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semicontinuity.

ESAIM: Control, Optimisation and Calculus of Variations (1)

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New convexity conditions in the calculus of variationsand compensated compactness theory
Krzysztof Chełmiński, Agnieszka Kałamajska
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 12 / Issue 1 / January 2006

Published online by Cambridge University Press:

15 December 2005, pp. 64-92

Print publication:

January 2006
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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.