The $BV$-energy of maps into a manifold : relaxation and density results

Giaquinta, Mariano; Mucci, Domenico

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 4, pp. 483-548.

Let $𝒴$  be a smooth compact oriented riemannian manifoldwithout boundary, and assume that its $1$-homology group has notorsion. Weak limits of graphs of smooth maps $u_k:B^n\to 𝒴$  with equibounded total variation give riseto equivalence classes of cartesian currents in ${\mathrm{cart}}^{1,1}\left(B^n𝒴\right)$  for which we introduce a natural$BV$-energy.Assume moreover that the first homotopy group of  $𝒴$  iscommutative. In any dimension  $n$  we prove that every element $T$  in  ${\mathrm{cart}}^{1,1}\left(B^n𝒴\right)$  can be approximatedweakly in the sense of currents by a sequence of graphs of smoothmaps  $u_k:B^n\to 𝒴$  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 $𝒴$.

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