Calculus of Variations and Geometric Measure Theory

M. Giaquinta - D. Mucci

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

created by mucci on 30 May 2005
modified on 15 Dec 2008


Published Paper

Inserted: 30 may 2005
Last Updated: 15 dec 2008

Journal: Ann. Scuola Norm. Sup. Pisa Cl. Sci (5)
Volume: 5
Number: 4
Pages: 483-548
Year: 2006

Erratum and Addendum: Ann. Scuola Norm. Sup. Pisa Cl. Sci (5) 6 (2007) 1, 185--194


Let \,${\cal Y}$\, be a smooth compact oriented Riemannian manifold without boundary, and assume that its $1$-homology group has no torsion. Weak limits of graphs of smooth maps \,$u_k:B^n\to{\cal Y}$\, with equibounded total variation give rise to equivalence classes of Cartesian currents in \,$cart^{1,1}(B^n\times{\cal Y})$\, for which we introduce natural a $BV$-energy. Assume moreover that the first homotopy group of \,${\cal Y}$\, is commutative. In any dimension \,$n$\, we prove that every element \,$T$\, in \,$cart^{1,1}(B^n\times{\cal Y})$\, can be approximated weakly in the sense of currents by a sequence of graphs of smooth maps \,$u_k:B^n\to{\cal Y}$\, with total variation converging to the $BV$-energy of \,$T$. As a consequence, we characterize the lower semicontinuous envelope of functions of bounded variations from $B^n$ into ${\cal Y}$.