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Mat. Sb., 2016, Volume 207, Number 3, Pages 111–136 (Mi msb8486)  

The Neumann problem for elliptic equations with multiscale coefficients: operator estimates for homogenization

S. E. Pastukhova

Moscow Technological University

Abstract: We prove an $L^2$-estimate for the homogenization of an elliptic operator $A_\varepsilon$ in a domain $\Omega$ with a Neumann boundary condition on the boundary $\partial\Omega$. The coefficients of the operator $A_\varepsilon$ are rapidly oscillating over different groups of variables with periods of different orders of smallness as $\varepsilon\to 0$. We assume minimal regularity of the data, which makes it possible to impart to the result the meaning of an estimate in the operator $(L^2(\Omega)\to L^2(\Omega))$-norm for the difference of the resolvents of the original and homogenized problems. We also find an approximation to the resolvent of the original problem in the operator $(L^2(\Omega)\to H^1(\Omega))$-norm.
Bibliography: 24 titles.

Keywords: multiscale homogenization, operator estimates for homogenization, Steklov smoothing.

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-00192
Russian Science Foundation 14-11-00398
This research was supported by the Russian Foundation for Basic Research (grant no. 14-01-00192) and the Russian Science Foundation (project no. 14-11-00398).


DOI: https://doi.org/10.4213/sm8486

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English version:
Sbornik: Mathematics, 2016, 207:3, 418–443

Bibliographic databases:

Document Type: Article
UDC: 517.956.8
MSC: Primary 35B27; Secondary 35J57
Received: 04.02.2015 and 24.05.2015

Citation: S. E. Pastukhova, “The Neumann problem for elliptic equations with multiscale coefficients: operator estimates for homogenization”, Mat. Sb., 207:3 (2016), 111–136; Sb. Math., 207:3 (2016), 418–443

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