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

Abstract

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.

Keywords:

• electromagnetic inhomogeneities
• Maxwell's equations
• electromagnetic energy
• asymptotic behaviour

AMS Subject Classifications::

• 78A25
• 35B20
• 35B40
• 78M35

Acknowledgements

The authors would like to thank Professor M. Georgescu Vladimir for his valuable suggestions and constructive comments.