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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
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

 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.
Citation: Giovanni Alessandrini, Eva Sincich, Sergio Vessella. Stable determination of surface impedance on a rough obstacle by far field data. Inverse Problems & 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. Google Scholar [2] V. Adolfsson and L. Escauriaza, $C^{1,\alpha}$ 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 and E. DiBenedetto, Determining 2-dimensional cracks in 3-dimensional bodies: uniqueness and stability,, Indiana Univ. Math. J., 46 (1997), 1. Google Scholar [4] G. Alessandrini, A. Morassi and E. Rosset, Detecting cavities by electrostatic boundary measurements,, Inverse Problems, 18 (2002), 1333. doi: 10.1088/0266-5611/18/5/308. Google Scholar [5] G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/12/123004. Google Scholar [6] A. Ballerini, Stable determination of an immersed body in a stationary Stokes fluid,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/12/125015. Google Scholar [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. Google Scholar [8] 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 [9] D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,", Appl. Math. Sci. 93, 93 (1992). Google Scholar [10] D. Gilbarg and N.S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Second edition, (1983). Google Scholar [11] Isakov, New stability results for soft obstacles in inverse scattering,, Inverse Problems, 9 (1993), 79. doi: 10.1088/0266-5611/9/5/003. Google Scholar [12] D.S. Jerison and C. E. Kenig, The Neumann problem on Lipschitz domains,, Bull. Amer. Math. Soc. (N.S), 4 (1981), 203. doi: 10.1090/S0273-0979-1981-14884-9. Google Scholar [13] 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 [14] L. E. Payne and H. F. Weinberger, New bounds in harmonic and biharmonic problems,, J. Math. Phys., 4 (1955), 291. Google Scholar [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. doi: 10.2140/pjm.1958.8.551. Google Scholar [16] F. Rellich, Darstellung der Eigenwerte von $\Delta u+\lambda u = 0$ durch ein Randintegral,, Math. Z., 46 (1940), 635. doi: 10.1007/BF01181459. Google Scholar [17] E. Sincich, "Stability and Reconstruction for the Determination of Boundary Terms by a Single Measurements,", PhD Thesis, (2005). Google Scholar [18] 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

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. Google Scholar [2] V. Adolfsson and L. Escauriaza, $C^{1,\alpha}$ 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 and E. DiBenedetto, Determining 2-dimensional cracks in 3-dimensional bodies: uniqueness and stability,, Indiana Univ. Math. J., 46 (1997), 1. Google Scholar [4] G. Alessandrini, A. Morassi and E. Rosset, Detecting cavities by electrostatic boundary measurements,, Inverse Problems, 18 (2002), 1333. doi: 10.1088/0266-5611/18/5/308. Google Scholar [5] G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/12/123004. Google Scholar [6] A. Ballerini, Stable determination of an immersed body in a stationary Stokes fluid,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/12/125015. Google Scholar [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. Google Scholar [8] 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 [9] D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,", Appl. Math. Sci. 93, 93 (1992). Google Scholar [10] D. Gilbarg and N.S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Second edition, (1983). Google Scholar [11] Isakov, New stability results for soft obstacles in inverse scattering,, Inverse Problems, 9 (1993), 79. doi: 10.1088/0266-5611/9/5/003. Google Scholar [12] D.S. Jerison and C. E. Kenig, The Neumann problem on Lipschitz domains,, Bull. Amer. Math. Soc. (N.S), 4 (1981), 203. doi: 10.1090/S0273-0979-1981-14884-9. Google Scholar [13] 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 [14] L. E. Payne and H. F. Weinberger, New bounds in harmonic and biharmonic problems,, J. Math. Phys., 4 (1955), 291. Google Scholar [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. doi: 10.2140/pjm.1958.8.551. Google Scholar [16] F. Rellich, Darstellung der Eigenwerte von $\Delta u+\lambda u = 0$ durch ein Randintegral,, Math. Z., 46 (1940), 635. doi: 10.1007/BF01181459. Google Scholar [17] E. Sincich, "Stability and Reconstruction for the Determination of Boundary Terms by a Single Measurements,", PhD Thesis, (2005). Google Scholar [18] 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
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