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New convexity conditions in the calculus of variationsand compensated compactness theory

Published online by Cambridge University Press:  15 December 2005

Krzysztof Chełmiński  and
Agnieszka Kałamajska
Krzysztof Chełmiński
Affiliation:
Cardinal Stefan Wyszyński University, ul. Dewajtis 5, 01-815 Warszawa, Poland; chelminski@uksw.edu.pl University of Constance, Universitätsstr. 10, 78464 Konstanz, Germany
Agnieszka Kałamajska
Affiliation:
Institute of Mathematics, Warsaw University, ul. Banacha 2, 02–097 Warszawa, Poland; kalamajs@mimuw.edu.pl
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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 Λ-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.

Keywords

Quasiconvexityrank-one convexitysemicontinuity.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 12 , Issue 1 , January 2006 , pp. 64 - 92
Copyright
© EDP Sciences, SMAI, 2006

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