Abstract
A strange term arising in the homogenization of elliptic (and parabolic) equations with dynamic boundary conditions given on some boundary parts of critical size is considered. A problem with dynamic boundary conditions given on the union of some boundary subsets of critical size arranged ε-periodically along the boundary and with homogeneous Neumann conditions given on the rest of the boundary is studied. It is proved that the homogenized boundary condition is a Robin-type containing a nonlocal term depending on the trace of the solution u(x, t) on the boundary \(\partial \Omega \).
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Funding
The studies of J.I. Diaz and D. Gomez-Castro were supported in part by project no. MTM2017-85449-P (Ministerio de Ciencia, Innovación y Universidades Agencia Estatal de Investigación (Spain)). In addition, the studies of D. Gomez-Castro were supported in part by grant no. PG 2018-098440-BI00 (Ministerio de Ciencia, Innovacin y Universidades Agencia Estatal de Investigacin (Spain)).
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Translated by N. Berestova
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Díaz, J.I., Gómez-Castro, D., Shaposhnikova, T.A. et al. A Time-Dependent Strange Term Arising in Homogenization of an Elliptic Problem with Rapidly Alternating Neumann and Dynamic Boundary Conditions Specified at the Domain Boundary: The Critical Case. Dokl. Math. 101, 96–101 (2020). https://doi.org/10.1134/S106456242002009X
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DOI: https://doi.org/10.1134/S106456242002009X