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

Mariano Giaquinta; Domenico Mucci

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2006)

  • Volume: 5, Issue: 4, page 483-548
  • ISSN: 0391-173X

Abstract

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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 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 → 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 .

How to cite

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Giaquinta, Mariano, and Mucci, Domenico. "The BV-energy of maps into a manifold : relaxation and density results." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 5.4 (2006): 483-548. <http://eudml.org/doc/242253>.

@article{Giaquinta2006,
abstract = {Let $\{\mathcal \{Y\}\}$  be a smooth compact oriented riemannian manifoldwithout boundary, and assume that its $1$-homology group has notorsion. Weak limits of graphs of smooth maps $u_k:B^n\rightarrow \{\mathcal \{Y\}\}$  with equibounded total variation give riseto equivalence classes of cartesian currents in $\mathop \{\rm cart\}\nolimits ^\{1,1\}(B^n\{\mathcal \{Y\}\})$  for which we introduce a natural$BV$-energy.Assume moreover that the first homotopy group of  $\{\mathcal \{Y\}\}$  iscommutative. In any dimension  $n$  we prove that every element $T$  in  $\mathop \{\rm cart\}\nolimits ^\{1,1\}(B^n\{\mathcal \{Y\}\})$  can be approximatedweakly in the sense of currents by a sequence of graphs of smoothmaps  $u_k:B^n\rightarrow \{\mathcal \{Y\}\}$  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 $\{\mathcal \{Y\}\}$.},
author = {Giaquinta, Mariano, Mucci, Domenico},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {smooth compact Riemannian manifolds; weak limit; Cartesian currents},
language = {eng},
number = {4},
pages = {483-548},
publisher = {Scuola Normale Superiore, Pisa},
title = {The BV-energy of maps into a manifold : relaxation and density results},
url = {http://eudml.org/doc/242253},
volume = {5},
year = {2006},
}

TY - JOUR
AU - Giaquinta, Mariano
AU - Mucci, Domenico
TI - The BV-energy of maps into a manifold : relaxation and density results
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2006
PB - Scuola Normale Superiore, Pisa
VL - 5
IS - 4
SP - 483
EP - 548
AB - Let ${\mathcal {Y}}$  be a smooth compact oriented riemannian manifoldwithout boundary, and assume that its $1$-homology group has notorsion. Weak limits of graphs of smooth maps $u_k:B^n\rightarrow {\mathcal {Y}}$  with equibounded total variation give riseto equivalence classes of cartesian currents in $\mathop {\rm cart}\nolimits ^{1,1}(B^n{\mathcal {Y}})$  for which we introduce a natural$BV$-energy.Assume moreover that the first homotopy group of  ${\mathcal {Y}}$  iscommutative. In any dimension  $n$  we prove that every element $T$  in  $\mathop {\rm cart}\nolimits ^{1,1}(B^n{\mathcal {Y}})$  can be approximatedweakly in the sense of currents by a sequence of graphs of smoothmaps  $u_k:B^n\rightarrow {\mathcal {Y}}$  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 ${\mathcal {Y}}$.
LA - eng
KW - smooth compact Riemannian manifolds; weak limit; Cartesian currents
UR - http://eudml.org/doc/242253
ER -

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