On operator-type homogenization estimates for elliptic equations with lower order terms
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S. E. Pastukhova and R. N. Tikhomirov
Translated by: S. E. Pastukhova - St. Petersburg Math. J. 29 (2018), 841-861
- DOI: https://doi.org/10.1090/spmj/1518
- 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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Bibliographic Information
- S. E. Pastukhova
- Affiliation: Moscow Technology University (MIREA), pr. Vernadskogo 78, 119454 Moscow, Russia
- Email: pas-se@yandex.ru
- R. N. Tikhomirov
- Affiliation: A. G. and N. G. Stoletov Vladimir state University, ul. Gorkogo 87, 600000 Vladimir, Russia
- Received by editor(s): October 25, 2016
- Published electronically: July 26, 2018
- © Copyright 2018 American Mathematical Society
- Journal: St. Petersburg Math. J. 29 (2018), 841-861
- MSC (2010): Primary 51B10, 53C50
- DOI: https://doi.org/10.1090/spmj/1518
- MathSciNet review: 3724642