JournalsjstVol. 11, No. 2pp. 587–660

On operator error estimates for homogenization of hyperbolic systems with periodic coefficients

  • Yulia M. Meshkova

    St. Petersburg State University, Russia
On operator error estimates for homogenization of hyperbolic systems with periodic coefficients cover

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Abstract

In L2(Rd;Cn)L_2(\mathbb{R}^d;\mathbb{C}^n), we consider a selfadjoint matrix strongly elliptic second order differential operator Aε\mathcal{A}_\varepsilon, ε>0\varepsilon > 0. The coefficients of the operator Aε\mathcal{A}_\varepsilon are periodic and depend on x/ε\mathbf{x}/\varepsilon. We study the asymptotic behavior of the operator Aε1/2sin(τAε1/2)\mathcal{A}_\varepsilon ^{-1/2}\sin (\tau \mathcal{A}_\varepsilon ^{1/2}), τR\tau\in\mathbb{R}, in the small period limit. The principal term of approximation in the (H1L2)(H^1 \to L_2)-norm for this operator is found. Approximation in the (H2H1)(H^2 \to H^1)-operator norm with the correction term taken into account is also established. The error estimates are of the sharp order O(ε)O(\varepsilon). The results are applied to homogenization for the solutions of the hyperbolic equation τ2uε=Aεuε+F\partial ^2_\tau \mathbf{u}_\varepsilon =-\mathcal{A}_\varepsilon \mathbf{u}_\varepsilon +\mathbf{F}. As examples, we consider the acoustics equation, the system of elasticity, and the model equation of electrodynamics.

Cite this article

Yulia M. Meshkova, On operator error estimates for homogenization of hyperbolic systems with periodic coefficients. J. Spectr. Theory 11 (2021), no. 2, pp. 587–660

DOI 10.4171/JST/350