# 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 & 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, (1975).   Google Scholar [2] V. Adolfsson and L. Escauriaza, $C^{1,\a}$ domains and unique continuation at the boundary,, Comm. Pure Appl. Math, 50 (1997), 935.  doi: 10.1002/(SICI)1097-0312(199710)50:10<935::AID-CPA1>3.0.CO;2-H.  Google Scholar [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.   Google Scholar [4] G. Alessandrini and A. Morassi, Strong unique continuation for the Lamè system of elasticity,, Comm. Partial Differential Equations, 26 (2001), 1787.  doi: 10.1081/PDE-100107459.  Google Scholar [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.   Google Scholar [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.  doi: 10.1137/S0036139996306468.  Google Scholar [7] I. Bushuyev, Stability of recovering the near-field wave from the scattering amplitude,, Inverse Problems, 12 (1996), 859.  doi: 10.1088/0266-5611/12/6/004.  Google Scholar [8] F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory,", Interaction of Mechanics and Mathematics, (2006).   Google Scholar [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.   Google Scholar [10] D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory,", Pure and Applied Mathematics (New York), (1983).   Google Scholar [11] D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,", Appl. Math. Sc. 93, (1992).   Google Scholar [12] D. Colton and B. D. Sleeman, Uniqueness theorems for the inverse problem of acoustic scattering,, IMA J. Appl. Math., 31 (1983), 253.  doi: 10.1093/imamat/31.3.253.  Google Scholar [13] J. Elschner and M. Yamamoto, Uniqueness in determining polygonal sound-hard obstacles with a single incoming wave,, Inverse Problems, 22 (2006), 355.  doi: 10.1088/0266-5611/22/1/019.  Google Scholar [14] P. R. Garabedian, "Partial Differential Equations,", Second edition, (1986).   Google Scholar [15] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Grundlehren der Mathematischen Wissenschaften, 224 (1977).   Google Scholar [16] D. Gintides, Local uniqueness for the inverse scattering problem in acoustics via the Faber-Krahn inequality,, Inverse Problems, 21 (2005), 1195.  doi: 10.1088/0266-5611/21/4/001.  Google Scholar [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).   Google Scholar [18] V. Isakov, Stability estimates for obstacles in inverse scattering,, J. Comp. Appl. Math., 42 (1991), 79.  doi: 10.1016/0377-0427(92)90164-S.  Google Scholar [19] V. Isakov, New stability results for soft obstacles in inverse scattering,, Inverse Problems, 9 (1993), 535.  doi: 10.1088/0266-5611/9/5/003.  Google Scholar [20] D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains,, J. Func. Anal, 130 (1995), 161.  doi: 10.1006/jfan.1995.1067.  Google Scholar [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.  doi: 10.1088/0266-5611/22/2/008.  Google Scholar [22] A. Morassi and E. Rosset, Stable determination of cavities in elastic bodies,, Inverse Problems, 20 (2004), 453.  doi: 10.1088/0266-5611/20/2/010.  Google Scholar [23] L. Rondi, Stable determination of sound-soft polyhedral scatterers by a single measurement,, to appear on Indiana Univ. Math. J., ().   Google Scholar [24] A. G. Ramm, "Inverse Problems, Mathematical and Analytical Techniques with Applications to Engineering,", Springer, (2004).   Google Scholar [25] 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 [26] E. Sincich, "Stability and Reconstruction for the Determination of Boundary Terms by a Single Measurements,", Ph.D. thesis, (2005).   Google Scholar [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.  doi: 10.1090/S0002-9939-03-07363-5.  Google Scholar

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##### References:
 [1] R. A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65,, Academic Press, (1975).   Google Scholar [2] V. Adolfsson and L. Escauriaza, $C^{1,\a}$ domains and unique continuation at the boundary,, Comm. Pure Appl. Math, 50 (1997), 935.  doi: 10.1002/(SICI)1097-0312(199710)50:10<935::AID-CPA1>3.0.CO;2-H.  Google Scholar [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.   Google Scholar [4] G. Alessandrini and A. Morassi, Strong unique continuation for the Lamè system of elasticity,, Comm. Partial Differential Equations, 26 (2001), 1787.  doi: 10.1081/PDE-100107459.  Google Scholar [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.   Google Scholar [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.  doi: 10.1137/S0036139996306468.  Google Scholar [7] I. Bushuyev, Stability of recovering the near-field wave from the scattering amplitude,, Inverse Problems, 12 (1996), 859.  doi: 10.1088/0266-5611/12/6/004.  Google Scholar [8] F. Cakoni and D. Colton, "Qualitative Methods in Inverse Scattering Theory,", Interaction of Mechanics and Mathematics, (2006).   Google Scholar [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.   Google Scholar [10] D. Colton and R. Kress, "Integral Equation Methods in Scattering Theory,", Pure and Applied Mathematics (New York), (1983).   Google Scholar [11] D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,", Appl. Math. Sc. 93, (1992).   Google Scholar [12] D. Colton and B. D. Sleeman, Uniqueness theorems for the inverse problem of acoustic scattering,, IMA J. Appl. Math., 31 (1983), 253.  doi: 10.1093/imamat/31.3.253.  Google Scholar [13] J. Elschner and M. Yamamoto, Uniqueness in determining polygonal sound-hard obstacles with a single incoming wave,, Inverse Problems, 22 (2006), 355.  doi: 10.1088/0266-5611/22/1/019.  Google Scholar [14] P. R. Garabedian, "Partial Differential Equations,", Second edition, (1986).   Google Scholar [15] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Grundlehren der Mathematischen Wissenschaften, 224 (1977).   Google Scholar [16] D. Gintides, Local uniqueness for the inverse scattering problem in acoustics via the Faber-Krahn inequality,, Inverse Problems, 21 (2005), 1195.  doi: 10.1088/0266-5611/21/4/001.  Google Scholar [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).   Google Scholar [18] V. Isakov, Stability estimates for obstacles in inverse scattering,, J. Comp. Appl. Math., 42 (1991), 79.  doi: 10.1016/0377-0427(92)90164-S.  Google Scholar [19] V. Isakov, New stability results for soft obstacles in inverse scattering,, Inverse Problems, 9 (1993), 535.  doi: 10.1088/0266-5611/9/5/003.  Google Scholar [20] D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains,, J. Func. Anal, 130 (1995), 161.  doi: 10.1006/jfan.1995.1067.  Google Scholar [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.  doi: 10.1088/0266-5611/22/2/008.  Google Scholar [22] A. Morassi and E. Rosset, Stable determination of cavities in elastic bodies,, Inverse Problems, 20 (2004), 453.  doi: 10.1088/0266-5611/20/2/010.  Google Scholar [23] L. Rondi, Stable determination of sound-soft polyhedral scatterers by a single measurement,, to appear on Indiana Univ. Math. J., ().   Google Scholar [24] A. G. Ramm, "Inverse Problems, Mathematical and Analytical Techniques with Applications to Engineering,", Springer, (2004).   Google Scholar [25] 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 [26] E. Sincich, "Stability and Reconstruction for the Determination of Boundary Terms by a Single Measurements,", Ph.D. thesis, (2005).   Google Scholar [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.  doi: 10.1090/S0002-9939-03-07363-5.  Google Scholar
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