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This article is cited in 3 scientific papers (total in 3 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.
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
Linking options:
https://www.mathnet.ru/eng/sm7793https://doi.org/10.1070/SM2011v202n09ABEH004192 https://www.mathnet.ru/eng/sm/v202/i9/p135
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Abstract page: | 591 | Russian version PDF: | 215 | English version PDF: | 12 | References: | 79 | First page: | 28 |
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