# 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [10] T. Brander, M. Kar and M. Salo, Enclosure method for the p-Laplace equation, Inverse Problems, 31 (2015), 045001, 16pp, arXiv: 1410.4048. Google Scholar [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.Google Scholar [12] M. Brühl, Gebietserkennung in der Elektrischen Impedanztomographie, PhD thesis, Universität Karlsruhe, 1999.Google Scholar [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. Google Scholar [14] F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Interaction of Mechanics and Mathematics, Springer-Verlag, Berlin, 2006. Google Scholar [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. Google Scholar [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. Google Scholar [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]. Google Scholar [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]. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [36] M. Ikehata, Reconstruction of obstacle from boundary measurements, Wave Motion, 30 (1999), 205-223. doi: 10.1016/S0165-2125(99)00006-2. Google Scholar [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. Google Scholar [38] M. Ikehata, {The probe and enclosure methods for inverse obstacle scattering problems. The past and present, RIMS Kôkyûroku, 1702 (2010), 1-22. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. 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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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [10] T. Brander, M. Kar and M. Salo, Enclosure method for the p-Laplace equation, Inverse Problems, 31 (2015), 045001, 16pp, arXiv: 1410.4048. Google Scholar [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.Google Scholar [12] M. Brühl, Gebietserkennung in der Elektrischen Impedanztomographie, PhD thesis, Universität Karlsruhe, 1999.Google Scholar [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. Google Scholar [14] F. Cakoni and D. Colton, Qualitative Methods in Inverse Scattering Theory, Interaction of Mechanics and Mathematics, Springer-Verlag, Berlin, 2006. Google Scholar [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. Google Scholar [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. Google Scholar [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]. Google Scholar [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]. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [36] M. Ikehata, Reconstruction of obstacle from boundary measurements, Wave Motion, 30 (1999), 205-223. doi: 10.1016/S0165-2125(99)00006-2. Google Scholar [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. Google Scholar [38] M. Ikehata, {The probe and enclosure methods for inverse obstacle scattering problems. The past and present, RIMS Kôkyûroku, 1702 (2010), 1-22. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. 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