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Articles

An integral equation method for the Helmholtz problem in the presence of small anisotropic inclusions

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Pages 384-400 | Received 09 Jun 2019, Accepted 04 Oct 2020, Published online: 02 Nov 2020

References

  • Colton D, Kress R. Inverse acoustic and electromagnetic scattering theory. 3rd ed. New York: Springer; 2013.
  • Buchuckuri T, Duduchava R, Kapanadze D, et al. Localisation of Helmholtz boundary value problem in a domain with a piecewise-smooth boundary. Proc A R Math Inst. 2013;62:37–44.
  • Gozzi M, Khelifi A. On the behavior of resonant frequencies in the presence of anisotropic imperfections. Indag Matematicae. 2017;28(6):1240–1257.
  • Kirsch A, Hettlich F. The mathematical theory of time-harmonic Maxwell's equations: expansion integral and variational methods. Heidelberg: Springer; 2015. (Appl. Math. Sc.; vol. 190).
  • Sauter SA, Schwaber C. Boundary elements methods. New York: Springer; 2011. (Series in Comp. Mathematics; vol. 39).
  • Ammari H, Iakovila H, Moskow S. Recovery of small inhomogeities from the scattering amplitude at a fixed frequency. SIAM J Math Anal. 2003;34(4):882–900.
  • Ammari H, Moskow S, Vogelius M. Boundary integral formulas for the reconstruction of electromagnetic imperfections of small diameter. ESAIM Con Opt Cal Var. 2003;9:49–66.
  • Asch M, Mefire SM. Numerical localization of electromagnetic imperfections from a perturbation formula in three dimensions. J Comp Math. 2008;26(2):149–195.
  • Coatléven J. Helmholtz equation in periodic media with a line defect. J Comp Phys. 2012;231:1675–1704.
  • Nédélec JC. Acoustic and electromagnetic equations. Integral representations for harmonic problem. New York: Springer; 2001. (Appl. Math. Sc.; 144).
  • Zribi H. Asymptotic expansions for currents caused by small interface changes of electromagnetic inclusions. Appl Anal. 2013;92(1):172–190.
  • Albrecht D, Duong X, Mc Intosh A. Operator theory and harmonic analysis. Workshop on analysis and geometry, part III. Proc Cen Math Appl Austral Nat Univ. 1996;34:77–136.
  • Amick CJ. Some remarks on Rellich's theorem and Poincaré inequality. J London Math. Soc. 1978;2:1–18.
  • Ammari H, Kang H. Reconstruction of small inhomogeneities for boundary measurements. Berlin: Springer; 2004. (Lecture Notes in Mathematics; vol. 1846).
  • Kato T. Perturbation theory for linear operators. Berlin: Springer; 1980.
  • Lax PD. Functional analysis. New York: Wiley-Interscience; 2002.
  • Melrose RB. Geometric scattering theory. Cambridge: Cambridge Univ. Press; 1995.
  • Apel T, Nicaise S, Sirch D. A posteriori error estimation of residual type for anisotropic diffusion-convection-reaction problems. J Comp Appl Math. 2011;235:2805–2820.
  • Bondarenko O, Kirsch A. The factorization method for inverse scattering by a penetrable anisotropic obstacle with conductive boundary condition. Inverse Probl. 2016;32:105011.
  • Fleck JA, Feit Jr MD. Beam propagation in uniaxial anisotropic media. J Opt Soc America. 1983;73(7):920–926.
  • Lakshtanov E, Vainberg B. Ellipcity in the interior transmission problem in anisotropic media. SIAM J Math Anal. 2012;44(2):1165–1174.
  • Zhu H. The method of approximate particular solutions for solving anisotropic elliptic problems. Engi Anal Boun Elem. 2014;40:123–127.
  • Kang H, Kim K. Anisotropic polarization tensors for ellipses and ellipsoids. J Comp Math. 2007;25(2):157–168.
  • Kang H, Kim E, Kim K. Anisotropic polarization tensors and detection of anisotropic inclusion,. SIAM J Appl Math. 2003;63(4):1276–1291.
  • Brezis H. Analyse fonctionnelle: théorie et applications. Paris: Masson; 1983.
  • Ammari H, Kang H, Kim K. Polarization tensors and effective properties of anisotropic composite materials. J Differ Equ. 2005;215:401–428.
  • Ammari H, Khelifi A. Electromagnetic scattering by small dielectric inhomogeneities. J Math Pures Appl. 2003;7:749–842.
  • Vogelius MS, Volkov D. Asymptotic formulas for perturbation in the electromagnetic fields due to the presence of inhomogeneities. Math Mod Num Anal. 2000;34:723–748.

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