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

¹ Cardinal Stefan Wyszyński 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: 22 March 2004
Revised: 10 August 2004

Abstract

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.