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Issue ESAIM: COCV
Volume 12, Number 1, January 2006
Page(s) 64 - 92
DOI 10.1051/cocv:2005034
Published online 15 December 2005

ESAIM: COCV, January 2006, Vol. 12, pp. 64-92
DOI: 10.1051/cocv:2005034

New convexity conditions in the calculus of variations and compensated compactness theory

Krzysztof Chelminski1, 2 and Agnieszka Kalamajska3

1  Cardinal Stefan Wyszynski University, ul. Dewajtis 5, 01-815 Warszawa, Poland; chelminski@uksw.edu.pl
2  University of Constance, Universitätsstr. 10, 78464 Konstanz, Germany
3  Institute of Mathematics, Warsaw University, ul. Banacha 2, 02-097 Warszawa, Poland; kalamajs@mimuw.edu.pl


(Received March 22, 2004. Revised August 10, 2004. / Published online: 15 December 2005)

Abstract
We consider the lower semicontinuous functional of the form $I_f(u)=\int_\Omega f(u){\rm d}x$ where u satisfies a given conservation law defined by differential operator of degree one with constant coefficients. We show that under certain constraints the well known Murat and Tartar's $\Lambda$-convexity condition for the integrand f extends to the new geometric conditions satisfied on four dimensional symplexes. Similar conditions on three dimensional symplexes were recently obtained by the second author. New conditions apply to quasiconvex functions.


Mathematics Subject Classification. 49J10, 49J45

Key words: Quasiconvexity, rank-one convexity, semicontinuity.


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