Homogenization of elliptic systems with periodic coefficients: operator error estimates in with corrector taken into account

Author:
T. A. Suslina

Translated by:
the author

Original publication:
Algebra i Analiz, tom **26** (2014), nomer 4.

Journal:
St. Petersburg Math. J. **26** (2015), 643-693

MSC (2010):
Primary 35B27

Published electronically:
May 6, 2015

MathSciNet review:
3289189

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Abstract | References | Similar Articles | Additional Information

Abstract: A matrix elliptic selfadjoint second order differential operator (DO) with rapidly oscillating coefficients is considered in . The principal part of this operator is given in a factorized form, where is a periodic, bounded, and positive definite matrix-valued function and is a matrix first order DO whose symbol is a matrix of maximal rank. The operator also includes first and zero order terms with unbounded coefficients. The problem of homogenization in the small period limit is studied. For the generalized resolvent of , approximation in the -operator norm with an error is obtained. The principal term of this approximation is given by the generalized resolvent of the effective operator with constant coefficients. The first order corrector is taken into account. The error estimate obtained is order sharp; the constants in estimates are controlled in terms of the problem data. General results are applied to homogenization problems for the Schrödinger operator and the two-dimensional Pauli operator with singular rapidly oscillating potentials.

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Additional Information

**T. A. Suslina**

Affiliation:
Department of Physics, St. Petersburg State University, Ul’yanovskaya 3, Petrodvoretz, 198504 St. Petersburg, Russia

Email:
suslina@list.ru

DOI:
https://doi.org/10.1090/spmj/1354

Keywords:
Homogenization,
effective operator,
corrector,
operator error estimates

Received by editor(s):
January 12, 2014

Published electronically:
May 6, 2015

Additional Notes:
Supported by RFBR (grant no.14-01-00760) and by St.Petersburg State University (grant no. 11.38.63.2012)

Article copyright:
© Copyright 2015
American Mathematical Society