New convexity conditions in the calculus of variations and compensated compactness theory
ESAIM: Control, Optimisation and Calculus of Variations, Volume 12 (2006) no. 1, p. 64-92
We consider the lower semicontinuous functional of the form I f (u)= Ω f(u)dx 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.
DOI : https://doi.org/10.1051/cocv:2005034
Classification:  49J10,  49J45
Keywords: quasiconvexity, rank-one convexity, semicontinuity
@article{COCV_2006__12_1_64_0,
     author = {Che\l mi\'nski, Krzysztof and Ka\l amajska, Agnieszka},
     title = {New convexity conditions in the calculus of variations and compensated compactness theory},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {12},
     number = {1},
     year = {2006},
     pages = {64-92},
     doi = {10.1051/cocv:2005034},
     zbl = {1114.49019},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2006__12_1_64_0}
}
Chełmiński, Krzysztof; Kałamajska, Agnieszka. New convexity conditions in the calculus of variations and compensated compactness theory. ESAIM: Control, Optimisation and Calculus of Variations, Volume 12 (2006) no. 1, pp. 64-92. doi : 10.1051/cocv:2005034. http://www.numdam.org/item/COCV_2006__12_1_64_0/

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