Calculus of Variations and Geometric Measure Theory

M. Giaquinta - D. Mucci

Density results relative to the Dirichlet energy of mappings into a manifold

created by mucci on 15 Mar 2005
modified on 22 Nov 2006


Published Paper

Inserted: 15 mar 2005
Last Updated: 22 nov 2006

Journal: Comm. Pure Appl. Math.
Volume: 59
Number: 12
Pages: 1791-1810
Year: 2006


Let \,${\cal Y}$\, be a smooth compact oriented Riemannian manifold without boundary. Weak limits of graphs of smooth maps \,$u_k$\, from \,$B^n$\, to \,${\cal Y}$\, with equibounded Dirichlet integral give rise to elements of the space \,${\rm{cart}}^{2,1}(B^n\times{\cal Y})$. Assume that \,${\cal Y}$\, is $1$-connected and that its $2$-homology group has no torsion. In any dimension \,$n$\, we prove that every element \,$T$\, in \,${\rm{cart}}^{2,1}(B^n\times{\cal Y})$\, with no singular vertical part can be approximated weakly in the sense of currents by a sequence of graphs of smooth maps \,$u_k$\, from \,$B^n$\, to \,${\cal Y}$\, with Dirichlet energies converging to the energy of \,$T$.