ESAIM: COCV 18 (2012) 1178–1206 ESAIM: Control, Optimisation and Calculus of Variations DOI: 10.1051/cocv/2011195 www.esaim-cocv.org A VARIATIONAL PROBLEM FOR COUPLES OF FUNCTIONS AND MULTIFUNCTIONS WITH INTERACTION BETWEEN LEAVES Emilio Acerbi 1 , Gianluca Crippa 1 and Domenico Mucci 1 Abstract. We discuss a variational problem defined on couples of functions that are constrained to take values into the 2-dimensional unit sphere. The energy functional contains, besides standard Dirichlet energies, a non-local interaction term that depends on the distance between the gradients of the two functions. Different gradients are preferred or penalized in this model, in dependence of the sign of the interaction term. In this paper we study the lower semicontinuity and the coercivity of the energy and we find an explicit representation formula for the relaxed energy. Moreover, we discuss the behavior of the energy in the case when we consider multifunctions with two leaves rather than couples of functions. Mathematics Subject Classification. 49Q20, 54C60. Received May 2, 2011. Published online 16 January 2012. 1. Introduction The Dirichlet energy In the last decades there has been a growing interest in variational problems for vector valued mappings with geometric constraints, as e.g. for mappings defined between smooth manifolds isometrically embedded in Euclidean spaces. The most studied one is perhaps the minimization problem of the Dirichlet energy D(u) := 1 2  B n |Du(x)| 2 dx (1.1) for maps u : B n → R 3 , where B n is the unit ball in R n , that are constrained to take values into the unit sphere S 2 of R 3 . The problem is naturally set in the Sobolev class W 1,2 ( B n , S 2 ) :=  u ∈ W 1,2 ( B n , R 3 ) : |u(x)| =1 for a.e. x ∈ B n  . (1.2) In the physical model n = 3, the above problem is related to the theory of liquid crystals [7], Vol. II, Section 5.1. Namely, the function u represents the direction of the symmetry axis of the rod-like molecules of the liquid crystal, and the minimization of the Dirichlet integral D(u) forces the molecules to organize Keywords and phrases. Relaxed energies, multifunctions, Cartesian currents. 1 Dipartimento di Matematica dell’Universit`a di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy. emilio.acerbi@unipr.it; gianluca.crippa@unipr.it; domenico.mucci@unipr.it Article published by EDP Sciences c  EDP Sciences, SMAI 2012