The BV-energy of maps into a manifold : relaxation and density results
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 5 (2006) no. 4, p. 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 →𝒴  with equibounded total variation give riseto equivalence classes of cartesian currents in  cart 1,1 (B n 𝒴)  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   cart 1,1 (B n 𝒴)  can be approximatedweakly in the sense of currents by a sequence of graphs of smoothmaps  u k :B n →𝒴  with total variation converging to theBV-energy of  T. As a consequence, we characterize the lowersemicontinuous envelope of functions of bounded variations fromB n into 𝒴.

Classification:  49Q15,  49Q20
@article{ASNSP_2006_5_5_4_483_0,
     author = {Giaquinta, Mariano and Mucci, Domenico},
     title = {The $BV$-energy of maps into a manifold : relaxation and density results},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 5},
     number = {4},
     year = {2006},
     pages = {483-548},
     zbl = {1150.49020},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2006_5_5_4_483_0}
}
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, Serie 5, Volume 5 (2006) no. 4, pp. 483-548. http://www.numdam.org/item/ASNSP_2006_5_5_4_483_0/

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