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Uniqueness on recovery of piecewise constant conductivity and inner core with one measurement
1. | School of Mathematics and Statistics, Hunan University of Commerce, School of Mathematics and Statistics, Central South University, Changsha, Hunan, China |
2. | School of Mathematics and Statistics, Central South University, Changsha, Hunan, China |
We consider the recovery of piecewise constant conductivity and an unknown inner core in inverse conductivity problem. We first show the unique recovery of the conductivity in a one layer structure without inner core by one measurement on any surface enclosing the unknown medium. Then we recover the unknown inner core in a one layer structure. We then show that in a two layer structure, the conductivity can be uniquely recovered by using one measurement.
References:
[1] |
G. Alessandrini and L. Rondi,
Determining a sound-soft polyhedral scatterer by a single far-field measurement, Proc. Amer. Math. Soc., 35 (2005), 1685-1691.
doi: 10.1090/S0002-9939-05-07810-X. |
[2] |
H. Ammari, G. Ciraolo, H. Kang, H. Lee and G. Milton,
Spectral analysis of a Neumann-Poincaré-type operator and analysis of cloaking due to anomalous localized resonance, Arch. Ration. Mech. Anal., 208 (2013), 667-692.
doi: 10.1007/s00205-012-0605-5. |
[3] |
H. Ammari and H. Kang,
Reconstruction of Small Inhomogeneities from Boundary Measurements Lecture Notes in Mathematics, 1846. Springer-Verlag, Berlin Heidelberg, 2004. |
[4] |
H. Ammari and H. Kang,
Polarization and Moment Tensors With Applications to Inverse Problems and Effective Medium Theory, Applied Mathematical Sciences, Springer-Verlag, Berlin Heidelberg, 2007. |
[5] |
H. Ammari, H. Kang, H. Lee and M. Lim,
Enhancement of near cloaking using generalized polarization tensors vanishing structures. Part Ⅰ: The conductivity problem, Comm. Math. Phys., 317 (2013), 253-266.
doi: 10.1007/s00220-012-1615-8. |
[6] |
H. Ammari, H. Kang, H. Lee and M. Lim,
Enhancement of near cloaking. Part Ⅱ: The Helmholtz equation, Comm. Math. Phys., 317 (2013), 485-502.
doi: 10.1007/s00220-012-1620-y. |
[7] |
H. Ammari, H. Kang, H. Lee, M. Lim and Y. Sanghyeon,
Enhancement of near cloaking for the full Maxwell equations, SIAM J. Appl. Math., 73 (2013), 2055-2076.
doi: 10.1137/120903610. |
[8] |
B. Barceló, E. Fabes and J. K. Seo,
The inverse conductivity problem with one measurement: Uniqueness for convex polyhedra, Proc. Am. Math. Soc., 122 (1994), 183-189.
doi: 10.1090/S0002-9939-1994-1195476-6. |
[9] |
E. Blåsten and H. Liu, Recovering piecewise constant refractive indices by a single far-field pattern, preprint, arXiv: 1705.00815 |
[10] |
D. Colton and B. D. Sleeman,
Uniqueness theorems for the inverse problem of acoustic scattering, IMA J. Appl. Math., 31 (1983), 253-259.
doi: 10.1093/imamat/31.3.253. |
[11] |
Y. Deng, X. Fang and J. Li,
Plasmon resonance and heat generation in nanostructures, Math. Method Appl. Sci., 38 (2015), 4663-4672.
doi: 10.1002/mma.3448. |
[12] |
L. Escauriaza, E. B. Fabes and G. Verchota,
On a regularity theorem for weak solutions to transmission problems with internal Lipschitz boundaries, Proc. Amer. Math. Soc., 115 (1992), 1069-1076.
doi: 10.1090/S0002-9939-1992-1092919-1. |
[13] |
A. Friedman and V. Isakov,
On the uniqueness in the inverse conductivity problem with one measurement, Indiana Univ. Math. J., 38 (1989), 553-579.
doi: 10.1512/iumj.1989.38.38027. |
[14] |
G. Hu, M. Salo and E. V. Vesalainen,
Shape identification in inverse medium scattering, SIAM J. Math. Anal., 48 (2016), 152-165.
doi: 10.1137/15M1032958. |
[15] |
V. Isakov,
Inverse Problems for Partial Differential Equations, Springer, New York, 1998. |
[16] |
V. Isakov and J. Powell,
On the inverse conductivity problem with one measurement, Inverse Problem, 6 (1990), 311-318.
doi: 10.1088/0266-5611/6/2/011. |
[17] |
H. Kang and J. K. Seo,
Inverse conductivity problem with one measurement: Uniqueness of balls in $ {\mathbb{R}}^3$, SIAM J. Appl. Math., 59 (1990), 1533-1539.
doi: 10.1137/S0036139997324595. |
[18] |
O. D. Kellogg,
Foundations of Potential Theory, Reprint from the first edition of 1929. Die Grundlehren der Mathematischen Wissenschaften, Band 31 Springer-Verlag, Berlin-New York, 1967. |
[19] |
D. Khavinson, M. Putinar and H. S. Shapiro,
Poincaré's variational problem in potential theory, Arch. Ration. Mech. Anal., 185 (2007), 143-184.
doi: 10.1007/s00205-006-0045-1. |
[20] |
J. Li, H. Liu, Z. Shang and H. Sun,
Two single-shot methods for locating multiple electromagnetic scatterers, SIAM J. Appl. Math., 73 (2013), 1721-1746.
doi: 10.1137/130907690. |
[21] |
H. Liu and X. Liu, Recovery of an embedded obstacle and its surrounding medium from formally determined scattering data,
Inverse Problems, 33(2017), 065001, 20pp. |
[22] |
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), 515-524.
doi: 10.1088/0266-5611/22/2/008. |
[23] |
J. K. Seo,
A uniqueness result on inverse conductivity problem with two measurements, J. Fourier Anal. Appl., 2 (1996), 227-235.
|
[24] |
J. Sylvester and G. Uhlmann,
A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.
doi: 10.2307/1971291. |
[25] |
J. Sylvester and G. Uhlmann, The Dirichlet to Neumann map and applications,
Inverse Problems in Partial Differential Equations(Arcata, CA, 1989), SIAM, Philadelphia, (1990), 101-139. |
[26] |
Inverse boundary value problems for partial differential equations,
Proceedings of the International Congress of Mathematicians, (Berlin, 1998) Documenta Mathematica, 3(1998), 77-86. |
[27] |
G. Uhlmann, Developments in inverse problems since Calderón's foundational paper,
Harmonic Analysis and Partial Differential Equations, M. Christ, C. Kenig and C. Sadosky, eds., University of Chicago Press, (1999), 295-345. |
show all references
References:
[1] |
G. Alessandrini and L. Rondi,
Determining a sound-soft polyhedral scatterer by a single far-field measurement, Proc. Amer. Math. Soc., 35 (2005), 1685-1691.
doi: 10.1090/S0002-9939-05-07810-X. |
[2] |
H. Ammari, G. Ciraolo, H. Kang, H. Lee and G. Milton,
Spectral analysis of a Neumann-Poincaré-type operator and analysis of cloaking due to anomalous localized resonance, Arch. Ration. Mech. Anal., 208 (2013), 667-692.
doi: 10.1007/s00205-012-0605-5. |
[3] |
H. Ammari and H. Kang,
Reconstruction of Small Inhomogeneities from Boundary Measurements Lecture Notes in Mathematics, 1846. Springer-Verlag, Berlin Heidelberg, 2004. |
[4] |
H. Ammari and H. Kang,
Polarization and Moment Tensors With Applications to Inverse Problems and Effective Medium Theory, Applied Mathematical Sciences, Springer-Verlag, Berlin Heidelberg, 2007. |
[5] |
H. Ammari, H. Kang, H. Lee and M. Lim,
Enhancement of near cloaking using generalized polarization tensors vanishing structures. Part Ⅰ: The conductivity problem, Comm. Math. Phys., 317 (2013), 253-266.
doi: 10.1007/s00220-012-1615-8. |
[6] |
H. Ammari, H. Kang, H. Lee and M. Lim,
Enhancement of near cloaking. Part Ⅱ: The Helmholtz equation, Comm. Math. Phys., 317 (2013), 485-502.
doi: 10.1007/s00220-012-1620-y. |
[7] |
H. Ammari, H. Kang, H. Lee, M. Lim and Y. Sanghyeon,
Enhancement of near cloaking for the full Maxwell equations, SIAM J. Appl. Math., 73 (2013), 2055-2076.
doi: 10.1137/120903610. |
[8] |
B. Barceló, E. Fabes and J. K. Seo,
The inverse conductivity problem with one measurement: Uniqueness for convex polyhedra, Proc. Am. Math. Soc., 122 (1994), 183-189.
doi: 10.1090/S0002-9939-1994-1195476-6. |
[9] |
E. Blåsten and H. Liu, Recovering piecewise constant refractive indices by a single far-field pattern, preprint, arXiv: 1705.00815 |
[10] |
D. Colton and B. D. Sleeman,
Uniqueness theorems for the inverse problem of acoustic scattering, IMA J. Appl. Math., 31 (1983), 253-259.
doi: 10.1093/imamat/31.3.253. |
[11] |
Y. Deng, X. Fang and J. Li,
Plasmon resonance and heat generation in nanostructures, Math. Method Appl. Sci., 38 (2015), 4663-4672.
doi: 10.1002/mma.3448. |
[12] |
L. Escauriaza, E. B. Fabes and G. Verchota,
On a regularity theorem for weak solutions to transmission problems with internal Lipschitz boundaries, Proc. Amer. Math. Soc., 115 (1992), 1069-1076.
doi: 10.1090/S0002-9939-1992-1092919-1. |
[13] |
A. Friedman and V. Isakov,
On the uniqueness in the inverse conductivity problem with one measurement, Indiana Univ. Math. J., 38 (1989), 553-579.
doi: 10.1512/iumj.1989.38.38027. |
[14] |
G. Hu, M. Salo and E. V. Vesalainen,
Shape identification in inverse medium scattering, SIAM J. Math. Anal., 48 (2016), 152-165.
doi: 10.1137/15M1032958. |
[15] |
V. Isakov,
Inverse Problems for Partial Differential Equations, Springer, New York, 1998. |
[16] |
V. Isakov and J. Powell,
On the inverse conductivity problem with one measurement, Inverse Problem, 6 (1990), 311-318.
doi: 10.1088/0266-5611/6/2/011. |
[17] |
H. Kang and J. K. Seo,
Inverse conductivity problem with one measurement: Uniqueness of balls in $ {\mathbb{R}}^3$, SIAM J. Appl. Math., 59 (1990), 1533-1539.
doi: 10.1137/S0036139997324595. |
[18] |
O. D. Kellogg,
Foundations of Potential Theory, Reprint from the first edition of 1929. Die Grundlehren der Mathematischen Wissenschaften, Band 31 Springer-Verlag, Berlin-New York, 1967. |
[19] |
D. Khavinson, M. Putinar and H. S. Shapiro,
Poincaré's variational problem in potential theory, Arch. Ration. Mech. Anal., 185 (2007), 143-184.
doi: 10.1007/s00205-006-0045-1. |
[20] |
J. Li, H. Liu, Z. Shang and H. Sun,
Two single-shot methods for locating multiple electromagnetic scatterers, SIAM J. Appl. Math., 73 (2013), 1721-1746.
doi: 10.1137/130907690. |
[21] |
H. Liu and X. Liu, Recovery of an embedded obstacle and its surrounding medium from formally determined scattering data,
Inverse Problems, 33(2017), 065001, 20pp. |
[22] |
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), 515-524.
doi: 10.1088/0266-5611/22/2/008. |
[23] |
J. K. Seo,
A uniqueness result on inverse conductivity problem with two measurements, J. Fourier Anal. Appl., 2 (1996), 227-235.
|
[24] |
J. Sylvester and G. Uhlmann,
A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.
doi: 10.2307/1971291. |
[25] |
J. Sylvester and G. Uhlmann, The Dirichlet to Neumann map and applications,
Inverse Problems in Partial Differential Equations(Arcata, CA, 1989), SIAM, Philadelphia, (1990), 101-139. |
[26] |
Inverse boundary value problems for partial differential equations,
Proceedings of the International Congress of Mathematicians, (Berlin, 1998) Documenta Mathematica, 3(1998), 77-86. |
[27] |
G. Uhlmann, Developments in inverse problems since Calderón's foundational paper,
Harmonic Analysis and Partial Differential Equations, M. Christ, C. Kenig and C. Sadosky, eds., University of Chicago Press, (1999), 295-345. |
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