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April  2013, 6(2): 501-515. doi: 10.3934/dcdss.2013.6.501

Estimating area of inclusions in anisotropic plates from boundary data

Antonino Morassi 1, , Edi Rosset 2, and  Sergio Vessella 3,

1. 

Dipartimento di Ingegneria Civile e Architettura, Università degli Studi di Udine, via Cotonificio 114, 33100 Udine
2. 

Dipartimento di Matematica e Geoscienze, Università degli Studi di Trieste, via Valerio 12/1, 34127 Trieste, Italy
3. 

DIMAD, Università degli Studi di Firenze, via Lombroso 6/17, 50134 Firenze

Received  October 2011 Revised  November 2011 Published  November 2012

We consider the inverse problem of determining the possible presence of an inclusion in a thin plate by boundary measurements. The plate is made by non-homogeneous linearly elastic material belonging to a general class of anisotropy. The inclusion is made by different elastic material. Under some a priori assumptions on the unknown inclusion, we prove constructive upper and lower estimates of the area of the unknown defect in terms of an easily expressed quantity related to work, which is given in terms of measurements of a couple field applied at the boundary and of the induced transversal displacement and its normal derivative taken at the boundary of the plate.
Keywords: Inverse problems, elastic plates, unique continuation., three spheres inequalities, size estimates, inclusions.
Mathematics Subject Classification: Primary: 35R30, 35R25, 74K2.
Citation: Antonino Morassi, Edi Rosset, Sergio Vessella. Estimating area of inclusions in anisotropic plates from boundary data. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 501-515. doi: 10.3934/dcdss.2013.6.501
References:
[1]

R. A. Adams, "Sobolev Spaces," Academic Press, New York, 1975.

[2]

G. Alessandrini, A. Morassi and E. Rosset, Size estimates, in "Inverse Problems: Theory and Applications" (eds. G. Alessandrini and G. Uhlmann), Contemp. Math., AMS, Providence, RI, 333 (2003), 33-75.

[3]

G. Alessandrini, A. Morassi and E. Rosset, Detecting an inclusion in an elastic body by boundary measurements, SIAM Rev., 46 (2004), 477-498. doi: 10.1137/S0036144504442098.

[4]

G. Alessandrini, A. Morassi, E. Rosset and S. Vessella, On doubling inequalities for elliptic systems, J. Math. Anal. Appl., 357 (2009), 349-355. doi: 10.1016/j.jmaa.2009.04.024.

[5]

G. Alessandrini and E. Rosset, The inverse conductivity problem with one measurement: bounds on the size of the unknown object, SIAM J. Appl. Math., 58 (1998), 1060-1071. doi: 10.1137/S0036139996306468.

[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, E. Rosset and J. K. Seo, Optimal size estimates for the inverse conductivity problem with one measurement, Proc. Amer. Math. Soc., 128 (2000), 53-64. doi: 10.1090/S0002-9939-99-05474-X.

[8]

S. Alinhac, Non-unicité pour des opérateurs différentiels à la caractéristiques complexes simples, Ann. Sci. École Norm. Sup., 13 (1980), 385-393.

[9]

M. Di Cristo, C. L. Lin and J. N. Wang, Quantitative uniqueness estimates for the shallow shell system and their application to an inverse problem, preprint (2011).

[10]

G. Fichera, Existence theorems in elasticity, in "Handbuch der Physik," VI, Springer-Verlag, Berlin (1972), 347-389.

[11]

M. E. Gurtin, The linear theory of elasticity, in "Handbuch der Physik," VI, Springer-Verlag, Berlin (1972), 1-295.

[12]

M. Ikehata, Size estimation of inclusion, J. Inverse Ill-Posed Probl., 6 (1998), 127-140.

[13]

H. Kang, J. K. Seo and D. Sheen, The inverse conductivity problem with one measurement: stability and estimation of size, SIAM J. Math. Anal., 28 (1997), 1389-1405. doi: 10.1137/S0036141096299375.

[14]

Y. Lei, M. Di Cristo and G. Nakamura, Size estimates in thermography, Appl. Anal., 88 (2009), 831-46. doi: 10.1080/00036810903042133.

[15]

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.

[16]

A. Morassi, E. Rosset and S. Vessella, Size estimates for inclusions in an elastic plate by boundary measurements, Indiana Univ. Math. J., 56 (2007), 2325-2384. doi: 10.1512/iumj.2007.56.2975.

[17]

A. Morassi, E. Rosset and S. Vessella, Detecting general inclusions in elastic plates, Inverse Problems, 25 (2009), Paper 045009. doi: 10.1088/0266-5611/25/4/045009.

[18]

A. Morassi, E. Rosset and S. Vessella, Stable determination of a rigid inclusion in an anisotropic plate, preprint (2011), arXiv:math/1111.0604.

[19]

A. Morassi, E. Rosset and S. Vessella, Sharp three sphere inequality for perturbations of a product of two second order elliptic operators and stability for the Cauchy problem for the anisotropic plate equation, J. Funct. Anal., 261 (2011), 1494-1541. doi: 10.1016/j.jfa.2011.05.011.

show all references

References:
[1]

R. A. Adams, "Sobolev Spaces," Academic Press, New York, 1975.

[2]

G. Alessandrini, A. Morassi and E. Rosset, Size estimates, in "Inverse Problems: Theory and Applications" (eds. G. Alessandrini and G. Uhlmann), Contemp. Math., AMS, Providence, RI, 333 (2003), 33-75.

[3]

G. Alessandrini, A. Morassi and E. Rosset, Detecting an inclusion in an elastic body by boundary measurements, SIAM Rev., 46 (2004), 477-498. doi: 10.1137/S0036144504442098.

[4]

G. Alessandrini, A. Morassi, E. Rosset and S. Vessella, On doubling inequalities for elliptic systems, J. Math. Anal. Appl., 357 (2009), 349-355. doi: 10.1016/j.jmaa.2009.04.024.

[5]

G. Alessandrini and E. Rosset, The inverse conductivity problem with one measurement: bounds on the size of the unknown object, SIAM J. Appl. Math., 58 (1998), 1060-1071. doi: 10.1137/S0036139996306468.

[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, E. Rosset and J. K. Seo, Optimal size estimates for the inverse conductivity problem with one measurement, Proc. Amer. Math. Soc., 128 (2000), 53-64. doi: 10.1090/S0002-9939-99-05474-X.

[8]

S. Alinhac, Non-unicité pour des opérateurs différentiels à la caractéristiques complexes simples, Ann. Sci. École Norm. Sup., 13 (1980), 385-393.

[9]

M. Di Cristo, C. L. Lin and J. N. Wang, Quantitative uniqueness estimates for the shallow shell system and their application to an inverse problem, preprint (2011).

[10]

G. Fichera, Existence theorems in elasticity, in "Handbuch der Physik," VI, Springer-Verlag, Berlin (1972), 347-389.

[11]

M. E. Gurtin, The linear theory of elasticity, in "Handbuch der Physik," VI, Springer-Verlag, Berlin (1972), 1-295.

[12]

M. Ikehata, Size estimation of inclusion, J. Inverse Ill-Posed Probl., 6 (1998), 127-140.

[13]

H. Kang, J. K. Seo and D. Sheen, The inverse conductivity problem with one measurement: stability and estimation of size, SIAM J. Math. Anal., 28 (1997), 1389-1405. doi: 10.1137/S0036141096299375.

[14]

Y. Lei, M. Di Cristo and G. Nakamura, Size estimates in thermography, Appl. Anal., 88 (2009), 831-46. doi: 10.1080/00036810903042133.

[15]

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.

[16]

A. Morassi, E. Rosset and S. Vessella, Size estimates for inclusions in an elastic plate by boundary measurements, Indiana Univ. Math. J., 56 (2007), 2325-2384. doi: 10.1512/iumj.2007.56.2975.

[17]

A. Morassi, E. Rosset and S. Vessella, Detecting general inclusions in elastic plates, Inverse Problems, 25 (2009), Paper 045009. doi: 10.1088/0266-5611/25/4/045009.

[18]

A. Morassi, E. Rosset and S. Vessella, Stable determination of a rigid inclusion in an anisotropic plate, preprint (2011), arXiv:math/1111.0604.

[19]

A. Morassi, E. Rosset and S. Vessella, Sharp three sphere inequality for perturbations of a product of two second order elliptic operators and stability for the Cauchy problem for the anisotropic plate equation, J. Funct. Anal., 261 (2011), 1494-1541. doi: 10.1016/j.jfa.2011.05.011.

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