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.

Kakutani's Theorem states that every point convex and use multifunction ϕ defined on a compact and convex set in a Euclidean space has at least one fixed point. Some necessary conditions are given here which ϕ must satisfy if c is the unique fixed point of ϕ. It is e.g. shown that if the width of ϕ(c) is greater than zero, then ϕ cannot be lsc at c, and if in addition c lies on the boundary of ϕ(c), then there exists a sequence {xk} which converges to c and for which the width of the sets ϕ(xk) converges to zero. If the width of ϕ(c) is zero, then the width of ϕ(xk) converges to zero whenever the sequence {xk} converges to c, but in this case ϕ can be lsc at c.