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

Chełmiński, Krzysztof; Kałamajska, Agnieszka

ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 1, pp. 64-92.

We consider the lower semicontinuous functional of the form $I_f\left(u\right)={\int }_{\Omega }f\left(u\right)\mathrm{d}x$ 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.

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