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May  2013, 7(2): 341-351. doi: 10.3934/ipi.2013.7.341

Stable determination of surface impedance on a rough obstacle by far field data

Giovanni Alessandrini 1, , Eva Sincich 1, and  Sergio Vessella 2,

1. 

Dipartimento di Matematica e Geoscienze, Università degli Studi di Trieste, Via Valerio 12/1, 34127 Trieste, Italy, Italy
2. 

Dipartimento di Matematica e Informatica "Ulisse Dini", Università degli Studi di Firenze, Viale Morgagni, 67/a - 50134 Firenze, Italy

Received  July 2012 Revised  January 2013 Published  May 2013

We treat the stability of determining the boundary impedance of an obstacle by scattering data, with a single incident field. A previous result by Sincich (SIAM J. Math. Anal. 38, (2006), 434-451) showed a log stability when the boundary of the obstacle is assumed to be $C^{1,1}$-smooth. We prove that, when the obstacle boundary is merely Lipschitz, a log-log type stability still holds.
Keywords: impedance boundary condition, stability., Inverse scattering problem.
Mathematics Subject Classification: Primary: 35R30, 35R25; Secondary: 31B2.
Citation: Giovanni Alessandrini, Eva Sincich, Sergio Vessella. Stable determination of surface impedance on a rough obstacle by far field data. Inverse Problems and Imaging, 2013, 7 (2) : 341-351. doi: 10.3934/ipi.2013.7.341
References:
[1]

G. Alessandrini, E. Beretta, E. Rosset and S. Vessella, Optimal stability for inverse elliptic boundary value problems with unknown boundaries, Ann. Sc. Norm. Super. Pisa - Scienze Fisiche e Matematiche - Serie IV, 29 (2000), 755-806.

[2]

V. Adolfsson and L. Escauriaza, $C^{1,\alpha}$ 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 and E. DiBenedetto, Determining 2-dimensional cracks in 3-dimensional bodies: uniqueness and stability, Indiana Univ. Math. J., 46 (1997), 1-82.

[4]

G. Alessandrini, A. Morassi and E. Rosset, Detecting cavities by electrostatic boundary measurements, Inverse Problems, 18 (2002), 1333-1353. doi: 10.1088/0266-5611/18/5/308.

[5]

G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 123004 (47pp). doi: 10.1088/0266-5611/25/12/123004.

[6]

A. Ballerini, Stable determination of an immersed body in a stationary Stokes fluid, Inverse Problems, 26 (2010), 125015. doi: 10.1088/0266-5611/26/12/125015.

[7]

M. Bellassoued, M. Choulli and A. Jbalia, "Stability of the Determination of the Surface Impedance of an Obstacle from the Scattering Amplitude,", 2012 Available from , ().  doi: 10.1002/mma.2762.

[8]

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.

[9]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," Appl. Math. Sci. 93, Springer-Verlag, Heidelberg, Germany, 1992.

[10]

D. Gilbarg and N.S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Second edition, Springer-Verlag, Berlin, Heidelberg, New York, 1983.

[11]

Isakov, New stability results for soft obstacles in inverse scattering, Inverse Problems, 9 (1993), 79-89. doi: 10.1088/0266-5611/9/5/003.

[12]

D.S. Jerison and C. E. Kenig, The Neumann problem on Lipschitz domains, Bull. Amer. Math. Soc. (N.S), 4 (1981), 203-207. doi: 10.1090/S0273-0979-1981-14884-9.

[13]

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.

[14]

L. E. Payne and H. F. Weinberger, New bounds in harmonic and biharmonic problems, J. Math. Phys., 4 (1955), 291-307.

[15]

L. E. Payne and H. F. Weinberger, New bounds for solutions of second order elliptic partial differential equations, Pacific J. Math., 8 (1958), 551-573. doi: 10.2140/pjm.1958.8.551.

[16]

F. Rellich, Darstellung der Eigenwerte von $\Delta u+\lambda u = 0$ durch ein Randintegral, Math. Z., 46 (1940), 635-636. doi: 10.1007/BF01181459.

[17]

E. Sincich, "Stability and Reconstruction for the Determination of Boundary Terms by a Single Measurements," PhD Thesis, SISSA-ISAS, Trieste, 2005. Available at http://digitallibrary.sissa.it/handle/1963/1973.

[18]

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.

show all references

References:
[1]

G. Alessandrini, E. Beretta, E. Rosset and S. Vessella, Optimal stability for inverse elliptic boundary value problems with unknown boundaries, Ann. Sc. Norm. Super. Pisa - Scienze Fisiche e Matematiche - Serie IV, 29 (2000), 755-806.

[2]

V. Adolfsson and L. Escauriaza, $C^{1,\alpha}$ 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 and E. DiBenedetto, Determining 2-dimensional cracks in 3-dimensional bodies: uniqueness and stability, Indiana Univ. Math. J., 46 (1997), 1-82.

[4]

G. Alessandrini, A. Morassi and E. Rosset, Detecting cavities by electrostatic boundary measurements, Inverse Problems, 18 (2002), 1333-1353. doi: 10.1088/0266-5611/18/5/308.

[5]

G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 123004 (47pp). doi: 10.1088/0266-5611/25/12/123004.

[6]

A. Ballerini, Stable determination of an immersed body in a stationary Stokes fluid, Inverse Problems, 26 (2010), 125015. doi: 10.1088/0266-5611/26/12/125015.

[7]

M. Bellassoued, M. Choulli and A. Jbalia, "Stability of the Determination of the Surface Impedance of an Obstacle from the Scattering Amplitude,", 2012 Available from , ().  doi: 10.1002/mma.2762.

[8]

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.

[9]

D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," Appl. Math. Sci. 93, Springer-Verlag, Heidelberg, Germany, 1992.

[10]

D. Gilbarg and N.S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Second edition, Springer-Verlag, Berlin, Heidelberg, New York, 1983.

[11]

Isakov, New stability results for soft obstacles in inverse scattering, Inverse Problems, 9 (1993), 79-89. doi: 10.1088/0266-5611/9/5/003.

[12]

D.S. Jerison and C. E. Kenig, The Neumann problem on Lipschitz domains, Bull. Amer. Math. Soc. (N.S), 4 (1981), 203-207. doi: 10.1090/S0273-0979-1981-14884-9.

[13]

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.

[14]

L. E. Payne and H. F. Weinberger, New bounds in harmonic and biharmonic problems, J. Math. Phys., 4 (1955), 291-307.

[15]

L. E. Payne and H. F. Weinberger, New bounds for solutions of second order elliptic partial differential equations, Pacific J. Math., 8 (1958), 551-573. doi: 10.2140/pjm.1958.8.551.

[16]

F. Rellich, Darstellung der Eigenwerte von $\Delta u+\lambda u = 0$ durch ein Randintegral, Math. Z., 46 (1940), 635-636. doi: 10.1007/BF01181459.

[17]

E. Sincich, "Stability and Reconstruction for the Determination of Boundary Terms by a Single Measurements," PhD Thesis, SISSA-ISAS, Trieste, 2005. Available at http://digitallibrary.sissa.it/handle/1963/1973.

[18]

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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