# American Institute of Mathematical Sciences

June  2016, 9(3): 643-650. doi: 10.3934/dcdss.2016018

## The problem of detecting corrosion by an electric measurement revisited

 1 Institut Élie Cartan de Lorraine, UMR CNRS 7502, Université de Lorraine-Metz, Boulevard des Aiguillettes, BP 70239, 54506 Vandoeuvre les Nancy cedex, Ile du Saulcy, 57045 Metz cedex 01, France 2 Faculté des Sciences de Bizerte, Département des Mathématiques, 7021 Jarzouna Bizerte, Tunisia

Received  February 2015 Revised  September 2015 Published  April 2016

We establish a logarithmic stability estimate for the problem of detecting corrosion by a single electric measurement. We give a proof based on an adaptation of the method initiated in [3] for solving the inverse problem of recovering the surface impedance of an obstacle from the scattering amplitude. The key idea consists in estimating accurately a lower bound of the local $L^2$-norm at the boundary, of the solution of the boundary value problem used in modeling the problem of detection corrosion by an electric measurement.
Citation: Mourad Choulli, Aymen Jbalia. The problem of detecting corrosion by an electric measurement revisited. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 643-650. doi: 10.3934/dcdss.2016018
##### References:
 [1] G. Alessandrini, L. Del Piero and L. Rondi, Stable determination of corrosion by a single electrostatic boundary measurement,, Inverse problems, 19 (2003), 973. doi: 10.1088/0266-5611/19/4/312. Google Scholar [2] G. Alessandrini, E. Sincich and S. Vessella, Stable determination of surface impedance on a rough obstacle by far field data,, Inverse Problems and Imaging, 7 (2013), 341. doi: 10.3934/ipi.2013.7.341. Google Scholar [3] M. Bellassoued, M. Choulli and A. Jbalia, Stability of the determination of the surface impedance of an obstacle from the scattering amplitude,, Math. Methods Appl. Sci., 36 (2013), 2429. doi: 10.1002/mma.2762. Google Scholar [4] M. Bellassoued, J. Cheng and M. Choulli, Stability estimate for an inverse boundary coefficient problem in thermal imaging,, J. Math Anal. Appl., 343 (2008), 328. doi: 10.1016/j.jmaa.2008.01.066. Google Scholar [5] L. Bourgeois, About stability and regularization of ill-posed elliptic Cauchy problems: the case of $C^{1,1}$ domains,, Math. Model. Numer. Anal., 44 (2010), 715. doi: 10.1051/m2an/2010016. Google Scholar [6] S. Chaabane, I. Fellah, M. Jaoua and J. Leblond, Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse problems,, Inverse Problems, 20 (2004), 47. doi: 10.1088/0266-5611/20/1/003. Google Scholar [7] S. Chaabane and M. Jaoua, Identification of Robin coefficients by the means of boundary measurements,, Inverse Problems, 15 (1999), 1425. doi: 10.1088/0266-5611/15/6/303. Google Scholar [8] J. Cheng, M. Choulli and J. Lin, Stable determination of a boundary coefficient in an elliptic equation,, Math. Models Methods Appl. Sci., 18 (2008), 107. doi: 10.1142/S0218202508002620. Google Scholar [9] J. Cheng, M. Choulli and X. Yang, An iterative BEM for the inverse problem of detecting corrosion in a pipe,, Numer. Math. J. Chinese Univ., 14 (2005), 252. Google Scholar [10] M. Choulli, Stability estimates for an inverse elliptic problem,, J. Inverse Ill-Posed Probl., 10 (2002), 601. doi: 10.1515/jiip.2002.10.6.601. Google Scholar [11] M. Choulli, An inverse problem in corrosion detection: Stability estimates,, J. Inverse Ill-Posed Probl., 12 (2004), 349. doi: 10.1515/1569394042248247. Google Scholar [12] M. Choulli, Une Introduction Aux Problèmes Inverses Elliptiques et Paraboliques,, SMAI-Springer, (2009). doi: 10.1007/978-3-642-02460-3. Google Scholar [13] M. Choulli and F. Triki, New stability estimates for the inverse medium problem with internal data,, SIAM J. Math. Anal., 47 (2015), 1778. doi: 10.1137/140988577. Google Scholar [14] M. Choulli, Applications of elliptic Carleman inequalities,, to appear in BCAM SpringerBriefs., (). Google Scholar [15] D. Fasino and G. Inglese, An inverse Robin problem for Laplace's equation: Theoretical and numerical methods},, Inverse Problems, 15 (1999), 41. doi: 10.1088/0266-5611/15/1/008. Google Scholar [16] G. Inglese, An inverse problem in corrosion detection,, Inverse Problems, 13 (1997), 977. doi: 10.1088/0266-5611/13/4/006. Google Scholar [17] E. Sincich, Stable determination of the surface impedance of an obstacle by far field measurements,, SIAM J. Math. Anal., 38 (2006), 434. doi: 10.1137/050631513. Google Scholar [18] E. Sincich, Lipschitz stability for the inverse Robin problem,, Inverse Problems, 23 (2007), 1311. doi: 10.1088/0266-5611/23/3/027. Google Scholar [19] E. Sincich, Smoothness dependent stability in corrosion detection,, J. Math. Anal. Appl., 426 (2015), 364. doi: 10.1016/j.jmaa.2014.10.036. Google Scholar

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##### References:
 [1] G. Alessandrini, L. Del Piero and L. Rondi, Stable determination of corrosion by a single electrostatic boundary measurement,, Inverse problems, 19 (2003), 973. doi: 10.1088/0266-5611/19/4/312. Google Scholar [2] G. Alessandrini, E. Sincich and S. Vessella, Stable determination of surface impedance on a rough obstacle by far field data,, Inverse Problems and Imaging, 7 (2013), 341. doi: 10.3934/ipi.2013.7.341. Google Scholar [3] M. Bellassoued, M. Choulli and A. Jbalia, Stability of the determination of the surface impedance of an obstacle from the scattering amplitude,, Math. Methods Appl. Sci., 36 (2013), 2429. doi: 10.1002/mma.2762. Google Scholar [4] M. Bellassoued, J. Cheng and M. Choulli, Stability estimate for an inverse boundary coefficient problem in thermal imaging,, J. Math Anal. Appl., 343 (2008), 328. doi: 10.1016/j.jmaa.2008.01.066. Google Scholar [5] L. Bourgeois, About stability and regularization of ill-posed elliptic Cauchy problems: the case of $C^{1,1}$ domains,, Math. Model. Numer. Anal., 44 (2010), 715. doi: 10.1051/m2an/2010016. Google Scholar [6] S. Chaabane, I. Fellah, M. Jaoua and J. Leblond, Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse problems,, Inverse Problems, 20 (2004), 47. doi: 10.1088/0266-5611/20/1/003. Google Scholar [7] S. Chaabane and M. Jaoua, Identification of Robin coefficients by the means of boundary measurements,, Inverse Problems, 15 (1999), 1425. doi: 10.1088/0266-5611/15/6/303. Google Scholar [8] J. Cheng, M. Choulli and J. Lin, Stable determination of a boundary coefficient in an elliptic equation,, Math. Models Methods Appl. Sci., 18 (2008), 107. doi: 10.1142/S0218202508002620. Google Scholar [9] J. Cheng, M. Choulli and X. Yang, An iterative BEM for the inverse problem of detecting corrosion in a pipe,, Numer. Math. J. Chinese Univ., 14 (2005), 252. Google Scholar [10] M. Choulli, Stability estimates for an inverse elliptic problem,, J. Inverse Ill-Posed Probl., 10 (2002), 601. doi: 10.1515/jiip.2002.10.6.601. Google Scholar [11] M. Choulli, An inverse problem in corrosion detection: Stability estimates,, J. Inverse Ill-Posed Probl., 12 (2004), 349. doi: 10.1515/1569394042248247. Google Scholar [12] M. Choulli, Une Introduction Aux Problèmes Inverses Elliptiques et Paraboliques,, SMAI-Springer, (2009). doi: 10.1007/978-3-642-02460-3. Google Scholar [13] M. Choulli and F. Triki, New stability estimates for the inverse medium problem with internal data,, SIAM J. Math. Anal., 47 (2015), 1778. doi: 10.1137/140988577. Google Scholar [14] M. Choulli, Applications of elliptic Carleman inequalities,, to appear in BCAM SpringerBriefs., (). Google Scholar [15] D. Fasino and G. Inglese, An inverse Robin problem for Laplace's equation: Theoretical and numerical methods},, Inverse Problems, 15 (1999), 41. doi: 10.1088/0266-5611/15/1/008. Google Scholar [16] G. Inglese, An inverse problem in corrosion detection,, Inverse Problems, 13 (1997), 977. doi: 10.1088/0266-5611/13/4/006. Google Scholar [17] E. Sincich, Stable determination of the surface impedance of an obstacle by far field measurements,, SIAM J. Math. Anal., 38 (2006), 434. doi: 10.1137/050631513. Google Scholar [18] E. Sincich, Lipschitz stability for the inverse Robin problem,, Inverse Problems, 23 (2007), 1311. doi: 10.1088/0266-5611/23/3/027. Google Scholar [19] E. Sincich, Smoothness dependent stability in corrosion detection,, J. Math. Anal. Appl., 426 (2015), 364. doi: 10.1016/j.jmaa.2014.10.036. Google Scholar
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