*Published Paper*

**Inserted:** 30 may 2005

**Last Updated:** 15 dec 2008

**Journal:** Ann. Scuola Norm. Sup. Pisa Cl. Sci (5)

**Volume:** 5

**Number:** 4

**Pages:** 483-548

**Year:** 2006

**Notes:**

Erratum and Addendum: Ann. Scuola Norm. Sup. Pisa Cl. Sci (5) 6 (2007) 1, 185--194

**Abstract:**

Let \,${\cal Y}$\, be a smooth compact oriented Riemannian manifold without boundary, and assume that its $1$-homology group has no torsion. Weak limits of graphs of smooth maps \,$u_k:B^n\to{\cal Y}$\, with equibounded total variation give rise to equivalence classes of Cartesian currents in \,$cart^{1,1}(B^n\times{\cal Y})$\, for which we introduce natural a $BV$-energy. Assume moreover that the first homotopy group of \,${\cal Y}$\, is commutative. In any dimension \,$n$\, we prove that every element \,$T$\, in \,$cart^{1,1}(B^n\times{\cal Y})$\, can be approximated weakly in the sense of currents by a sequence of graphs of smooth maps \,$u_k:B^n\to{\cal Y}$\, with total variation converging to the $BV$-energy of \,$T$. As a consequence, we characterize the lower semicontinuous envelope of functions of bounded variations from $B^n$ into ${\cal Y}$.

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