# American Institute of Mathematical Sciences

January 2018, 12(1): 91-123. doi: 10.3934/ipi.2018004

## Superconductive and insulating inclusions for linear and non-linear conductivity equations

 Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland

* Corresponding author: Joonas Ilmavirta

Received  April 2016 Revised  August 2017 Published  December 2017

We detect an inclusion with infinite conductivity from boundary measurements represented by the Dirichlet-to-Neumann map for the conductivity equation. We use both the enclosure method and the probe method. We use the enclosure method to prove partial results when the underlying equation is the quasilinear $p$-Laplace equation. Further, we rigorously treat the forward problem for the partial differential equation $\operatorname{div}(σ\lvert\nabla u\rvert^{p-2}\nabla u) = 0$ where the measurable conductivity $σ\colonΩ\to[0,∞]$ is zero or infinity in large sets and $1<p<∞$.

Citation: Tommi Brander, Joonas Ilmavirta, Manas Kar. Superconductive and insulating inclusions for linear and non-linear conductivity equations. Inverse Problems & Imaging, 2018, 12 (1) : 91-123. doi: 10.3934/ipi.2018004
##### References:
 [1] G. Alessandrini and A. D. Valenzuela, Unique determination of multiple cracks by two measurements, SIAM Journal on Control and Optimization, 34 (1996), 913-921. doi: 10.1137/S0363012994262853. [2] S. N. Antontsev and J. F. Rodrigues, On stationary thermo-rheological viscous flows, Annali dell'Universita di Ferrara, 52 (2006), 19-36. doi: 10.1007/s11565-006-0002-9. [3] G. Aronsson, On p-harmonic functions, convex duality and an asymptotic formula for injection mould filling, European Journal of Applied Mathematics, 7 (1996), 417-437. [4] K. Astala, M. Lassas and L. Päivärinta, The borderlines of invisibility and visibility in Calderón's inverse problem, Anal. PDE, 9 (2016), 43-98. doi: 10.2140/apde.2016.9.43. [5] K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Annals of Mathematics, 163 (2006), 265-299. doi: 10.4007/annals.2006.163.265. [6] C. Atkinson and C. R. Champion, Some boundary-value problems for the equation $\nabla · (|\nabla φ|^N \nabla φ)$, The Quarterly Journal of Mechanics and Applied Mathematics, 37 (1984), 401-419. [7] L. C. Berselli, L. Diening and M. Růžička, Existence of strong solutions for incompressible fluids with shear dependent viscosities, Journal of Mathematical Fluid Mechanics, 12 (2010), 101-132. doi: 10.1007/s00021-008-0277-y. [8] D. Borman, D. B. Ingham, B. T. Johansson and D. Lesnic, The method of fundamental solutions for detection of cavities in EIT, J. Integral Equations Applications, 21 (2009), 381-404. doi: 10.1216/JIE-2009-21-3-383. [9] T. Brander, Calderón problem for the p-Laplacian: First order derivative of conductivity on the boundary, Proceedings of American mathematical society, 144 (2016), 177-189, arXiv: 1403.0428. [10] T. Brander, M. Kar and M. Salo, Enclosure method for the p-Laplace equation, Inverse Problems, 31 (2015), 045001, 16pp, arXiv: 1410.4048. [11] T. Brander, B. von Harrach, M. Kar and M. Salo, Monotonicity and enclosure methods for the p-Laplace equation, ArXiv e-prints, Preprint arXiv: /1703.02814. [12] M. Brühl, Gebietserkennung in der Elektrischen Impedanztomographie, PhD thesis, Universität Karlsruhe, 1999. [13] P. R. Bueno, J. A. Varela and E. Longo, SnO2, ZnO and related polycrystalline compound semiconductors: An overview and review on the voltage-dependent resistance (non-ohmic) feature, Journal of the European Ceramic Society, 28 (2008), 505-529. doi: 10.1016/j.jeurceramsoc.2007.06.011. [14] F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Interaction of Mechanics and Mathematics, Springer-Verlag, Berlin, 2006. [15] F. Cakoni, M. Fares and H. Haddar, Analysis of two linear sampling methods applied to electromagnetic imaging of buried objects, Inverse Problems, 22 (2006), 845-867. doi: 10.1088/0266-5611/22/3/007. [16] A. P. Calderón, Lebesgue spaces of differentiable functions and distributions, in Partial Differential Equations, vol. 4 of Proceedings of Symposia in Pure Mathematics, American mathematical society, Providence, Rhode Island, USA, 1961, 33-49. [17] A. P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics (eds. W. Meyer and M. Raupp), Sociedade Brasileira de Matematica, 1980, 65-73, URL http://www.maths.manchester.ac.uk/~bl/Calderon/, Reprinted as [18]. [18] A. P. Calder´on, On an inverse boundary problem, Computation and applied mathematics, 25 (2006), 133-138, URL http://www.scielo.br/pdf/cam/v25n2-3/a02v2523.pdf, Reprint of [17]. [19] P. Caro and K. M. Rogers, Global uniqueness for the Calderón problem with Lipschitz conductivities, Forum of Mathematics, Pi, 4 (2016), e2, 28pp. doi: 10.1017/fmp.2015.9. [20] D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393. doi: 10.1088/0266-5611/12/4/003. [21] A. Friedman and M. Vogelius, Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence, Archive for Rational Mechanics and Analysis, 105 (1989), 299-326. doi: 10.1007/BF00281494. [22] A. Garroni and R. V. Kohn, Some three--dimensional problems related to dielectric breakdown and polycrystal plasticity, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 459 (2003), 2613-2625. doi: 10.1098/rspa.2003.1152. [23] A. Garroni, V. Nesi and M. Ponsiglione, Dielectric breakdown: Optimal bounds, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 457 (2001), 2317-2335. doi: 10.1098/rspa.2001.0803. [24] B. Gebauer, M. Hanke, A. Kirsch, W. Muniz and C. Schneider, A sampling method for detecting buried objects using electromagnetic scattering, Inverse Problems, 21 (2005), 2035-2050. doi: 10.1088/0266-5611/21/6/015. [25] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 2nd edition, Springer-Verlag, Berlin, 1983. [26] R. Glowinski and J. Rappaz, Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid flow model in glaciology, ESAIM: Mathematical Modelling and Numerical Analysis, 37 (2003), 175-186. doi: 10.1051/m2an:2003012. [27] Y. Gorb and A. Novikov, Blow-up of solutions to a p-Laplace equation, Multiscale Model. Simul., 10 (2012), 727-743. doi: 10.1137/110857167. [28] C.-Y. Guo, M. Kar and M. Salo, Inverse problems for p-Laplace type equations under monotonicity assumptions, Rend. Istit. Mat. Univ. Trieste, 48 (2016), 79-99. [29] B. Harrach, Recent progress on the factorization method for electrical impedance tomography Comput. Math. Methods Med. , 2013 (2013), Art. ID 425184, 8pp. doi: 10.1155/2013/425184. [30] B. Harrach and M. Ullrich, Monotonicity-based shape reconstruction in electrical impedance tomography, SIAM J. Math. Anal., 45 (2013), 3382-3403. doi: 10.1137/120886984. [31] D. Hauer, The p-Dirichlet-to-Neumann operator with applications to elliptic and parabolic problems, Journal of Differential Equations, 259 (2015), 3615-3655. doi: 10.1016/j.jde.2015.04.030. [32] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, Oxford, 1993, Oxford Science Publications. [33] M. I. Idiart, The macroscopic behavior of power-law and ideally plastic materials with elliptical distribution of porosity, Mechanics Research Communications, 35 (2008), 583-588. doi: 10.1016/j.mechrescom.2008.06.002. [34] M. Ikehata, Reconstruction of the shape of the inclusion by boundary measurements, Comm. Partial Differential Equations, 23 (1998), 1459-1474. doi: 10.1080/03605309808821390. [35] M. Ikehata, How to draw a picture of an unknown inclusion from boundary measurements. Two mathematical inversion algorithms, J. Inverse Ill-Posed Probl., 7 (1999), 255-271. doi: 10.1515/jiip.1999.7.3.255. [36] M. Ikehata, Reconstruction of obstacle from boundary measurements, Wave Motion, 30 (1999), 205-223. doi: 10.1016/S0165-2125(99)00006-2. [37] M. Ikehata, A new formulation of the probe method and related problems, Inverse Problems, 21 (2005), 413-426. doi: 10.1088/0266-5611/21/1/025. [38] M. Ikehata, {The probe and enclosure methods for inverse obstacle scattering problems. The past and present, RIMS Kôkyûroku, 1702 (2010), 1-22. [39] V. Isakov, On uniqueness of recovery of a discontinuous conductivity coefficient, Comm. Pure Appl. Math., 41 (1988), 865-877. doi: 10.1002/cpa.3160410702. [40] H. Kang, M. Lim and K. Yun, Asymptotics and computation of the solution to the conductivity equation in the presence of adjacent inclusions with extreme conductivities, J. Math. Pures Appl.(9), 99 (2013), 234-249. doi: 10.1016/j.matpur.2012.06.013. [41] M. Kar and M. Sini, Reconstruction of interfaces from the elastic farfield measurements using CGO solutions, SIAM J. Math. Anal., 46 (2014), 2650-2691. doi: 10.1137/120903130. [42] M. Kar and M. Sini, Reconstruction of interfaces using CGO solutions for the Maxwell equations, J. Inverse Ill-Posed Probl., 22 (2014), 169-208. doi: 10.1515/jip-2012-0054. [43] J. King and G. Richardson, The Hele-Shaw injection problem for an extremely shear-thinning fluid, European Journal of Applied Mathematics, 26 (2015), 563-594. doi: 10.1017/S095679251500039X. [44] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, vol. 36 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2008. [45] R. Kress and W. Rundell, Nonlinear integral equations and the iterative solution for an inverse boundary value problem, Inverse Problems, 21 (2005), 1207-1223. doi: 10.1088/0266-5611/21/4/002. [46] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Mathematics in Science and Engineering : a series of monographs and textbooks, Academic Press, 1968. [47] O. Levy and R. V. Kohn, Duality relations for non-Ohmic composites, with applications to behavior near percolation, Journal of Statistical Physics, 90 (1998), 159-189. doi: 10.1023/A:1023251701546. [48] J. L. Lewis, Note on a theorem of Wolff, in Holomorphic Functions and Moduli I, vol. 10 of Mathematical Sciences Research Institute Publications, Springer US, 1988, 93-100. [49] P. Lindqvist, Notes on the p-Laplace Equation, vol. 102 of Reports of University of Jyväskylä Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland, 2006. [50] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. [51] A. Moradifam, A. Nachman and A. Tamasan, Conductivity imaging from one interior measurement in the presence of perfectly conducting and insulating inclusions, SIAM Journal on Mathematical Analysis, 44 (2012), 3969-3990. doi: 10.1137/120866701. [52] A. Munnier and K. Ramdani, Conformal mapping for cavity inverse problem: An explicit reconstruction formula, Appl. Anal., 96 (2017), 108-129. doi: 10.1080/00036811.2016.1208816. [53] S. Nagayasu, G. Uhlmann and J.-N. 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##### References:
 [1] G. Alessandrini and A. D. Valenzuela, Unique determination of multiple cracks by two measurements, SIAM Journal on Control and Optimization, 34 (1996), 913-921. doi: 10.1137/S0363012994262853. [2] S. N. Antontsev and J. F. Rodrigues, On stationary thermo-rheological viscous flows, Annali dell'Universita di Ferrara, 52 (2006), 19-36. doi: 10.1007/s11565-006-0002-9. [3] G. Aronsson, On p-harmonic functions, convex duality and an asymptotic formula for injection mould filling, European Journal of Applied Mathematics, 7 (1996), 417-437. [4] K. Astala, M. Lassas and L. Päivärinta, The borderlines of invisibility and visibility in Calderón's inverse problem, Anal. PDE, 9 (2016), 43-98. doi: 10.2140/apde.2016.9.43. [5] K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Annals of Mathematics, 163 (2006), 265-299. doi: 10.4007/annals.2006.163.265. [6] C. Atkinson and C. R. Champion, Some boundary-value problems for the equation $\nabla · (|\nabla φ|^N \nabla φ)$, The Quarterly Journal of Mechanics and Applied Mathematics, 37 (1984), 401-419. [7] L. C. Berselli, L. Diening and M. Růžička, Existence of strong solutions for incompressible fluids with shear dependent viscosities, Journal of Mathematical Fluid Mechanics, 12 (2010), 101-132. doi: 10.1007/s00021-008-0277-y. [8] D. Borman, D. B. Ingham, B. T. Johansson and D. Lesnic, The method of fundamental solutions for detection of cavities in EIT, J. Integral Equations Applications, 21 (2009), 381-404. doi: 10.1216/JIE-2009-21-3-383. [9] T. Brander, Calderón problem for the p-Laplacian: First order derivative of conductivity on the boundary, Proceedings of American mathematical society, 144 (2016), 177-189, arXiv: 1403.0428. [10] T. Brander, M. Kar and M. Salo, Enclosure method for the p-Laplace equation, Inverse Problems, 31 (2015), 045001, 16pp, arXiv: 1410.4048. [11] T. Brander, B. von Harrach, M. Kar and M. Salo, Monotonicity and enclosure methods for the p-Laplace equation, ArXiv e-prints, Preprint arXiv: /1703.02814. [12] M. Brühl, Gebietserkennung in der Elektrischen Impedanztomographie, PhD thesis, Universität Karlsruhe, 1999. [13] P. R. Bueno, J. A. Varela and E. Longo, SnO2, ZnO and related polycrystalline compound semiconductors: An overview and review on the voltage-dependent resistance (non-ohmic) feature, Journal of the European Ceramic Society, 28 (2008), 505-529. doi: 10.1016/j.jeurceramsoc.2007.06.011. [14] F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Interaction of Mechanics and Mathematics, Springer-Verlag, Berlin, 2006. [15] F. Cakoni, M. Fares and H. Haddar, Analysis of two linear sampling methods applied to electromagnetic imaging of buried objects, Inverse Problems, 22 (2006), 845-867. doi: 10.1088/0266-5611/22/3/007. [16] A. P. Calderón, Lebesgue spaces of differentiable functions and distributions, in Partial Differential Equations, vol. 4 of Proceedings of Symposia in Pure Mathematics, American mathematical society, Providence, Rhode Island, USA, 1961, 33-49. [17] A. P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics (eds. W. Meyer and M. Raupp), Sociedade Brasileira de Matematica, 1980, 65-73, URL http://www.maths.manchester.ac.uk/~bl/Calderon/, Reprinted as [18]. [18] A. P. Calder´on, On an inverse boundary problem, Computation and applied mathematics, 25 (2006), 133-138, URL http://www.scielo.br/pdf/cam/v25n2-3/a02v2523.pdf, Reprint of [17]. [19] P. Caro and K. M. Rogers, Global uniqueness for the Calderón problem with Lipschitz conductivities, Forum of Mathematics, Pi, 4 (2016), e2, 28pp. doi: 10.1017/fmp.2015.9. [20] D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393. doi: 10.1088/0266-5611/12/4/003. [21] A. Friedman and M. Vogelius, Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence, Archive for Rational Mechanics and Analysis, 105 (1989), 299-326. doi: 10.1007/BF00281494. [22] A. Garroni and R. V. Kohn, Some three--dimensional problems related to dielectric breakdown and polycrystal plasticity, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 459 (2003), 2613-2625. doi: 10.1098/rspa.2003.1152. [23] A. Garroni, V. Nesi and M. Ponsiglione, Dielectric breakdown: Optimal bounds, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 457 (2001), 2317-2335. doi: 10.1098/rspa.2001.0803. [24] B. Gebauer, M. Hanke, A. Kirsch, W. Muniz and C. Schneider, A sampling method for detecting buried objects using electromagnetic scattering, Inverse Problems, 21 (2005), 2035-2050. doi: 10.1088/0266-5611/21/6/015. [25] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, vol. 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 2nd edition, Springer-Verlag, Berlin, 1983. [26] R. Glowinski and J. Rappaz, Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid flow model in glaciology, ESAIM: Mathematical Modelling and Numerical Analysis, 37 (2003), 175-186. doi: 10.1051/m2an:2003012. [27] Y. Gorb and A. Novikov, Blow-up of solutions to a p-Laplace equation, Multiscale Model. Simul., 10 (2012), 727-743. doi: 10.1137/110857167. [28] C.-Y. Guo, M. Kar and M. Salo, Inverse problems for p-Laplace type equations under monotonicity assumptions, Rend. Istit. Mat. Univ. Trieste, 48 (2016), 79-99. [29] B. Harrach, Recent progress on the factorization method for electrical impedance tomography Comput. Math. Methods Med. , 2013 (2013), Art. ID 425184, 8pp. doi: 10.1155/2013/425184. [30] B. Harrach and M. Ullrich, Monotonicity-based shape reconstruction in electrical impedance tomography, SIAM J. Math. Anal., 45 (2013), 3382-3403. doi: 10.1137/120886984. [31] D. Hauer, The p-Dirichlet-to-Neumann operator with applications to elliptic and parabolic problems, Journal of Differential Equations, 259 (2015), 3615-3655. doi: 10.1016/j.jde.2015.04.030. [32] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, Oxford, 1993, Oxford Science Publications. [33] M. I. Idiart, The macroscopic behavior of power-law and ideally plastic materials with elliptical distribution of porosity, Mechanics Research Communications, 35 (2008), 583-588. doi: 10.1016/j.mechrescom.2008.06.002. [34] M. Ikehata, Reconstruction of the shape of the inclusion by boundary measurements, Comm. Partial Differential Equations, 23 (1998), 1459-1474. doi: 10.1080/03605309808821390. [35] M. Ikehata, How to draw a picture of an unknown inclusion from boundary measurements. Two mathematical inversion algorithms, J. Inverse Ill-Posed Probl., 7 (1999), 255-271. doi: 10.1515/jiip.1999.7.3.255. [36] M. Ikehata, Reconstruction of obstacle from boundary measurements, Wave Motion, 30 (1999), 205-223. doi: 10.1016/S0165-2125(99)00006-2. [37] M. Ikehata, A new formulation of the probe method and related problems, Inverse Problems, 21 (2005), 413-426. doi: 10.1088/0266-5611/21/1/025. [38] M. Ikehata, {The probe and enclosure methods for inverse obstacle scattering problems. The past and present, RIMS Kôkyûroku, 1702 (2010), 1-22. [39] V. Isakov, On uniqueness of recovery of a discontinuous conductivity coefficient, Comm. Pure Appl. Math., 41 (1988), 865-877. doi: 10.1002/cpa.3160410702. [40] H. Kang, M. Lim and K. Yun, Asymptotics and computation of the solution to the conductivity equation in the presence of adjacent inclusions with extreme conductivities, J. Math. Pures Appl.(9), 99 (2013), 234-249. doi: 10.1016/j.matpur.2012.06.013. [41] M. Kar and M. Sini, Reconstruction of interfaces from the elastic farfield measurements using CGO solutions, SIAM J. Math. Anal., 46 (2014), 2650-2691. doi: 10.1137/120903130. [42] M. Kar and M. Sini, Reconstruction of interfaces using CGO solutions for the Maxwell equations, J. Inverse Ill-Posed Probl., 22 (2014), 169-208. doi: 10.1515/jip-2012-0054. [43] J. King and G. Richardson, The Hele-Shaw injection problem for an extremely shear-thinning fluid, European Journal of Applied Mathematics, 26 (2015), 563-594. doi: 10.1017/S095679251500039X. [44] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, vol. 36 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2008. [45] R. Kress and W. Rundell, Nonlinear integral equations and the iterative solution for an inverse boundary value problem, Inverse Problems, 21 (2005), 1207-1223. doi: 10.1088/0266-5611/21/4/002. [46] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Mathematics in Science and Engineering : a series of monographs and textbooks, Academic Press, 1968. [47] O. Levy and R. V. Kohn, Duality relations for non-Ohmic composites, with applications to behavior near percolation, Journal of Statistical Physics, 90 (1998), 159-189. doi: 10.1023/A:1023251701546. [48] J. L. 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