 
Summary: A variational problem for couples of functions and
multifunctions with interaction between leaves
Emilio Acerbi, Gianluca Crippa and Domenico Mucci
Abstract. We discuss a variational problem defined on couples of functions that are constrained to take values
into the 2dimensional 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.
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 Bn
Du(x)2
dx (0.1)
