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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
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 |
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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