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Efficient tensor tomography in fan-beam coordinates. Ⅱ: Attenuated transforms
Cloaking for a quasi-linear elliptic partial differential equation
1. | Jockey Club Institute for Advanced Study, Hong Kong University of Science and Technology, Hong Kong, China |
2. | Department of Mathematics, University of Washington, Seattle, WA 98195-4350, USA |
In this article we consider cloaking for a quasi-linear elliptic partial differential equation of divergence type defined on a bounded domain in $\mathbb{R}^N$ for $N = 2, 3$. We show that a perfect cloak can be obtained via a singular change of variables scheme and an approximate cloak can be achieved via a regular change of variables scheme. These approximate cloaks, though non-degenerate, are anisotropic. We also show, within the framework of homogenization, that it is possible to get isotropic regular approximate cloaks. This work generalizes to quasi-linear settings previous work on cloaking in the context of Electrical Impedance Tomography for the conductivity equation.
References:
[1] |
G. Allaire, Shape Optimization by the Homogenization Method, vol. 146 of Applied Mathematical Sciences, Springer-Verlag, New York, 2002. |
[2] |
H. Ammari, H. Kang, H. Lee and M. Lim,
Enhancement of near cloaking using generalized polarization tensors vanishing structures. part Ⅰ: The conductivity problem, Communications in Mathematical Physics, 317 (2013), 253-266.
doi: 10.1007/s00220-012-1615-8. |
[3] |
K. Astala and L. Päivärinta,
Calderón's inverse conductivity problem in the plane, Ann. of Math. (2), 163 (2006), 265-299.
doi: 10.4007/annals.2006.163.265. |
[4] |
K. Astala, L. Päivärinta and M. Lassas,
Calderón's inverse problem for anisotropic conductivity in the plane, Comm. Partial Differential Equations, 30 (2005), 207-224.
doi: 10.1081/PDE-200044485. |
[5] |
J. Bear and Y. Bachmat, Introduction to Modelling of Transport Phenomena in Porous Media, 1991. |
[6] |
A. Bensoussan, J. -L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, vol. 5 of Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam-New York, 1978. |
[7] |
L. Boccardo and F. Murat,
Remarques sur l'homogénéisation de certains problémes quasi-linéaires, Portugal. Math., 41 (1982), 535-562 (1984).
|
[8] |
R. M. Brown and G. A. Uhlmann,
Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Comm. Partial Differential Equations, 22 (1997), 1009-1027.
doi: 10.1080/03605309708821292. |
[9] |
A. -P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), Soc. Brasil. Mat., Rio de Janeiro, 1980, 65-73. |
[10] |
P. Caro and K. M. Rogers, Global uniqueness for the Calderón problem with Lipschitz conductivities Forum Math. Pi, 4 (2016), e2, 28pp.
doi: 10.1017/fmp.2015.9. |
[11] |
M. Chipot, Elliptic Equations: An Introductory Course, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2009.
doi: 10.1007/978-3-7643-9982-5. |
[12] |
D. Cioranescu and P. Donato, An Introduction to Homogenization, vol. 17 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York, 1999. |
[13] |
Y. Deng, H. Liu and G. Uhlmann,
Full and partial cloaking in electromagnetic scattering, Archive for Rational Mechanics and Analysis, 223 (2017), 265-299.
doi: 10.1007/s00205-016-1035-6. |
[14] |
Y. Deng, H. Liu and G. Uhlmann,
On regularized full-and partial-cloaks in acoustic scattering, Comm. Partial Differential Equations, 42 (2017), 821-851.
doi: 10.1080/03605302.2017.1286673. |
[15] |
R. Fleury, F. Monticone and A. Alú,
Invisibility and cloaking: Origins, present, and future perspectives, Phys. Rev. Applied, 4 (2015), 037001.
doi: 10.1103/PhysRevApplied.4.037001. |
[16] |
N. Fusco and G. Moscariello,
On the homogenization of quasilinear divergence structure operators, Ann. Mat. Pura Appl. (4), 146 (1987), 1-13.
doi: 10.1007/BF01762357. |
[17] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition. |
[18] |
A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann,
Improvement of cylindrical cloaking with the shs lining, Opt. Express, 15 (2007), 12717-12734.
|
[19] |
A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann,
Full-wave invisibility of active devices at all frequencies, Comm. Math. Phys., 275 (2007), 749-789.
doi: 10.1007/s00220-007-0311-6. |
[20] |
A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann,
Electromagnetic wormholes via handlebody constructions, Comm. Math. Phys., 281 (2008), 369-385.
doi: 10.1007/s00220-008-0492-7. |
[21] |
A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann,
Isotropic transformation optics: Approximate acoustic and quantum cloaking, New Journal of Physics, 10 (2008), 115024.
|
[22] |
A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann,
Cloaking devices, electromagnetic wormholes, and transformation optics, SIAM Rev., 51 (2009), 3-33.
doi: 10.1137/080716827. |
[23] |
A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann,
Invisibility and inverse problems, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 55-97.
doi: 10.1090/S0273-0979-08-01232-9. |
[24] |
A. Greenleaf, M. Lassas and G. Uhlmann, Anisotropic conductivities that cannot be detected by eit,
Physiological measurement, 24.
doi: 10.1088/0967-3334/24/2/353. |
[25] |
A. Greenleaf, M. Lassas and G. Uhlmann,
On nonuniqueness for Calderón's inverse problem, Math. Res. Lett., 10 (2003), 685-693.
doi: 10.4310/MRL.2003.v10.n5.a11. |
[26] |
B. Haberman,
Uniqueness in Calderón's problem for conductivities with unbounded gradient, Comm. Math. Phys., 340 (2015), 639-659.
doi: 10.1007/s00220-015-2460-3. |
[27] |
B. Haberman and D. Tataru,
Uniqueness in Calderón's problem with Lipschitz conductivities, Duke Math. J., 162 (2013), 496-516.
doi: 10.1215/00127094-2019591. |
[28] |
G. Hu and H. Liu,
Nearly cloaking the elastic wave fields, J. Math. Pures Appl. (9), 104 (2015), 1045-1074.
doi: 10.1016/j.matpur.2015.07.004. |
[29] |
V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994, Translated from the Russian by G. A. Yosifian [G. A. Iosifýan].
doi: 10.1007/978-3-642-84659-5. |
[30] |
A. Karageorghis and D. Lesnic,
Steady-state nonlinear heat conduction in composite materials using the method of fundamental solutions, Comput. Methods Appl. Mech. Engrg., 197 (2008), 3122-3137.
doi: 10.1016/j.cma.2008.02.011. |
[31] |
R. V. Kohn, H. Shen, M. S. Vogelius and M. I. Weinstein, Cloaking via change of variables in electric impedance tomography Inverse Problems, 24 (2008), 015016, 21pp.
doi: 10.1088/0266-5611/24/1/015016. |
[32] |
O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. |
[33] |
J. M. Lee and G. Uhlmann,
Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math., 42 (1989), 1097-1112.
doi: 10.1002/cpa.3160420804. |
[34] |
U. Leonhardt,
Optical conformal mapping, Science, 312 (2006), 1777-1780.
doi: 10.1126/science.1126493. |
[35] |
J. Li and J. B. Pendry,
Hiding under the carpet: A new strategy for cloaking, Phys. Rev. Lett., 101 (2008), 203901.
doi: 10.1103/PhysRevLett.101.203901. |
[36] |
Y. -H. Lin, Nearly cloaking for the elasticity system with residual stress, Asymptotic Analysis,
106 (2018), 1-23, arXiv: 1611.05151v2
doi: 10.3233/ASY-171439. |
[37] |
H. Liu and H. Sun,
Enhanced near-cloak by FSH lining, J. Math. Pures Appl. (9), 99 (2013), 17-42.
doi: 10.1016/j.matpur.2012.06.001. |
[38] |
H. Liu and G. Uhlmann, Regularized transformation-optics cloaking in acoustic and electromagnetic scattering, in Inverse problems and imaging, vol. 44 of Panor. Synth`eses, Soc. Math. France, Paris, 2015,111-136. |
[39] |
H. Liu and T. Zhou,
On approximate electromagnetic cloaking by transformation media, SIAM J. Appl. Math., 71 (2011), 218-241.
doi: 10.1137/10081112X. |
[40] |
J. Malík, On homogenization of a quasilinear elliptic equation connected with heat conductivity, URL http://www2.cs.cas.cz/mweb/download/publi/MaII2006.pdf. |
[41] |
A. I. Nachman,
Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. (2), 143 (1996), 71-96.
doi: 10.2307/2118653. |
[42] |
J. B. Pendry, D. Schurig and D. R. Smith,
Controlling electromagnetic fields, Science, 312 (2006), 1780-1782.
doi: 10.1126/science.1125907. |
[43] |
Z. Ruan, M. Yan, C. W. Neff and M. Qiu,
Ideal cylindrical cloak: Perfect but sensitive to tiny perturbations, Phys. Rev. Lett., 99 (2007), 113903.
doi: 10.1103/PhysRevLett.99.113903. |
[44] |
Z. Sun,
On a quasilinear inverse boundary value problem, Math. Z., 221 (1996), 293-305.
doi: 10.1007/BF02622117. |
[45] |
Z. Sun and G. Uhlmann, Inverse problems in quasilinear anisotropic media, Amer. J. Math., 119 (1997), 771-797, URL http://muse.jhu.edu/journals/american_journal_of_mathematics/v119/119.4sun.pdf.
doi: 10.1353/ajm.1997.0027. |
[46] |
Z. Sun and G. Uhlmann,
Anisotropic inverse problems in two dimensions, Inverse Problems, 19 (2003), 1001-1010.
doi: 10.1088/0266-5611/19/5/301. |
[47] |
J. Sylvester and G. Uhlmann,
A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2), 125 (1987), 153-169.
doi: 10.2307/1971291. |
[48] |
L. Tartar,
The General Theory of Homogenization, vol. 7 of Lecture Notes of the Unione Matematica Italiana, Springer-Verlag, Berlin; UMI, Bologna, 2009, A personalized introduction.
doi: 10.1007/978-3-642-05195-1. |
[49] |
G. Uhlmann,
Inverse problems: seeing the unseen, Bull. Math. Sci., 4 (2014), 209-279.
doi: 10.1007/s13373-014-0051-9. |
[50] |
A. Whittington, A. Hofmeister and P. Nabelek,
Temperature-dependent thermal diffusivity of the earths crust and implications for magmatism, Nature, 458 (2009), 319-321.
|
show all references
References:
[1] |
G. Allaire, Shape Optimization by the Homogenization Method, vol. 146 of Applied Mathematical Sciences, Springer-Verlag, New York, 2002. |
[2] |
H. Ammari, H. Kang, H. Lee and M. Lim,
Enhancement of near cloaking using generalized polarization tensors vanishing structures. part Ⅰ: The conductivity problem, Communications in Mathematical Physics, 317 (2013), 253-266.
doi: 10.1007/s00220-012-1615-8. |
[3] |
K. Astala and L. Päivärinta,
Calderón's inverse conductivity problem in the plane, Ann. of Math. (2), 163 (2006), 265-299.
doi: 10.4007/annals.2006.163.265. |
[4] |
K. Astala, L. Päivärinta and M. Lassas,
Calderón's inverse problem for anisotropic conductivity in the plane, Comm. Partial Differential Equations, 30 (2005), 207-224.
doi: 10.1081/PDE-200044485. |
[5] |
J. Bear and Y. Bachmat, Introduction to Modelling of Transport Phenomena in Porous Media, 1991. |
[6] |
A. Bensoussan, J. -L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, vol. 5 of Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam-New York, 1978. |
[7] |
L. Boccardo and F. Murat,
Remarques sur l'homogénéisation de certains problémes quasi-linéaires, Portugal. Math., 41 (1982), 535-562 (1984).
|
[8] |
R. M. Brown and G. A. Uhlmann,
Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Comm. Partial Differential Equations, 22 (1997), 1009-1027.
doi: 10.1080/03605309708821292. |
[9] |
A. -P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), Soc. Brasil. Mat., Rio de Janeiro, 1980, 65-73. |
[10] |
P. Caro and K. M. Rogers, Global uniqueness for the Calderón problem with Lipschitz conductivities Forum Math. Pi, 4 (2016), e2, 28pp.
doi: 10.1017/fmp.2015.9. |
[11] |
M. Chipot, Elliptic Equations: An Introductory Course, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2009.
doi: 10.1007/978-3-7643-9982-5. |
[12] |
D. Cioranescu and P. Donato, An Introduction to Homogenization, vol. 17 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York, 1999. |
[13] |
Y. Deng, H. Liu and G. Uhlmann,
Full and partial cloaking in electromagnetic scattering, Archive for Rational Mechanics and Analysis, 223 (2017), 265-299.
doi: 10.1007/s00205-016-1035-6. |
[14] |
Y. Deng, H. Liu and G. Uhlmann,
On regularized full-and partial-cloaks in acoustic scattering, Comm. Partial Differential Equations, 42 (2017), 821-851.
doi: 10.1080/03605302.2017.1286673. |
[15] |
R. Fleury, F. Monticone and A. Alú,
Invisibility and cloaking: Origins, present, and future perspectives, Phys. Rev. Applied, 4 (2015), 037001.
doi: 10.1103/PhysRevApplied.4.037001. |
[16] |
N. Fusco and G. Moscariello,
On the homogenization of quasilinear divergence structure operators, Ann. Mat. Pura Appl. (4), 146 (1987), 1-13.
doi: 10.1007/BF01762357. |
[17] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition. |
[18] |
A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann,
Improvement of cylindrical cloaking with the shs lining, Opt. Express, 15 (2007), 12717-12734.
|
[19] |
A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann,
Full-wave invisibility of active devices at all frequencies, Comm. Math. Phys., 275 (2007), 749-789.
doi: 10.1007/s00220-007-0311-6. |
[20] |
A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann,
Electromagnetic wormholes via handlebody constructions, Comm. Math. Phys., 281 (2008), 369-385.
doi: 10.1007/s00220-008-0492-7. |
[21] |
A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann,
Isotropic transformation optics: Approximate acoustic and quantum cloaking, New Journal of Physics, 10 (2008), 115024.
|
[22] |
A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann,
Cloaking devices, electromagnetic wormholes, and transformation optics, SIAM Rev., 51 (2009), 3-33.
doi: 10.1137/080716827. |
[23] |
A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann,
Invisibility and inverse problems, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 55-97.
doi: 10.1090/S0273-0979-08-01232-9. |
[24] |
A. Greenleaf, M. Lassas and G. Uhlmann, Anisotropic conductivities that cannot be detected by eit,
Physiological measurement, 24.
doi: 10.1088/0967-3334/24/2/353. |
[25] |
A. Greenleaf, M. Lassas and G. Uhlmann,
On nonuniqueness for Calderón's inverse problem, Math. Res. Lett., 10 (2003), 685-693.
doi: 10.4310/MRL.2003.v10.n5.a11. |
[26] |
B. Haberman,
Uniqueness in Calderón's problem for conductivities with unbounded gradient, Comm. Math. Phys., 340 (2015), 639-659.
doi: 10.1007/s00220-015-2460-3. |
[27] |
B. Haberman and D. Tataru,
Uniqueness in Calderón's problem with Lipschitz conductivities, Duke Math. J., 162 (2013), 496-516.
doi: 10.1215/00127094-2019591. |
[28] |
G. Hu and H. Liu,
Nearly cloaking the elastic wave fields, J. Math. Pures Appl. (9), 104 (2015), 1045-1074.
doi: 10.1016/j.matpur.2015.07.004. |
[29] |
V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994, Translated from the Russian by G. A. Yosifian [G. A. Iosifýan].
doi: 10.1007/978-3-642-84659-5. |
[30] |
A. Karageorghis and D. Lesnic,
Steady-state nonlinear heat conduction in composite materials using the method of fundamental solutions, Comput. Methods Appl. Mech. Engrg., 197 (2008), 3122-3137.
doi: 10.1016/j.cma.2008.02.011. |
[31] |
R. V. Kohn, H. Shen, M. S. Vogelius and M. I. Weinstein, Cloaking via change of variables in electric impedance tomography Inverse Problems, 24 (2008), 015016, 21pp.
doi: 10.1088/0266-5611/24/1/015016. |
[32] |
O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. |
[33] |
J. M. Lee and G. Uhlmann,
Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math., 42 (1989), 1097-1112.
doi: 10.1002/cpa.3160420804. |
[34] |
U. Leonhardt,
Optical conformal mapping, Science, 312 (2006), 1777-1780.
doi: 10.1126/science.1126493. |
[35] |
J. Li and J. B. Pendry,
Hiding under the carpet: A new strategy for cloaking, Phys. Rev. Lett., 101 (2008), 203901.
doi: 10.1103/PhysRevLett.101.203901. |
[36] |
Y. -H. Lin, Nearly cloaking for the elasticity system with residual stress, Asymptotic Analysis,
106 (2018), 1-23, arXiv: 1611.05151v2
doi: 10.3233/ASY-171439. |
[37] |
H. Liu and H. Sun,
Enhanced near-cloak by FSH lining, J. Math. Pures Appl. (9), 99 (2013), 17-42.
doi: 10.1016/j.matpur.2012.06.001. |
[38] |
H. Liu and G. Uhlmann, Regularized transformation-optics cloaking in acoustic and electromagnetic scattering, in Inverse problems and imaging, vol. 44 of Panor. Synth`eses, Soc. Math. France, Paris, 2015,111-136. |
[39] |
H. Liu and T. Zhou,
On approximate electromagnetic cloaking by transformation media, SIAM J. Appl. Math., 71 (2011), 218-241.
doi: 10.1137/10081112X. |
[40] |
J. Malík, On homogenization of a quasilinear elliptic equation connected with heat conductivity, URL http://www2.cs.cas.cz/mweb/download/publi/MaII2006.pdf. |
[41] |
A. I. Nachman,
Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. (2), 143 (1996), 71-96.
doi: 10.2307/2118653. |
[42] |
J. B. Pendry, D. Schurig and D. R. Smith,
Controlling electromagnetic fields, Science, 312 (2006), 1780-1782.
doi: 10.1126/science.1125907. |
[43] |
Z. Ruan, M. Yan, C. W. Neff and M. Qiu,
Ideal cylindrical cloak: Perfect but sensitive to tiny perturbations, Phys. Rev. Lett., 99 (2007), 113903.
doi: 10.1103/PhysRevLett.99.113903. |
[44] |
Z. Sun,
On a quasilinear inverse boundary value problem, Math. Z., 221 (1996), 293-305.
doi: 10.1007/BF02622117. |
[45] |
Z. Sun and G. Uhlmann, Inverse problems in quasilinear anisotropic media, Amer. J. Math., 119 (1997), 771-797, URL http://muse.jhu.edu/journals/american_journal_of_mathematics/v119/119.4sun.pdf.
doi: 10.1353/ajm.1997.0027. |
[46] |
Z. Sun and G. Uhlmann,
Anisotropic inverse problems in two dimensions, Inverse Problems, 19 (2003), 1001-1010.
doi: 10.1088/0266-5611/19/5/301. |
[47] |
J. Sylvester and G. Uhlmann,
A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2), 125 (1987), 153-169.
doi: 10.2307/1971291. |
[48] |
L. Tartar,
The General Theory of Homogenization, vol. 7 of Lecture Notes of the Unione Matematica Italiana, Springer-Verlag, Berlin; UMI, Bologna, 2009, A personalized introduction.
doi: 10.1007/978-3-642-05195-1. |
[49] |
G. Uhlmann,
Inverse problems: seeing the unseen, Bull. Math. Sci., 4 (2014), 209-279.
doi: 10.1007/s13373-014-0051-9. |
[50] |
A. Whittington, A. Hofmeister and P. Nabelek,
Temperature-dependent thermal diffusivity of the earths crust and implications for magmatism, Nature, 458 (2009), 319-321.
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