Abstract

Let ${\cal Y}$ be a smooth compact oriented riemannian manifoldwithout boundary, and assume that its $1$-homology group has notorsion. Weak limits of graphs of smooth maps $u_k:B^n\rightarrow {\cal Y}$ with equibounded total variation give riseto equivalence classes of cartesian currents in $\mathop {\rm cart}\nolimits ^{1,1}$ for which we introduce a natural$BV$-energy.Assume moreover that the first homotopy group of ${\cal Y}$ iscommutative. In any dimension $n$ we prove that every element $T$ in $\mathop {\rm cart}\nolimits ^{1,1}$ can be approximatedweakly in the sense of currents by a sequence of graphs of smoothmaps $u_k:B^n\rightarrow {\cal Y}$ with total variation converging to the$BV$-energy of $T$. As a consequence, we characterize the lowersemicontinuous envelope of functions of bounded variations from$B^n$ into ${\cal Y}$