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

Giaquinta, Mariano; Mucci, Domenico

Geodesic

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B V

-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

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Résumé

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 ${cart}^{1, 1} (B^{n} 𝒴)$ for which we introduce a natural $B V$ -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 $u_{k} : B^{n} \to 𝒴$ with total variation converging to the $B V$ -energy of $T$ . As a consequence, we characterize the lowersemicontinuous envelope of functions of bounded variations from $B^{n}$ into $𝒴$ .

Zbl

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},
     pages = {483--548},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 5},
     number = {4},
     year = {2006},
     zbl = {1150.49020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ASNSP_2006_5_5_4_483_0/}
}

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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. http://geodesic.mathdoc.fr/item/ASNSP_2006_5_5_4_483_0/