The BV-energy of maps into a manifold : relaxation and density results
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 uk:Bn→𝒴  with equibounded total variation give riseto equivalence classes of cartesian currents in cart1,1(Bn𝒴)  for which we introduce a naturalBV-energy.Assume moreover that the first homotopy group of  𝒴  iscommutative. In any dimension  n  we prove that every element T  in  cart1,1(Bn𝒴)  can be approximatedweakly in the sense of currents by a sequence of graphs of smoothmaps  uk:Bn→𝒴  with total variation converging to theBV-energy of  T. As a consequence, we characterize the lowersemicontinuous envelope of functions of bounded variations fromBn into 𝒴.

Classification : 49Q15, 49Q20 
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     title = {The $BV$-energy of maps into a manifold : relaxation and density results},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {483--548},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 5},
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Giaquinta, Mariano; Mucci, Domenico. The $BV$-energy of maps into a manifold : relaxation and density results. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 4, pp. 483-548. https://www.numdam.org/item/ASNSP_2006_5_5_4_483_0/

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