*Published Paper*

**Inserted:** 15 mar 2005

**Last Updated:** 22 nov 2006

**Journal:** Comm. Pure Appl. Math.

**Volume:** 59

**Number:** 12

**Pages:** 1791-1810

**Year:** 2006

**Abstract:**

Let \,${\cal Y}$\, be a smooth compact oriented Riemannian manifold without boundary. Weak limits of graphs of smooth maps \,$u_k$\, from \,$B^n$\, to \,${\cal Y}$\, with equibounded Dirichlet integral give rise to elements of the space \,${\rm{cart}}^{2,1}(B^n\times{\cal Y})$. Assume that \,${\cal Y}$\, is $1$-connected and that its $2$-homology group has no torsion. In any dimension \,$n$\, we prove that every element \,$T$\, in \,${\rm{cart}}^{2,1}(B^n\times{\cal Y})$\, with no singular vertical part can be approximated weakly in the sense of currents by a sequence of graphs of smooth maps \,$u_k$\, from \,$B^n$\, to \,${\cal Y}$\, with Dirichlet energies converging to the energy of \,$T$.

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