# American Institute of Mathematical Sciences

August  2014, 8(3): 611-644. doi: 10.3934/ipi.2014.8.611

## Uniqueness and Lipschitz stability for the identification of Lamé parameters from boundary measurements

 1 Dipartimento di Matematica Francesco Brioschi, Piazza Leonardo da Vinci, 32, 20133 Milano, Italy 2 Dipartimento di Matematica e Informatica, Università di Firenze, viale Morgagni, 67A, 50134 Firenze, Italy, Italy

Received  December 2013 Revised  May 2014 Published  August 2014

In this paper we consider the problem of determining an unknown pair $\lambda$, $\mu$ of piecewise constant Lamé parameters inside a three dimensional body from the Dirichlet to Neumann map. We prove uniqueness and Lipschitz continuous dependence of $\lambda$ and $\mu$ from the Dirichlet to Neumann map.
Citation: Elena Beretta, Elisa Francini, Sergio Vessella. Uniqueness and Lipschitz stability for the identification of Lamé parameters from boundary measurements. Inverse Problems & Imaging, 2014, 8 (3) : 611-644. doi: 10.3934/ipi.2014.8.611
##### References:
 [1] M. Akamatsu, G. Nakamura and S. Steinberg, Identification of Lamé coefficients from boundary observations,, Inverse Problems, 7 (1991), 335. doi: 10.1088/0266-5611/7/3/003. Google Scholar [2] G. Alessandrini, Stable determination of conductivity by boundary measurements,, Applicable Analysis, 27 (1988), 153. doi: 10.1080/00036818808839730. Google Scholar [3] G. Alessandrini, E. Beretta and S. Vessella, Determining cracks by boundary measurements-Lipschitz Stability,, SIAM J. Math. Anal., 27 (1996), 361. doi: 10.1137/S0036141094265791. Google Scholar [4] G. Alessandrini, M. Di Cristo, A. Morassi and E. Rosset, Stable determination of an inclusion in an elastic body by boundary measurements,, SIAM J. Math. Anal., 46 (2014), 2692. doi: 10.1137/130946307. Google Scholar [5] G. Alessandrini and A. Morassi, Strong unique continuation for the Lamé system of elasticity,, Comm. PDE, 26 (2001), 1787. doi: 10.1081/PDE-100107459. Google Scholar [6] G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations,, Inverse Problems, 25 (2009), 1. doi: 10.1088/0266-5611/25/12/123004. Google Scholar [7] G. Alessandrini and S. Vessella, Lipschitz stability for the inverse conductivity problem,, Adv. in Appl. Math., 35 (2005), 207. doi: 10.1016/j.aam.2004.12.002. Google Scholar [8] H. Ammari, E. Beretta and E. Francini, Reconstruction of thin conducting inhomogeneities from electrostatic measurements, II. The case of multiple segments,, Applicable Analysis, 85 (2006), 87. doi: 10.1080/00036810500277736. Google Scholar [9] K. Astala and L. Paivarinta, Calderón's inverse conductivity problem in the plane,, Ann. of Math., 163 (2006), 265. doi: 10.4007/annals.2006.163.265. Google Scholar [10] V. Bacchelli and S. Vessella, Lipschitz stability for a stationary 2D inverse problem with unknown polygonal boundary,, Inverse Problems, 22 (2006), 1627. doi: 10.1088/0266-5611/22/5/007. Google Scholar [11] E. Beretta, E. Bonnetier, E. Francini and A. Mazzucato, An asymptotic formula for the displacement field in the presence of small anisotropic elastic inclusions,, Inverse Problems and Imaging, 6 (2012), 1. doi: 10.3934/ipi.2012.6.1. Google Scholar [12] E. Beretta, M. V de Hoop and L. Qiu, Lipschitz stability of an inverse boundary value problem for a Schrödinger type equation,, SIAM J. Math. Anal., 45 (2013), 679. doi: 10.1137/120869201. Google Scholar [13] E. Beretta and E. Francini, Lipschitz stability for the impedance tomography problem. The complex case,, Comm. PDE, 36 (2011), 1723. doi: 10.1080/03605302.2011.552930. Google Scholar [14] E. Beretta, E. Francini and S. Vessella, Determination of a linear crack in an elastic body from boundary measurements. Lipschitz stability,, SIAM J. Math. Anal., 40 (2008), 984. doi: 10.1137/070698397. Google Scholar [15] M. Bonnet and A. Constantinescu, Inverse problems in elasticity,, Inverse problems, 21 (2005). doi: 10.1088/0266-5611/21/2/R01. Google Scholar [16] B. M. Brown, M. Jais and I. W. Knowles, A variational approach to an elastic inverse problem,, Inverse Problems, 21 (2005), 1953. doi: 10.1088/0266-5611/21/6/010. Google Scholar [17] M. Chipot, D. Kinderlehrer and G. Vergara-Caffarelli, Smoothness of linear laminates,, Arch. Rational Mech. Anal., 96 (1986), 81. doi: 10.1007/BF00251414. Google Scholar [18] M. V. de Hoop, L. Qiu and O. Scherzer, Local analysis of inverse problems: Höelder stability and iterative reconstruction,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/4/045001. Google Scholar [19] M. V. de Hoop, L. Qiu and O. Scherzer, A convergence analysis of a multi-level projected steepest descent iteration for nonlinear inverse problems in Banach spaces subject to stability constraints,, to appear in Numerische Mathematik, (). Google Scholar [20] G. Eskin and J. Ralston, On the inverse boundary value problem for linear isotropic elasticity,, Inverse Problems, 18 (2002), 907. doi: 10.1088/0266-5611/18/3/324. Google Scholar [21] L. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (2010). Google Scholar [22] S. Hofmann and S. Kim, The Green function estimates for strongly elliptic systems of second order,, Manuscripta Math., 124 (2007), 139. doi: 10.1007/s00229-007-0107-1. Google Scholar [23] M. Ikehata, Inversion formulas for the linearized problem for an inverse boundary value problem in elastic prospection,, SIAM J. Appl. Math., 50 (1990), 1635. doi: 10.1137/0150097. Google Scholar [24] V. Isakov, Inverse Problems for Partial Differential Equations,, $2^{nd}$ edition, (2006). Google Scholar [25] Y. Y. Li and L. Nirenberg, Estimates for elliptic systems from composite materials,, Comm. Pure Appl. Math., 56 (2003), 892. doi: 10.1002/cpa.10079. Google Scholar [26] N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation,, Inverse Problems, 17 (2001), 1435. doi: 10.1088/0266-5611/17/5/313. Google Scholar [27] G. Milton, The Theory of Composites,, Cambridge University Press, (2002). doi: 10.1017/CBO9780511613357. Google Scholar [28] C. Mengcheng and T. Renji, An explicit tensor expression for the fundamental solutions of a bimaterial space problem,, Applied Mathematics and Mechanics, 18 (1997), 331. doi: 10.1007/BF02457547. Google Scholar [29] A. Morassi and E. Rosset, Stable determination of cavities in elastic bodies,, Inverse Problems, 20 (2004), 453. doi: 10.1088/0266-5611/20/2/010. Google Scholar [30] G. Nakamura, Inverse problems for elasticity,, in Selected Papers on Analysis and Differential Equations, (2003), 71. Google Scholar [31] G. Nakamura and G. Uhlmann, Identification of Lamé parameters by boundary measurements,, American Journal of Mathematics, 115 (1993), 1161. doi: 10.2307/2375069. Google Scholar [32] G. Nakamura and G. Uhlmann, Erratum: Global uniqueness for an inverse boundary value problem arising in elasticity,, Invent. Math., 118 (1994), 457. doi: 10.1007/BF01231541. Google Scholar [33] G. Nakamura and G. Uhlmann, Inverse boundary problems at the boundary for an elastic system,, SIAM J. Math. Anal., 26 (1995), 263. doi: 10.1137/S0036141093247494. Google Scholar [34] L. Rongved, Force interior to one of two joined semi-infinite solids,, in Proc. 2nd Midwestern Conf. Solid Mech, (1955), 1. Google Scholar [35] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem,, Ann. of Math., 125 (1987), 153. doi: 10.2307/1971291. Google Scholar [36] S. Vessella, Locations and strengths of point sources: Stability estimates,, Inverse Problems, 8 (1992), 911. doi: 10.1088/0266-5611/8/6/008. Google Scholar

show all references

##### References:
 [1] M. Akamatsu, G. Nakamura and S. Steinberg, Identification of Lamé coefficients from boundary observations,, Inverse Problems, 7 (1991), 335. doi: 10.1088/0266-5611/7/3/003. Google Scholar [2] G. Alessandrini, Stable determination of conductivity by boundary measurements,, Applicable Analysis, 27 (1988), 153. doi: 10.1080/00036818808839730. Google Scholar [3] G. Alessandrini, E. Beretta and S. Vessella, Determining cracks by boundary measurements-Lipschitz Stability,, SIAM J. Math. Anal., 27 (1996), 361. doi: 10.1137/S0036141094265791. Google Scholar [4] G. Alessandrini, M. Di Cristo, A. Morassi and E. Rosset, Stable determination of an inclusion in an elastic body by boundary measurements,, SIAM J. Math. Anal., 46 (2014), 2692. doi: 10.1137/130946307. Google Scholar [5] G. Alessandrini and A. Morassi, Strong unique continuation for the Lamé system of elasticity,, Comm. PDE, 26 (2001), 1787. doi: 10.1081/PDE-100107459. Google Scholar [6] G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations,, Inverse Problems, 25 (2009), 1. doi: 10.1088/0266-5611/25/12/123004. Google Scholar [7] G. Alessandrini and S. Vessella, Lipschitz stability for the inverse conductivity problem,, Adv. in Appl. Math., 35 (2005), 207. doi: 10.1016/j.aam.2004.12.002. Google Scholar [8] H. Ammari, E. Beretta and E. Francini, Reconstruction of thin conducting inhomogeneities from electrostatic measurements, II. The case of multiple segments,, Applicable Analysis, 85 (2006), 87. doi: 10.1080/00036810500277736. Google Scholar [9] K. Astala and L. Paivarinta, Calderón's inverse conductivity problem in the plane,, Ann. of Math., 163 (2006), 265. doi: 10.4007/annals.2006.163.265. Google Scholar [10] V. Bacchelli and S. Vessella, Lipschitz stability for a stationary 2D inverse problem with unknown polygonal boundary,, Inverse Problems, 22 (2006), 1627. doi: 10.1088/0266-5611/22/5/007. Google Scholar [11] E. Beretta, E. Bonnetier, E. Francini and A. Mazzucato, An asymptotic formula for the displacement field in the presence of small anisotropic elastic inclusions,, Inverse Problems and Imaging, 6 (2012), 1. doi: 10.3934/ipi.2012.6.1. Google Scholar [12] E. Beretta, M. V de Hoop and L. Qiu, Lipschitz stability of an inverse boundary value problem for a Schrödinger type equation,, SIAM J. Math. Anal., 45 (2013), 679. doi: 10.1137/120869201. Google Scholar [13] E. Beretta and E. Francini, Lipschitz stability for the impedance tomography problem. The complex case,, Comm. PDE, 36 (2011), 1723. doi: 10.1080/03605302.2011.552930. Google Scholar [14] E. Beretta, E. Francini and S. Vessella, Determination of a linear crack in an elastic body from boundary measurements. Lipschitz stability,, SIAM J. Math. Anal., 40 (2008), 984. doi: 10.1137/070698397. Google Scholar [15] M. Bonnet and A. Constantinescu, Inverse problems in elasticity,, Inverse problems, 21 (2005). doi: 10.1088/0266-5611/21/2/R01. Google Scholar [16] B. M. Brown, M. Jais and I. W. Knowles, A variational approach to an elastic inverse problem,, Inverse Problems, 21 (2005), 1953. doi: 10.1088/0266-5611/21/6/010. Google Scholar [17] M. Chipot, D. Kinderlehrer and G. Vergara-Caffarelli, Smoothness of linear laminates,, Arch. Rational Mech. Anal., 96 (1986), 81. doi: 10.1007/BF00251414. Google Scholar [18] M. V. de Hoop, L. Qiu and O. Scherzer, Local analysis of inverse problems: Höelder stability and iterative reconstruction,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/4/045001. Google Scholar [19] M. V. de Hoop, L. Qiu and O. Scherzer, A convergence analysis of a multi-level projected steepest descent iteration for nonlinear inverse problems in Banach spaces subject to stability constraints,, to appear in Numerische Mathematik, (). Google Scholar [20] G. Eskin and J. Ralston, On the inverse boundary value problem for linear isotropic elasticity,, Inverse Problems, 18 (2002), 907. doi: 10.1088/0266-5611/18/3/324. Google Scholar [21] L. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (2010). Google Scholar [22] S. Hofmann and S. Kim, The Green function estimates for strongly elliptic systems of second order,, Manuscripta Math., 124 (2007), 139. doi: 10.1007/s00229-007-0107-1. Google Scholar [23] M. Ikehata, Inversion formulas for the linearized problem for an inverse boundary value problem in elastic prospection,, SIAM J. Appl. Math., 50 (1990), 1635. doi: 10.1137/0150097. Google Scholar [24] V. Isakov, Inverse Problems for Partial Differential Equations,, $2^{nd}$ edition, (2006). Google Scholar [25] Y. Y. Li and L. Nirenberg, Estimates for elliptic systems from composite materials,, Comm. Pure Appl. Math., 56 (2003), 892. doi: 10.1002/cpa.10079. Google Scholar [26] N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation,, Inverse Problems, 17 (2001), 1435. doi: 10.1088/0266-5611/17/5/313. Google Scholar [27] G. Milton, The Theory of Composites,, Cambridge University Press, (2002). doi: 10.1017/CBO9780511613357. Google Scholar [28] C. Mengcheng and T. Renji, An explicit tensor expression for the fundamental solutions of a bimaterial space problem,, Applied Mathematics and Mechanics, 18 (1997), 331. doi: 10.1007/BF02457547. Google Scholar [29] A. Morassi and E. Rosset, Stable determination of cavities in elastic bodies,, Inverse Problems, 20 (2004), 453. doi: 10.1088/0266-5611/20/2/010. Google Scholar [30] G. Nakamura, Inverse problems for elasticity,, in Selected Papers on Analysis and Differential Equations, (2003), 71. Google Scholar [31] G. Nakamura and G. Uhlmann, Identification of Lamé parameters by boundary measurements,, American Journal of Mathematics, 115 (1993), 1161. doi: 10.2307/2375069. Google Scholar [32] G. Nakamura and G. Uhlmann, Erratum: Global uniqueness for an inverse boundary value problem arising in elasticity,, Invent. Math., 118 (1994), 457. doi: 10.1007/BF01231541. Google Scholar [33] G. Nakamura and G. Uhlmann, Inverse boundary problems at the boundary for an elastic system,, SIAM J. Math. Anal., 26 (1995), 263. doi: 10.1137/S0036141093247494. Google Scholar [34] L. Rongved, Force interior to one of two joined semi-infinite solids,, in Proc. 2nd Midwestern Conf. Solid Mech, (1955), 1. Google Scholar [35] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem,, Ann. of Math., 125 (1987), 153. doi: 10.2307/1971291. Google Scholar [36] S. Vessella, Locations and strengths of point sources: Stability estimates,, Inverse Problems, 8 (1992), 911. doi: 10.1088/0266-5611/8/6/008. Google Scholar
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