Abstract
We study the essential spectrum of operator pencils associated with anisotropic Maxwell equations, with permittivity \(\varepsilon \), permeability \(\mu \) and conductivity \(\sigma \), on finitely connected unbounded domains. The main result is that the essential spectrum of the Maxwell pencil is the union of two sets: namely, the spectrum of the pencil \({{\,\mathrm{div}\,}}((\omega \varepsilon + i \sigma ) \nabla \,\cdot \,)\), and the essential spectrum of the Maxwell pencil with constant coefficients. We expect the analysis to be of more general interest and to open avenues to investigation of other questions concerning Maxwell’s and related systems.
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Acknowledgements
The authors express their sincere thanks to Dr. Pedro Caro of BCAM, who visited us on several occasions and provided a lot of helpful comments and useful insights. We are also very grateful to the two referees whose exceptionally careful reading of our first draft enabled us to make substantial improvements. We gratefully acknowledge the financial support of the UK Engineering and Physical Sciences Research Council under Grant EP/K024078/1 and the support of the LMS and EPSRC for our participation in the Durham Symposium on Mathematical and Computational Aspects of Maxwell’s Equations (Grant EP/K040154/1).
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Communicated by Jan Derezinski.
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Alberti, G.S., Brown, M., Marletta, M. et al. Essential Spectrum for Maxwell’s Equations. Ann. Henri Poincaré 20, 1471–1499 (2019). https://doi.org/10.1007/s00023-019-00762-x
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DOI: https://doi.org/10.1007/s00023-019-00762-x