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 Mat. Zametki: Year: Volume: Issue: Page: Find

 Mat. Zametki, 2013, Volume 94, Issue 1, Pages 130–150 (Mi mz10105)

Approximations of the Resolvent for a Non–Self-Adjoint Diffusion Operator with Rapidly Oscillating Coefficients

S. E. Pastukhova

Moscow State Institute of Radio-Engineering, Electronics and Automation (Technical University)

Abstract: A strongly inhomogeneous diffusion operator with drift depending on a small parameter $\varepsilon$ is studied in the space $L^2(\mathbb R^n)$. The strong inhomogeneity consists in that the coefficients of the operator are $\varepsilon$-periodic and, in addition, the drift vector is of the order of $\varepsilon^{-1}$. As $\varepsilon\to 0$, approximations in the operator $L^2$‑norm of order $\varepsilon$ and $\varepsilon^2$ are constructed for the resolvent of the operator. For each of these orders of approximation, an averaged diffusion operator is obtained. A spectral method based on the Bloch representation for an operator with periodic coefficients is used.

Keywords: diffusion operator with drift, resolvent of an operator, averaged diffusion operator, Bloch representation for an operator, Sobolev space, Gelfand transformation.

DOI: https://doi.org/10.4213/mzm10105

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English version:
Mathematical Notes, 2013, 94:1, 127–145

Bibliographic databases:

Document Type: Article
UDC: 517.956.8

Citation: S. E. Pastukhova, “Approximations of the Resolvent for a Non–Self-Adjoint Diffusion Operator with Rapidly Oscillating Coefficients”, Mat. Zametki, 94:1 (2013), 130–150; Math. Notes, 94:1 (2013), 127–145

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz10105
• https://doi.org/10.4213/mzm10105
• http://mi.mathnet.ru/eng/mz/v94/i1/p130

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This publication is cited in the following articles:
1. N. N. Senik, “On Homogenization for Non-Self-Adjoint Periodic Elliptic Operators on an Infinite Cylinder”, Funct. Anal. Appl., 50:1 (2016), 71–75
2. S. E. Pastukhova, R. N. Tikhomirov, “Operator-type estimates in homogenization of elliptic equations with lower terms”, St. Petersburg Math. J., 29:5 (2018), 841–861
3. N. N. Senik, “Homogenization for non-self-adjoint periodic elliptic operators on an infinite cylinder”, SIAM J. Math. Anal., 49:2 (2017), 874–898
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