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The problem of detecting corrosion by an electric measurement revisited

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  • 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.
    Mathematics Subject Classification: Primary: 35R30.

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    [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), 2429-2448.doi: 10.1002/mma.2762.

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    M. Bellassoued, J. Cheng and M. Choulli, Stability estimate for an inverse boundary coefficient problem in thermal imaging, J. Math Anal. Appl., 343 (2008), 328-336.doi: 10.1016/j.jmaa.2008.01.066.

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    J. Cheng, M. Choulli and J. Lin, Stable determination of a boundary coefficient in an elliptic equation, Math. Models Methods Appl. Sci., 18 (2008), 107-123.doi: 10.1142/S0218202508002620.

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    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), 252-266.

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    M. Choulli, Stability estimates for an inverse elliptic problem, J. Inverse Ill-Posed Probl., 10 (2002), 601-610.doi: 10.1515/jiip.2002.10.6.601.

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    M. Choulli, An inverse problem in corrosion detection: Stability estimates, J. Inverse Ill-Posed Probl., 12 (2004), 349-367.doi: 10.1515/1569394042248247.

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    M. Choulli, Une Introduction Aux Problèmes Inverses Elliptiques et Paraboliques, SMAI-Springer, Berlin, 2009.doi: 10.1007/978-3-642-02460-3.

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    E. Sincich, Stable determination of the surface impedance of an obstacle by far field measurements, SIAM J. Math. Anal., 38 (2006), 434-451.doi: 10.1137/050631513.

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    E. Sincich, Lipschitz stability for the inverse Robin problem, Inverse Problems, 23 (2007), 1311-1326.doi: 10.1088/0266-5611/23/3/027.

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    E. Sincich, Smoothness dependent stability in corrosion detection, J. Math. Anal. Appl., 426 (2015), 364-379.doi: 10.1016/j.jmaa.2014.10.036.

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