Abstract
Homogenization of the Neumann problem for a differential equation in a periodically broken multidimensional cylinder leads to a second-order ordinary differential equation. We study asymptotics for the coefficient of the averaged operator in the case of small transverse cross-sections. The main asymptotic term depends on the “area” of cross-sections of the links, their lengths, and the coefficient matrix of the original operator. We find the characteristics of kink zones which affect correction terms, while the asymptotic remainder becomes exponentially small. The justification of the asymptotics is based on Friedrichs’s inequality with a coefficient independent of both small parameters: the period of fractures and the relative diameter of cross-sections.
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Notes
The accuracy just as impressive was attained in [10] while modeling a Stokes flow in a branching thin blood vessel.
For brevity, it is convenient to denote by \( \mathbf{A}^{\pm} \) the truncated \( (n-1)\times(n-1) \)-matrix and decorate the full one with \( \square \).
\( {\mathcal{C}}^{\circ}_{0} \) is a continuous linear functional of \( \mathbf{F}\in{W}^{1}_{-\beta}(\Pi^{\circ})^{\ast} \); see Remark 2.
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The authors were financially supported by the Russian Science Foundation (Project 22–11–00046).
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Translated from Sibirskii Matematicheskii Zhurnal, 2024, Vol. 65, No. 2, pp. 374–393. https://doi.org/10.33048/smzh.2024.65.211
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Nazarov, S.A., Slutskii, A.S. Homogenization of the Scalar Boundary Value Problem in a Thin Periodically Broken Cylinder. Sib Math J 65, 363–380 (2024). https://doi.org/10.1134/S0037446624020113
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DOI: https://doi.org/10.1134/S0037446624020113