Weak and strong density results for the Dirichlet energy

Let Y be a smooth oriented Riemannian manifold which is compact, connected, without boundary and with second homology group without torsion. In this paper we characterize the sequential weak closure of smooth graphs in Bn ×Y with equibounded Dirichlet energies, Bn being the unit ball in Rn. More precisely, weak limits of graphs of smooth maps uk : Bn ! Y with equibounded Dirichlet integral give rise to elements of the space cart2,1(Bn ×Y) (cf. [4], [5], [6]). In this paper we prove that every element T in cart2,1(Bn×Y) is the weak limit of a sequence {uk} of smooth graphs with equibounded Dirichlet energies. Moreover, in dimension n = 2, we show that the sequence {uk} can be chosen in such a way that the energy of uk converges to the energy of T .