Advanced Search
Article Contents
Article Contents

Explicit characterization of the support of non-linear inclusions

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 35R30, 31B20, 78A46; Secondary: 45Q05.


    \begin{equation} \\ \end{equation}
  • [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), 820-850.doi: 10.1016/j.jcp.2007.08.022.


    M. Brühl, Explicit characterization of inclusions in electrical impedance tomography, SIAM J. Math. Anal., 32 (2001), 1327-1341.doi: 10.1137/S003614100036656X.


    F. Cakoni, H. Haddar and D. Gintides, The existence of an infinite discrete set of transmission eigenvalues, SIAM J. Math. Anal., 42 (2010), 237-255.doi: 10.1137/090769338.


    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), 112-114.doi: 10.1016/0022-247X(67)90185-0.


    D. L. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, 1992.


    L. C. Evans, "Partial Differential Equations," Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.


    G. Fibich and B. Ilan, Vectorial and random effects in self-focusing and in multiple filamentation, Physica D, 157 (2001), 112-146.doi: 10.1016/S0167-2789(01)00293-7.


    M. Friesen and A. Gurevich, Nonlinear current flow in superconductors with restricted geometries, Phys. Rev. B, 63 (2001), 064521.doi: 10.1103/PhysRevB.63.064521.


    D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Second edition, Grundlehren der Mathematischen Wissenschaften, 224, Springer-Verlag, Berlin, 1983.


    N. Grinberg and A. Kirsch, The linear sampling method in inverse obstacle scattering for impedance boundary conditions, J. Inverse Ill-Posed Probl., 10 (2002), 171-185.


    M. Hanke and A. Kirsch, Sampling methods, in "Handbook of Mathematical Methods in Imaging" (ed. O. Scherzer), Springer, 2011.


    V. Isakov and J. Sylvester, Global uniqueness for a semilinear elliptic inverse problem, Comm. Pure. Appl. Math., 47 (1994), 1403-1410.doi: 10.1002/cpa.3160471005.


    V. Isakov and A. I. Nachman, Global uniqueness for a two-dimensional semilinear elliptic inverse problem, Trans. Amer. Math. Soc., 347 (1995), 3375-3390.doi: 10.2307/2155015.


    E. Jalade, Inverse problem for a nonlinear Helmholtz equation, Ann. I. H. Poincaré Anal. Non Linéaire, 21 (2004), 517-531.


    H. Kang and G. Nakamura, Identification of nonlinearity in a conductivity equation via the Dirichlet-to-Neumann map, Inverse Problems, 18 (2002), 1079-1088.doi: 10.1088/0266-5611/18/4/309.


    A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), 1489-1512.doi: 10.1088/0266-5611/14/6/009.


    A. Kirsch, New characterizations of solutions in inverse scattering theory, Applicable Analysis, 76 (2000), 319-350.doi: 10.1080/00036810008840888.


    A. Kirsch and N. I. Grinberg, "The Factorization Method for Inverse Problems," Oxford Lecture Series in Mathematics and its Applications, 36, Oxford University Press, Oxford, 2008.


    P. Kügler, Identification of a temperature dependent heat conductivity from single boundary measurements, SIAM J. Num. Anal., 41 (2003), 1543-1563.doi: 10.1137/S0036142902415900.


    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), 751-757.doi: 10.1364/JOSAB.17.000751.


    W. Rundell, Recovering an obstacle and a nonlinear conductivity from Cauchy data, Inverse Problems, 24 (2008), 12 pp.doi: 10.1088/0266-5611/24/5/055015.


    V. Serov and M. Harju, A uniqueness theorem and reconstruction of singularities for a two-dimensional nonlinear Schrödinger equation, Nonlinearity, 21 (2008), 1323-1337.doi: 10.1088/0951-7715/21/6/010.


    V. Serov, Inverse born approximation for the nonlinear two-dimensional Schrödinger operator, Inverse Problems, 23 (2007), 1259-1270.doi: 10.1088/0266-5611/23/3/024.


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


    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), 3637-3645.doi: 10.1090/S0002-9939-01-06016-6.

  • 加载中

Article Metrics

HTML views() PDF downloads(123) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint