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

Giovanni Alessandrini; Eva Sincich; Sergio Vessella

doi:10.3934/ipi.2013.7.341

Article Contents

2013, Volume 7, Issue 2: 341-351. Doi: 10.3934/ipi.2013.7.341

This issue Previous Article On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines Next Article Inverse diffusion from knowledge of power densities

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

1.
Dipartimento di Matematica e Geoscienze, Università degli Studi di Trieste, Via Valerio 12/1, 34127 Trieste
2.
Dipartimento di Matematica e Informatica "Ulisse Dini", Università degli Studi di Firenze, Viale Morgagni, 67/a - 50134 Firenze

Received: July 2012

Revised: January 2013

Published: May 2013

Abstract / Introduction Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 35R30, 35R25; Secondary: 31B20.

Citation:

Related Papers

Cited by

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 http://arxiv.org/pdf/1201.2552v3. 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.