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Asymptotics of solutions of the stationary and nonstationary Maxwell systems in a domain with small cavities


Authors: D. V. Korikov and B. A. Plamenevskiĭ
Translated by: B. A. Plamenevskiĭ
Original publication: Algebra i Analiz, tom 28 (2016), nomer 4.
Journal: St. Petersburg Math. J. 28 (2017), 507-554
MSC (2010): Primary 35Q61
DOI: https://doi.org/10.1090/spmj/1462
Published electronically: May 4, 2017
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Abstract: The nonstationary Maxwell system is considered, for all times $ t\in \mathbb{R}$, in a bounded domain $ \Omega (\varepsilon )\subset \mathbb{R}^3$ with finitely many small cavities; the cavity diameters are proportional to a small parameter $ \varepsilon $. The perfect conductivity conditions or the impedance conditions are prescribed on the boundary of $ \Omega (\varepsilon )$. The asymptotics of solutions are derived as $ \varepsilon $ tends to zero. The cavities are ``singular'' perturbations of the domain $ \Omega (0)$: they are collapsing into points as $ \varepsilon \to 0$. The presented mathematical model describes the electromagnetic field behavior inside a conductive resonator with metallic inclusions of small size. This model can be of use for the diagnostics of plasma filling a resonator and containing such inclusions.

To describe the asymptotics of solutions, the method of compound asymptotic expansions is employed. The asymptotics in question is formed by solutions of the ``limit'' problems independent of $ \varepsilon $. One of such problems turns out to be a non-stationary problem in a bounded domain with singular points on the boundary. The other limit problems are stationary in the complements of bounded domains. The method of compound asymptotics makes it possible to study the behavior of waves with length greater than the diameters of cavities. It is shown that the contribution of the short waves into the asymptotics of solutions is negligible, due to the smoothness of the right-hand side of the Maxwell system in time.


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Additional Information

D. V. Korikov
Affiliation: St. Petersburg State University, Universitetskaya nab. 7/9, 199034 St. Petersburg, Russia
Email: st020856@student.spbu.ru

B. A. Plamenevskiĭ
Affiliation: St. Petersburg State University, Universitetskaya nab. 7/9, 199034 St. Petersburg, Russia
Email: boris.plamen@gmail.com

DOI: https://doi.org/10.1090/spmj/1462
Keywords: Maxwell dynamical system, singularly perturbed domains, impedance boundary conditions, asymptotics of solutions
Received by editor(s): February 29, 2016
Published electronically: May 4, 2017
Additional Notes: The first author was supported by the fund “Dynasty”, a Russian Federation Government stipend, and a Rokhlin stipend; the second author was supported by St. Petersburg State University grant no. 11.42.1075.2016.
Article copyright: © Copyright 2017 American Mathematical Society