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April  2018, 12(2): 461-491. doi: 10.3934/ipi.2018020

Cloaking for a quasi-linear elliptic partial differential equation

Tuhin Ghosh 1, and  Karthik Iyer 2,

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

Received  June 2017 Revised  September 2017 Published  February 2018

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.

Keywords: Quasilinear elliptic, homogenization, cloaking, inverse problems.
Mathematics Subject Classification: Primary: 35R30; Secondary: 35J62.
Citation: Tuhin Ghosh, Karthik Iyer. Cloaking for a quasi-linear elliptic partial differential equation. Inverse Problems & Imaging, 2018, 12 (2) : 461-491. doi: 10.3934/ipi.2018020
References:
[1]

G. Allaire, Shape Optimization by the Homogenization Method, vol. 146 of Applied Mathematical Sciences, Springer-Verlag, New York, 2002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

J. Bear and Y. Bachmat, Introduction to Modelling of Transport Phenomena in Porous Media, 1991. Google Scholar

[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.  Google Scholar

[7]

L. Boccardo and F. Murat, Remarques sur l'homogénéisation de certains problémes quasi-linéaires, Portugal. Math., 41 (1982), 535-562 (1984).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Improvement of cylindrical cloaking with the shs lining, Opt. Express, 15 (2007), 12717-12734.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

U. Leonhardt, Optical conformal mapping, Science, 312 (2006), 1777-1780.  doi: 10.1126/science.1126493.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[42]

J. B. Pendry, D. Schurig and D. R. Smith, Controlling electromagnetic fields, Science, 312 (2006), 1780-1782.  doi: 10.1126/science.1125907.  Google Scholar

[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.  Google Scholar

[44]

Z. Sun, On a quasilinear inverse boundary value problem, Math. Z., 221 (1996), 293-305.  doi: 10.1007/BF02622117.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[49]

G. Uhlmann, Inverse problems: seeing the unseen, Bull. Math. Sci., 4 (2014), 209-279.  doi: 10.1007/s13373-014-0051-9.  Google Scholar

[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.   Google Scholar

show all references

References:
[1]

G. Allaire, Shape Optimization by the Homogenization Method, vol. 146 of Applied Mathematical Sciences, Springer-Verlag, New York, 2002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

J. Bear and Y. Bachmat, Introduction to Modelling of Transport Phenomena in Porous Media, 1991. Google Scholar

[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.  Google Scholar

[7]

L. Boccardo and F. Murat, Remarques sur l'homogénéisation de certains problémes quasi-linéaires, Portugal. Math., 41 (1982), 535-562 (1984).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Improvement of cylindrical cloaking with the shs lining, Opt. Express, 15 (2007), 12717-12734.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

U. Leonhardt, Optical conformal mapping, Science, 312 (2006), 1777-1780.  doi: 10.1126/science.1126493.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[42]

J. B. Pendry, D. Schurig and D. R. Smith, Controlling electromagnetic fields, Science, 312 (2006), 1780-1782.  doi: 10.1126/science.1125907.  Google Scholar

[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.  Google Scholar

[44]

Z. Sun, On a quasilinear inverse boundary value problem, Math. Z., 221 (1996), 293-305.  doi: 10.1007/BF02622117.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[49]

G. Uhlmann, Inverse problems: seeing the unseen, Bull. Math. Sci., 4 (2014), 209-279.  doi: 10.1007/s13373-014-0051-9.  Google Scholar

[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.   Google Scholar

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