On new geometric conditions for some weakly lower semicontinuous functionals with applications to the rank-one conjecture of Morrey

Abstract

We obtain new geometric necessary conditions for a function f to define a lower semicontinuous functional of the form I_f(u) = ∫_Ωf(u)dx, where u satisfies a given conservation law, Pu = 0, defined by a differential operator P of degree one with constant coefficients. Those conditions imply the so-called Λ-convexity condition known as the rank-one condition when we deal with a functional of the calculus of variations. In particular, we derive some new geometric properties of quasi-convex functions and state some new questions related to the rank-one conjecture of Morrey.