Weak and strong density results for the Dirichlet energy

Giaquinta, Mariano, and Mucci, Domenico. "Weak and strong density results for the Dirichlet energy." Journal of the European Mathematical Society 006.1 (2004): 95-117. <http://eudml.org/doc/277577>.

@article{Giaquinta2004,
abstract = {Let $\mathcal \{Y\}$ be a smooth oriented Riemannian manifold which is compact, connected, without boundary and with second homology group without torsion. In this paper we characterize the sequential weak closure of smooth graphs in $B^n\times \mathcal \{Y\}$ with equibounded Dirichlet energies, $B^n$ being the unit ball in $\mathbb \{R\}^n$. More precisely, weak limits of graphs of smooth maps $u_k:B^n\rightarrow \mathcal \{Y\}$ with equibounded Dirichlet integral give rise to elements of the space $\operatorname\{cart\}^\{2,1\}(B^n\times \mathcal \{Y\})$ (cf. [4], [5], [6]). In this paper we prove that every element $T$ in $\operatorname\{cart\}^\{2,1\}(B^n\times \mathcal \{Y\})$ is the weak limit of a sequence $\lbrace u_k\rbrace $ of smooth graphs with equibounded Dirichlet energies. Moreover, in dimension $n=2$, we show that the sequence $\lbrace u_k\rbrace $ can be chosen in such a way that the energy of $u_k$ converges to the energy of $T$.},
author = {Giaquinta, Mariano, Mucci, Domenico},
journal = {Journal of the European Mathematical Society},
keywords = {sequential weak closure; smooth graphs; Dirichlet energy; energy minimizing currents; weak limit; Dirichlet energy; energy minimizing currents; weak limits},
language = {eng},
number = {1},
pages = {95-117},
publisher = {European Mathematical Society Publishing House},
title = {Weak and strong density results for the Dirichlet energy},
url = {http://eudml.org/doc/277577},
volume = {006},
year = {2004},
}

TY - JOUR
AU - Giaquinta, Mariano
AU - Mucci, Domenico
TI - Weak and strong density results for the Dirichlet energy
JO - Journal of the European Mathematical Society
PY - 2004
PB - European Mathematical Society Publishing House
VL - 006
IS - 1
SP - 95
EP - 117
AB - Let $\mathcal {Y}$ be a smooth oriented Riemannian manifold which is compact, connected, without boundary and with second homology group without torsion. In this paper we characterize the sequential weak closure of smooth graphs in $B^n\times \mathcal {Y}$ with equibounded Dirichlet energies, $B^n$ being the unit ball in $\mathbb {R}^n$. More precisely, weak limits of graphs of smooth maps $u_k:B^n\rightarrow \mathcal {Y}$ with equibounded Dirichlet integral give rise to elements of the space $\operatorname{cart}^{2,1}(B^n\times \mathcal {Y})$ (cf. [4], [5], [6]). In this paper we prove that every element $T$ in $\operatorname{cart}^{2,1}(B^n\times \mathcal {Y})$ is the weak limit of a sequence $\lbrace u_k\rbrace $ of smooth graphs with equibounded Dirichlet energies. Moreover, in dimension $n=2$, we show that the sequence $\lbrace u_k\rbrace $ can be chosen in such a way that the energy of $u_k$ converges to the energy of $T$.
LA - eng
KW - sequential weak closure; smooth graphs; Dirichlet energy; energy minimizing currents; weak limit; Dirichlet energy; energy minimizing currents; weak limits
UR - http://eudml.org/doc/277577
ER -