# American Institute of Mathematical Sciences

May  2011, 5(2): 431-464. doi: 10.3934/ipi.2011.5.431

## Recovering two Lamé kernels in a viscoelastic system

 1 Dipartimento di Matematica “F. Enriques”, Universitá di Milano, via C. Saldini 50, 20133 Milano, Italy 2 Sobolev Institute of Mathematics, Siberian branch of Russian Academy of Sciences, Acad. Koptyug prosp., 4, Novosibirsk, 630090, Russian Federation

Received  September 2009 Revised  February 2011 Published  May 2011

Let $\mathcal B$ be a viscoelastic body with a (smooth) bounded open reference set $\Omega$ in $\mathbb R^3$, with the equation of motion being described by the Lamé coefficients $\lambda_0$ and $\mu_0$ and the related viscoelastic coefficients $\lambda_1$ and $\mu_1$. The latter are assumed to be factorized with the same temporal part, i.e. $\lambda_1(t,x)=k(t)p(x)$ and $\mu_1(t,x)=k(t)q(x)$. Furthermore, it is assumed that the spatial parts $p$ and $q$ of $\lambda_1$ and $\mu_1$ are unknown and the three additional measurements $\sum_{j=1}^3\sigma_{i,j}^0(t,x)$n$_j(x) = g_i(t,x)$, $i=1,2,3$, are available on $(0,T)\times \partial \Omega$ for some (sufficiently large) subset $\Gamma\subset \partial \Omega$.
The fundamental task of this paper is to show the uniqueness of the pair $(p,q)$ as well as its continuous dependence on the boundary conditions, the initial data being kept fixed and the initial velocity being suitably related to the initial displacement.
Citation: Alfredo Lorenzi, Vladimir G. Romanov. Recovering two Lamé kernels in a viscoelastic system. Inverse Problems & Imaging, 2011, 5 (2) : 431-464. doi: 10.3934/ipi.2011.5.431
##### References:
 [1] R. A. Adams, "Sobolev Spaces,", Academic Press, (1975). [2] A. L. Bukhgeim and M. V. Klibanov, Global uniqueness of class of multidimensional inverse problems,, Soviet Math. Dokl., 24 (1981), 244. [3] C. Cavaterra, A. Lorenzi and M. Yamamoto, A stability result via Carleman estimates for an inverse source problem related to a hyperbolic integro-differential equation,, Comput. Appl. Math., 25 (2006), 229. [4] L. Hörmander, "Linear Partial Differential Operators,", Springer-Verlag, (1963). [5] O. Yu. Imanuvilov, On Carleman estimates for hyperbolic equations,, Asymptot. Anal., 32 (2002), 185. [6] O. Yu. Imanuvilov and M. Yamamoto, Global uniqueness and stability in determining coefficients of wave equations,, Comm. Partial Differential Equations, 26 (2001), 1409. doi: 10.1081/PDE-100106139. [7] O. Yu. Imanuvilov and M. Yamamoto, Global Lipschitz stability in an inverse hyperbolic problem by interior observations,, Inverse Problems, 17 (2001), 717. doi: 10.1088/0266-5611/17/4/310. [8] O. Yu. Imanuvilov and M. Yamamoto, Carleman estimates for the non-stationary Lamé system and the application to an inverse problem,, ESAIM Control Optim. Calc. Var., 11 (2005), 1. doi: 10.1051/cocv:2004030. [9] O. Yu. Imanuvilov and M. Yamamoto, Carleman estimates for the three-dimensional non-stationary Lamé system and application to an inverse problem,, Lect. Notes Pure Appl. Math., 242 (2005), 337. [10] V. Isakov, "Inverse Source Problems,", American Mathematical Society, (1990). [11] V. Isakov, Carleman type estimates in an anisotropic case and applications,, J. Differential Equations, 105 (1993), 217. doi: 10.1006/jdeq.1993.1088. [12] V. Isakov, Carleman estimates and applications to inverse problems,, Milan J. Math., 72 (2004), 249. doi: 10.1007/s00032-004-0033-6. [13] V. Isakov, "Inverse Problems for Partial Differential Equations,", Springer-Verlag, (2005). [14] V. Isakov and M. Yamamoto, "Carleman Estimate with the Neumann Boundary Condition and its Applications to the Observability Inquality and Inverse Hyperbolic Problems,", Differential geometric methods in the control of partial differential equations, 268 (2000). [15] M. V. Klibanov, Inverse problems and Carleman estimates,, Inverse Problems, 8 (1992), 575. doi: 10.1088/0266-5611/8/4/009. [16] M. V. Klibanov and M. Yamamoto, Lipschitz stability of an inverse problem for an acoustic equation,, Appl. Anal., 85 (2006), 515. doi: 10.1080/00036810500474788. [17] M. V. Klibanov and A. Timonov, "Carleman Estimates for Coefficient Inverse Problems and Numerical Applications,", VSP, (2004). [18] M. M. Lavrent'ev, V. G. Romanov and S. P. Shishat'skiĭ, "Ill-posed Problems of Mathematics Physics and Analysis,", vol. 64, 64 (1986). [19] A. Lorenzi, F. Messina and V. G. Romanov, Recovering a Lamé kernel in a viscoelastic system,, Applicable Analysis, 86 (2007), 1375. doi: 10.1080/00036810701675183. [20] J. Nečas, "Les Methodes Directes en Theorie des Equations Elliptiques,", Masson, (1967). [21] J. Nečas and I. Hlaváček, "Mathematical Theory Of Elastic And Elasto-Plastic Bodies: An Introduction,", Elsevier, (1981). [22] V. G. Romanov, Carleman estimates for second-order hyperbolic equation,, Siberian Math. J., 47 (2006), 135. doi: 10.1007/s11202-006-0014-9. [23] V. G. Romanov, Stability estimates in inverse problems for hyperbolic equations,, Milan J. Math., 74 (2006), 357. doi: 10.1007/s00032-006-0056-2. [24] V. G. Romanov and M. Yamamoto, Recovering a Lamé kernel in a viscoelastic equation by a single boundary measurement,, Appl. Anal., 89 (2010), 377. doi: 10.1080/00036810903518975. [25] M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems,, J. Math. Pures Appl., 78 (1999), 65. doi: 10.1016/S0021-7824(99)80010-5.

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##### References:
 [1] R. A. Adams, "Sobolev Spaces,", Academic Press, (1975). [2] A. L. Bukhgeim and M. V. Klibanov, Global uniqueness of class of multidimensional inverse problems,, Soviet Math. Dokl., 24 (1981), 244. [3] C. Cavaterra, A. Lorenzi and M. Yamamoto, A stability result via Carleman estimates for an inverse source problem related to a hyperbolic integro-differential equation,, Comput. Appl. Math., 25 (2006), 229. [4] L. Hörmander, "Linear Partial Differential Operators,", Springer-Verlag, (1963). [5] O. Yu. Imanuvilov, On Carleman estimates for hyperbolic equations,, Asymptot. Anal., 32 (2002), 185. [6] O. Yu. Imanuvilov and M. Yamamoto, Global uniqueness and stability in determining coefficients of wave equations,, Comm. Partial Differential Equations, 26 (2001), 1409. doi: 10.1081/PDE-100106139. [7] O. Yu. Imanuvilov and M. Yamamoto, Global Lipschitz stability in an inverse hyperbolic problem by interior observations,, Inverse Problems, 17 (2001), 717. doi: 10.1088/0266-5611/17/4/310. [8] O. Yu. Imanuvilov and M. Yamamoto, Carleman estimates for the non-stationary Lamé system and the application to an inverse problem,, ESAIM Control Optim. Calc. Var., 11 (2005), 1. doi: 10.1051/cocv:2004030. [9] O. Yu. Imanuvilov and M. Yamamoto, Carleman estimates for the three-dimensional non-stationary Lamé system and application to an inverse problem,, Lect. Notes Pure Appl. Math., 242 (2005), 337. [10] V. Isakov, "Inverse Source Problems,", American Mathematical Society, (1990). [11] V. Isakov, Carleman type estimates in an anisotropic case and applications,, J. Differential Equations, 105 (1993), 217. doi: 10.1006/jdeq.1993.1088. [12] V. Isakov, Carleman estimates and applications to inverse problems,, Milan J. Math., 72 (2004), 249. doi: 10.1007/s00032-004-0033-6. [13] V. Isakov, "Inverse Problems for Partial Differential Equations,", Springer-Verlag, (2005). [14] V. Isakov and M. Yamamoto, "Carleman Estimate with the Neumann Boundary Condition and its Applications to the Observability Inquality and Inverse Hyperbolic Problems,", Differential geometric methods in the control of partial differential equations, 268 (2000). [15] M. V. Klibanov, Inverse problems and Carleman estimates,, Inverse Problems, 8 (1992), 575. doi: 10.1088/0266-5611/8/4/009. [16] M. V. Klibanov and M. Yamamoto, Lipschitz stability of an inverse problem for an acoustic equation,, Appl. Anal., 85 (2006), 515. doi: 10.1080/00036810500474788. [17] M. V. Klibanov and A. Timonov, "Carleman Estimates for Coefficient Inverse Problems and Numerical Applications,", VSP, (2004). [18] M. M. Lavrent'ev, V. G. Romanov and S. P. Shishat'skiĭ, "Ill-posed Problems of Mathematics Physics and Analysis,", vol. 64, 64 (1986). [19] A. Lorenzi, F. Messina and V. G. Romanov, Recovering a Lamé kernel in a viscoelastic system,, Applicable Analysis, 86 (2007), 1375. doi: 10.1080/00036810701675183. [20] J. Nečas, "Les Methodes Directes en Theorie des Equations Elliptiques,", Masson, (1967). [21] J. Nečas and I. Hlaváček, "Mathematical Theory Of Elastic And Elasto-Plastic Bodies: An Introduction,", Elsevier, (1981). [22] V. G. Romanov, Carleman estimates for second-order hyperbolic equation,, Siberian Math. J., 47 (2006), 135. doi: 10.1007/s11202-006-0014-9. [23] V. G. Romanov, Stability estimates in inverse problems for hyperbolic equations,, Milan J. Math., 74 (2006), 357. doi: 10.1007/s00032-006-0056-2. [24] V. G. Romanov and M. Yamamoto, Recovering a Lamé kernel in a viscoelastic equation by a single boundary measurement,, Appl. Anal., 89 (2010), 377. doi: 10.1080/00036810903518975. [25] M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems,, J. Math. Pures Appl., 78 (1999), 65. doi: 10.1016/S0021-7824(99)80010-5.
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