Asymptotic behaviour of the energy for electromagnetic systems in the presence of small inhomogeneities

In this article we consider solutions to the time-harmonic and time-dependent Maxwell's systems with piecewise constant coefficients with a finite number of small inhomogeneities in ℝ3. In time-harmonic case and for such solutions, we derive the asymptotic expansions due to the presence of small inhomogeneities embedded in the entire space. Further, we analyse the behaviour of the electromagnetic energy caused by the presence of these inhomogeneities. For a general time-dependent case, we show that the local electromagnetic energy, trapped in the total collection of these well-separated inhomogeneities, decays towards zero as the shape parameter decreases to zero or as time increases.

[1]  Jaime E. Muñoz Rivera,et al.  Asymptotic behaviour of the energy for electromagnetic systems with memory , 2004 .

[2]  A. M. Ilʹin,et al.  Matching of Asymptotic Expansions of Solutions of Boundary Value Problems , 1992 .

[3]  K.A. Michalski,et al.  Electromagnetic wave theory , 1987, Proceedings of the IEEE.

[4]  Ekaterina Iakovleva,et al.  MUSIC-Type Electromagnetic Imaging of a Collection of Small Three-Dimensional Inclusions , 2007, SIAM J. Sci. Comput..

[5]  M. Lockwood,et al.  The correct application of Poynting’s theorem to the time-dependent magnetosphere: reply to Heikkila , 1999 .

[6]  J. Jiménez,et al.  About Poynting's theorem , 1992 .

[7]  Habib Ammari,et al.  Correction of order three for the expansion of two dimensional electromagnetic fields perturbed by the presence of inhomogeneities of small diameter , 2003 .

[8]  A. Khelifi,et al.  On the perturbation of the electromagnetic energy due to the presence of small inhomogeneities , 2008 .

[9]  G. Szegö,et al.  Virtual mass and polarization , 1949 .

[10]  Michael Vogelius,et al.  Identification of conductivity imperfections of small diameter by boundary measurements. Continuous , 1998 .

[11]  Avner Friedman,et al.  Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence , 1989 .

[12]  Ziolkowski,et al.  Localized transmission of electromagnetic energy. , 1989, Physical review. A, General physics.

[13]  H. Ammari,et al.  Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter II. The full Maxwell equations , 2001 .

[14]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .

[15]  Christophe Hazard,et al.  On the solution of time-harmonic scattering problems for Maxwell's equations , 1996 .

[16]  M. Vogelius,et al.  Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogen , 2000 .

[17]  Cathleen S. Morawetz,et al.  Decay for solutions of the exterior problem for the wave equation , 1975 .

[18]  B. Ducomet.  Decay of electromagnetic energy in a perturbed half-space , 1995 .

[19]  H. Ammari,et al.  Electromagnetic scattering by small dielectric inhomogeneities , 2003 .