Homogenization of a nonstationary model equation of electrodynamics
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In L 2(ℝ3;ℂ3), we consider a self-adjoint operator ℒ ε , ε > 0, generated by the differential expression curl η(x/ε)−1 curl−∇ν(x/ε) div. Here the matrix function η(x) with real entries and the real function ν(x) are periodic with respect to some lattice, are positive definite, and are bounded. We study the behavior of the operators cos(τℒ ε 1/2 ) and ℒ ε −1/2 sin(τℒ ε 1/2 ) for τ ∈ ℝ and small ε. It is shown that these operators converge to cos(τ(ℒ0)1/2) and (ℒ0)−1/2 sin(τ(ℒ0)1/2), respectively, in the norm of the operators acting from the Sobolev space H s (with a suitable s) to ℒ2. Here ℒ0 is an effective operator with constant coefficients. Error estimates are obtained and the sharpness of the result with respect to the type of operator norm is studied. The results are used for homogenizing the Cauchy problem for the model hyperbolic equation ∂ τ 2 v ε = −ℒ ε v ε , div v ε = 0, appearing in electrodynamics. We study the application to a nonstationary Maxwell system for the case in which the magnetic permeability is equal to 1 and the dielectric permittivity is given by the matrix η(x/ε).
Keywordsperiodic differential operator homogenization operator error estimate nonstationary Maxwell system
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- 1.A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, in Stud. Math. Appl. (North-Holland Publ., Amsterdam, 1978), Vol. 5.Google Scholar
- 3.V. V. Zhikov, S. M. Kozlov, and O. A. Olejnik (Oleinik), Homogenization of Differential Operators (Fizmatlit, Moscow, 1993; Springer-Verlag, Berlin, 1994).Google Scholar
- 6.M. Sh. Birman and T. A. Suslina, “Homogenization with corrector term for periodic differential operators. Approximation of solutions in the Sobolev class H1(Rd),” Algebra Anal. 18 (6), 1–130 (2006) [St. Petersburg Math. J. 18 (6), 857–955 (2007)].Google Scholar
- 7.M. Sh. Birman and T. A. Suslina, “Operator error estimates in the homogenization problem for nonstationary periodic equations,” Algebra Anal. 20 (6), 30–107 (2008) [St. PetersburgMath. J. 20 (6), 873–928 (2009)].Google Scholar
- 15.Yu. M. Meshkova, On Operator Error Estimates for Homogenization of Hyperbolic Systems with Periodic Coefficients, arXiv: https://arxiv.org/abs/1705.02531v3 (2017).Google Scholar
- 17.M. A. Dorodnyi and T. A. Suslina, Spectral Approach to Homogenization of Hyperbolic Equations with Periodic Coefficients, arXiv: https://arxiv.org/abs/1708.00859.Google Scholar
- 18.T. A. Suslina, “Homogenization of a stationary periodic Maxwell system,” Algebra Anal. 16 (5), 162–244 (2004) [St. PetersburgMath. J. 16 (5), 863–922 (2004)].Google Scholar