# Weak and strong density results for the Dirichlet energy

Mariano Giaquinta; Domenico Mucci

Journal of the European Mathematical Society (2004)

- Volume: 006, Issue: 1, page 95-117
- ISSN: 1435-9855

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topGiaquinta, 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 -

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