New convexity conditions in the calculus of variations and compensated compactness theory
Cardinal Stefan Wyszyński University,
ul. Dewajtis 5, 01-815 Warszawa, Poland;
2 University of Constance, Universitätsstr. 10, 78464 Konstanz, Germany
3 Institute of Mathematics, Warsaw University, ul. Banacha 2, 02–097 Warszawa, Poland; firstname.lastname@example.org
Revised: 10 August 2004
We consider the lower semicontinuous functional of the form 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.
Mathematics Subject Classification: 49J10 / 49J45
Key words: Quasiconvexity / rank-one convexity / semicontinuity.
© EDP Sciences, SMAI, 2006