto, New realization on the pseudoconvexity and its application to an inverse problem (preprint). Zbl1181.35141
  21. [21] O. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problems by the Carleman estimate. Inverse Problems 14 (1998) 1229-1245. Zbl0992.35110MR1654631
  22. [22] O. Imanuvilov and M. Yamamoto, Global Lipschitz stability in an inverse hyperbolic problem by interior observations. Inverse Problems 17 (2001) 717-728. Zbl0983.35151MR1861478
  23. [23] O. Imanuvilov and M. Yamamoto, Global uniqueness and stability in determining coefficients of wave equations. Commun. Partial Differ. Equations 26 (2001) 1409-1425. Zbl0985.35108MR1855284
  24. [24] O. Imanuvilov and M. Yamamoto, Determination of a coefficient in an acoustic equation with a single measurement. Inverse Problems 19 (2003) 151-171. Zbl1020.35117MR1964256
  25. [25] O. Imanuvilov and M. Yamamoto, Remarks on Carleman estimates and controllability for the Lamé system. Journées Équations aux Dérivées Partielles, Forges-les-Eaux, 3-7 juin 2002, GDR 2434 (CNRS) 1-19. MR1968201
  26. [26] O. Imanuvilov and M. Yamamoto, Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations. Publ. Res. Inst. Math. Sci. 39 (2003) 227-274. Zbl1065.35079MR1987865
  27. [27] O. Imanuvilov and M. Yamamoto, Carleman estimate for a stationary isotropic Lamé system and the applications. Appl. Anal. 83 (2004) 243-270. Zbl1055.35030MR2033238
  28. [28] V. Isakov, A nonhyperbolic Cauchy problem for b c and its applications to elasticity theory. Comm. Pure Appl. Math. 39 (1986) 747-767. Zbl0649.35015MR859272
  29. [29] V. Isakov, Inverse Source Problems. American Mathematical Society, Providence, Rhode Island (1990). Zbl0721.31002MR1071181
  30. [30] V. Isakov, Inverse Problems for Partial Differential Equations. Springer-Verlag, Berlin (1998). Zbl0908.35134MR1482521
  31. [31] V. Isakov and M. Yamamoto, Carleman estimate with the Neumann boundary condition and its applications to the observability inequality and inverse hyperbolic problems. Contem. Math. 268 (2000) 191-225. Zbl1004.35028MR1804796
  32. [32] M.A. Kazemi and M.V. Klibanov, Stability estimates for ill-posed Cauchy problems involving hyperbolic equations and inequalities. Appl. Anal. 50 (1993) 93-102. Zbl0795.35134MR1281205
  33. [33] A. Khaĭdarov, Carleman estimates and inverse problems for second order hyperbolic equations. Math. USSR Sbornik 58 (1987) 267-277. Zbl0656.35146MR854975
  34. [34] A. Khaĭdarov, On stability estimates in multidimensional inverse problems for differential equations. Soviet Math. Dokl. 38 (1989) 614-617. Zbl0679.35085MR984643
  35. [35] M.V. Klibanov, Inverse problems and Carleman estimates. Inverse Problems 8 (1992) 575-596. Zbl0755.35151MR1178231
  36. [36] H. Kumano-go, Pseudo-differential Operators. MIT Press, Cambrige (1981). Zbl0489.35003
  37. [37] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Cambridge University Press, Cambridge (2000). Zbl0961.93003
  38. [38] J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, Berlin (1971). Zbl0203.09001MR271512
  39. [39] J.L. Lions, Contrôlabilité exacte perturbations et stabilisation de systèmes distribués. Masson, Paris (1988). Zbl0653.93003
  40. [40] J.-P. Puel and M. Yamamoto, On a global estimate in a linear inverse hyperbolic problem. Inverse Problems 12 (1996) 995-1002. Zbl0862.35141MR1421661
  41. [41] J.-P. Puel and M. Yamamoto, Generic well-posedness in a multidimensional hyperbolic inverse problem. J. Inverse Ill-posed Problems 5 (1997) 55-83. Zbl0867.35115MR1434926
  42. [42] L. Rachele, An inverse problem in elastodynamics: uniqueness of the wave speeds in the interior. J. Differ. Equations 162 (2000) 300-325. Zbl0968.74040MR1751708
  43. [43] A. Ruiz, Unique continuation for weak solutions of the wave equation plus a potential. J. Math. Pures. Appl. 71 (1992) 455-467. Zbl0832.35084MR1191585
  44. [44] D. Tataru, Carleman estimates and unique continuation for solutions to boundary value problems. J. Math. Pures. Appl. 75 (1996) 367-408. Zbl0896.35023MR1411157
  45. [45] D. Tataru, A priori estimates of Carleman’s type in domains with boundary. J. Math. Pures. Appl. 73 (1994) 355-387. Zbl0835.35031
  46. [46] M. Taylor, Pseudodifferential Operators. Princeton University Press, Princeton, New Jersey (1981). Zbl0453.47026MR618463
  47. [47] M. Taylor, Pseudodifferential Operators and Nonlinear PDE. Birkhäuser, Boston (1991). Zbl0746.35062MR1121019
  48. [48] V.G. Yakhno, Inverse Problems for Differential Equations of Elasticity. Nauka, Novosibirsk (1990). Zbl0787.35124MR1071385
  49. [49] K. Yamamoto, Singularities of solutions to the boundary value problems for elastic and Maxwell’s equations. Japan J. Math. 14 (1988) 119-163. Zbl0669.73017
  50. [50] M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems. J. Math. Pures Appl. 78 (1999) 65-98. Zbl0923.35200MR1671221
  51. [51] X. Zhang, Explicit observability inequalities for the wave equation with lower order terms by means of Carleman inequalities. SIAM J. Control Optim. 39 (2001) 812-834. Zbl0982.35059MR1786331
  52. [52] C. Zuily, Uniqueness and Non-uniqueness in the Cauchy Problem. Birkhäuser, Boston, Basel, Berlin, (1983). Zbl0521.35003

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