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Mat. Sb., 2011, Volume 202, Number 9, Pages 135–160 (Mi msb7793)  

This article is cited in 2 scientific papers (total in 2 papers)

Several versions of the compensated compactness principle

S. E. Pastukhovaa, A. S. Khripunovab

a Moscow State Institute of Radio-Engineering, Electronics and Automation (Technical University)
b Vladimir State Humanitarian University

Abstract: The convergence of the product of a solenoidal vector $w_\varepsilon$ and a gradient $\nabla u_\varepsilon$ in $L^1(\Omega)$ (where $\Omega$ is a region in $\mathbb R^d$) is investigated in the case when the factors converge weakly in the spaces $L^\gamma(\Omega)^d$ and $L^\alpha(\Omega)^d$, 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 $\Omega$)
$$ \lim_{\varepsilon\to0}w_\varepsilon\cdot\nabla u_\varepsilon =\lim_{\varepsilon\to0}w_\varepsilon\cdot\lim_{\varepsilon\to0} \nabla u_\varepsilon=w\cdot\nabla u $$
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.
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DOI: https://doi.org/10.4213/sm7793

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English version:
Sbornik: Mathematics, 2011, 202:9, 1387–1412

Bibliographic databases:

UDC: 517.956.4
MSC: Primary 46E40; Secondary 49J45
Received: 29.09.2010 and 14.01.2011

Citation: S. E. Pastukhova, A. S. Khripunova, “Several versions of the compensated compactness principle”, Mat. Sb., 202:9 (2011), 135–160; Sb. Math., 202:9 (2011), 1387–1412

Citation in format AMSBIB
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\vol 202
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Pastukhova S.E., Khripunova A.S., “Gamma-closure of some classes of nonstandard convex integrands”, J. Math. Sci. (N. Y.), 177:1 (2011), 83–108  crossref  mathscinet  zmath  elib  scopus
    2. V. V. Zhikov, S. E. Pastukhova, “Uniform convexity and variational convergence”, Trans. Moscow Math. Soc., 75 (2014), 205–231  mathnet  crossref  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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