# American Institute of Mathematical Sciences

May  2008, 2(2): 301-315. doi: 10.3934/ipi.2008.2.301

## Local stability for soft obstacles by a single measurement

 1 RICAM, Altenbergerstrasse 69, A4040, Linz, Austria, Austria

Received  October 2007 Revised  March 2008 Published  April 2008

We consider an inverse scattering problem arising in target identification. We prove a local stability result of logarithmic type for the determination of a sound soft obstacle from the far field measurements associated to one single incident wave.
Citation: Eva Sincich, Mourad Sini. Local stability for soft obstacles by a single measurement. Inverse Problems and Imaging, 2008, 2 (2) : 301-315. doi: 10.3934/ipi.2008.2.301
##### References:
 [1] R. A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press, New York, San Francisco, London, 1975. [2] V. Adolfsson and L. Escauriaza, $C^{1,\a}$ domains and unique continuation at the boundary, Comm. Pure Appl. Math, 50 (1997), 935-969. doi: 10.1002/(SICI)1097-0312(199710)50:10<935::AID-CPA1>3.0.CO;2-H. [3] G. Alessandrini, E. Beretta, E. Rosset and S. Vessella, "Optimal Stability for Inverse Elliptic Boundary Value Problems with Unknown Boundaries," Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 29 (2000), 755-806. [4] G. Alessandrini and A. Morassi, Strong unique continuation for the Lamè system of elasticity, Comm. Partial Differential Equations, 26 (2001), 1787-1810. doi: 10.1081/PDE-100107459. [5] G. Alessandrini and L. Rondi, Determining a sound-soft polyhedral scatterer by a single far-field measurement, Proc. Amer. Math. Soc., 133 (2005), 1685-1691. Corrigendum, preprint, 2006 (down-loadable at http://www.arxiv.org/archive/math/). arXiv:0601406 [6] 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. [7] I. Bushuyev, Stability of recovering the near-field wave from the scattering amplitude, Inverse Problems, 12 (1996), 859-867. doi: 10.1088/0266-5611/12/6/004. [8] F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory," Interaction of Mechanics and Mathematics, Springer-Verlag, Berlin, 2006. [9] J. Cheng and M. Yamamoto, Uniqueness in an inverse scattering problem within non-trapping polygonal obstacles with at most two incoming waves, [Inverse Problems, 19 (2003), 1361-1384; MR2036535], Inverse Problems, 21 (2005). [10] D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory," Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983. [11] D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," Appl. Math. Sc. 93, Springer-Verlag, Berlin, 1992. [12] D. Colton and B. D. Sleeman, Uniqueness theorems for the inverse problem of acoustic scattering, IMA J. Appl. Math., 31 (1983), 253-259. doi: 10.1093/imamat/31.3.253. [13] J. Elschner and M. Yamamoto, Uniqueness in determining polygonal sound-hard obstacles with a single incoming wave, Inverse Problems, 22 (2006), 355-364. doi: 10.1088/0266-5611/22/1/019. [14] P. R. Garabedian, "Partial Differential Equations," Second edition, Chelsea Publishing Co., New York, 1986. [15] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. [16] D. Gintides, Local uniqueness for the inverse scattering problem in acoustics via the Faber-Krahn inequality, Inverse Problems, 21 (2005), 1195-1205. doi: 10.1088/0266-5611/21/4/001. [17] N. Honda, G. Nakamura and M. Sini, Analytic extention and reconstruction of obstacles from few measurements for elliptic second order operators, RICAM Preprint series, (2008). [18] V. Isakov, Stability estimates for obstacles in inverse scattering, J. Comp. Appl. Math., 42 (1991), 79-89. doi: 10.1016/0377-0427(92)90164-S. [19] V. Isakov, New stability results for soft obstacles in inverse scattering, Inverse Problems, 9 (1993), 535-543. doi: 10.1088/0266-5611/9/5/003. [20] D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Func. Anal, 130 (1995), 161-219. doi: 10.1006/jfan.1995.1067. [21] H. Liu and J. Zou, Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers, Inverse Problems, 22 (2006), 515-524. doi: 10.1088/0266-5611/22/2/008. [22] 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. [23] L. Rondi, Stable determination of sound-soft polyhedral scatterers by a single measurement, to appear on Indiana Univ. Math. J., (available on line at http://www.iumj.indiana.edu/Preprints/3217.pdf). [24] A. G. Ramm, "Inverse Problems, Mathematical and Analytical Techniques with Applications to Engineering," Springer, 2004. [25] E. Sincich, Stable determination of the surface impedance of an obstacle by far field measurements, SIAM J. Math. Anal., 38 (2006), 434-451 (electronic). doi: 10.1137/050631513. [26] E. Sincich, "Stability and Reconstruction for the Determination of Boundary Terms by a Single Measurements," Ph.D. thesis, S.I.S.S.A./I.S.A.S., Trieste, Italy, 2005; available on line at http://www.sissa.it/library/. [27] P. Stefanov and G. Uhlmann, Local uniqueness for the fixed energy fixed angle inverse problem in obstacle scattering, Proc. Amer. Math. Soc., 132 (2004), 1351-1354. doi: 10.1090/S0002-9939-03-07363-5.

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##### References:
 [1] R. A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press, New York, San Francisco, London, 1975. [2] V. Adolfsson and L. Escauriaza, $C^{1,\a}$ domains and unique continuation at the boundary, Comm. Pure Appl. Math, 50 (1997), 935-969. doi: 10.1002/(SICI)1097-0312(199710)50:10<935::AID-CPA1>3.0.CO;2-H. [3] G. Alessandrini, E. Beretta, E. Rosset and S. Vessella, "Optimal Stability for Inverse Elliptic Boundary Value Problems with Unknown Boundaries," Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 29 (2000), 755-806. [4] G. Alessandrini and A. Morassi, Strong unique continuation for the Lamè system of elasticity, Comm. Partial Differential Equations, 26 (2001), 1787-1810. doi: 10.1081/PDE-100107459. [5] G. Alessandrini and L. Rondi, Determining a sound-soft polyhedral scatterer by a single far-field measurement, Proc. Amer. Math. Soc., 133 (2005), 1685-1691. Corrigendum, preprint, 2006 (down-loadable at http://www.arxiv.org/archive/math/). arXiv:0601406 [6] 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. [7] I. Bushuyev, Stability of recovering the near-field wave from the scattering amplitude, Inverse Problems, 12 (1996), 859-867. doi: 10.1088/0266-5611/12/6/004. [8] F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory," Interaction of Mechanics and Mathematics, Springer-Verlag, Berlin, 2006. [9] J. Cheng and M. Yamamoto, Uniqueness in an inverse scattering problem within non-trapping polygonal obstacles with at most two incoming waves, [Inverse Problems, 19 (2003), 1361-1384; MR2036535], Inverse Problems, 21 (2005). [10] D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory," Pure and Applied Mathematics (New York), A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983. [11] D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," Appl. Math. Sc. 93, Springer-Verlag, Berlin, 1992. [12] D. Colton and B. D. Sleeman, Uniqueness theorems for the inverse problem of acoustic scattering, IMA J. Appl. Math., 31 (1983), 253-259. doi: 10.1093/imamat/31.3.253. [13] J. Elschner and M. Yamamoto, Uniqueness in determining polygonal sound-hard obstacles with a single incoming wave, Inverse Problems, 22 (2006), 355-364. doi: 10.1088/0266-5611/22/1/019. [14] P. R. Garabedian, "Partial Differential Equations," Second edition, Chelsea Publishing Co., New York, 1986. [15] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. [16] D. Gintides, Local uniqueness for the inverse scattering problem in acoustics via the Faber-Krahn inequality, Inverse Problems, 21 (2005), 1195-1205. doi: 10.1088/0266-5611/21/4/001. [17] N. Honda, G. Nakamura and M. Sini, Analytic extention and reconstruction of obstacles from few measurements for elliptic second order operators, RICAM Preprint series, (2008). [18] V. Isakov, Stability estimates for obstacles in inverse scattering, J. Comp. Appl. Math., 42 (1991), 79-89. doi: 10.1016/0377-0427(92)90164-S. [19] V. Isakov, New stability results for soft obstacles in inverse scattering, Inverse Problems, 9 (1993), 535-543. doi: 10.1088/0266-5611/9/5/003. [20] D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Func. Anal, 130 (1995), 161-219. doi: 10.1006/jfan.1995.1067. [21] H. Liu and J. Zou, Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers, Inverse Problems, 22 (2006), 515-524. doi: 10.1088/0266-5611/22/2/008. [22] 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. [23] L. Rondi, Stable determination of sound-soft polyhedral scatterers by a single measurement, to appear on Indiana Univ. Math. J., (available on line at http://www.iumj.indiana.edu/Preprints/3217.pdf). [24] A. G. Ramm, "Inverse Problems, Mathematical and Analytical Techniques with Applications to Engineering," Springer, 2004. [25] E. Sincich, Stable determination of the surface impedance of an obstacle by far field measurements, SIAM J. Math. Anal., 38 (2006), 434-451 (electronic). doi: 10.1137/050631513. [26] E. Sincich, "Stability and Reconstruction for the Determination of Boundary Terms by a Single Measurements," Ph.D. thesis, S.I.S.S.A./I.S.A.S., Trieste, Italy, 2005; available on line at http://www.sissa.it/library/. [27] P. Stefanov and G. Uhlmann, Local uniqueness for the fixed energy fixed angle inverse problem in obstacle scattering, Proc. Amer. Math. Soc., 132 (2004), 1351-1354. doi: 10.1090/S0002-9939-03-07363-5.
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