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

Weak limits of graphs of smooth maps u_k:B^n\to Y with equibounded Dirichlet integral give rise to elements of the space cart^{2,1}(B^n x Y). We assume that the 2-homology group of Y has no torsion and that the Hurewicz homomorphism p_2(Y)\to H_2(Y,Q) is injective. Then, in dimension n=3, we prove that every element T in cart^{2,1}(B^3 x Y), 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.