# American Institute of Mathematical Sciences

August  2011, 5(3): 675-694. doi: 10.3934/ipi.2011.5.675

## Explicit characterization of the support of non-linear inclusions

 1 INRIA Saclay–Ile-de-France and CMAP, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France

Received  September 2010 Revised  June 2011 Published  August 2011

We study inverse problems for non-linear penetrable media in the context of scattering theory and impedance tomography. Using a general description of the range of the non-linear far-field operator we show an explicit characterization of the support of a weakly non-linear inhomogeneous scattering object. Application of the same technique to the impedance tomography problem for a monotonic non-linear inclusion yields a characterization of the inclusion's support from the non-linear Neumann-to-Dirichlet operator.
Citation: Armin Lechleiter. Explicit characterization of the support of non-linear inclusions. Inverse Problems & Imaging, 2011, 5 (3) : 675-694. doi: 10.3934/ipi.2011.5.675
##### References:
 [1] G. Baruch, G. Fibich and S. Tsynkov, High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension,, J. Comput. Phys., 227 (2007), 820. doi: 10.1016/j.jcp.2007.08.022. Google Scholar [2] M. Brühl, Explicit characterization of inclusions in electrical impedance tomography,, SIAM J. Math. Anal., 32 (2001), 1327. doi: 10.1137/S003614100036656X. Google Scholar [3] F. Cakoni, H. Haddar and D. Gintides, The existence of an infinite discrete set of transmission eigenvalues,, SIAM J. Math. Anal., 42 (2010), 237. doi: 10.1137/090769338. Google Scholar [4] J. R. Cannon, Determination of the unknown coefficient $k(u)$ in the equation $\nabla k(u) \nabla u = 0$ from overspecified boundary data,, J. Math. Anal. Appl., 18 (1967), 112. doi: 10.1016/0022-247X(67)90185-0. Google Scholar [5] D. L. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,", Applied Mathematical Sciences, 93 (1992). Google Scholar [6] L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (1998). Google Scholar [7] G. Fibich and B. Ilan, Vectorial and random effects in self-focusing and in multiple filamentation,, Physica D, 157 (2001), 112. doi: 10.1016/S0167-2789(01)00293-7. Google Scholar [8] M. Friesen and A. Gurevich, Nonlinear current flow in superconductors with restricted geometries,, Phys. Rev. B, 63 (2001). doi: 10.1103/PhysRevB.63.064521. Google Scholar [9] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Second edition, 224 (1983). Google Scholar [10] N. Grinberg and A. Kirsch, The linear sampling method in inverse obstacle scattering for impedance boundary conditions,, J. Inverse Ill-Posed Probl., 10 (2002), 171. Google Scholar [11] M. Hanke and A. Kirsch, Sampling methods,, in, (2011). Google Scholar [12] V. Isakov and J. Sylvester, Global uniqueness for a semilinear elliptic inverse problem,, Comm. Pure. Appl. Math., 47 (1994), 1403. doi: 10.1002/cpa.3160471005. Google Scholar [13] V. Isakov and A. I. Nachman, Global uniqueness for a two-dimensional semilinear elliptic inverse problem,, Trans. Amer. Math. Soc., 347 (1995), 3375. doi: 10.2307/2155015. Google Scholar [14] E. Jalade, Inverse problem for a nonlinear Helmholtz equation,, Ann. I. H. Poincaré Anal. Non Linéaire, 21 (2004), 517. Google Scholar [15] H. Kang and G. Nakamura, Identification of nonlinearity in a conductivity equation via the Dirichlet-to-Neumann map,, Inverse Problems, 18 (2002), 1079. doi: 10.1088/0266-5611/18/4/309. Google Scholar [16] A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator,, Inverse Problems, 14 (1998), 1489. doi: 10.1088/0266-5611/14/6/009. Google Scholar [17] A. Kirsch, New characterizations of solutions in inverse scattering theory,, Applicable Analysis, 76 (2000), 319. doi: 10.1080/00036810008840888. Google Scholar [18] A. Kirsch and N. I. Grinberg, "The Factorization Method for Inverse Problems,", Oxford Lecture Series in Mathematics and its Applications, 36 (2008). Google Scholar [19] P. Kügler, Identification of a temperature dependent heat conductivity from single boundary measurements,, SIAM J. Num. Anal., 41 (2003), 1543. doi: 10.1137/S0036142902415900. Google Scholar [20] T. A. Laine and A. T. Friberg, Self-guided waves and exact solutions of the nonlinear Helmholtz equation,, J. Opt. Soc. Am. B Opt.Phys., 17 (2000), 751. doi: 10.1364/JOSAB.17.000751. Google Scholar [21] W. Rundell, Recovering an obstacle and a nonlinear conductivity from Cauchy data,, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/5/055015. Google Scholar [22] V. Serov and M. Harju, A uniqueness theorem and reconstruction of singularities for a two-dimensional nonlinear Schrödinger equation,, Nonlinearity, 21 (2008), 1323. doi: 10.1088/0951-7715/21/6/010. Google Scholar [23] V. Serov, Inverse born approximation for the nonlinear two-dimensional Schrödinger operator,, Inverse Problems, 23 (2007), 1259. doi: 10.1088/0266-5611/23/3/024. Google Scholar [24] Z. Sun, On a quasilinear inverse boundary value problem,, Mathematische Zeitschrift, 221 (1996), 293. doi: 10.1007/BF02622117. Google Scholar [25] R. Weder, Inverse scattering for the nonlinear Schrödinger equation. II. Reconstruction of the potential and the nonlinearity in the multidimensional case,, Proc. Amer. Math. Soc., 129 (2001), 3637. doi: 10.1090/S0002-9939-01-06016-6. Google Scholar

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##### References:
 [1] G. Baruch, G. Fibich and S. Tsynkov, High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension,, J. Comput. Phys., 227 (2007), 820. doi: 10.1016/j.jcp.2007.08.022. Google Scholar [2] M. Brühl, Explicit characterization of inclusions in electrical impedance tomography,, SIAM J. Math. Anal., 32 (2001), 1327. doi: 10.1137/S003614100036656X. Google Scholar [3] F. Cakoni, H. Haddar and D. Gintides, The existence of an infinite discrete set of transmission eigenvalues,, SIAM J. Math. Anal., 42 (2010), 237. doi: 10.1137/090769338. Google Scholar [4] J. R. Cannon, Determination of the unknown coefficient $k(u)$ in the equation $\nabla k(u) \nabla u = 0$ from overspecified boundary data,, J. Math. Anal. Appl., 18 (1967), 112. doi: 10.1016/0022-247X(67)90185-0. Google Scholar [5] D. L. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,", Applied Mathematical Sciences, 93 (1992). Google Scholar [6] L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, 19 (1998). Google Scholar [7] G. Fibich and B. Ilan, Vectorial and random effects in self-focusing and in multiple filamentation,, Physica D, 157 (2001), 112. doi: 10.1016/S0167-2789(01)00293-7. Google Scholar [8] M. Friesen and A. Gurevich, Nonlinear current flow in superconductors with restricted geometries,, Phys. Rev. B, 63 (2001). doi: 10.1103/PhysRevB.63.064521. Google Scholar [9] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Second edition, 224 (1983). Google Scholar [10] N. Grinberg and A. Kirsch, The linear sampling method in inverse obstacle scattering for impedance boundary conditions,, J. Inverse Ill-Posed Probl., 10 (2002), 171. Google Scholar [11] M. Hanke and A. Kirsch, Sampling methods,, in, (2011). Google Scholar [12] V. Isakov and J. Sylvester, Global uniqueness for a semilinear elliptic inverse problem,, Comm. Pure. Appl. Math., 47 (1994), 1403. doi: 10.1002/cpa.3160471005. Google Scholar [13] V. Isakov and A. I. Nachman, Global uniqueness for a two-dimensional semilinear elliptic inverse problem,, Trans. Amer. Math. Soc., 347 (1995), 3375. doi: 10.2307/2155015. Google Scholar [14] E. Jalade, Inverse problem for a nonlinear Helmholtz equation,, Ann. I. H. Poincaré Anal. Non Linéaire, 21 (2004), 517. Google Scholar [15] H. Kang and G. Nakamura, Identification of nonlinearity in a conductivity equation via the Dirichlet-to-Neumann map,, Inverse Problems, 18 (2002), 1079. doi: 10.1088/0266-5611/18/4/309. Google Scholar [16] A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator,, Inverse Problems, 14 (1998), 1489. doi: 10.1088/0266-5611/14/6/009. Google Scholar [17] A. Kirsch, New characterizations of solutions in inverse scattering theory,, Applicable Analysis, 76 (2000), 319. doi: 10.1080/00036810008840888. Google Scholar [18] A. Kirsch and N. I. Grinberg, "The Factorization Method for Inverse Problems,", Oxford Lecture Series in Mathematics and its Applications, 36 (2008). Google Scholar [19] P. Kügler, Identification of a temperature dependent heat conductivity from single boundary measurements,, SIAM J. Num. Anal., 41 (2003), 1543. doi: 10.1137/S0036142902415900. Google Scholar [20] T. A. Laine and A. T. Friberg, Self-guided waves and exact solutions of the nonlinear Helmholtz equation,, J. Opt. Soc. Am. B Opt.Phys., 17 (2000), 751. doi: 10.1364/JOSAB.17.000751. Google Scholar [21] W. Rundell, Recovering an obstacle and a nonlinear conductivity from Cauchy data,, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/5/055015. Google Scholar [22] V. Serov and M. Harju, A uniqueness theorem and reconstruction of singularities for a two-dimensional nonlinear Schrödinger equation,, Nonlinearity, 21 (2008), 1323. doi: 10.1088/0951-7715/21/6/010. Google Scholar [23] V. Serov, Inverse born approximation for the nonlinear two-dimensional Schrödinger operator,, Inverse Problems, 23 (2007), 1259. doi: 10.1088/0266-5611/23/3/024. Google Scholar [24] Z. Sun, On a quasilinear inverse boundary value problem,, Mathematische Zeitschrift, 221 (1996), 293. doi: 10.1007/BF02622117. Google Scholar [25] R. Weder, Inverse scattering for the nonlinear Schrödinger equation. II. Reconstruction of the potential and the nonlinearity in the multidimensional case,, Proc. Amer. Math. Soc., 129 (2001), 3637. doi: 10.1090/S0002-9939-01-06016-6. Google Scholar
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