Annales Henri Poincaré

, Volume 20, Issue 5, pp 1471–1499 | Cite as

Essential Spectrum for Maxwell’s Equations

  • Giovanni S. Alberti
  • Malcolm Brown
  • Marco MarlettaEmail author
  • Ian Wood
Open Access


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.

Mathematics Subject Classification

35Q61 35P05 35J46 78A25 



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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© The Author(s) 2019

OpenAccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Giovanni S. Alberti
    • 1
  • Malcolm Brown
    • 2
  • Marco Marletta
    • 3
    Email author
  • Ian Wood
    • 4
  1. 1.Department of MathematicsUniversity of GenoaGenoaItaly
  2. 2.Cardiff School of Computer Science and InformaticsCardiffUK
  3. 3.Cardiff School of MathematicsCardiffUK
  4. 4.School of Mathematics, Statistics and Actuarial ScienceUniversity of KentCanterburyUK

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