Homogenization in the Sobolev class for second order periodic elliptic operators with the inclusion of first order terms

Author:
T. A. Suslina

Translated by:
the author

Original publication:
Algebra i Analiz, tom **22** (2010), nomer 1.

Journal:
St. Petersburg Math. J. **22** (2011), 81-162

MSC (2010):
Primary 35B27

DOI:
https://doi.org/10.1090/S1061-0022-2010-01135-X

Published electronically:
November 17, 2010

MathSciNet review:
2641084

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Abstract: Matrix periodic elliptic second order differential operators in with rapidly oscillating coefficients (depending on ) are studied. The principal part of the operator is given in a factorized form , where is a periodic, bounded and positive definite matrix-valued function and is a matrix first order operator whose symbol is a matrix of maximal rank. The operator also has zero and first order terms with unbounded coefficients. The problem of homogenization in the small period limit is considered. Approximation for the generalized resolvent of the operator is obtained in the operator norm in with error term . Also, approximation for this resolvent is obtained in the norm of operators acting from to with error term of order and with the corrector taken into account. The general results are applied to homogenization problems for the Schrödinger operator and the two-dimensional Pauli operator with potentials involving singular terms.

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Additional Information

**T. A. Suslina**

Affiliation:
Physics Department, St. Petersburg State University, Ulyanovskaya 3, Petrodvorets, St. Petersburg 198504, Russia

Email:
suslina@list.ru

DOI:
https://doi.org/10.1090/S1061-0022-2010-01135-X

Keywords:
Periodic differential operators,
homogenization,
effective operator,
corrector

Received by editor(s):
July 20, 2009

Published electronically:
November 17, 2010

Additional Notes:
Supported by RFBR (grant no. 08-01-00209-a), by a “Scientific schools” grant no. 816.2008.1, and by a “Development of scientific potential of high school” grant no. 2.1.1/2501

Article copyright:
© Copyright 2010
American Mathematical Society