We study geometric conditions for integrand $f$ to define lower semicontinuous functional of the form ${I}_{f}\left(u\right){=}_{\Omega}^{\int}f\left(u\right)dx$, where $u$ satisfies certain conservation law. Of our particular interest is tetrahedral convexity condition introduced by the first author in 2003, which is the variant of maximum principle expressed on tetrahedrons, and the new condition which we call tetrahedral polyconvexity. We prove that second condition is sufficient but it is not necessary for lower semicontinuity of ${I}_{f}$, tetrahedral polyconvexity condition is non-local and both conditions are not equivalent. Problems we discuss are strongly connected with the rank-one conjecture of Morrey known in the multidimensional calculus of variations.

Accepted:

DOI: 10.1051/cocv/2015057

Keywords: Quasiconvexity, compensated compactness, calculus of variations

^{1, 2}; Kozarzewski, Piotr

^{1, 3}

@article{COCV_2017__23_2_475_0, author = {Ka{\l}amajska, Agnieszka and Kozarzewski, Piotr}, title = {On the condition of tetrahedral polyconvexity, arising from calculus of variations}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {475--495}, publisher = {EDP-Sciences}, volume = {23}, number = {2}, year = {2017}, doi = {10.1051/cocv/2015057}, zbl = {1358.49001}, mrnumber = {3608090}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2015057/} }

TY - JOUR AU - Kałamajska, Agnieszka AU - Kozarzewski, Piotr TI - On the condition of tetrahedral polyconvexity, arising from calculus of variations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 475 EP - 495 VL - 23 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2015057/ DO - 10.1051/cocv/2015057 LA - en ID - COCV_2017__23_2_475_0 ER -

%0 Journal Article %A Kałamajska, Agnieszka %A Kozarzewski, Piotr %T On the condition of tetrahedral polyconvexity, arising from calculus of variations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 475-495 %V 23 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2015057/ %R 10.1051/cocv/2015057 %G en %F COCV_2017__23_2_475_0

Kałamajska, Agnieszka; Kozarzewski, Piotr. On the condition of tetrahedral polyconvexity, arising from calculus of variations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 23 (2017) no. 2, pp. 475-495. doi : 10.1051/cocv/2015057. http://archive.numdam.org/articles/10.1051/cocv/2015057/

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