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June  2018, 12(3): 733-743. doi: 10.3934/ipi.2018031

Uniqueness on recovery of piecewise constant conductivity and inner core with one measurement

Xiaoping Fang 1, and  Youjun Deng 2,,

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

* Corresponding author: Youjun Deng, dengyijun_001@163.com

Received  September 2017 Revised  November 2017 Published  March 2018

Fund Project: The work is supported by NSF grant of China No. 11601528, NSF grant of Hunan No. 2017JJ3432 and No. 2018JJ3622, Innovation-Driven Project of Central South University, No. 2018CX041, Mathematics and Interdisciplinary Sciences Project of Central South University, Major Project for National Natural Science Foundation of China(71790615).

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.

Keywords: Inverse medium problem, uniqueness, layer potential, transmission condition, unique continuation.
Mathematics Subject Classification: Primary: 35J05, 35R30; Secondary: 31B10.
Citation: Xiaoping Fang, Youjun Deng. Uniqueness on recovery of piecewise constant conductivity and inner core with one measurement. Inverse Problems & Imaging, 2018, 12 (3) : 733-743. doi: 10.3934/ipi.2018031
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.  Google Scholar

[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.  Google Scholar

[3]

H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements Lecture Notes in Mathematics, 1846. Springer-Verlag, Berlin Heidelberg, 2004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

E. Blåsten and H. Liu, Recovering piecewise constant refractive indices by a single far-field pattern, preprint, arXiv: 1705.00815 Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

V. Isakov, Inverse Problems for Partial Differential Equations, Springer, New York, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

J. K. Seo, A uniqueness result on inverse conductivity problem with two measurements, J. Fourier Anal. Appl., 2 (1996), 227-235.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

Inverse boundary value problems for partial differential equations, Proceedings of the International Congress of Mathematicians, (Berlin, 1998) Documenta Mathematica, 3(1998), 77-86.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

H. Ammari and H. Kang, Reconstruction of Small Inhomogeneities from Boundary Measurements Lecture Notes in Mathematics, 1846. Springer-Verlag, Berlin Heidelberg, 2004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

E. Blåsten and H. Liu, Recovering piecewise constant refractive indices by a single far-field pattern, preprint, arXiv: 1705.00815 Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

V. Isakov, Inverse Problems for Partial Differential Equations, Springer, New York, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

J. K. Seo, A uniqueness result on inverse conductivity problem with two measurements, J. Fourier Anal. Appl., 2 (1996), 227-235.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

Inverse boundary value problems for partial differential equations, Proceedings of the International Congress of Mathematicians, (Berlin, 1998) Documenta Mathematica, 3(1998), 77-86.  Google Scholar

[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.  Google Scholar

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