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 and 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), 335-354. doi: 10.1088/0266-5611/7/3/003.

[2]

G. Alessandrini, Stable determination of conductivity by boundary measurements, Applicable Analysis, 27 (1988), 153-172. doi: 10.1080/00036818808839730.

[3]

G. Alessandrini, E. Beretta and S. Vessella, Determining cracks by boundary measurements-Lipschitz Stability, SIAM J. Math. Anal., 27 (1996), 361-375. doi: 10.1137/S0036141094265791.

[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), 2692-2729. doi: 10.1137/130946307.

[5]

G. Alessandrini and A. Morassi, Strong unique continuation for the Lamé system of elasticity, Comm. PDE, 26 (2001), 1787-1810. doi: 10.1081/PDE-100107459.

[6]

G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 1-47. doi: 10.1088/0266-5611/25/12/123004.

[7]

G. Alessandrini and S. Vessella, Lipschitz stability for the inverse conductivity problem, Adv. in Appl. Math., 35 (2005), 207-241. doi: 10.1016/j.aam.2004.12.002.

[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), 87-105. doi: 10.1080/00036810500277736.

[9]

K. Astala and L. Paivarinta, Calderón's inverse conductivity problem in the plane, Ann. of Math., 163 (2006), 265-299. doi: 10.4007/annals.2006.163.265.

[10]

V. Bacchelli and S. Vessella, Lipschitz stability for a stationary 2D inverse problem with unknown polygonal boundary, Inverse Problems, 22 (2006), 1627-1658. doi: 10.1088/0266-5611/22/5/007.

[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), 1-23. doi: 10.3934/ipi.2012.6.1.

[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), 679-699. doi: 10.1137/120869201.

[13]

E. Beretta and E. Francini, Lipschitz stability for the impedance tomography problem. The complex case, Comm. PDE, 36 (2011), 1723-1749. doi: 10.1080/03605302.2011.552930.

[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), 984-1002. doi: 10.1137/070698397.

[15]

M. Bonnet and A. Constantinescu, Inverse problems in elasticity, Inverse problems, 21 (2005), R1-R50. doi: 10.1088/0266-5611/21/2/R01.

[16]

B. M. Brown, M. Jais and I. W. Knowles, A variational approach to an elastic inverse problem, Inverse Problems, 21 (2005), 1953-1973. doi: 10.1088/0266-5611/21/6/010.

[17]

M. Chipot, D. Kinderlehrer and G. Vergara-Caffarelli, Smoothness of linear laminates, Arch. Rational Mech. Anal., 96 (1986), 81-96. doi: 10.1007/BF00251414.

[18]

M. V. de Hoop, L. Qiu and O. Scherzer, Local analysis of inverse problems: Höelder stability and iterative reconstruction, Inverse Problems, 28 (2012), 045001, 16 pp. doi: 10.1088/0266-5611/28/4/045001.

[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, (). 

[20]

G. Eskin and J. Ralston, On the inverse boundary value problem for linear isotropic elasticity, Inverse Problems, 18 (2002), 907-921. doi: 10.1088/0266-5611/18/3/324.

[21]

L. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010.

[22]

S. Hofmann and S. Kim, The Green function estimates for strongly elliptic systems of second order, Manuscripta Math., 124 (2007), 139-172. doi: 10.1007/s00229-007-0107-1.

[23]

M. Ikehata, Inversion formulas for the linearized problem for an inverse boundary value problem in elastic prospection, SIAM J. Appl. Math., 50 (1990), 1635-1644. doi: 10.1137/0150097.

[24]

V. Isakov, Inverse Problems for Partial Differential Equations, $2^{nd}$ edition, Springer, 2006.

[25]

Y. Y. Li and L. Nirenberg, Estimates for elliptic systems from composite materials, Comm. Pure Appl. Math., 56 (2003), 892-925. doi: 10.1002/cpa.10079.

[26]

N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems, 17 (2001), 1435-1444. doi: 10.1088/0266-5611/17/5/313.

[27]

G. Milton, The Theory of Composites, Cambridge University Press, 2002. doi: 10.1017/CBO9780511613357.

[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), 331-340. doi: 10.1007/BF02457547.

[29]

A. Morassi and E. Rosset, Stable determination of cavities in elastic bodies, Inverse Problems, 20 (2004), 453-480. doi: 10.1088/0266-5611/20/2/010.

[30]

G. Nakamura, Inverse problems for elasticity, in Selected Papers on Analysis and Differential Equations, Amer. Math. Soc. Transl. Ser. 2, 211, Amer. Math. Soc., Providence, RI, 2003, 71-85.

[31]

G. Nakamura and G. Uhlmann, Identification of Lamé parameters by boundary measurements, American Journal of Mathematics, 115 (1993), 1161-1187. doi: 10.2307/2375069.

[32]

G. Nakamura and G. Uhlmann, Erratum: Global uniqueness for an inverse boundary value problem arising in elasticity, Invent. Math., 118 (1994), 457-474. doi: 10.1007/BF01231541.

[33]

G. Nakamura and G. Uhlmann, Inverse boundary problems at the boundary for an elastic system, SIAM J. Math. Anal., 26 (1995), 263-279. doi: 10.1137/S0036141093247494.

[34]

L. Rongved, Force interior to one of two joined semi-infinite solids, in Proc. 2nd Midwestern Conf. Solid Mech, Purdue, 1955, 1-13.

[35]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169. doi: 10.2307/1971291.

[36]

S. Vessella, Locations and strengths of point sources: Stability estimates, Inverse Problems, 8 (1992), 911-917. doi: 10.1088/0266-5611/8/6/008.

show all references

References:
[1]

M. Akamatsu, G. Nakamura and S. Steinberg, Identification of Lamé coefficients from boundary observations, Inverse Problems, 7 (1991), 335-354. doi: 10.1088/0266-5611/7/3/003.

[2]

G. Alessandrini, Stable determination of conductivity by boundary measurements, Applicable Analysis, 27 (1988), 153-172. doi: 10.1080/00036818808839730.

[3]

G. Alessandrini, E. Beretta and S. Vessella, Determining cracks by boundary measurements-Lipschitz Stability, SIAM J. Math. Anal., 27 (1996), 361-375. doi: 10.1137/S0036141094265791.

[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), 2692-2729. doi: 10.1137/130946307.

[5]

G. Alessandrini and A. Morassi, Strong unique continuation for the Lamé system of elasticity, Comm. PDE, 26 (2001), 1787-1810. doi: 10.1081/PDE-100107459.

[6]

G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 1-47. doi: 10.1088/0266-5611/25/12/123004.

[7]

G. Alessandrini and S. Vessella, Lipschitz stability for the inverse conductivity problem, Adv. in Appl. Math., 35 (2005), 207-241. doi: 10.1016/j.aam.2004.12.002.

[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), 87-105. doi: 10.1080/00036810500277736.

[9]

K. Astala and L. Paivarinta, Calderón's inverse conductivity problem in the plane, Ann. of Math., 163 (2006), 265-299. doi: 10.4007/annals.2006.163.265.

[10]

V. Bacchelli and S. Vessella, Lipschitz stability for a stationary 2D inverse problem with unknown polygonal boundary, Inverse Problems, 22 (2006), 1627-1658. doi: 10.1088/0266-5611/22/5/007.

[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), 1-23. doi: 10.3934/ipi.2012.6.1.

[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), 679-699. doi: 10.1137/120869201.

[13]

E. Beretta and E. Francini, Lipschitz stability for the impedance tomography problem. The complex case, Comm. PDE, 36 (2011), 1723-1749. doi: 10.1080/03605302.2011.552930.

[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), 984-1002. doi: 10.1137/070698397.

[15]

M. Bonnet and A. Constantinescu, Inverse problems in elasticity, Inverse problems, 21 (2005), R1-R50. doi: 10.1088/0266-5611/21/2/R01.

[16]

B. M. Brown, M. Jais and I. W. Knowles, A variational approach to an elastic inverse problem, Inverse Problems, 21 (2005), 1953-1973. doi: 10.1088/0266-5611/21/6/010.

[17]

M. Chipot, D. Kinderlehrer and G. Vergara-Caffarelli, Smoothness of linear laminates, Arch. Rational Mech. Anal., 96 (1986), 81-96. doi: 10.1007/BF00251414.

[18]

M. V. de Hoop, L. Qiu and O. Scherzer, Local analysis of inverse problems: Höelder stability and iterative reconstruction, Inverse Problems, 28 (2012), 045001, 16 pp. doi: 10.1088/0266-5611/28/4/045001.

[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, (). 

[20]

G. Eskin and J. Ralston, On the inverse boundary value problem for linear isotropic elasticity, Inverse Problems, 18 (2002), 907-921. doi: 10.1088/0266-5611/18/3/324.

[21]

L. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010.

[22]

S. Hofmann and S. Kim, The Green function estimates for strongly elliptic systems of second order, Manuscripta Math., 124 (2007), 139-172. doi: 10.1007/s00229-007-0107-1.

[23]

M. Ikehata, Inversion formulas for the linearized problem for an inverse boundary value problem in elastic prospection, SIAM J. Appl. Math., 50 (1990), 1635-1644. doi: 10.1137/0150097.

[24]

V. Isakov, Inverse Problems for Partial Differential Equations, $2^{nd}$ edition, Springer, 2006.

[25]

Y. Y. Li and L. Nirenberg, Estimates for elliptic systems from composite materials, Comm. Pure Appl. Math., 56 (2003), 892-925. doi: 10.1002/cpa.10079.

[26]

N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems, 17 (2001), 1435-1444. doi: 10.1088/0266-5611/17/5/313.

[27]

G. Milton, The Theory of Composites, Cambridge University Press, 2002. doi: 10.1017/CBO9780511613357.

[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), 331-340. doi: 10.1007/BF02457547.

[29]

A. Morassi and E. Rosset, Stable determination of cavities in elastic bodies, Inverse Problems, 20 (2004), 453-480. doi: 10.1088/0266-5611/20/2/010.

[30]

G. Nakamura, Inverse problems for elasticity, in Selected Papers on Analysis and Differential Equations, Amer. Math. Soc. Transl. Ser. 2, 211, Amer. Math. Soc., Providence, RI, 2003, 71-85.

[31]

G. Nakamura and G. Uhlmann, Identification of Lamé parameters by boundary measurements, American Journal of Mathematics, 115 (1993), 1161-1187. doi: 10.2307/2375069.

[32]

G. Nakamura and G. Uhlmann, Erratum: Global uniqueness for an inverse boundary value problem arising in elasticity, Invent. Math., 118 (1994), 457-474. doi: 10.1007/BF01231541.

[33]

G. Nakamura and G. Uhlmann, Inverse boundary problems at the boundary for an elastic system, SIAM J. Math. Anal., 26 (1995), 263-279. doi: 10.1137/S0036141093247494.

[34]

L. Rongved, Force interior to one of two joined semi-infinite solids, in Proc. 2nd Midwestern Conf. Solid Mech, Purdue, 1955, 1-13.

[35]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169. doi: 10.2307/1971291.

[36]

S. Vessella, Locations and strengths of point sources: Stability estimates, Inverse Problems, 8 (1992), 911-917. doi: 10.1088/0266-5611/8/6/008.

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