Abstract
An elliptic second-order differential operator \(A_\varepsilon=b(\mathbf{D})^*g(\mathbf{x}/\varepsilon)b(\mathbf{D})\) on \(L_2(\mathbb{R}^d)\) is considered, where \(\varepsilon >0\), \(g(\mathbf{x})\) is a positive definite and bounded matrix-valued function periodic with respect to some lattice, and \(b(\mathbf{D})\) is a matrix first-order differential operator. Approximations for small \(\varepsilon\) of the operator-functions \(\cos(\tau A_\varepsilon^{1/2})\) and \(A_\varepsilon^{-1/2} \sin (\tau A_\varepsilon^{1/2})\) in various operator norms are obtained. The results can be applied to study the behavior of the solution of the Cauchy problem for the hyperbolic equation \(\partial^2_\tau \mathbf{u}_\varepsilon(\mathbf{x},\tau) = - A_\varepsilon \mathbf{u}_\varepsilon(\mathbf{x},\tau)\).
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Funding
This work was supported by the Russian Science Foundation under grant no. 22-11- 00092, https://rscf.ru/project/22-11-00092/.
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Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2023, Vol. 57, pp. 123–129 https://doi.org/10.4213/faa4149.
To the memory of Izrael Moiseevich Gelfand
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Dorodnyi, M.A., Suslina, T.A. Homogenization of Hyperbolic Equations: Operator Estimates with Correctors Taken into Account. Funct Anal Its Appl 57, 364–370 (2023). https://doi.org/10.1134/S0016266323040093
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DOI: https://doi.org/10.1134/S0016266323040093