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Title: Several versions of the compensated compactness principle

Abstract

The convergence of the product of a solenoidal vector w{sub {epsilon}} and a gradient {nabla}u{sub {epsilon}} in L{sup 1}({Omega}) (where {Omega} is a region in R{sup d}) is investigated in the case when the factors converge weakly in the spaces L{sup {gamma}({Omega})}{sup d} and L{sup {alpha}({Omega})}{sup d}, respectively, with 1/{gamma}+1/{alpha}>1, which means that the main assumption of the classical div-curl lemma fails. Nevertheless, the same convergence (in the sense of distributions in {Omega}) as in the framework of the div-curl lemma, survives under certain additional assumptions. The new versions of the compensated compactness principle proved in the paper can be used in homogenization and in the theory of G-convergence of monotone operators with non-standard coercivity and growth properties, for instance, some degenerate operators. Bibliography: 20 titles.

Authors:
;
Publication Date:
OSTI Identifier:
21612604
Resource Type:
Journal Article
Journal Name:
Sbornik. Mathematics
Additional Journal Information:
Journal Volume: 202; Journal Issue: 9; Other Information: DOI: 10.1070/SM2011v202n09ABEH004192; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1064-5616
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICAL METHODS AND COMPUTING; COERCIVE FORCE; CONVERGENCE; DISTRIBUTION; MATHEMATICAL SPACE; VECTORS; SPACE; TENSORS

Citation Formats

Pastukhova, Svetlana E, and Khripunova, Anna S. Several versions of the compensated compactness principle. United States: N. p., 2011. Web. doi:10.1070/SM2011V202N09ABEH004192; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
Pastukhova, Svetlana E, & Khripunova, Anna S. Several versions of the compensated compactness principle. United States. doi:10.1070/SM2011V202N09ABEH004192; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
Pastukhova, Svetlana E, and Khripunova, Anna S. Fri . "Several versions of the compensated compactness principle". United States. doi:10.1070/SM2011V202N09ABEH004192; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
@article{osti_21612604,
title = {Several versions of the compensated compactness principle},
author = {Pastukhova, Svetlana E and Khripunova, Anna S},
abstractNote = {The convergence of the product of a solenoidal vector w{sub {epsilon}} and a gradient {nabla}u{sub {epsilon}} in L{sup 1}({Omega}) (where {Omega} is a region in R{sup d}) is investigated in the case when the factors converge weakly in the spaces L{sup {gamma}({Omega})}{sup d} and L{sup {alpha}({Omega})}{sup d}, respectively, with 1/{gamma}+1/{alpha}>1, which means that the main assumption of the classical div-curl lemma fails. Nevertheless, the same convergence (in the sense of distributions in {Omega}) as in the framework of the div-curl lemma, survives under certain additional assumptions. The new versions of the compensated compactness principle proved in the paper can be used in homogenization and in the theory of G-convergence of monotone operators with non-standard coercivity and growth properties, for instance, some degenerate operators. Bibliography: 20 titles.},
doi = {10.1070/SM2011V202N09ABEH004192; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)},
journal = {Sbornik. Mathematics},
issn = {1064-5616},
number = 9,
volume = 202,
place = {United States},
year = {2011},
month = {9}
}