The Dirichlet energy of mappings from B^3 into a manifold: density results and gap phenomenon

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). We assume that the 2- homology group of Y has no torsion and that the Hurewicz homomorphism π2(Y) → H2(Y,Q) is injective. Then, in dimension n = 3, we prove that every element T in cart2,1(B3 × Y), which has no singular vertical part, can be approximated weakly in the sense of currents by a sequence of smooth graphs {uk} 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.