Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T10:30:55.709Z Has data issue: false hasContentIssue false
  • English
  • Français

Carleman estimates for the non-stationary Lamé systemand theapplication to an inverse problem

Published online by Cambridge University Press:  15 December 2004

Oleg Yu. Imanuvilov  and
Masahiro Yamamoto
Oleg Yu. Imanuvilov
Affiliation:
Department of Mathematics, Iowa State University, 400 Carver Hall Ames IA 50011-2064 USA; vika@iastate.edu
Masahiro Yamamoto
Affiliation:
Department of Mathematical Sciences, The University of Tokyo, Komaba Meguro Tokyo 153-8914 Japan; myama@ms.u-tokyo.ac.jp
Get access

Abstract

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

Keywords

Carleman estimateLamé systeminverse problem.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 11 , Issue 1 , January 2005 , pp. 1 - 56
Copyright
© EDP Sciences, SMAI, 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bardos, C., Lebeau, G. and Rauch, J., Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim. 30 (1992) 1024-1065. CrossRef
M. Bellassoued, Distribution of resonances and decay of the local energy for the elastic wave equations. Comm. Math. Phys. 215 (2000) 375-408.
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.
Bellassoued, M., Unicité et contrôle pour le système de Lamé. ESAIM: COCV 6 (2001) 561-592. CrossRef
Baudouin, L. and Puel, J.-P., Uniqueness and stability in an inverse problem for the Schrödinger equation. Inverse Problems 18 (2002) 1537-1554. CrossRef
A.L. Bukhgeim, Introduction to the Theory of Inverse Problems. VSP, Utrecht (2000).
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.
Bukhgeim, A.L. and Klibanov, M.V., Global uniqueness of a class of multidimensional inverse problems. Soviet Math. Dokl. 24 (1981) 244-247.
Carleman, T., 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.
Dehman, B. and L.Robbiano, La propriété du prolongement unique pour un système elliptique. Le système de Lamé. J. Math. Pures Appl. 72 (1993) 475-492.
G. Duvaut and J.L. Lions, Inequalities in Mechanics and Physics. Springer-Verlag, Berlin (1976).
Yu.V. Egorov, Linear Differential Equations of Principal Type. Consultants Bureau New York (1986).
Eller, M., Isakov, V., Nakamura, G. and Tataru, D., 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.
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).
L. Hörmander, Linear Partial Differential Operators. Springer-Verlag, Berlin (1963).
Ikehata, M., Nakamura, G. and Yamamoto, M., Uniqueness in inverse problems for the isotropic Lamé system. J. Math. Sci. Univ. Tokyo 5 (1998) 627-692.
Imanuvilov, O., Controllability of parabolic equations. Mat. Sbornik 6 (1995) 109-132.
O. Imanuvilov, On Carleman estimates for hyperbolic equations. Asymptotic Analysis (2002) 32 185-220.
Imanuvilov, O., Isakov, V. and Yamamoto, M., An inverse problem for the dynamical Lamé system with two sets of boundary data. Commun. Pure Appl. Math. 56 (2003) 1366-1382. CrossRef
O. Imanuvilov, V. Isakov and M. Yamamoto, New realization on the pseudoconvexity and its application to an inverse problem (preprint).
Imanuvilov, O. and Yamamoto, M., Lipschitz stability in inverse parabolic problems by the Carleman estimate. Inverse Problems 14 (1998) 1229-1245. CrossRef
Imanuvilov, O. and Yamamoto, M., Global Lipschitz stability in an inverse hyperbolic problem by interior observations. Inverse Problems 17 (2001) 717-728. CrossRef
Imanuvilov, O. and Yamamoto, M., Global uniqueness and stability in determining coefficients of wave equations. Commun. Partial Differ. Equations 26 (2001) 1409-1425. CrossRef
Imanuvilov, O. and Yamamoto, M., Determination of a coefficient in an acoustic equation with a single measurement. Inverse Problems 19 (2003) 151-171. CrossRef
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.
Imanuvilov, O. and Yamamoto, M., 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. CrossRef
Imanuvilov, O. and Yamamoto, M., Carleman estimate for a stationary isotropic Lamé system and the applications. Appl. Anal. 83 (2004) 243-270. CrossRef
Isakov, V., A nonhyperbolic Cauchy problem for $\square_b\square_c$ and its applications to elasticity theory. Comm. Pure Appl. Math. 39 (1986) 747-767. CrossRef
V. Isakov, Inverse Source Problems. American Mathematical Society, Providence, Rhode Island (1990).
V. Isakov, Inverse Problems for Partial Differential Equations. Springer-Verlag, Berlin (1998).
Isakov, V. and Yamamoto, M., Carleman estimate with the Neumann boundary condition and its applications to the observability inequality and inverse hyperbolic problems. Contem. Math. 268 (2000) 191-225. CrossRef
Kazemi, M.A. and Klibanov, M.V., Stability estimates for ill-posed Cauchy problems involving hyperbolic equations and inequalities. Appl. Anal. 50 (1993) 93-102. CrossRef
Khaĭdarov, A., Carleman estimates and inverse problems for second order hyperbolic equations. Math. USSR Sbornik 58 (1987) 267-277. CrossRef
Khaĭdarov, A., On stability estimates in multidimensional inverse problems for differential equations. Soviet Math. Dokl. 38 (1989) 614-617.
Klibanov, M.V., Inverse problems and Carleman estimates. Inverse Problems 8 (1992) 575-596. CrossRef
H. Kumano-go, Pseudo-differential Operators. MIT Press, Cambrige (1981).
I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Cambridge University Press, Cambridge (2000).
J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, Berlin (1971).
J.L. Lions, Contrôlabilité exacte perturbations et stabilisation de systèmes distribués. Masson, Paris (1988).
Puel, J.-P. and Yamamoto, M., On a global estimate in a linear inverse hyperbolic problem. Inverse Problems 12 (1996) 995-1002. CrossRef
Puel, J.-P. and Yamamoto, M., Generic well-posedness in a multidimensional hyperbolic inverse problem. J. Inverse Ill-posed Problems 5 (1997) 55-83. CrossRef
Rachele, L., An inverse problem in elastodynamics: uniqueness of the wave speeds in the interior. J. Differ. Equations 162 (2000) 300-325. CrossRef
Ruiz, A., Unique continuation for weak solutions of the wave equation plus a potential. J. Math. Pures. Appl. 71 (1992) 455-467.
Tataru, D., Carleman estimates and unique continuation for solutions to boundary value problems. J. Math. Pures. Appl. 75 (1996) 367-408.
Tataru, D., A priori estimates of Carleman's type in domains with boundary. J. Math. Pures. Appl. 73 (1994) 355-387.
M. Taylor, Pseudodifferential Operators. Princeton University Press, Princeton, New Jersey (1981).
M. Taylor, Pseudodifferential Operators and Nonlinear PDE. Birkhäuser, Boston (1991).
V.G. Yakhno, Inverse Problems for Differential Equations of Elasticity. Nauka, Novosibirsk (1990).
Yamamoto, K., Singularities of solutions to the boundary value problems for elastic and Maxwell's equations. Japan J. Math. 14 (1988) 119-163. CrossRef
Yamamoto, M., Uniqueness and stability in multidimensional hyperbolic inverse problems. J. Math. Pures Appl. 78 (1999) 65-98. CrossRef
Zhang, X., Explicit observability inequalities for the wave equation with lower order terms by means of Carleman inequalities. SIAM J. Control Optim. 39 (2001) 812-834. CrossRef
C. Zuily, Uniqueness and Non-uniqueness in the Cauchy Problem. Birkhäuser, Boston, Basel, Berlin, (1983).