We discuss a variational problem defined on couples of functions that are constrained totake values into the 2-dimensional unit sphere. The energy functional contains, besidesstandard Dirichlet energies, a non-local interaction term that depends on the distancebetween the gradients of the two functions. Different gradients are preferred or penalizedin this model, in dependence of the sign of the interaction term. In this paper we studythe lower semicontinuity and the coercivity of the energy and we find an explicitrepresentation formula for the relaxed energy. Moreover, we discuss the behavior of theenergy in the case when we consider multifunctions with two leaves rather than couples offunctions.

A unified framework for analyzing the existence of ground states in wideclasses of elastic complex bodies is presented here. The approach makes useof classical semicontinuity results, Sobolev mappings and Cartesiancurrents. Weak diffeomorphisms are used to represent macroscopicdeformations. Sobolev maps and Cartesian currents describe the innersubstructure of the material elements. Balance equations for irregularminimizers are derived. A contribution to the debate about the role of thebalance of configurational actions follows. After describing a list ofpossible applications of the general results collected here, a concretediscussion of the existence of ground states in thermodynamically stablequasicrystals is presented at the end.