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

¹ Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland
² Institute of Mathematics Polish Academy of Sciences, ul. Śniadeckich 8, P.O. Box 21, 00-956 Warszawa, Poland.
kalamajs@mimuw.edu.pl
³ Institute of Mathematics and Cryptology, Military University of Technology, ul. gen. Sylwestra Kaliskiego 2, 00-908 Warszawa, Poland.
piotr.kozarzewski@wat.edu.pl

Received: 10 July 2015
Revised: 9 December 2015
Accepted: 15 December 2015

Abstract

We study geometric conditions for integrand f to define lower semicontinuous functional of the form I_f(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 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.