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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



On operator-type homogenization estimates for elliptic equations with lower order terms

Authors: S. E. Pastukhova and R. N. Tikhomirov
Translated by: S. E. Pastukhova
Original publication: Algebra i Analiz, tom 29 (2017), nomer 5.
Journal: St. Petersburg Math. J. 29 (2018), 841-861
MSC (2010): Primary 51B10, 53C50
Published electronically: July 26, 2018
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Abstract: In the space $ \mathbb{R}^d$, a divergent-type second order elliptic equation in a nonselfadjoint form is studied. The coefficients of the equation oscillate with a period $ \varepsilon {\to } 0$. They can be unbounded in lower order terms. In this case, they are subordinate to some integrability conditions over the unit periodicity cell. An $ L^2$-estimate of order of $ O(\varepsilon )$ is proved for the difference of solutions of the original and homogenized problems. The estimate is of operator type. It can be stated as an estimate for the difference of the corresponding resolvents in the operator ( $ L^2(\mathbb{R}^d){\to } L^2(\mathbb{R}^d)$)-norm. Also, an $ H^1$-approximation is found for the original solution with error estimate of order of $ O(\varepsilon )$. This estimate, also of operator type, implies that an appropriate approximation of order of $ O(\varepsilon )$ is found for the original resolvent in the operator ( $ L^2(\mathbb{R}^d){\to } H^1(\mathbb{R}^d)$)-norm.

The results are obtained with the help of the so-called shift method, first suggested by V. V. Zhikov.

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Additional Information

S. E. Pastukhova
Affiliation: Moscow Technology University (MIREA), pr. Vernadskogo 78, 119454 Moscow, Russia

R. N. Tikhomirov
Affiliation: A. G. and N. G. Stoletov Vladimir state University, ul. Gorkogo 87, 600000 Vladimir, Russia

Keywords: Homogenization, error estimate, first approximation, integration over an additional parameter, Steklov`s smoothing
Received by editor(s): October 25, 2016
Published electronically: July 26, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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