On the condition of tetrahedral polyconvexity, arising from calculus of variations∗
1 Institute of Mathematics, University
of Warsaw, ul. Banacha
2 Institute of Mathematics Polish Academy of Sciences, ul. Śniadeckich 8, P.O. Box 21, 00-956 Warszawa, Poland.
3 Institute of Mathematics and Cryptology, Military University of Technology, ul. gen. Sylwestra Kaliskiego 2, 00-908 Warszawa, Poland.
Revised: 9 December 2015
Accepted: 15 December 2015
We study geometric conditions for integrand f to define lower semicontinuous functional of the form If(u) = ∫Ωf(u)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 If, 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.
Mathematics Subject Classification: 49J10 / 49J45
Key words: Quasiconvexity / compensated compactness / calculus of variations
© EDP Sciences, SMAI 2017