|
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.
Author to whom correspondence should be addressed
DOI:
https://doi.org/10.4213/sm7793
Full text:
PDF file (658 kB)
References:
PDF file
HTML file
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
\Bibitem{PasKhr11}
\by S.~E.~Pastukhova, A.~S.~Khripunova
\paper Several versions of the compensated compactness principle
\jour Mat. Sb.
\yr 2011
\vol 202
\issue 9
\pages 135--160
\mathnet{http://mi.mathnet.ru/msb7793}
\crossref{https://doi.org/10.4213/sm7793}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2884368}
\zmath{https://zbmath.org/?q=an:1246.46027}
\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2011SbMat.202.1387P}
\elib{https://elibrary.ru/item.asp?id=19066308}
\transl
\jour Sb. Math.
\yr 2011
\vol 202
\issue 9
\pages 1387--1412
\crossref{https://doi.org/10.1070/SM2011v202n09ABEH004192}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000296920400007}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-79961029425}
Linking options:
http://mi.mathnet.ru/eng/msb7793https://doi.org/10.4213/sm7793 http://mi.mathnet.ru/eng/msb/v202/i9/p135
Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles
This publication is cited in the following articles:
-
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
-
V. V. Zhikov, S. E. Pastukhova, “Uniform convexity and variational convergence”, Trans. Moscow Math. Soc., 75 (2014), 205–231
|
Number of views: |
This page: | 438 | Full text: | 126 | References: | 51 | First page: | 28 |
|