New Convexity Conditions in the Calculus of Variations and Compensated Compactness Theory

Abstract

We consider the lower semicontinuous functional term of the form I (u) = ∫Ωf(u)dx where u satisfies a given conservation law defined by differential operator of degree one with constant coefficientsWe show that under certain constraints the well known Murat and Tartar's Λ-convexity condition for the integrand extends to the new geometric conditions satisfied on four dimensional simplexes. Similar conditions on three dimensional simplexes were recently obtained by the second author. New conditions apply to quasiconvex functions.