# American Institute of Mathematical Sciences

June  2012, 1(1): 141-154. doi: 10.3934/eect.2012.1.141

## Carleman estimates for some anisotropic elasticity systems and applications

 1 Wichita State University, 1845 Fairmount, Wichita, KS, 67260-0033

Received  October 2011 Revised  February 2012 Published  March 2012

We show that under some conditions one can obtain Carleman type estimates for the transversely isotropic elasticity system with residual stress. We consider both time dependent and static cases. The main idea is to reduce this system to a principally upper triangular one and the main technical tool is Carleman estimates with two large parameters for general second order partial differential operators.
Citation: Victor Isakov. Carleman estimates for some anisotropic elasticity systems and applications. Evolution Equations & Control Theory, 2012, 1 (1) : 141-154. doi: 10.3934/eect.2012.1.141
##### References:
 [1] P. Albano and D. Tataru, Carleman estimates and boundary observability for a coupled parabolic-hyperbolic system,, Electr. J. Diff. Equat., 2000 ().   Google Scholar [2] H. Ding, W. Chen and L. Zhang, "Elasticity of Transversely Isotropic Materials,", Solid Mechanics and its Applications, 126 (2006).   Google Scholar [3] M. Eller and V. Isakov, Carleman estimates with two large parameters and applications,, in, 268 (2000), 117.   Google Scholar [4] M. Eller, V. Isakov, G. Nakamura and D. Tataru, Uniqueness and stability in the Cauchy problem for Maxwell and elasticity system, in, Nonlinear Partial Differential Equations. Collège de France Seminar, Vol. XIV, 31 (2002), 329.   Google Scholar [5] L. Hörmander, "Linear Partial Differential Operators,", Springer-Verlag, (1976).   Google Scholar [6] O. Imanuvilov, V. Isakov and M. Yamamoto, An inverse problem for the dynamical Lamé system with two sets of boundary data,, Comm. Pure Appl. Math., 56 (2003), 1366.  doi: 10.1002/cpa.10097.  Google Scholar [7] V. Isakov, A nonhyperbolic Cauchy problem for $\square_b\square_c$ and its applications to elasticity theory,, Comm. Pure and Applied Math., 39 (1986), 747.  doi: 10.1002/cpa.3160390603.  Google Scholar [8] V. Isakov, Carleman type estimates in an anisotropic case and applications,, J. Differential Equations, 105 (1993), 217.   Google Scholar [9] V. Isakov, On the uniqueness of the continuation for a thermoelasticity system,, SIAM J. Math. Anal., 33 (2001), 509.  doi: 10.1137/S0036141000366509.  Google Scholar [10] V. Isakov, "Inverse Problems for Partial Differential Equation," Second edition,, Applied Mathematical Sciences, 127 (2006).   Google Scholar [11] V. Isakov and N. Kim, Carleman estimates with second large parameter for second order operators,, in, 10 (2009), 135.   Google Scholar [12] V. Isakov and N. Kim, Weak Carleman estimates with two large parameters for second order operators and applications to elasticity with residual stress,, Discr. Cont. Dyn. Syst., 27 (2010), 799.  doi: 10.3934/dcds.2010.27.799.  Google Scholar [13] V. Isakov, G. Nakamura and J.-N. Wang, Uniqueness and stability in the Cauchy problem for the elasticity system with residual stress,, in, 333 (2003), 99.   Google Scholar [14] V. Isakov, J.-N. Wang and M. Yamamoto, Uniqueness and stability of determining the residual stress by one measurement,, Comm. Part. Diff. Equat., 23 (2007), 833.   Google Scholar [15] A. Khaidarov, Carleman estimates and inverse problems for second order hyperbolic equations,, Math. USSR Sbornik, 58 (1987), 267.  doi: 10.1070/SM1987v058n01ABEH003103.  Google Scholar [16] I. Lasiecka, R. Triggiani and P.-F. Yao, Inverse/observability estimates for second order hyperbolic equations with variable coefficients,, J. Math. Anal. Appl., 235 (1999), 13.  doi: 10.1006/jmaa.1999.6348.  Google Scholar [17] C.-L. Lin, G. Nakamura and M. Sini, Unique continuation for the elastic transversely isotropic dynamical systems and its application,, J. Diff. Equat., 245 (2008), 3008.  doi: 10.1016/j.jde.2008.07.021.  Google Scholar [18] C.-S. Man, Hartig's law and linear elasticity with initial stress,, Inverse Problems, 14 (1998), 313.  doi: 10.1088/0266-5611/14/2/007.  Google Scholar [19] A. Mazzucato and L. Rachele, On transversely isotropic elastic media with ellipsoidal slowness surfaces,, Mathematics and Mechanics of Solids, 13 (2008), 611.  doi: 10.1177/1081286507078498.  Google Scholar [20] R. Payton, "Elastic Wave Propagation in Transversely Isotropic Media,", Kluwer, (1983).  doi: 10.1007/978-94-009-6866-0.  Google Scholar [21] V. Romanov, Carleman estimates for second-order hyperbolic equations,, Sib. Math. J., 47 (2006), 135.  doi: 10.1007/s11202-006-0014-9.  Google Scholar

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##### References:
 [1] P. Albano and D. Tataru, Carleman estimates and boundary observability for a coupled parabolic-hyperbolic system,, Electr. J. Diff. Equat., 2000 ().   Google Scholar [2] H. Ding, W. Chen and L. Zhang, "Elasticity of Transversely Isotropic Materials,", Solid Mechanics and its Applications, 126 (2006).   Google Scholar [3] M. Eller and V. Isakov, Carleman estimates with two large parameters and applications,, in, 268 (2000), 117.   Google Scholar [4] M. Eller, V. Isakov, G. Nakamura and D. Tataru, Uniqueness and stability in the Cauchy problem for Maxwell and elasticity system, in, Nonlinear Partial Differential Equations. Collège de France Seminar, Vol. XIV, 31 (2002), 329.   Google Scholar [5] L. Hörmander, "Linear Partial Differential Operators,", Springer-Verlag, (1976).   Google Scholar [6] O. Imanuvilov, V. Isakov and M. Yamamoto, An inverse problem for the dynamical Lamé system with two sets of boundary data,, Comm. Pure Appl. Math., 56 (2003), 1366.  doi: 10.1002/cpa.10097.  Google Scholar [7] V. Isakov, A nonhyperbolic Cauchy problem for $\square_b\square_c$ and its applications to elasticity theory,, Comm. Pure and Applied Math., 39 (1986), 747.  doi: 10.1002/cpa.3160390603.  Google Scholar [8] V. Isakov, Carleman type estimates in an anisotropic case and applications,, J. Differential Equations, 105 (1993), 217.   Google Scholar [9] V. Isakov, On the uniqueness of the continuation for a thermoelasticity system,, SIAM J. Math. Anal., 33 (2001), 509.  doi: 10.1137/S0036141000366509.  Google Scholar [10] V. Isakov, "Inverse Problems for Partial Differential Equation," Second edition,, Applied Mathematical Sciences, 127 (2006).   Google Scholar [11] V. Isakov and N. Kim, Carleman estimates with second large parameter for second order operators,, in, 10 (2009), 135.   Google Scholar [12] V. Isakov and N. Kim, Weak Carleman estimates with two large parameters for second order operators and applications to elasticity with residual stress,, Discr. Cont. Dyn. Syst., 27 (2010), 799.  doi: 10.3934/dcds.2010.27.799.  Google Scholar [13] V. Isakov, G. Nakamura and J.-N. Wang, Uniqueness and stability in the Cauchy problem for the elasticity system with residual stress,, in, 333 (2003), 99.   Google Scholar [14] V. Isakov, J.-N. Wang and M. Yamamoto, Uniqueness and stability of determining the residual stress by one measurement,, Comm. Part. Diff. Equat., 23 (2007), 833.   Google Scholar [15] A. Khaidarov, Carleman estimates and inverse problems for second order hyperbolic equations,, Math. USSR Sbornik, 58 (1987), 267.  doi: 10.1070/SM1987v058n01ABEH003103.  Google Scholar [16] I. Lasiecka, R. Triggiani and P.-F. Yao, Inverse/observability estimates for second order hyperbolic equations with variable coefficients,, J. Math. Anal. Appl., 235 (1999), 13.  doi: 10.1006/jmaa.1999.6348.  Google Scholar [17] C.-L. Lin, G. Nakamura and M. Sini, Unique continuation for the elastic transversely isotropic dynamical systems and its application,, J. Diff. Equat., 245 (2008), 3008.  doi: 10.1016/j.jde.2008.07.021.  Google Scholar [18] C.-S. Man, Hartig's law and linear elasticity with initial stress,, Inverse Problems, 14 (1998), 313.  doi: 10.1088/0266-5611/14/2/007.  Google Scholar [19] A. Mazzucato and L. Rachele, On transversely isotropic elastic media with ellipsoidal slowness surfaces,, Mathematics and Mechanics of Solids, 13 (2008), 611.  doi: 10.1177/1081286507078498.  Google Scholar [20] R. Payton, "Elastic Wave Propagation in Transversely Isotropic Media,", Kluwer, (1983).  doi: 10.1007/978-94-009-6866-0.  Google Scholar [21] V. Romanov, Carleman estimates for second-order hyperbolic equations,, Sib. Math. J., 47 (2006), 135.  doi: 10.1007/s11202-006-0014-9.  Google Scholar
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