Calculus of Variations and Geometric Measure Theory

M. Giaquinta - D. Mucci

The Dirichlet energy of mappings from {\mbox{$*B^3*$}} into a manifold: density results and gap phenomenon

created on 25 Jun 2003
modified by mucci on 09 Nov 2004


Published Paper

Inserted: 25 jun 2003
Last Updated: 9 nov 2004

Journal: Calc. Var.
Volume: 20
Pages: 367-397
Year: 2004


Weak limits of graphs of smooth maps \,$u_k:B^n\to{\cal Y}$\, with equibounded Dirichlet integral give rise to elements of the space \,${\mbox{\rm cart}}^{2,1}(B^n\times{\cal Y})$. We assume that the $2$-homology group of ${\cal Y}$ has no torsion and that the Hurewicz homomorphism \,$\pi_2({\cal Y})\to H_2({\cal Y},Q)$\, is injective. Then, in dimension \,$n=3$, we prove that every element \,$T$\, in \,${\mbox{\rm cart}}^{2,1}(B^3\times{\cal Y})$, which has no singular vertical part, can be approximated weakly in the sense of currents by a sequence of smooth graphs \,$\{u_k\}$\, with Dirichlet energies converging to the energy of \,$T$. We also show that the injectivity hypothesis on the Hurewicz map cannot be removed. We finally show that a similar topological obstruction on the target manifold holds for the approximation problem of the area functional.