The Dirichlet energy of mappings from ${\bf B^3}$ into a manifold: density results and gap phenomenon

Weak limits of graphs of smooth maps with equibounded Dirichlet integral give rise to elements of the space . We assume that the 2-homology group of has no torsion and that the Hurewicz homomorphism is injective. Then, in dimension n = 3, we prove that every element T in , which has no singular vertical part, can be approximated weakly in the sense of currents by a sequence of smooth graphs {u _k } with Dirichlet energies converging to the energy of T. We also show that the injectivity hypothesis on the Hurewicz map cannot be removed. We finally show that a similar topological obstruction on the target manifold holds for the approximation problem of the area functional.Received: 9 May 2003, Accepted: 5 June 2003, Published online: 25 February 2004