On the condition of tetrahedral polyconvexity, arising from calculus of variations

Kałamajska, Agnieszka; Kozarzewski, Piotr

ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 2, pp. 475-495.

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.

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