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. AstalaM. 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. BerselliL. 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. BormanD. B. InghamB. 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. BuenoJ. 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. CakoniM. 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. GarroniV. 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. GebauerM. HankeA. KirschW. 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. GuoM. 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. KangM. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[50]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. Google Scholar

[51]

A. MoradifamA. 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. Google Scholar

[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. Google Scholar

[53]

S. NagayasuG. Uhlmann and J.-N. Wang, Reconstruction of penetrable obstacles in acoustic scattering, SIAM J. Math. Anal., 43 (2011), 189-211. doi: 10.1137/09076218X. Google Scholar

[54]

G. NakamuraG. Uhlmann and J.-N. Wang, Oscillating-decaying solutions, Runge approximation property for the anisotropic elasticity system and their applications to inverse problems, J. Math. Pures Appl.(9), 84 (2005), 21-54. doi: 10.1016/j.matpur.2004.09.002. Google Scholar

[55]

G. Nakamura and K. Yoshida, Identification of a non-convex obstacle for acoustical scattering, J. Inverse Ill-Posed Probl., 15 (2007), 611-624. doi: 10.1515/jiip.2007.034. Google Scholar

[56]

P. Ponte Castañeda and P. Suquet, Nonlinear composites, Advances in Applied Mechanics, 34 (1998), 171-302. Google Scholar

[57]

P. Ponte Castañeda and J. R. Willis, Variational second-order estimates for nonlinear composites, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 455 (1999), 1799-1811. doi: 10.1098/rspa.1999.0380. Google Scholar

[58]

R. Potthast, Sampling and probe methods-an algorithmical view, Computing, 75 (2005), 215-235. doi: 10.1007/s00607-004-0084-0. Google Scholar

[59]

M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, no. 1748 in Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000. Google Scholar

[60]

M. Salo and X. Zhong, An inverse problem for the p-Laplacian: Boundary determination, SIAM J. Math. Anal., 44 (2012), 2474-2495. doi: 10.1137/110838224. Google Scholar

[61]

S. Schmitt, Detection and Characterization of Inclusions in Impedance Tomography, PhD thesis, Karlsruher Institut für Technologie, 2010.Google Scholar

[62]

M. Sini and K. Yoshida, On the reconstruction of interfaces using complex geometrical optics solutions for the acoustic case Inverse Problems, 28 (2012), 055013, 22pp. doi: 10.1088/0266-5611/28/5/055013. Google Scholar

[63]

P. Suquet, Overall potentials and extremal surfaces of power law or ideally plastic composites, Journal of the Mechanics and Physics of Solids, 41 (1993), 981-1002. doi: 10.1016/0022-5096(93)90051-G. Google Scholar

[64]

D. R. S. Talbot and J. R. Willis, Upper and lower bounds for the overall properties of a nonlinear composite dielectric. I. Random microgeometry, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 447 (1994), 365-384, With second part [65]. doi: 10.1098/rspa.1994.0145. Google Scholar

[65]

D. R. S. Talbot and J. R. Willis, Upper and lower bounds for the overall properties of a nonlinear composite dielectric. II. Periodic microgeometry, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 447 (1994), 385-396, With first part [64].Google Scholar

[66]

G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse problems, 25 (2009), 123011, 39pp, URL http://www.dim.uchile.cl/~axosses/calderoniprevised.pdf. Google Scholar

[67]

G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains, J. Funct. Anal., 59 (1984), 572-611. doi: 10.1016/0022-1236(84)90066-1. Google Scholar

[68]

T. H. Wolff, Gap series constructions for the p-Laplacian, Journal d'Analyse Mathematique, 102 (2007), 371-394, Preprint written in 1984. doi: 10.1007/s11854-007-0026-9. Google Scholar

[69]

T. Zhou, Reconstructing electromagnetic obstacles by the enclosure method, Inverse Probl. Imaging, 4 (2010), 547-569. doi: 10.3934/ipi.2010.4.547. Google Scholar

show all references

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. AstalaM. 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. BerselliL. 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. BormanD. B. InghamB. 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. BuenoJ. 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. CakoniM. 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. GarroniV. 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. GebauerM. HankeA. KirschW. 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. GuoM. 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. KangM. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[50]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. Google Scholar

[51]

A. MoradifamA. 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. Google Scholar

[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. Google Scholar

[53]

S. NagayasuG. Uhlmann and J.-N. Wang, Reconstruction of penetrable obstacles in acoustic scattering, SIAM J. Math. Anal., 43 (2011), 189-211. doi: 10.1137/09076218X. Google Scholar

[54]

G. NakamuraG. Uhlmann and J.-N. Wang, Oscillating-decaying solutions, Runge approximation property for the anisotropic elasticity system and their applications to inverse problems, J. Math. Pures Appl.(9), 84 (2005), 21-54. doi: 10.1016/j.matpur.2004.09.002. Google Scholar

[55]

G. Nakamura and K. Yoshida, Identification of a non-convex obstacle for acoustical scattering, J. Inverse Ill-Posed Probl., 15 (2007), 611-624. doi: 10.1515/jiip.2007.034. Google Scholar

[56]

P. Ponte Castañeda and P. Suquet, Nonlinear composites, Advances in Applied Mechanics, 34 (1998), 171-302. Google Scholar

[57]

P. Ponte Castañeda and J. R. Willis, Variational second-order estimates for nonlinear composites, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 455 (1999), 1799-1811. doi: 10.1098/rspa.1999.0380. Google Scholar

[58]

R. Potthast, Sampling and probe methods-an algorithmical view, Computing, 75 (2005), 215-235. doi: 10.1007/s00607-004-0084-0. Google Scholar

[59]

M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, no. 1748 in Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000. Google Scholar

[60]

M. Salo and X. Zhong, An inverse problem for the p-Laplacian: Boundary determination, SIAM J. Math. Anal., 44 (2012), 2474-2495. doi: 10.1137/110838224. Google Scholar

[61]

S. Schmitt, Detection and Characterization of Inclusions in Impedance Tomography, PhD thesis, Karlsruher Institut für Technologie, 2010.Google Scholar

[62]

M. Sini and K. Yoshida, On the reconstruction of interfaces using complex geometrical optics solutions for the acoustic case Inverse Problems, 28 (2012), 055013, 22pp. doi: 10.1088/0266-5611/28/5/055013. Google Scholar

[63]

P. Suquet, Overall potentials and extremal surfaces of power law or ideally plastic composites, Journal of the Mechanics and Physics of Solids, 41 (1993), 981-1002. doi: 10.1016/0022-5096(93)90051-G. Google Scholar

[64]

D. R. S. Talbot and J. R. Willis, Upper and lower bounds for the overall properties of a nonlinear composite dielectric. I. Random microgeometry, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 447 (1994), 365-384, With second part [65]. doi: 10.1098/rspa.1994.0145. Google Scholar

[65]

D. R. S. Talbot and J. R. Willis, Upper and lower bounds for the overall properties of a nonlinear composite dielectric. II. Periodic microgeometry, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 447 (1994), 385-396, With first part [64].Google Scholar

[66]

G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse problems, 25 (2009), 123011, 39pp, URL http://www.dim.uchile.cl/~axosses/calderoniprevised.pdf. Google Scholar

[67]

G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains, J. Funct. Anal., 59 (1984), 572-611. doi: 10.1016/0022-1236(84)90066-1. Google Scholar

[68]

T. H. Wolff, Gap series constructions for the p-Laplacian, Journal d'Analyse Mathematique, 102 (2007), 371-394, Preprint written in 1984. doi: 10.1007/s11854-007-0026-9. Google Scholar

[69]

T. Zhou, Reconstructing electromagnetic obstacles by the enclosure method, Inverse Probl. Imaging, 4 (2010), 547-569. doi: 10.3934/ipi.2010.4.547. Google Scholar

[1]

Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems & Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355

[2]

Masaru Ikehata, Mishio Kawashita. An inverse problem for a three-dimensional heat equation in thermal imaging and the enclosure method. Inverse Problems & Imaging, 2014, 8 (4) : 1073-1116. doi: 10.3934/ipi.2014.8.1073

[3]

Zhousheng Ruan, Sen Zhang, Sican Xiong. Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method. Evolution Equations & Control Theory, 2018, 7 (4) : 669-682. doi: 10.3934/eect.2018032

[4]

Albert Clop, Daniel Faraco, Alberto Ruiz. Stability of Calderón's inverse conductivity problem in the plane for discontinuous conductivities. Inverse Problems & Imaging, 2010, 4 (1) : 49-91. doi: 10.3934/ipi.2010.4.49

[5]

Matteo Santacesaria. Note on Calderón's inverse problem for measurable conductivities. Inverse Problems & Imaging, 2019, 13 (1) : 149-157. doi: 10.3934/ipi.2019008

[6]

Petteri Piiroinen, Martin Simon. Probabilistic interpretation of the Calderón problem. Inverse Problems & Imaging, 2017, 11 (3) : 553-575. doi: 10.3934/ipi.2017026

[7]

Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569

[8]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271

[9]

Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems & Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121

[10]

Pedro Caro, Mikko Salo. Stability of the Calderón problem in admissible geometries. Inverse Problems & Imaging, 2014, 8 (4) : 939-957. doi: 10.3934/ipi.2014.8.939

[11]

Mohsen Tadi. A computational method for an inverse problem in a parabolic system. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 205-218. doi: 10.3934/dcdsb.2009.12.205

[12]

Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems & Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291

[13]

Yacheng Liu, Runzhang Xu. Potential well method for initial boundary value problem of the generalized double dispersion equations. Communications on Pure & Applied Analysis, 2008, 7 (1) : 63-81. doi: 10.3934/cpaa.2008.7.63

[14]

Yavar Kian, Morgan Morancey, Lauri Oksanen. Application of the boundary control method to partial data Borg-Levinson inverse spectral problem. Mathematical Control & Related Fields, 2019, 9 (2) : 289-312. doi: 10.3934/mcrf.2019015

[15]

Fausto Ferrari, Qing Liu, Juan Manfredi. On the characterization of $p$-harmonic functions on the Heisenberg group by mean value properties. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2779-2793. doi: 10.3934/dcds.2014.34.2779

[16]

Shenzhou Zheng, Laping Zhang, Zhaosheng Feng. Everywhere regularity for P-harmonic type systems under the subcritical growth. Communications on Pure & Applied Analysis, 2008, 7 (1) : 107-117. doi: 10.3934/cpaa.2008.7.107

[17]

Masaru Ikehata. On finding an obstacle with the Leontovich boundary condition via the time domain enclosure method. Inverse Problems & Imaging, 2017, 11 (1) : 99-123. doi: 10.3934/ipi.2017006

[18]

Oleg Yu. Imanuvilov, Masahiro Yamamoto. Calderón problem for Maxwell's equations in cylindrical domain. Inverse Problems & Imaging, 2014, 8 (4) : 1117-1137. doi: 10.3934/ipi.2014.8.1117

[19]

Yernat M. Assylbekov. Reconstruction in the partial data Calderón problem on admissible manifolds. Inverse Problems & Imaging, 2017, 11 (3) : 455-476. doi: 10.3934/ipi.2017021

[20]

Pasquale Candito, Giovanni Molica Bisci. Multiple solutions for a Navier boundary value problem involving the $p$--biharmonic operator. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 741-751. doi: 10.3934/dcdss.2012.5.741

2018 Impact Factor: 1.469

Metrics

  • PDF downloads (41)
  • HTML views (288)
  • Cited by (1)

Other articles
by authors

[Back to Top]